Unsteady-State Fluid Flow: Analysis and Applications to Petroleum Reservoir Behavior [1st ed.] 9780444501844, 0444501843

Citation preview

Unsteady-State Fluid Flow: Analysis and Applications to Petroleum Reservoir Behavior by E. J. Hoffman

• ISBN: 0444501843 • Publisher: Elsevier Science & Technology Books • Pub. Date: August 1999

PREFACE The partial differential equations for the unsteady-state flow of compressible fluids through porous media are deceptively simple in their statement. Their solution to yield pressure and density in terms of position and time, and subject to the requisite boundary conditions, is another matter entirely. In the case of compressible liquids, the compressibility behavior may be so represented that the differential equations become analogous to those for heat conduction. And for this circumstance there are purportedly analytical solutions for either linear flow or radial flow, subject to one or another of several different combinations of boundary conditions. This will of course bring up the matter of whether solutions can in fact exist for the particular occasion. Or more precisely, does or can the differential form properly result from a differentiation of some primal relationship between the dependent variable and the independent variables? Implicit in the question is the assignment of boundary conditions. There is first the difficulty of whether these conditions are physically real. There is also a problem of whether too many boundary conditions have been specified. There is even the question of the validity of the radial form, assuming the linear form is valid. For compressible gases, the compressibility relationship is such that the resulting partial differential equations do not yield to analytical solutions, and numerical techniques are required. But here again, physically-real situations may call for an over-specification of the degrees of freedom or number of boundary conditions, resulting in numerical solutions which have to be "forced." The perplexities encountered are such that extra-audacious measures are in order. Paradoxically, these measures can result in simplifications, and at the same time encompass a wider range of flow geometries and systems. One such measure is to express unsteady-state behavior in terms of volume and surface integrals, and to "synthesize" a solution or correlating form. Another is to consider that at any given instant the unsteady-state profile can be approximated by the corresponding steady-state profile, and which is justified by the evidence. Both approaches will be developed and utilized. The preeminent embodiment of unsteady-state fluid flow through porous media is in the production of gases and/or liquids from petroleum reservoirs. In tum the reservoir may be closed or open (e.g., water-drive), and more than one fluid phase may exist (e.g., gas, oil, and/or water). Thus simultaneous two-phase or multiphase flow may be encountered, or there may be flow by displacement. Moreover, there is the concomitant flow up through the wellbore, and its effect on

vi

PREFACE

the bottom-hole flowing pressure. All these considerations are to be examined as related to the unsteady-state behavior of the reservoir, not the least of which is to estimate the reservoir extent and the reserves from well performance data. The underlying objective, in sum, will be to simplify and unify or bring to order the problems in predicting reservoir size or extent, and reservoir performance or depletion. While reservoir extent, for instance, can be estimated by areal means based on geological information, such is not always reliable or in evidence. There is a need for alternative or confirming evidence such as obtained from well-test data (and which may be limited), and from production histories. While a degree of accommodation or correlation can be made utilizing the relationships for steady-state or established production, in final analysis reservoir performance is that of the unsteady or transient state. Thus there is the occasion for definitive examination of unsteady-state predictions~based on drawdown or other longer-term flow tests~and backed by a comparison or correlation (e.g., by analogy with other known production and reserve reformation from similarly producing wells and fields. The difficulty may be overviewed as that of providing "solutions" for complex problems which in reality may not have solutions in the strict sense of the word, and of translating the effort into down-to-earth terms which will have meaning to a diverse readership. There comes a point beyond which further mathematical manipulation (read juggling) becomes futile and self-defeating, another instance of diminishing returns. The expectation is to generate behavior patterns for situations where there can be no rigorous representation, only approximation in varying degree. It is in this spirit therefore that this work is commenced.

Chapter 1 PETROLEUM RESERVES AND THEIR ESTIMATION

Oil and gas resources or reserves exist in what are uniformly referred to as reservoirs. A reservoir in turn is delineated by a common field and formation. The term "field" refers to the land-surface designation, whereas "formation" denotes the particular oil- and/or gas-bearing strata. Furthermore, in this case, these strata will be of sedimentary origin, in the mare composed of sandstone or limestone. Such sedimentary rock may be further viewed as consolidated or unconsolidated, and may also be called a "sand." Thus the same geologic formation may exist under any of several distract fields, which are separated by discontinuities in the formation (e.g., faults, traps or stratigraphic zones of zero porosity or permeability). On the other hand, the same field may be underlain with one or more -- even multitudinous -- oil- and/or gasbearing strata or formations, also called sands, each distinct from the other, and one above the other, and commonly called production zones. There is the possibility also that one or more of the oil- or gas-bearing strata in multiple zones may extend from one field to another, or be more widespread. Hence the field designator assigned applies to a particular strata or formation. In this sense, the field designator and formation designator both pertain to the same common circumstance, and can be used mterchangeably. It may inferred, however, that the designation of common reservoirs by the names of the field and formation is not always clear-cut. Each specific strata or formation, or sand, is identified and named according to its geologic history and makeup -- both the inorganic composition and the type and concentration of organic content. Moreover, a given set of formations with a common geologic history may be further identified by the basin or province of occurrence, and which may encompass one or more fields. When produced, the flow behavior can be regarded as steady-state at least over short-term intervals, aside from being unsteady-state over the initial drawdown, and being unsteady-state as viewed over the long-term. The steadystate appellation will in fact seem more appropriate for the transition intermediate between drawdown and long-term depletion, for a closed reservoir. The picture is

4

UNSTEADY-STATE FLUID FLOW

beclouded, however, by the presence of a gas-drive and/or water drive, which can serve to sustain a near steady-state condition. The spectrum is further reviewed as follows. For additional reformation about terms and terminology, consult the Glossary.

1.1 CHARACTERIZATION BY UNSTEADY-STATE BEHAVIOR The more usual considerations of fluid flow involve the steady-state. These considerations are the energy balances, and the correlations and calculations for flow through pipes and channels, through process equipment, and of pumping and compression. Gases and liquids are involved, as are flow systems of two or more distract phases. The considerations of unsteady-state behavior have largely been confined to conductive heat transfer, which is a manifestation of the broader field of potential theory, and whose fundamental relationships are amenable to certain analytic solutions subject to the prescribed boundary conditions or statements. It was in fact the earlier investigations of Jean-Baptiste-Joseph Fourier (1746-1830) into the series solutions for linear conductive heat transfer boundary problems which launched and vitalized the study of partial differential equations. Fortuitously, the same sort of representation can sometimes be applied to the flow of fluids through porous media, whereby velocity gradients are not recurred normal to the direction of flow, as would be in characterizing the concept for shear viscosity. In fact, for compressible liquids, the compressibility behavior may be so assum~ that the fundamental partial differential equation is identical in form to the corresponding heat conduction equation. Pressure (or density) becomes the dependent variable rather than temperature. The same sort of solutions may therefore be utilized, subject to the qualifying boundary conditions, for certain select cases. (But which, unfortunately, are not physically real.) For what are considered gases, however, the compressibility behavior is such that the resulting partial differential equation involves pressure to a higher degree -- to the second degree rather than the first. This obviates the possibility of analytic solutions, requiring instead the utilization of numerical or computer solutions. For the above and other reasons, the study of unsteady-state flow behavior is rather the more interesting and compelling. Moreover, there is the incentive for using other and more generic approaches to effect solutions or correlations. Among these, to be developed in detail, is the use of volume and surface integrals

PETROLEUM RESERVES

5

to describe flow, as an option to the classical approach based on the differential increment of volume. In application, the methodology can be applied to either or both gaseous and liquid flow systems, and to irregular (and threedimensional) geometries. Finally and significantly, the unsteady-state flow of fluids through porous media has as its most ubiquitous and pervasive application in the study of petroleum reservoir performance, and the references thereof. These are the oiland gas-bearing rock or sand formations containing the earth's supply of these fossil fuels. These formations may be closed (or geopressured, or abnormally pressured) or otherwise connected to other formations, and notably to the aquifers containing fresh and brine waters. For the last-cited reason, a higher percentage of petroleum reserves will exhibit an initial (shut-in) pressure which corresponds to that of the hydraulic gradient of the earth (about 62.4/144 = 0.43 psi per foot, or about 0.46 psi per foot if dissolved brine solids are also involved). Always of prime interest is the extent of the reservoir and the amount of the initial oil and gas originally in place. This provides the economic incentive for proceeding with production. Beyond this are such matters of performance as predicting how the formation will produce, for how long and how much. That is, its deliverability expressed in production/decline at the well. While these matters of size and performance may be approached in a number of ways, it is from the standpoint of unsteady-state behavior that the subject will be emphasized in the treatment herein. This applies both to short-term behavior -- e.g., well-testing, involving drawdown and the stabilized initial flow -and to long-term depletion. In the way of introduction, the origins of petroleum will be briefly reviewed, followed by an overview of alternative techniques used in evaluating reserves.

1.2 ORIGINS OF PETROLEUM The origins of petroleum (oil and gas) are traceable back to the beginnings of the Cambrian Period of the Paleozoic Era, some 600 million years ago in geologic time. Though Precambrian rock formations have on occasion yielded petroleum, the petroleum is generally thought to have migrated from a later

6

UNSTEADY-STATE FLUID FLOW

TABLE 1.1 Geologic Time Scale (Adapted from McGraw-Hill Encyclopedia of Science and Technology, Vol. 6 Copyright 9 1960 by McGraw-HiU Book Co. Reprintedby Permission) Period ..Systems of rocks CENOZOIC ERA Quaternary Tertiary

Epoch Lifeforms Series(r~ks). ........... Holocene Pleistocene Pliocene

Approx. occurrence millions of~s.a_g 9. . . . .

Modem man Early man Large carnivores

10,000 yrs 0. 6 12

Miocene

Whales, apes, herbivores

20

Oligocene

Large browsing animals

35

Eocene

Rise of modem floras

55

Paleocene

First placental mammals

65

Last of dinosaurs

90

MESOZOIC ERA

Cretaceous Jurassic

Rise of flowering plants

140

Toothed birds

155

Flying reptiles First primitive mammals

170 185

Triassic

Rise of dinosaurs

200

PALEOZOIC ERA Permian

Primitive reptiles

245

Spread of amphibians Great coal forest

310

End of spore-bearing plants Abundant sharks

350

Carboniferous Periods U~ Pennnsylvanian) Lower (Mississippian) Devonian

Silurian

First forests

365

Rise of ferns

385

Earliest known amphibians

400

Appearance of land plants First know scorpions

(420)

Brachiopods and corals Ordovician

Primitive fishes

440

Cambrian

Shelled invertebrates

540

PRECAMBRIAN No basis for division

Marine algae, worm burrows, simple forms Abtmdant organic carbon Earth's crust formed .

.

.

.

.

.

.

.

.

.

1,000-1,400 1,800

4:50o .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

PETROLEUM RESERVES

7

source. The occurrences continue on through the Permian Period of the Paleozoic Era. The geologic time scale is reproduced in Table 1.1 In tum, petroleum is also found in later eras, the Mesozoic and the Cenozoic, and in their subdivisions the various periods ranging from the Triassic through the Quatemary of recent times. Petroleum has been found in the rocks of each period, in varying degree, either as the result of metamorphosis of the organic material contained in the rock itself, or as the result of migrations, or both. Further details on the geology of petroleum may be encountered in the several books and monographs extent, some of the best of which have been around awhile, e.g., References 1, 2, and 3. The stratigraphic distribution for petroleum production is indicated in Table 1.2. A feature of the origins of petroleum is that the organic source must have been deposited in a diffuse manner, as organic particles or particulates of one sort or another, and laid down concomitantly and more or less uniformly throughout the sediment. After metamorphosing or being converted to liquids and gases, by whatever earth processes are involved, and if the sediments were suitably permeable, the resulting fluid products migrated to be collected in entrapments of various sorts, also sometimes called "pools" -- a misnomer, since the petroleum will exist in the pores or interstices of the formation rock which constitutes the entrapment. The term "reservoir" is also appropriate, and is used synonymously. By comparison, deposits of coal occur in beds, from myriad plant (and animal) life forms which are laid down in a concentrated fashion, and ultimately undergo metamorphosis or coalification. The coal may be interspersed with veins or particles or minute occlusions of soil or rock, the principal source for the ash content. The first great deposits are of the Pennsylvania Period of the Paleozoic Era. some 350 million years ago. For further reformation on the origins of coal, which also has bearing on petroleum origins, Reference 4 is authoritative. Reference 5 examines the structure and chemistry. Coal also being of organic origins, an reference at least can be made that there may be a further in situ transformation to liquids and gases, under appropriate geologic conditions and sufficient time, which thereby contributes to the formation and makeup of petroleum sources. The organic sources may be regarded as fundamentally of the same composition or basic chemical makeup, organically speaking, being of carbon, hydrogen, and oxygen. (Though the liquid and gaseous conversion products are increasingly of higher hydrogen content and lower oxygen content -- becoming in the end, hydrocarbons.) Therefore it may be more a matter of how the organic material was originally distributed or laid down -- uniformly or in concentrations.

8

UNSTEADY-STATE FLUID FLOW

TABLE 1.2 Stratigraphic Distribution for Petroleum Production (1)

PERIOD Tertiary

Cretaceous Jurassic Triassic Permian Pennsylvanian Mississippian Devonian Silurian Ordovician Cambrian Pre-Cambrian

EPOCH Pliocene Miocene Oligocene Eocene

PER CENT 20 21 7 5 16 1

0.05 5 10 5 3 1

5 1

0.004

PETROLEUM RESERVES

9

And while the d,~ting of geologic time has in the past been based on sedimentation rates, the modem reference is via radioactive dating. Nevertheless, a distraction can be made, or a modifier used, to distinguish between "geologic" time and ordinary time, or, in other words, to differentiate between the time scales so designated. Ordinary time being based on planetary motion, to which radioactive time may be calibrated. The genesis of petroleum is viewed as being of a sedimentary nature -that of microscopic sea life (e.g., plankton), intermixed with the mineral sediments laid down. The sediments proper are in tum classified as elastic if they are of purely inorganic first-origins, or are classified otherwise, e.g., the skeletal remains which become the limestone sediments. A definitive modem treatise on the origins of petroleum is that of Hunt (6), though there have been other forays, and there are still options for speculation (7) -- for example, uniformitarianism versus nonuniformitarianism, or a continuous metamorphosis versus cataclysmic change. The subject is further reviewed as follows. An average sedimentary rock formation, for instance, may contain as much as 0.5% organic material. Some sedimentary rocks in fact may have as much as several percent organic content. And after metamorphosis, migration and entrapment in the pores or interstices of a petroleum-bearing formation, the organic concentration becomes higher yet. Whether the metamorphosis or conversion to oil and gas is of a biological nature due mainly to biological action, and/or a more purely chemical transformation remains partly conjectural, as does the time interval for transition. That is, is the metamorphosis or conversion epochal, or cataclysmic, or both? The metamorphosis may involve reaction with water or steam (as a hydrogen source), or with the mineral content of the rock or other deposits, and may in part be catalytic in nature. As to biochemical action, in some quarters this is viewed more as occurring during the early stages of transformation, and that the succeeding stages are of a dynamochemical nature, at least as applies to coalification (4), and tolerably to the formation of petroleum as well. Other things have been going on as well. Thus natural gas tends to be at deeper levels, whereas the heavier oils and tars tend to be shallow -- and are increasingly oxygenated, suggesting contact in some way with the atmosphere. Moreover, it has been inferred that deep methane may be of abiogenic origin, from inorganic sources or existing as a fundamental constituent of the earth's mantle. Whatever, the processes of metamorphosis or conversion result in the formation of oil and gas, which tend to migrate ever upward, and which in part may collect in geologic entrapments, or be collected at the earth's surface (e.g.,

10

UNSTEADY-STATE FLUID FLOW

asphalt deposits), or be dissipated. For instance, natural gas which migrates to the earth's surface is lost to the atmosphere, a process called outgassmg. It is these phenomena of migration and geologic entrapment, and the associated features of entrapment and other accoutrements, plus losses to the surroundings, which provide clues to the occurrence and size of oil and gas reservoirs. A structure contour map provides an indication, e.g., Reference 1. The entrapments themselves are variously categorized as anticlines, domes, synclines, monoclmes, stratigraphic traps, or by other designators. A schematic comparison is shown in Figure 1.1. An anticline is convex upward, and a syncline is convex downward. A dome is a circular- or oval-shaped anticline, and it has been noted that anticlines and especially domes are the most common entrapments for petroleum. A special case is the salt domes of the Gulf Coast. Needless to say, the entrapments which actually occur are far from ideal in their geometry. Thus representations as either purely linear flow systems or radial flow systems may be far from the mark -- though a necessary simplification from a mathematical standpoint. Such nonidealities call for more abstract or generic representations, whereby correlation (and prediction) maybe instituted so as to accommodate any circumstance. About this approach there will be more to say in forthcoming chapters. Ordinarily the reservoirs or entrapments are connected to an aquifer in some way or another, evidenced by the fact that the reservoir or formation pressure will be the same as would be calculated from the depth using the hydraulic gradient (circa 0.46 psi per foot). Such a reservoir or entrapment would be an open system -- that is the outer boundaries are subject to encroachment by the aquifer as production ensues. Closed systems, on the other hand may exhibit pressures above or below that of the hydraulic head, and are called geopressured or abnormally-pressured reservoirs (8). Among the more unusual of these closed systems are the geopressured aquifers or brines, as encountered in the southem United States along the Gulf Coast, and which contain substantial quantities of natural gas and even oil dissolved in the aqueous phase. These geopressured systems may exist at substantial temperatures, and afford the possibility also of geothermal energy recovery. Another example of an unusual kind of entrapment lies in the occurrence of natural gas hydrates in the arctic regions. The lower hydrocarbons form a solid ice-like phase with water at lower temperatures, which are also called clathrates due to the "crab-like" or "claw-like" nature of their chemical structure. The

PETROLEUM RESERVES

{

i

~

~

11

~ . W A T EIR-i~'_~----"~'L"~''~"~''~ TM

i....~_

1. l a Displacement of rock layers along a fracture

...

1. l b Faults with oil traps

~-:-~----~~~-:~~ y~~---~~~ _~____~~-~_-

1. lc Anticline or dome with trap for oil and gas

__

=---I --=_

_-. ;-~-,~~__~ z.__~~i.~..~~-

1. ld Syncline with oil and gas trap

Fig. 1.1 Schematic comparison of various types of entrapments. (From Oil and Gas Production: An Introductory Guide to Production Techniques and Conservation Methods, compiled by the Engineering Committee, Interstate Oil Compact Commission, University of Oklahoma Press, Norman, 1951). Some of the referenced figures were redrawn from reformation supplied by Professor Leslie W. LeRoy, then of the Colorado School of Mines, Golden CO. Although long out of date, the reference has schematics of a number of other entrapments of interest.

('luo~)) I'I t!,.4

pues o+ OleqS tuo~ uop,~eaO .['I'I Oleqs o~, pu~s tuo~ uop,epea 0 TI'I

d~a~,jooa ouo~ou~."l qI'l

,(~,!ttuojuootm ue ~,OlOq ptre OAOqe sdea~, I!0 :~I'I

offp!a pounq ptre ,(~,.muojuootm ue mOlOq ptm OAOqe sdea~, I!0 JI'I

OmlOOUOlAI o [ I

-~s ~ +~6, ~ x - ~ ~ _ . ~ . . . . . ~ ~ /-/~

A~O"L4G I P I ~ I .q.LV.IS-Af~~LSNTI

[ ~',~ / ~ / - ' ]

ZI

PETROLEUM RESERVES

13

subject is further discussed and analyzed in References 9 and 10. An overview of these unconventional sources, both geopressured aquifers and gas hydrates, is presented in Reference 11.

1.3

TECHNIQUES FOR ESTIMATING RESERVES

Techniques for assessing petroleum reserves range from the inferences of exploration to the realization of discovery, and then to measurement and calculation. The subject of the classification of reserves is reviewed for instance in an article by Garb (12). This particular article classifies reserve estimating methods first by analogy with known and similar fields, and then by the use of volumetric and performance characteristics. An earlier presentation of methods is contained in a collection of papers published by the Society of Petroleum Engineers (13). It may be added that the subject of reserve estimation methods is somewhat veiled, inasmuch as who owns what reserves, and how much, is preferably kept proprietary, for a variety of reasons. More than this, the extent of reserves is also a matter of opinion -- depending upon what methods are used, if any, and depending upon who all is doing the estimating, say the buyer or the seller, the lessor or the lessee, or the owner or the government. The compilation of oil and gas reserve figures for the United States for instance was formerly under the auspices of the Bureau of Mines of the U. S. Department of the Interior. It is now conducted more broadly by the U. S. Department of Energy. The data compilation methods are outlined for example in the publication U. S. Crude OiL Natural Gas, and Natural Gas Liquids Reserves 1983 Annual Report (DOE-EIA 1216 83) published by the Energy Information Administration of DOE. Information is furnished by individual companies, and is compiled by DOE. There are, or have been, such other sources as NRG Associates of Colorado Springs CO and the Potential Gas Agency at the Colorado School of Mines, Golden CO. The American Association of Petroleum Geologists, located at Tulsa OK, maintains a computer data bank titled the AAPG Exploratory Well File Data, with the API well number provided. (The oil and gas commissions of the oil-producing states keep written files of the individual well reports.) There is the USGS Petroleum Data System, or PDS, of the U.S. Geological Survey, and the University of Oklahoma's own Petroleum Data System or PDS called TOTL. Additionally, there is the U. S. Bureau of Mines extensive data compilation, published as "Analyses of Natural Gases" (14, 15). There has, however, been a resulting consolidation, incorporation and crosslinking such that all the afore-cited

14

UNSTEADY-STATE FLUID FLOW

computerized reformation, and more, can be obtained through such large-scale filing and retrieval specialists as Dwight's EnergyData at Dallas, and the Petroleum Information Corporation at Denver. It will also be found, nevertheless, that quantitative reserve information for specific fields (and/or reservoirs) is not otten listed. Moreover, such data as there may exist are for known fields (and/or reservoirs). It is desirable to establish ready and easily-calculated estimates about the reserves for new wells and new fields. Whether this new information need be reported, and where and how, is of course another question. For the purposes here, volumetric and performance techniques are of prevailing interest, and the actual numerical evaluation is assigned to one or another of the following estimation methods: acreage and formation thickness, well tests (flow and pressure), and pressure-production decline. The first mentioned is of a geological nature, whereas the latter are based on physical principles and material balances. Volumetric techniques are generally based on the areal extent of the field as estimated from its geology, and in conjunction with the formation thickness and porosity. In tum there is the matter of the density or specific volume of the reservoir fluid(s) at reservoir conditions, as obtained by sampling the well and by evaluation in the laboratory. This enabling completes the determination of the reserves in place. Such may be classified as a volumetric balance. Performance techniques include first the use of a material balance, which utilizes the equation of state of the reservoir fluid to refer the original fluid in place from the amount of reservoir fluid produced. It is/most applicable to single-phase gaseous reservoirs with closed outer boundaries, whose volume or extent remains constant, whereby the change in average or shut-in reservoir pressure leads to the deduction of reserves, past and present. Beyond the simple material balance are other references which can be made from the flow behavior of the reservoir -- e.g., well-test data which involves patently transient effects and/or long-term pseudo steady-state behavior. Of particular interest are production-decline behavior and pressure-decline behavior, which will provide an empirical introduction to the subject of unsteady-state performance, prior to the mathematical derivations and correlations for unsteadystate flow.

The Role of the Geosciences The geosciences are more closely associated with exploration, i.e., the search for oil- and gas-bearing formations. The methods are both direct and

PETROLEUM RESERVES

15

redirect, varying from the observations and mappings of and at the surface, to the detection of subsurface reformation and depositions, and to the micro-detection of chemical aberrations.

Geology. The first of the earth sciences, geology may be viewed unto itself or as overlapping with geophysics and geochemistry. To begin with, the topography and character of the surface rocks of a region can suggest the existence of monoclmes, anticlines, synclines, and domes -- all of which may indicate the entrapment of petroleum. In addition to these geologic structures, a surface inspection can also indicate faults, also a source for entrapment. The foregoing may be augmented by aerial photography and supported by closer observation. Furthermore, there is the matter of stratigraphic differences, which may be suggested by outcroppings. An impermeable layer of rock above a permeable layer may indicate that petroleum could be entrapped. Contouring and petrographic inspection, however, can only indicate the possibility for occurrences. Such must be backed up by other means, and finally, by drilling. Nevertheless, surface exploration methods provide a guideline, a point of departure, and to some degree give supportive evidence for the magnitude of the reserve if it is there, all by inference from the surface rock deposits and outcroppings and the geologic history, and from the general lay of the land. It is where intuition also plays a hand. Geophysics. As applied to petroleum exploration, geophysical methods include the detection of differences in the behavior of the earth's gravitational and magnetic fields. These differences may be correlated to the presence (and magnitude) of potential petroleum-bearing formations. Both ground-level and aerial reconnaissance can be used. Electrical methods are also used, which generate currents by conduction or induction. It is also a tool used in well logging, to assess what already has been drilled through and into. Perhaps most notable is the use of seismic (sound) waves to indicate subsurface features, either by refraction or reflection. While radioactive emissions may fumish some sort of indirect correlatable reformation of use, it is more a tool of well logging. As an exploration method, it is more akin to geochemical methods. Geochemistry. Geochemistry, among other things, employs minute measurements of hydrocarbons or other compounds to indicate subsurface sources. These indications may occur by migration or leakage. On a larger scale, oil seepages or other macroscopic indications are sometimes a sure thing. Asphalt deposits

16

UNSTEADY-STATE FLUID FLOW

(presumably from the evaporation and/or air oxidation of oil) provide another inference. The technology is rapidly advancing, moreover, and may be expected also to provide an indication not only of a larger presence, but of its magnitude. Inasmuch as it is a newer subject area, an overview of geochemical techniques is shown in Table 1.3, as presented in Reference 6.

Core Samples. The only sure way to determine if oil and gas are present is to drill a well. This may be augmented by taking core samples during drilling, which along with drill cuttings, fumishes an indicator for the presence or absence of petroleum, and also shows the geology for the successive formations encountered - which may likewise give an indication of approaching success or failure. And enough drilling and core sampling through a region will establish the thickness of the formation throughout, and its extent. Furthermore, this will permit determining the reservoir rock properties of porosity and permeability, and establishing the percent saturation with petrolemn. The above information serves to define the field or reservoir and its reserves. Otherwise, if the foregoing is not sufficiently known, well-test data can be used to infer the magnitude of the reserves.

Well Logging. The well may be "logged" during and after drilling in order to identify or substantiate the properties of the various rock formations -- in particular, petroleum bearing formations, if any. These logging methods will include electrical measurements and radioactive measurements (gamma rays). The appearance of the logged vertical section designates not only the rock type but allows for the presence of oil and gas. The relationship is to rocks and their properties -- and to the extent or concentration of hydrocarbons. In final analysis, however, well logs must be correlated against actual core data -- and only then can be used to predict. Nevertheless, in spite of all the foregoing signs being favorable, sometimes the well is simply not a producer.

1.4 RESERVOIRS AND GEOLOGIC PROVINCES As a guide to the geology of the oil and gas producing areas of the United States, for instance, the country has been divided up into tectonic regions, each designated a geologic province and assigned a code number.

PETROLEUM RESERVES

17

TABLE 1.3 Summary of Geochemical Techniques (6) 1. Gas logging (mud logging) 2. Organic carbon 3. Pyrolysis-FID (Flame Ionization Detector) 4. Pyrolysis-Gas Chromatography- Mass Spectrometry (P-GCMS) 5. Pyrolysis-fluorescence 6. C~ - C7 hydrocarbon extraction a. Ca - C7 gas chromatography 7. C~5+ hydrocarbon extraction a. Carbon Preference Index (CPI). Odd-Even Predominance (OEP) b. n-Paraf~ distribution c. Saturate-aromatic distribution d. Mass spectrometry on fractions (polycondensed naphthenes, aromatics, and so forth e. Optical activity of fractions s

Carbon isotopes of fractions

8. Vitrmite reflectance 9. Kerogen color 10. Elemental analysis of kerogen and solid bitumens

11. C4- C~0 gas chromatography analysis of oil 12. GCMS analysis of oil (terpane distributions, C27+hydrocarbons, and so forth) 13. Hempel analysis 14. Carbon, oxygen, hydrogen, and sulfur isotopes of gases and oil fractions 15. Vanadium/nickel ratios

18

UNSTEADY-STATE FLUID FLOW

These petroliferous provinces include the large basins and uplifts in the United States, with the fields in each province related geologically as the age of the producing formations and the time of folding of the rocks. This alternative representation is embodied in the U. S. Bureau of Mines compilations titled "Analyses of Natural Gases" (14), which is also available on magnetic tape or disks (15). Nearly 20,000 gas samples have so far been analyzed, and for each sample the following reformation items are supplied, where known or made available: state, cotmty, field, well name, location, owner, completion date, sample date, formation, geological province code number, well depth, wellhead pressure, projected open flow rate, plus the component analysis of the gas and its heating value and specific gravity (as compared to air). The network of geologic provinces of the United States is shown in Figure 1.2, with the assigned code numbers in Table 1.4 as defined by the Committee on Statistics of Drilling of the American Association of Petroleum Geologists. Computer methodology for applying the code is described in Reference 16. As an example for utilizing the foregoing information, the USBM data previously cited can be used to make an areal estimate of the gas reserves for each common reservoir, defined as follows: A petroleum reservoir will here be defined as having the same field and the same formation--and will automatically have the same province code number. This will be the accepted definition used elsewhere in this work. A common reservoir, altemately, could be defined as that which has the same geologic province code, same state, same cotmty, same field, and the same formation designator, as supplied by the data. In a computer study by Tongish (17), common reservoirs were so delineated. These were established by determining the gas samples which had the same common designators as cited. Furthermore, the gas samples common to each reservoir were averaged to provide a representative gas analysis for each such reservoir. These analyses were further broken down into composition limits. There is an obvious complication, above, that a field and formation, or reservoir, may extend across state and county lines, thus the previous definition is somewhat restrictive, though not frequently so. Accordingly, the use only of common field and formation designators is in general less restrictive. The foregoing procedure can in tum be taken a step further. Each gas sample in a common reservoir will also have a location designation in terms of county (and field), and more specifically, in terms usually of section, township, and range. This will affix the boundaries of a common reservoir in terms of the

PETROLEUM RESERVES

19

sections which encompass a given reservoir. This procedure, then, will more closely identify the location of the reservoir. Moreover, within limits, by knowing the common sections which overlay a reservoir, an estimate can be made of the areal extent of the reservoir. That is, for example, the number of common overlaying sections multiplied by 640 acres per section multiplied by 43,560 square feet per acre will provide an estimated areal extent in square feet. This in tum multiplied by the averaged formation thickness (as obtained from the respective gas sample data), and multiplied by an assumed porosity, will provide an estimate of the total pore volume of the reservoir. The pore volume multiplied by the initial gas density (as determined from the initial formation pressure and temperature) will fumish a value for the initial gas reserves in place. (It is assumed that there is 100% saturation by the gas -- that is, the gas occupies the entire pore volume -- but which may be adjusted.) Thus, continuing in this fashion, the reformation so procured about reserves can be compared say with estimates as obtained from flow procedures and calculations, a methodology which will be formalized in Chapter 13. The foregoing reformation, in part, has in fact been incorporated into a report on subquality natural gas (18), in which the potential of the resource is examined. As outlined, it serves as a method of screening the reserves and composition limits of natural gases containing significant quantities of nonhydrocarbons. An update is provided in Reference 19.

t~

7X)O 963, 964, 965

960, 961,961 395 ~ ) 0 ~ 900, 901,902 L~, 903, 904, 905 725'

'~ 909. 910, 911 :w 912, 913, 914' 915, 916, 917 918, 919, 920 921,922, 923 924, 925, 926 927, 928, 929

325 625

73~ 957, 958, 959 " ~

d~

750J 755 / 76o

930, 931,932 765

/

933, 934, 935

C

936, 937, 938

9 939, 949, 941 951,952, 953 ,P l~ 954, 955, 956 ~o'~ qo "~ on' 0

942,943, 944

Fig. 1.2 Geologic Provinces of the United States [from Moore and Sigler (14)].

PETROLEUM RESERVES TABLE 1.4 Code Numbers of U.S. Geologic Provinces

Code 100 110 120 130

Province

New England Province Adirondack UpliR Atlantic Coast Basin South Georgia-North Florida Sedimentary Province 140 South Florida Province 150 Piedmont-Blue Ridge Province 160 Applachian Basin 200 Warrior Basin 210 Mid-Gulf Coast Basin 220 GulkfCoast Basin 230 Arkla Basin 240 Desha Basin 250 Upper Mississippi Embayment 260 East Texas Basin 300 Cindnnati Arch 305 Michigan Basin 310 Wisconsin Arch 315 Illinois Basin 320 Sioux Uplitt 325 Iowa Shelf 330 Lincoln Aanticline 335 Flrest City Basin 340 Ozark Uplift 345 Arkoma Basin 350 Sooth Oklahoma Folded Belt Province 355 Chautauqua Platform 360 Anadarko Basin 365 Cherokee Basin 370 Nemaha Anticline 375 Sedgwick Basin 380 Salina Basin 385 Central Kansas Uplift 390 Chadron Arch 395 Williston Arch 400 Ouachita Tectonic Belt Province 405 Kerr Basin 410 Llano UpliR 415 Strawn Basin 420 Fort Worth Syndine 425 Bead Arch 430 Permian Basin 435 Palo Duro Basin 440 Amarillo Arch 445 Sierra Grande Uplift 450 Las Animas Arch 455 Las Vegas-Raton Basin 460 Estanci Basin 465 Orogrande Basin 470 Pedrogosa Basin 475 Basin-and Range Province 500 Sweetgrass Arrch

Code 505 510 515 520 525 530 535 540 545 550 555 560 565 570 575 580 585 590 595 600 605 610 615 620 625 630 635 640 645 650 700 705 710 715 720 725 730 735 740 745 750 755 760 765 800 805 810 815 820 830 835

Province

Montana Folded Belt Province Central Montana Uplift Powder River Basin Big Horn Basin Yellowstone Province Wind River Basin Green River Basin Denver Basin North Park Basin South Park Basin Eagle Basin San Luis Basin San Juan Mountain Province Uinta Uplift Uinta Basin San Juan Basin Paradox Basin Black Mesa Basin Piceance Basin Northern Cascade Range-Okanagan Province Eatem Columbia Basin Idaho Mountains Province Snake River Basin Southern Oregon Basin Great Basin Province Wasatch Uplift Plateau Sedimentary Province Mojave Basin Salton Basin Sierra Nevada Province Bellingham Basin Puget Sound Province Western Columbia Basin Klamath Mountains Province Eel River Basin Northem Coast Range Province Sacramento Basin Santa Cruz Basin Coastal Basin San Juaquin Basin Santa Maria Basin Ventura Basin Los Angeles Basin Capistrano Basin Heceta Island Area Keku Islands Area Gulf of Alaska Basin Copper River Basin Cook Inlet Basin Kandik Province Kobuk Province

21

22

UNSTEADY-STATE FLUID FLOW TABLE 1.4 (cont.) U.S. Geologic Provinces

Code 840 845 850 855 860 863 865 867 870 873 875 877 880 885 890 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936

Province

Koyukuk Province Bristol Bay Basin Bethel Basin Norton Basin Selawik Basin Yukon Flats Basin Lower Tanana Basin Middle Tanana Basin Upper Tanana Basin Galena Basin Innoka Basin Minchumina Basin Holitna Basin Attic Foothills Province Artic Slope Basin Maine Atlantic Offshore-general Maine Atlantic Offshore-State Maine Atlantic OffshoreFederal New Hampshire Atlantic-general New Hampshire Atlantic.state New Hampshire Atlantic-Federa Massachusetts Atlantic Offshore-general Massachusetts Atlantic Offshore-state Massachusetts Atlantic Offshore-Federal Rhode Island Atlantic Offshore-general Rhode Island Atlantic Offshore-State Rhode Island Atlantic Offshore-Federal Connecti~xa Atlantic Offshore-general Connecticut Atlantic Offshore-State Connecticut Atlantic Offshore-Federal New York Atlantic Offshore-general New York Atlantic Offshore-State New York Atlantic Offshore-Federal New Jersey Atlantic Offshore-general New Jersey Atlantic Offshore-State New Jersey Atlantic Offshore-Federal Delaware Atlantic Offshore-general Delaware Atlantic Offshore-State Delaware Atlantic Offshore-Federal Maryland Atlantic Offshore-general Maryland Atlantic Offshore-State Maryland Atlantic Offshore-Federal Virginia Atlantic Offshore-general Virginia Atlantic Offshore-State Virginia Atlantic Offshore-Federal North Carolina Atlantic Offshore-general North Carolina Atlantic Offshore-State North Carolina Atlantic Offshore-Federal South Carolina Atlantic Offshore-general South Carolina Atlantic Offshore-State South Carolina Atlantic Offshore-Federal Georgia Atlantic Offshore-general

Code 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 972 973 974 975 976 977 978 979 980 987 988 989 990 991 992 993 994 995 996 997 998 999

Province

Georgia Atlantic Offshore-State Georgia Atlantic Offshore-Federal Florida Atlantic Offshore-general Florida Atlantic Offshore-State Florida Atlantic Offshore-Federal Florida Gulf of Mexico Offshore-general Florida Gulf of Mexico Offshore-State Florida Gulf of Mexico Offshore-Federal Alabama Gulf of Mexico Offshore-general Alabama Gulf of Mexico Offshore-State Alabama Gulf of Mexico Offshore-Federal Mississippi Gulf of Mexico Offshore-general Mississippi Gulf of Mexico Offshore-state Mississippi Gulf of Mexico Offshore-Federal Louisiana Gulf of Mexico Offshore-general Louisiana Gulf of Mexico Offshore-State Louisiana Gulf of Mexico Offshore-Federal Texas Gulf of Mexico Offshore-general Texas Gulf of Mexico Offshore-State Texas Gulf of Mexico Offshore-Federal Califomia Pacific Offshore-general California Pacific Offshore-State California Pacific Offshore-Federal Oregon Pacific Offshore-general Oregon Pacific Offshore-State Oregon Pacific Offshore-Federal Washington Pacific Offshore-general Washington Pacific Offshore-State Washington Pacific Offshore-Federal Alaska Arctic Offshore-general Alaska Arctic Offshore-State Alaska Arctic Offshore-Federal Alaska Bering Sea Offshore-general Alaska Bering Sea Offshore-State Alaska Bering Sea Offshore-Federal Alaska Pacific Offshore-general Alaska Pacific Offshore-State Alaska Pacific Offshore-Federal Minnesota Lake Superior Offshore Wisconsin Lake Superior Offshore Michigan Lake Superior Offshore Indiana Lake Michigan Offshore Illinois Lake Michigan Offshore Wisconsin Lake Michigan Offshore Michigan Lake Michigan Offshore Michigan Lake Huron Offshore Michigan Lake Erie Offshore Ohio Lake Erie Offshore Pennsylvania Lake Erie Offshore New York Lake Erie Offshore New York Lake Ontario Offshore

PETROLEUM RESERVES

23

REFERENCES 1. Lalicker, C. G., Principles of Petroleum Geology, Appleton-Century-Crofts, New York, 1949. 2. Emmons, W. H., Geology of Petroleum, Second Edition, McGraw-Hill, New York, 1931. 3. Oil and Gas Production: An Introductory Guide to Production Techniques and Conservation Methods, Compiled by the Engineering Committee, Interstate Oil Compact Commission, University of Oklahoma Press, Norman, 1951. 4. Moore, E. S., Coal: Its Properties, Analysis, Classification, Geology, Extraction, Uses and Distribution, Wiley, New York, 1940. 5. Hoffman, E. J., Coal Conversion, Energon, Laramie WY, 1978. 6. Hunt, J. M., Petroleum Geochemistry and Geology, Freeman, San Francisco, 1979. 7. Hoffman, E. J., Synfuels: The Problems and the Promise, Energon, Laramie WY, 1982. 8. Fertl, W. H., Abnormal l~brmation Pressures." Implications to Exploration, Drilling, and Production of Oil and Gas Resources, Elsevier, Amsterdam, 1976. 9. Makogen, Y. F., Hydrates of Natural Gas, PennWell, Tulsa OK, 1981. Translated by W. J. Cieslewicz. 10. Natural Gas Hydrates: Properties, Occurrence and Recovery, John L. Cox Ed., Butterworth, Boston, 1983. 11. Hoffinan, E. J., Phase and Flow Behavior in Petroleum Production, Energon, Laramie WY, 1981. 12. Garb, F. A., "Oil and Gas Reserves Classification, Estimation, and Evaluation," Journal of Petroleum Technology, 37(3), 373 (March, 1985). 13. Oil and Gas Property Evaluation and Reserve Estimates, SPE Reprint Series No. 3, Society of Petroleum Engineers of AIME, Dallas, 1970. 14. Moore, B. J. and S. Sigler, "Analyses of Natural Gases, 1917-85," U. S. Bureau of Mines Information Circular 9129, 1987. Updates consist of IC 9167 (1986), ... Information is added annually or biennially. 15. Tape No. PB89-158661, National Technical Information Service. 16. Meyer, R. F., "Geologic Provinces Code Map for Computer Use," AAPG Bull., 54(7), 1301-1305 (July, 1970). 17. Tongish, C. A., "Helium -- Its Relation to Geologic Systems and Its Occurrence with the Natural Gases, Nitrogen, Carbon Dioxide, and Argon," U. S. Bureau of Mines Report of Investigations 8444, 1980, p.6.

24

UNSTEADY-STATE FLUID FLOW

18. Hoffman, E. J., "Subquality Natural Gas: The Resource and Its Potential," on

Investigation of Structure Permeability Relationships of Silicone Membranes, Final Report, by Chi-long Lee, S. A. Stem, J. E. Mark and E. Hoffman, Gas Research Institute, Chicago, 1987. Report No. GRI-87/0037. GRI Contract No. 5082-260-0666. GRI Contract Managers: J. L. Cox, F. Little, and H. Meyer. Report available form the National Technical Information Service. 19. Hoffinan, E. J., "Subquality Natural Gas Reserves," Energy Sources, 10(4), 239-245 (1988).

Chapter 2 PRESSURE/PRODUCTION BEHAVIOR PATTERNS

The production of oil and/or natural gas may, narrowly speaking, be conducted at a constant rate or at a constant bottom-hole or sandface pressure. The one will cause the other to vary. From a practical standpoint, it is much more likely that both will vary over the long-term. Which brings up the point that the production of an oil or gas well will revolve an initial drawdown, followed by stabilization and long-term depletion. The behavior, moreover, will depend upon whether the fluid is a liquid or a gas, or both, and whether internal or external, or endogenous or exogenous, drives are revolved---e.g., inherent compressibility, or a gas cap or exsolution of a gas phase, or an aquifer, or gaseous or aqueous injection. The patterns and factors involved during production will be examined descriptively in this chapter. The complexity of the problem leads to the use of empirical methods for correlations and predictions, and which will also be examined in the following chapter (Chapter 3). The remaining chapters will be devoted to more theoretical approaches to unsteady-state production behavior. It may be added that Morris Muskat's Physical Principles q)c Oil Production, published in 1949 and reprinted in 1981, remains a viable source of practical field reformation (1). It was published about the time that unsteady-state fluid flow studies were beginning, as applied to petroleum reservoirs.

2.1 LIQUIDS VERSUS GASES The production behavior of liquids versus gases will depend to a great extent on the differences in their compressibilities. With a gas, the compressibility is such that production can be sustained in a (geopressured) reservoir whose outer boundaries are closed--albeit an exception to the more usual circumstance of being connected to an aquifer. With a liquid, while there is some degree of compressibility, the situation is such that some sort of drive becomes necessary~such as a gas cap or the exsolution of dissolved gases, or via connections to an aquifer. These driving media in fact are part of, or become an extension of the reservoir.

26

UNSTEADY-STATE FLUID FLOW

Compressibility Efjects The short-term or initial production of an oil or gas well may be described in terms of a drawdown curve. This is a plot of the decline in wellhead or bottomhole pressure with time as the well is first produced, usually at a more or less stable rate. Ordinarily the flowing or producing pressure will tend to level off with time to a more or less constant value, denoting a steady-state or quasi steady-state condition. The effect is perhaps more applicable to gases than liquids, in as much as gases exhibit more pronotmced variations in pressure due to a greater compressibility. Moreover, the effect can occur with gases in a reservoir system whereby the outer boundary or boundaries are closed, whereas with liquids there is the matter of a lesser degree of compressibility, and which to an extent may require an extemal driving force or pressure, in one form or another---e.g., a gas drive or water drive. In the extreme case, with petroleum liquids which would be largely incompressible, there would be no flow unless there was a gaseous or liquid drive by displacement. That is, by a gas cap or exsolved gas, and/or by the hydraulic head of an aquifer.

Pressure Profiles The long-term production of an oil or gas well will generally result in a falling off of the production rate, accompanied by a decline in the average or mean reservoir pressure. These manifestations, however, must be qualified. With liquids, where the density tends to be largely independent of pressure (incompressible), the depletion profile or gradient by definition will be more of a linear nature, thus permitting an extrapolation and interpolation for a mean reservoir pressure at any given time. For the long-term production of gases, there are significant non-linear pressure gradients across or through the reservoir which may preclude an accurate assessment of average reservoir pressure. The obtaining of successive pressuredistance profiles in fact is a principal object of the forays made into unsteady-state flow. The average or mean reservoir pressure would be "after the fact." Such studies, moreover, generally assume a constant production rate. In sum, therefore, the use of pressure/production decline for gaseous reservoir is suspect, even at a constant product rate, and expectedly so at a varying or declining rate. Only if a sufficient stabilization occurs after shut-in can an accurate estimate be made of the overall mean reservoir pressure. Nevertheless, the methodology has its uses.

PRESSURE/PRODUCTION BEHAVIOR

2'/

2.2 MAINTENANCE OF PRODUCTION It may also be noted, that with fluids of low mobility', and/or with little driving pressure in the formation, it may be necessary to reduce the bottom-hole or sand-face pressure to near-atmospheric or sub-atmospheric levels by pumping or compression~where the pump or compressor inlet pressure corresponds to the bottom-hole pressure. Another significant difference between oil and gas wells is due to the fluid densities in the well bore. That is, the gas density will be relatively slight, whereas the density of petroleum liquids may approach the density of the aquifer water or brine. This will of course also relate to the pressure and temperature, and to the gas and liquid compositions. There is moreover the matter of two-phase or multiple phase flow. Nevertheless, in the opposing limits, the pressure-drop or pressure difference between the wellhead and bottom-hole pressures will be relatively low for gaseous systems, and relatively higher for liquid systems. This means that for practical purposes, the wellhead pressure for purely liquid systems may not be radically different from atmospheric pressure. And the bottom-hole pressure will not be radically different from the initial formation pressure. Hence the occasion for pumping.

Pressurization

The foregoing assumes a connection with an aquifer. If such is not the case, and the outer producing boundary or boundaries are closed, then gaseous injection would be required in order to produce the well. Such may of course be used in any event as a means of pressurizing the formation and promoting flow.

Multi- Well Production

The behavior and patterns are of course affected by other wells which are producing from the same formation or reservoir. This is particularly so over the long term. Inasmuch as multi-well behavior can severely complicate the analysis, the methodology will be confined hereto to a single well -- as if it were independent of all others. Moreover, injection wells will not be considered. Even for a single producing well, the analysis is complex enough.

28

UNSTEADY-STATE FLUID FLOW

Shut-In

Interrupted production or shut-in will in tum affect future performance, depending upon the length of shut-in and upon the degree of stabilization of the pressure profile across or throughout the reservoir. With gases, particularly, there is a pronounced transient behavior. Saturation The presence of original or connate water in the reservoir proper can contribute to and change the flow characteristics. In tum, the encroachment of water or brines can have a significant effect, also causing a change in the flow characteristics, even bypassing the hydrocarbons. The phenomena are characterized under the general term of degree of saturation (the relative proportions of hydrocarbons versus water).

2.3 RESERVOIR PRESSURES Pressure Within and across a reservoir will show variation, depending upon shut-in and flowing conditions.

Bottom-Hole Pressure

The bottom-hole or sandface pressure will depend on whether the well is shut-in or flowing. The shut-in pressure difference is purely a function of pressure-density or pressure-volume behavior and depth, as related to the wellhead pressure. The bottom-hole flowing pressure will in part depend upon the deliverability of the reservoir and in part upon the flowing head in the well bore. The flowing head depends not only upon pressure-volume behavior and depth but upon velocity and frictional effects. The maintenance of flow may be natural or may be reduced. And flow may be reduced by lowering the bottom-hole pressure by pumping or compression of the product stream(s), or by the injection of a gas or liquid (water) under pressure at one or more points in the formation. For the purposes here, the use of injection wells will not be considered, albeit they can be regarded as a means for sustaining the formation pressure, preferably at the outer boundary or radius of drainage. In the case of water injection or flooding, the processes of injection can be viewed as a means for contributing to the effects of the aquifer, as will be developed in Part IV.

PRESSURE/PRODUCTION BEHAVIOR

29

Hydraulic Head In the production of gases or liquids via a water drive, natural or reduced, the pressure at the interface between the petroleum fluid(s) and the water drive may be equal to or less than the hydraulic head. This will depend upon the mobility of the water drive. There is also the consideration that the flowing well works against the head of the fluid reside the well bore. This can mean that the pressure during flow can be adjusted to the head at the bottom of the hole. Depending upon the density of the flowing fluid in the well-bore, and upon frictional losses, this can be very nearly equal to the hydraulic head. Thus in the extreme, the actual reservoir pressure minus the hydraulic head or well-bore fluid head can become the driving pressure. This is of significance in the production of liquids, but of less significance in production of liquids, but of less significance in production of gases due to the much lower density of gases (except conceivably at very high-pressures).

Geopressured or Abnormally Pressured Systems Petroleum gases and liquids, however, may be geopressured~that is, not connected to an aquifer~whereby the formation pressure may be greater or may be less than that represented by the hydraulic head (2, 3). The behavior will therefore be independent of hydraulic or aquifer pressure. There may also be the circumstance where the pressure is originally greater than the hydraulic head, but as production ensues a connection or connections are made with the natural aquifer, and encroachment ensues. This conceivably can occur by the release of blockages which act as check valves. Or the normal processes of drilling and completion may cause a connection of sorts to be set up. There is also the situation where encroachment is so slow (or even negligible) whereby the interface pressures at the reservoir boundary may start out equal to the hydraulic head, then fall to less than the hydraulic head. In the limit is the case where the outer boundary or boundaries remain closed.

Average of Mean Pressure,/Production Methods The extent of a reservoir in terms of its average or mean pressure can be assessed via the P-V-T behavior of the reservoir fluid(s) and an overall material

30

UNSTEADY-STATE FLUID FLOW

balance. At the same time a record is produced of production versus the average or mean pressure of the reservoir.

Production Decline Methods

The long-term production decline behavior can be projected by an empirical curve fit of the data to any one of several forms, the best fit assumably providing the most accurate projection of prediction behavior. To complete the circle, the production decline may then be related to the average or mean reservoir pressure via the overall material balance. In substance, the combined methodologies more closely represents empiricism than the methods to be developed from the concepts of flow. Finally, it can be anticipated that the behavior of pressure and production with time will somehow exhibit the warp and sweep characteristic of exponentialtype or logarithmic-type curves. That is, in one fashion or another, straight line plots ca be expected on semi-log or log-log plots of one variable or combination against another. These plots, apparently purely empirical in nature, nevertheless are of considerable use in the interpolation and extrapolation of decline behavior. Of the potentially may possibilities, three methodologies will be examined in the next chapter: geometric production-decline, production-time decline, and production-loss ratio. All relate to production decline behavior, and extensions can be made to pressure decline. More rigorous representations will be made in subsequent chapters, based on the equations or relationships for unsteady-state fluid flow through porous media.

2.4 RESERVES AND DEPLETION TIMES In a classic study by M. King Hubbert (4), it has been predicated that the discovery and production rates from a resource tend to follow a Gaussian-type distribution with time, whereby the cumulative discoveries and cumulative production tend to behave as the corresponding ogives. This is shown schematically in Figure 2.1. The difference between the cumulative discoveries and the cumulative production is the (net) proved reserves at any given time, and presumably will also follow a distribution curve, as indicated. If the discovery rate and production rate can in fact be approximated by a Gaussian or normal distribution, it may be written that for the rate, here designated as q,

PRESSURE/PRODUCTION BEHAVIOR

CUMULATIVE DISCOVERIES,

31

Qd

8

o

0

Time

+

o

Fig. 2.1 Distribution curves [from Hubbert (4)]. From" RESOURCES AND MAN by Cloud 9 1969 by W.H. Freeman and Company. Used with Permission.

32

UNSTEADY-STATE FLUID FLOW dQ q-~=ae dt

_h(t_trn)2

: a exp[-h(t - t,. )2 ]

where here Q is the cumulative discoveries or the cumulative production, as the case may be, and tm is the corresponding median value. The distribution constant r will take on different values, whereby tt = Cta~r and r = %rod , as will the distribution constant h , whereby h = hd~ and h = hp~od. Furthermore, tm takes on the values tm= (tm)di~ and tm = (tm)p~od. When t = tin, the rate will have a maximum value. Note that at t = 0, - a e x p [ ( O - t~) 2 ]

q -

whereby a-

qt:o - qt:o exp[h(-tr.)

2]

Therefore, if (-tin)2 : t m 2 , q - q,:o explh(-t.,) 21 exp[-h( t2 - 2ttm +tm 2 )]-- qt:o e x p [ h t ( 2 t m - t)l

If 2tin>> t , then the rate increases exponentially with time. Integrating, t

Q - ~ a exp[-h(t - t , . ) 2 l d t 0

where at t = t f , the depletion time, the cumulative discoveries will equal the cumulative production. That is, Qtotal =

(Qdi~o)tot.1 =

(Qprod)total

or Qf:

(Qdtsc)f- (Qprod)f

That is, the total of the reservoir fluids originally in place will by definition equal the total cumulative discoveries and will also equal the total cumulative production. Note that the total reservoir fluids originally in place are always greater than, and not equal to, the cumulative proved reserves. Proved reserves represent the difference between cumulative discoveries and cumulative production.

PRESSURE/PRODUCTION BEHAVIOR

33 I

I

120~

i 8O

,-~ x~ 0 0

40~ t

1900

1920

l

~

1940

1960

l

'980

Years

Fig. 2.2 Distribution data for cumulative production (Qp), cumulative proved discoveries (Od), and proven reserves (Q0 for the United States up into the mid1960's [from Hubbert (4).]. From: RESOURCES AND MAN by Cloud 9 by W.H. Freeman and Company. Used with Permission. The Gaussian distribution may be further operated upon as follows: Q-

i

a exp[-h(t

- t~)2 ]dt -

0

z=t~tm a

exp[-h(z) 2 ]dz

0

1 2 / 4 ~ ~--'{~z 2 4%2/,f~ o expI-h(8)]ao

~

a erf[x/h (t - t~)]

Thus the cumulative discoveries and the cumulative production can be expressed in terms of error functions. The cumulative behavior for crude oil in the United States is shown in Figure 2.2, up until the mid 1960's, and the rate behavior is shown and projected in Figures 2.3 and 2.4.

34

UNSTEADY-STATE FLUID FLOW I

I

I

I

I

I

I dOd~dr 4-

I

I

I

I

IA

3 ~-

%

_

'LPmoo.,

,-

"

"F~o~.

L/s'~ '"

~.,;,..~..

I

"'-...... ]

"~,

o t

,

1

1900

I2

l

19 0

,

I

1

1940 Yeors

I

1960

1980

Fig. 2.3 Rates for proven discoveries (dQd/dt), production (dQp/dt), and reserves (dQd/st) derived from Figure 2.2 [from Hubbert (4)]. From: RESOURCES A N D MAN by Cloud 9 by W.H. Freeman and Company. Used with Permission.

i2o•

, 5-~, 4

-I

,

/

i Q~ = 165 X 109 bbls it

|

~

o

80 PERCENT(65 YEARS)

I I

I

t

I

l

,

ii

I

~.1~,~ "//////,.v/~/,

i],,,

i

~.-~X.~,~}~BlXlO9bbls]i bbls ! o

1860

. _

1880

~

,_ _ .~...,~--. ~

1900

1920

1940

,

,1

t t

,4~..--~~.

-J,

I'

,

-

I

~

1960

~

1980

'| ""-. 29 X I09 bbls

,,

2000

I

---' 2020 2040 ~

.

.

.

.

.

2060

Years

Fig. 2.4 Cycle of U.S. crude production [from Hubbert (4)]. From: RESOURCES AND MAN by Cloud 9by W.H. Freeman and Company. Used with Permission.

PRESSURE/PRODUCTION BEHAVIOR

35

Exponential Increases It has been previously developed that at leas~ initially the rate of production can be represented by

dt

- q ,:-o explht(2t~ )l - q,:o explk tl

where k = h(2tm) and t is small. There is the indication that such an exponential rate of increase will be sustained, and Mokhtar M. El-Gassier has carried out the calculation to its logical conclusion (5). This is further developed as follows. Thus rearranging and integrating between limits, t

Q- qt=o~eUdt- ..qt=~ k 0

- 11

or

Q,o,o, k -

[e k' - II

Qt = 0

Altemately,

1

Q

t - -- In[ k + 1] k q,:o or

1

tI - ~ l n [

Qtot.l

k + 1]

qt =o The units of k are reciprocal time. The value of k may be determined at some point in the production history. The calculation will be trial-and-error. If, however,

q, io k a first approximation for the logarithm is

p

>l and Ct2/Ro >> 1, 1-C

1-C

This expression may be further utilized to characterize the production behavior.

Example 3.3 From Table 3.1, at the corresponding years, the cumulative production is as follows: Wl = 1000 bbls WE = 1500 W3 = 1833 Therefore, 1-C

15 oo - 1 ooo

1833 - 1500

= 1.5 o -

(1)

1-C

c -(2) 1-C

(2)

c 1-C

- (3)--

where C is theoretically to be positive and less than one. By trial-and-error it will be found that in this particular ease C -~ 1.25, which indicates a deviation from the original assumptions. Assume, however, for the purposes of simplicity that C ~ 1, such that

W_(~_t___] Ro lnR0 +_______ft \dt) o Ro

66

UNSTEADY-STATE FLUID FLOW

whereby In 1+

w~ w,

( 1t

In l + R o If Ro is very small, then

w~ and 1500 ~ = 1000

1.5~

Solving the above, Ro = 0.25. Lastly, since W R o In R~ + 1 Ro

0

then say at t = 1, 1000

0

0.25+ 1 0.251n ~ ~ 0.25

= 2485 bbls per year

Based on t - 2 ,

1500

I.~tW~ o -0.251n

0.25+ 2

= 2731

0.25

At t = 3 , 1800 _

0

0.251n

0.25+ 3 0.25

=2859

PRESSURE/PRODUCTION DECLINE

67

As a check, at say t = 6 years,

W6 - 2485 (0.25)In

0.25 + 6

= 2485 ( 0 . 8 0 4 7 ) - 2 0 0 0

bbls

0.25

The given value is 2450 bbls. Adjustment could be made say by calculating (dW/dt)0 based on t = 2 or t = 3 for the starting time.. Finally, since C is close to unity, the calculation for Wf becomes largely indeterminate.

3.5 PRESSURE DECLINE The material balance for a closed single-phase reservoir (no water drive) in consistent traits may be expressed for instance as W = VXp o - VXp

(41 )

where W = cumulative production at time t. V = reservoir volume X = porosity po = initial density (in consistent W units) p = density at time t ( in consistent W units) k is assumed, moreover, that the density is tmiform throughout the reservoir. That is, p = p. The density in tum may be expressed as a function of the pressure P . A formula used for compressible liquids is p = A e ~v where A is a characteristic constant and c is the liquid compressibility. The equation of state for a gas can be written as

p

MP ~

zRT where, in consistent units, M = molecular weight z = compressibility factor R = gas constant T = absolute temperature

68

UNSTEADY-STATE FLUID FLOW

Thus in a manner of fashion the pressure can be related to W, the cumulative production. For compressible petroleum liquids, it follows that

Ae ~Po _ A e ~ =

W VX

or

(42)

p _ l In [e ~P~ _ W__W_] c

ViVA

For a gas, at constant z, utilizing the material balance, W

zRT

VX

M

P0 w p ~ ~ or

P - Po

W

VX

.

zRT

M

(43)

Thus, for a gas, the pressure decline would tend to be linear with cumulative production. In the instances above, the reservoir fluid is assumed either a liquid or a gas, but is not comprised of both. The symbol W denotes the cumulative mass production of either the liquid or the gas, and could be appropriately subscripted, say WL for liquid production and Wv for gaseous or vapor production. The analysis becomes more complicated if two phases coexist in the reservoir, say a liquid phase L and a gas or vapor phase V. The cases of interest are for the production of both phases simultaneously, and for the production of the liquid phase alone. Of less interest would be the production of the gas phase alone. For the purposes here, liquid phase density will be regarded as constant. The material balance for the gas phase will read:

M

M

X(I,~, )o zR T Po - XVv zt('-"V P - ~" where (Vv)0 = initial reservoir volume occupied by the gas phase Vv = final reservoir volume occupied by the gas phase Wv = cumulative gas production of the gas phase

(44)

PRESSURE/PRODUCTION DECLINE

69

The material balance for the liquid phase will read: x ( ~ ~ )o ( ; ~ ) o - x ~ ~p,~ - w~.

(44a)

where

(VL)0 = initial reservoir volume occupied by the liquid phase VL = final reservoir volume occupied by the liquid phase (PL)0 = initial liquid density PL = final liquid density For the purpose here, PL = (PL)0.

Furthermore, for the total reservoir volume, which is a constant,

That is, as VL becomes smaller with production, Vv becomes larger. It readily follows that, on solving for VL,

Vv -VT

-

[x(V)o x(PL)o

(45)

From the gas phase material balance, solving for P,

P =

M zR-T Po M

(46)

Xt~, zRT where Vv is defined above. This expression represents the pressure-decline curve for the simultaneous production of gas and liquid. The special cases follow. Thus Wv = 0 for a gas cap drive. Whereas W L= 0 if no liquid is produced. Etc.

70

UNSTEADY-STATE FLUID FLOW REFERENCES

1. Calhoun, J. C., Fundamentals of Reservoir Engineering, University of Oklahoma Press, Norman, 1953, 1960, p. 282 ft. Reprinted from the Oil and Gas Journal. 2. Lalieker, C. G., Principles of Petroleum Geology, Apple~on-Century-Crot~s, New York, 1949. 3. Johnson, R. H. and A. L. Bollens, "The Loss Ratio Method of Extrapolating Oil Well Decline Curves," Pet. Dev. and Tech., 771-778, 1927. American Institute of Mining and Metallurgical Engineers. 4. Emmons, W. H., Geology of Petroleum, Second Edition, McGraw-Hill, New York, 1931, p. 147.

Chapter 4 CONCEPTS OF FLOW

4.1 UNSTEADY-STATE FLOW AND COMPRESSIBILITY Unsteady-state fluid flow, in practice, generally revolves what are referred to as compressible fluids, whether liquids or gases or some combination. Even what are thought of as liquids contain a degree of compressibility, and which will be of significance in the flow behavior of a large body of liquid. With gases of course, the effects are much more pronounced.

Compressibility By compressibility will be meant changes in density or specific volume caused by changes in pressure (at constant temperature). These variations in density or specific volume can be viewed as what cause the liquid to expand and "flow" when the pressure is reduced at some point. At the same time an unsteadystate or transient behavior will be set up if there is no corresponding and equal mass inflow to maintain a steady-state condition. "Incompressible" fluids are to be excluded, by definition, there being no change in density with pressure, and in this sense cannot "flow" under a pressure gradient. If density truly remains constant with a change in pressure, then the substance is not a "fluid." (Even liquid mercury has a slight compressibility.) The designation steady-state is used to denote that the density or pressure does not change with time at any particular point in space. It does not refer, however, that the density or pressure does not change with position. Although for the purposes of derivation or calculation a constant or mean value for fluid density may be assumed, nevertheless the concept of flow or fluidity requires that density vary with pressure. For density difference is as much the driving force for flow as is pressure difference---though the density difference may be minutely small. If the density is defined as absolutely constant, then there can be no variation in velocity with position nor with time, and no flow in the strict meaning of the word. The notion of viscosity will in fact vanish. Enter the idealization of a completely rigid body.

76

UNSTEADY-STATE FLUID FLOW

Gases versus Liquids

Strictly speaking, the terms "liquid" and "gas" or "vapor" do not have a distinguishing meaning unless in juxtaposition or better yet in equilibrium with one another. For by a change in pressure and temperature, the phase behavior of what is considered a gas may be changed to that of what is considered a liquid-without the system undergoing any phase change. (this is accomplished by circumventing the two-phase envelope.) Thus, when referring to the single-phase region, it is more appropriate to speak of a "less-dense single phase" (thought of as a gas) and "a more-dense single phase" (thought of as a liquid). In tum, the compressibility behavior or P-V-T behavior will change from the less-dense region to the more-dense single phase region. The former is more appropriately described or correlated with the several forms used for the equations of sate of a gas---e.g., the concept of the compressibility factor~whereas the latter is more readily correlated with the exponential form generally used revolving the compressibility coefficient (for a liquid). In principle, however, either or all could be used for either the less-dense (gaseous) regions or the more-dense (liquid) regions, though the compressibility factor "z" or the compressibility coefficient "c" would exhibit wider than usual variations or departures from the more ideal circumstances.

Petroleum Gases and Liquids

For petroleum reservoirs, the gases revolved are various hydrocarbons and non-hydrocarbons (such as nitrogen, carbon dioxide, and hydrogen sulfide), and the gases may exist in the gaseous state proper and/or be dissolved in any liquid phase(s) present. The degree of solution or exsolution will depend upon the temperature, pressure, and composition of the system (the phase behavior), and will expectedly change during flow. The presence of a gas phase is allied with the maintenance of pressure during production, and is usually encouraged, by limiting the production rate and/or injecting or re-rejecting gases. The petroleum liquids found will be principally hydrocarbons of diverse molecular weight and structure, and with heavier crudes, oxygenated constituents increasingly appear. Trace elements also may appear, such as sulfur and even metals or metallic compounds such as vanadium, free or combined. Also, more than likely water or a water-rich phase will be present, usually carrying dissolved solids and referred to as a brine. This water or water-rich

CONCEPTS OF FLOW

77

phase may coexist originally in the reservoir proper as connate water), or may encroach or flow into the reservoir during production, either from a natural by a reduction of the reservoir pressure during production, or may be reduced artificially by injection. All of the foregoing contribute to the complexity of correlating and predicting well and reservoir performance, which here is intended also to convey information about the reservoir extent and reserves.

4.2 FLOW SYSTEMS AND DISSIPATIVE EFFECTS The flow of a fluid may be characterized as Eulerian or Lagrangian. These concepts are further discussed in References 1 and 2, but in brief may be distinguished as follows.. In Eulerian flow, the fluid is regarded as a continuum, and flow is referenced to a spatial element of volume. Flow occurs in and/or out of the element of volume, and fluid may accumulate or be depleted within the element of volume. It is the basis for the so-called equation of continuity. The flow geometry may be viewed from the perspective of a differential element or from an integral element of volume. In Lagrangian or point particle flow, the reference system is the flowing element of fluid itself. It is akin to the point particle concept as developed in analytical mechanics, and as pertains to the laws of motion for a rigid body or point particle.

Energy Balance

The generalization of point particle behavior to fluids leads to the classic energy balance, representable in the differential form as follows for a flow system of unit mass: dq-dw

- ~dv2 + g dz + V dP + dlw 2g~ g~

where the terms are in consistent pressure-volume work units, or mass-distance units, and where it may be noted that multiplying through by go will convert the units to force-distance. Dividing by the mechanical equivalent of heat (or multiplying by the heat equivalent of work), will convert the units to thermal units. (Point particle behavior is Lagrangean flow, whereas Eulerian flow applies to continuity balances for mass or energy.) The variables are defined as follows:

78

UNSTEADY-STATE FLUID FLOW

q = heat added to system (per unit mass) w = work done by system (per unit mass) v = velocity z = elevation g = acceleration of gravity g~ - gravitational constant V = specific volume P = pressure It may be noted, furthermore, that dividing the above equation by the mechanical equivalent of heat will convert the units to heat units. The quantity lw is called the unit lost work (lost work per unit mass), and may be expressed with a bar above the symbol. It is also called the intrinsic energy o~ (or f2), in consistent units. For a differential change, d l w = dco - T d S

in consistent units, and where the above constitutes the definition for the entropy change dS (per unit mass). It is a mandate in the study of fluid flow behavior to correlate lw to the flow geometry and fluid properties, in particular viscosity.

Viscosity

The concept of shear viscosity arises from the flow of fluid tangentially to a surface (2). The coefficient of viscosity IX relates the velocity gradient normal to the surface to the applied force causing flow. Thus

where in free flow FT will revolve the gravitational force and density, and may be further related to forced flow (2). The solution of this relationship subject to the geometry of the contiguous surface or confining surface will relate velocity to the dimensions of the particular

CONCEPTS OF FLOW

79

system---e.g., Stokes' law for fluid flow relative to a sphere and Poiseuille's law for fluid flow through a cylinder. Etc. Ideally the viscosity coefficient should be independent of the contiguous or confining geometry. Viscous behavior, however, is a function of whether the fluid is a gas or liquid, and of temperature, pressure and composition. Moreover, with regard to liquids, there is a transition to what may be considered "glassy" or viscoelastic materials, in which a liquid on cooling becomes increasingly viscous without crystallization occurring. This rheological region, in which the material displays both elastic and ~ascous properties and is cohesive, is perhaps best characterized by the concept of elongation flow and elongational viscosity (2). Here, the coefficient of elongational or Trouton viscosity relates the applied force to the velocity gradient in the direction of flow or elongation.

Correlation o f Dissipative Effects

The correlation of lost work or dissipative effects, or by whatever name called, is commonly based on some form of a friction factor relationship. This generalization is intended to encompass both turbulent and laminar or viscous flows, or flow regimes. Moreover, on simplification to the conditions for laminar flow, the relationship is required to reproduce the particular viscous relation for the system geometry -- e.g., derivations based on Stokes' law or Poiseuille's law. A friction factor relationship commonly used is (2, 3) 1)

2

dlw = f ~ dL 2goD

where f = friction factor L = distance in the direction of flow D = characteristic system dimension Other friction factors are also used, e.g., such that f = 2f' = 4f'. For flow through packed beds of particles, D would be the particle diameter D o . For flow though a pipe or tubing, D would be the reside diameter. The behavior of f would be correlated accordingly. Either of the above can suffice as the model for flow through porous media. Flow through packed beds would perhaps more closely correspond to flow through the interstices of unconsolidated media. Whereas flow through tubing or capillaries would more closely correspond to flow through the pore spaces of consolidated media.

80

U N S T E A D Y - S T A T E FLUID F L O W

In any event, for laminar or viscous flow, the friction factor can be correlated to c Re where c is a constant depending on the flow system configuration, and the Reynolds number Re is given by

Dvp

Re-

,u

DG ,u

where p is the fluid density and G is the mass flux. Thus it will follow that, for laminar or viscous flow, C

dlw -

V

2

. ~ d L 2g~D

Dvp

/a

_- ~ c 2g~D z

_l't vdL P

or

2 g ~D 2

1

c

l.t

1

-K.--

dlw p ~ dL

dlw p ~ dL

,u

where K can be designated as the permeability. In turbulent flow, the relationship between the friction factor and Reynolds number maybe expressed as the function, f = f (Re). Over an interval, the approximation a f

=

Re b

c or

Re n

may be used, where "a" (or c) and "b" (or n) are constant over the interval. The relationship for velocity will thus be more complicated in the turbulent region. This may be accommodated by adjusting the value for K or for 9 . Since the viscosity tx is ordinarily determined independently by a separate

CONCEPTS OF FLOW

81

test under laminar conditions, it would seem more appropriate to adjust K. However, K is also determinable independently (for laminar conditions), and it is the preferable practice therefore to speak of changes in viscosity with the onset of turbulence. More explicitly, or substituting, the relation f = e/Re" , then

dlw -

C

V

2

~ dL 2g~D

C

~n ~

2 g ~ D "+1

p"

v2--ndZ

or

1

V-

~t~

C

where for laminar flow n = 1 , and for horizontal flow dlw = - (l/p) d P . With this interpretation, the generalization of Darcy's law to turbulent flows becomes considerably more complex.

4.3 DARCY'S LAW The derived expression v-~

K it.

p

d/w dL

relates velocity to dissipative effects via a coefficient K which can be defined as the permeability, with the ratio K/~t called the mobility. In general, the above form can be assumed applicable to any flow system geometry. By substituting for dlw from the energy balance and neglecting kinetic energy effects or changes (which are generally relatively small), K v-rap.

It

82

UNSTEADY-STATE FLUID FLOW

If the system is regarded as isolated, then K v

-

~

- g / g~ d z - VdP p.

p

dL

n

where V = l/p and the ratio g/go is the multiplier or coefficient for dz. If flow is also horizontal, then K

dP

p

dL

1J--

This may be regarded as the special form that is Darcy's law (d'Arcy). The above form, which refers the steady-state, can be generalized to the tmsteady-state, whereby V----~"

P If flow is opposite in direction to increasing L , and dropping the multiplier sign, the convention can be used that K

dP dL

or

V -- 7

t

Multiplication by the density will yield the mass flux G , and multiplication by both density and crossectional area normal to flow will yield the mass flow rate Q or Qm. Substitution into the differential equation for continuity will yield the classical equations for fluid flow, e.g., through porous media. The derivations will be presented in the next chapter.

Radial bbrm The form of Darcy's law for radial flow is of prevailing interest here. The mass rate of flow is given by

CONCEPTS OF FLOW

83

Q - K (2nr)AhpdP p dr where ~ or h is the thickness normal to flow, and flow is inward toward the radial origin (dP/dr is positive). At the steady-state, Q is constant. Thus for the steady-state condition, integrating between limits for a fluid or liquid of assumed near-constant density, the mass rate of flow would be linear with the pressure difference or pressure-drop AP 9

- (;aP where C represents a proportionality constant obtained via the integration. For an ideal gas at steady-state, the mass flow rate would theoretically be proportional to the difference in the squares of the pressures:

Q _ c(

o2)

where C again represents a proportionality constant, albeit of a different value than for liquid flow.

Back-Pressure Tests More generally, as the result of so-called back-pressure tests on gas wells, it is found that the data for successive differences between the squares of the initial reservoir pressure and bottom-hole flowing pressure are more aptly represented by

where

a ? ' -P0 Here, the subscript "o" denotes the initial condition, and "a" the bottom-hole or sandface.(of radius r, ). Extrapolation to open flow is made whereby P~ -~ 0. The above relationship presumably can account for deviations from laminar flow via the exponent n. The observation is also made that the constants (or coefficients) will vary with the time of the test (3, 4, 5, 6). That is, for low permeability formations, the bottom-hole flowing pressure can be very slow to stabilize, with the pressure disturbance or effective radius of drainage (r0 moving outward only very slowly.

84

UNSTEADY-STATE FLUID FLOW

(This corresponds to the intercept of the profile along p0 or Po, as will be shown in a later chapter). The above indicates that the transient drawdown behavior would more properly serve to determine reservoir characteristics than would the back-pressure determmation~smce the back-pressure behavior is also transient.

Units The steady-state linear flow relation for Darcy's law in terlns of mass flow may be written as follows: K dP 0 - --Ap p dx where, in consistent units, he variables are defined as follows, with preference given to English units: Q = mass flow rate (lb/hr) K - permeability (ff3/hr2) = fluid viscosity (lb/ff-hr) p = fluid density (lb/R 3) A = area (if2) P = pressure (lb/R 2) x = distance (feet) opposite to the direction of flow Flow is assumed to occur in the (-x) direction (dP/dx is positive). At the steadystate, Q is constant. Alternately, in terms of the mass flux G , or

G - 0"-" - - -kp - - dP A It dx

(lb / h r - ft 2 )

or in terms of the velocity v , v-

K dP

(ft / hr)

pdx Etc. The conversion between the viscosity in centipoises and in lb/ff-hr is ~t (in lb/ff-hr) = 2.42 tmaes ~' (in centipoises)

CONCEPTS OF FLOW

85

Note also that P (in lb/ff 2) = 144 times pressure in lb/in z For gaseous flow systems measured in MMSCF/D (millions of standard cubic feet per day), then O(inlb/hr) -

=

MMSCF/D 24

06 1 (1)379 M

where M (or MW) is the gas molecular weight (or gas specific gravity times 29). The term K/~ may be incorporated as the mobility, and can also be called the modulus of flow or flow modulus. Alternately, Kp/~t can be incorporated into a single term. Darcy's law, as customarily used in the petroleum and natural gas industry, employs other units. That is,

K' dP'

,11' dx' where v' = velocity (in cm/sec) K' = permeability (in darcy's or darcies) ~t' = viscosity (m centipoises) P' = pressure (atm) x' = distance (cm) opposite to the direction of flow The above relationship in fact serves as the definition for measuring the permeability in darcies. The more common unit, however, is the millidarcy, whereby K" (in millidarcies)= 1000 K' (in darcies) In terms of the mass flux, where p is in gwJcm,

K'

G'- ~

/a'

dP'

p'~

(gm / sec - cm 2 )

dx'

and in terms of the mass flow rate, here designated as Q' in cgs units, file above equation may be rewritten as

86

UNSTEADY-STATE FLUID FLOW K' O'- ~p' ,u

dP' A'~ dx'

(gm / sec)

where A' is in cm 2, and p'in gm/cm3. Etc The conversion between the permeability in R3/hr2 and in darcies has been derived elsewhere (7): K (in ft3/hr2) = 0.00442 K' (in darcies) This conversion is convenient m the field application of Darcy's law. [The performance of membranes in the separation of gases is correlated in terms of a membrane permeability "P", defined by AP v-" P"~ Ax Thus membrane permeability is the equivalent of reservoir or formation mobility: t! p t !

_..

K

where K denotes the reservoir or formation permeability. Whereas the formation mobility K/I~, in darcies per g/cm-sec, will have the units of cm3/sec-cm2-atm/cm the membrane permeability "P" customarily has the units of 10-9 times cm3/sec-cm2-cm Hg/cm This is a much smaller unit of measure, the multiplier between the above sets of units being 10 -9 (1/76) - 1.316 x 1011

That is, if"P" is measured in 10-9 times cm3/sec-cm2-cmHg/cm, then multiplying its value by 1.316 x 10~1 will yield the formation mobility in darcies per g/cm-sec. And multiplying the mobility by the fluid viscosity ~t m g/cm-sec (or poises) will yield the formation permeability in darcies.]

CONCEPTS OF FLOW

87

Liquids For liquids, since QorQL -

K

dP

Ap(-) ;,

t/L

P

where L is positive in the direction of flow (dL = -dx), then it will follow that the following relationship will be obtained:

Q _ K Ap(_) AP t.t AL = (PI)(-AP)

where (PI) becomes a correlating constant or coefficient, and may be designated the productivity index, and the concept may be extended to radial flow.

Gases For gases, if it is substituted that by virtue of the equation of state,

p - MP zRT where z is the compressibility factor and M the molecular weight, then

Q or Qr - K A M P

7 zRr(-)

dP

At constant z, then p2 becomes the variable:

Q_KA1

.

M

- zRT

dp 2 dL

On integrating, at steady-state,

Q_KA1 M ( _ ) ~Ap 2 - C ( - ~ I.t 2 zRT " AL

2)

88

UNSTEADY-STATE FLUID FLOW

where C becomes a correlating coefficient or constant for linear flow. The foregoing may be generalized to the form Q = C ( - A P ~ )" where n is a constant. This is of the same form as the back-pressure equation used to correlate back-pressure tests on gas wells, as previously noted. Presumably the exponent n will serve to accommodate velocity and frictional effects, or other effects, as will be discussed further in Chapter 8 for radial flow behavior.

REFERENCES 1 2. 3.

4. 5.

6. 7.

Hoffman, E. J., The Concept qf Energy." An Inquiry into Origins and Applications, Ann Arbor Science, Ann Arbor MI, 1977. Hoffman, E. J., Heat Transfer Rate Analysis, PennWell, Tulsa OK, 1980. Brown, G. G., A. S. Foust, D. L. Katz, R. Schneidewind, R. R. White, W. P. Wood, G. M. Brown, L. E. Brownell, J. J. Martin, G. B. Williams, J. T. Banchero and J. L. York, Unit Operations, Wiley, New York, 1950. Cullender, M. H., "The Isochronal Performance Method of Determining the Flow Characteristics of Gas Wells," Trans. ASME, 204, 137 (1955). Katz, D. L., D. Comell, R. Kobayashi, F. H. Poettmann, J. A. Vary, K. R. Elenbaas and C. F. Wemaug, Handbook of Natural Gas Engineering, Wiley, New York, 1959. Smith, R. V., Practical Natural Gas Engineering, PennWell, Tulsa OK, 1983. Hoffman, E. J., Phase and blow Behavior in Petroleum Production, Energon, Laramie WY, 198 l, pp. 553ft.

Chapter 5 THE CLASSIC DIFFERENTIAL EQUATIONS FOR FLOW THOUGH POROUS MEDIA

The classical equations of flow in a number of versions or variants have been presented in detail in Reference 1 and other references. The principal applicable features are reviewed as follows.

5.1 CONTINUITY EQUATION Based upon a cubic differential element of volume, the continuity equation may be written as

+

Y+

= X

(1)

where the mass velocity components are Gx = pv~ G,, = pv~, G~ = pv~ The initial density 90 is a ftmction of position or a constant. The set (Vx, vv, v~) is the velocity; and the members of the set are the velocity components. In another choice of words, the velocity is the vector v , and its magnitude is determined from the familiar Pythagorean relationship extended to three spatial dimensions based on the velocity components. The set (Gx, Gv, G~) is the mass velocity; and the members are the components of the mass velocity. The porosity X is introduced to allow for the fraction void space. The initial density p0 is included to denote that at some initial time "to" the density function is known~and is most likely a constant value in the case of the depletion problem.

90

UNSTEADY-STATE FLUID FLOW

The mass velocity can be related to some sort of potential fimction 9 to include the effect mainly of pressure, but also of temperature and other variables. In conformance with Darcy's law, however, the mass velocity components become as follows"

KX Gx - - ~ P - p

c79 &

gy 0]0 Gy - - - - p - p K: cT G: - - - - p - p

where the potential function becomes the pressure P and the K-components denote the respective permeabilities in the respective directions. If the permeability is isotropic (2), then K = K~-

Ky- K,~

Substituting for the mass velocity components into the continuity equation, assuming constant and isotropic permeability,

+

,7x

+

~

--

~

(2) Kilt

gt

where the convention is that flow is in the x-y-z direction. The initial density function p0 is a constant or a fimction of position only, and accordingly can be disregarded in the partial derivative, as is usually the case.

Incompressible Fluid If the fluid is incompressible, P is a constant, and the Laplacean equation is obtained

THE CLASSIC DIFFERENTIAL EQUATIONS

0 2p C 2P 3 2p Vp 2 -- & 2 F +~=0

91

(3)

It would thus be required that flow be steady-state, that is, independent of time. Unsteady-state depletion, therefore, is inadmissible. As indicated in the previous chapter, the concept of an incompressible fluid is in itself inadmissible in that velocity gradients cannot occur since there can be no change in density. Nor can density or pressure gradients exist. The analogy of course is to a (cyrstallme) solid phase which in the conventional sense does not "flow" -- albeit we may speak of amorphous solids which creep or exhibit elongational flow.

Compressible Liquids For compressible liquids, the density function may be assumed exponential with pressure. That is,

p_po

e ....c(Po-P) =[Po e-CPo]e~P _ Ao e c P

or

p_lln c

P

Ao

where c is the liquid compressibility and the subscript "0" or "o" refers to some reference condition. Substitution for P will yield

V2 p

- c - X -

Op

(4)

K / ,u c~ This is of the form of the heat conduction equation. Finally, for slightly compressible fluids, approximation for the exponential yields p ~ po[l-c (P0-P)]

92

UNSTEADY-STATE FLUID FLOW

or

P ~ P0 [(1-cP0)+cP)] or

p ~ p0[B + cP] That is, p is linear with P. For this special circumstance, Equation 4 becomes

X

c7'

Klp

c~

V 2P - c

(5)

This is again the form for the heat conduction equation. The foregoing may be generalized to read X V 2 @ - 7/K / p c~

(6)

where 11 is the correlating coefficient. The potential function 9 is most likely the density, but may also be the pressure, and 11 = c, the liquid compressibility.

Compressible Gas For a compressible gas, of molecular weight M,

P V - zRT or

p-

MP zRT

where "z" here is the compressibility factor. This relation works satisfactorily for the more ideal gases, well away from the critical. For many purposes the compressibility factor z may be held constant and even assumed to have a value of unity for the more ideal gases (3). That is, the P-T phase region revolved is such that the gas behaves more or less ideally. If an average value for the compressibility factor is assumed (z = z), then it will follow that

THE CLASSIC DIFFERENTIAL EQUATIONS

VP 2 =

a c~2

t-

a

q-~ c~2 ~z

=-2

x

4P0-e)

K / ,u

a

93

= 2

~

x

(7)

K / ,u c?

which is the commonly-recognized form for three-dimensional gaseous unsteadystate flow in terms of pressure as the dependent variable. Whether or not this is an allowable representation is a matter to be discussed in Chapter 11.

Laplacean The Laplacean may be transformed to cylindrical coordinates or spherical coordinates. Thus for cylindrical coordinates, in terms of some potential function

V2r =

32r

1 c~ ~---~q r c~

#2

1 c92r

02

r 2 302

02"

where x = r cos 0 y=rsm0 In spherical coordinates,

V2~ -

1 ~r27 r 2

(TP

1

1

+ r 2

sin v

6 ~ s i n v ~~P-1 2V

1

+ r 2

C 2(I)

s i n 2 ~ r (70 2

where here the spherical coordinates are defined by the transformations x = r sm ~g cos 0 y = r sm ~g sm 0 Z = r COS

and here

94

UNSTEADY-STATE

(X2+y 2 )

FLUID

FLOW

+ z 2 _ 192 + z 2 _ r 2

where p = r sm ~g. The distance "p" is also the "r" of cylindrical coordinates.

Continuity, Balances, and Tautologies The idea of continuity merely expresses the fact that what is originally present in an element of volume less what is left must be balanced by what flows out minus what flows in -- whether mathematically expressed in the form of Stokes' theorem or the divergence theorem, or Green's theorem, or whatever. These various forms must be tautological with each other, else one or another (or all) are in error.

5.2 STEADY-STATE SOLUTIONS The equations of flow readily integrate at a constant gradient to yield the following representations for single-dimensional flow.

Compressible Liquids The mass flow rate Q may be expressed in temls of the gradient of the generalized potential function O , in terms of the density p, or in terms of the pressure P , as follows:

Linear Flow: Q_KAIO la

2 -0, q x 2 - x~

K A 1 P2 - Pl C

X 2 -- X 1

K

---A /a

Po

P -P, X 2 -- X 1

(8)

TIlE CLASSIC DIFFERENTIAL EQUATIONS

95

Radial Flow."

Q_ K(2rr)Ah(n) ~b--~ /.t In rb r~

(9)

K (2z)Ah(c 1 Pb--Po ~t i In r~ r~

K ( 2 r c ) A h p ~ Pb - P,, /.t In rb r~ In both linear and radial flow, P ~ po is assumed essentially constant for varying pressure. Also, Q or Qt = liquid mass flow rate, a constant A = crossectional area normal to flow h or z~h = reservoir thickness The direction of flow is such that PE > P~ for x2 > x~ Pb > P. for rb > r, That is, the direction of flow is determined by the gradient, or vice versa.

Compressible Gases

In terms of pressure, the relationships are as follows' Linear Flow

Q _ K A 1 M Pz2-P~2 t.t

2 zRT

x 2 - x~

(lo)

96

UNSTEADY-STATE FLUID FLOW

Radial Flow Q_K(rc)

M__M__P2 2 _ 1 , 2

l.t

zRT

(11)

ln rb r~

where P2 > Pl for x2 > x~ Pb > P, for r, > rb

and where the compressibility factor z and temperature T assume constant or mean values The results for steady-state gaseous flow are of special interest with regard to back-pressure flow tests, to be discussed further in a subsequent chapter.

5.3 ANALYTICAL SOLUTIONS FOR UNSTEADY-STATE FLOW If the flow equation to be considered is in the form for heat conduction,

V20 -

lc~ xc~

solutions have long been available for unidirectional flow. The function 9 would .denote either pressure or temperature, say, and ~c a mobility or conductance term. Carslaw and Jaeger detail the subject for heat transfer (4). The operational calculus, or Laplace transform, has been employed by Van Everdingen and Hurst (5) to achieve solutions for flow through porous media. For the purposes here, the discussion will be confined to an examination of the classical treatment based on the separation of variables.

Linear Flow For the linear flow of a compressible liquid,

320

X

c~

cgx2 - rl K / /.t 3t

lc~ xc~

(13)

THE CLASSIC DIFFERENTIAL EQUATIONS

97

where 9 represents the density or prossure and q = c, the liquid compressibility. This may also be written conveniently in terms of a reduced coordinate such that 020

x

~2

c~

K/p

c~ = c~

(14)

where t is defined by the substitution,

_

t x

K/Pt

Kt

rl~ K/p Solutions are attained by the method of separation of variables and specifically apply to boundary conditions whereby = Oo, a constant at t = 0 (I)f, a constant at t = oc (I) = l:I)f, at the point of efflux (I) =

The function @, as previously indicated, may be the density or pressure. For the record, the familiar solution for linear flow using Fourier series may be expressed as

9 - O y - ~ b, e x p ( - c , 2xt) sin c, x where tl

n = 1,2,3,4,5,...

Cyl ~ ~

~D

and the Fourier coefficients b. are given by 2 D

-

Do] ( O o

- Ol)sin

c. x dx

= 2 (Oo_Ol)[cosnzc_l]

2

-

98

UNSTEADY-STATE FLUID FLOW

The coefficients b, are zero for even numbers of n. Therefore,

9 - Oy 90 - O :

_ 4/_.,x-' 1 e x - ' - c rc m

2 ~ ) sin

C

X

(15)

where m

Cm--~

zrD

m--

1,3,5,...

The linear solution has the further characteristic that it is synuuetric over the distance D. That is, at

x

=

-0

~

2

D/2

at x = 0

O=Of

at

(I) =

x = D

(I)f

In other words, depletion would occur from both ends of the system such that the one half is the mirror image of the other. The solution brings up the matter that, knowing O, t and x, it is hypothetically possible to calculate D (or D/2) by trial-and-error using the parameters X and K/~t. This is perhaps more readily detenumed from the flow rate as will be shown subsequently (e.g., in Chapter 10).

RadialFlow For the radial flow of a compressible liquid, which is of primary interest here, C20 1 c7~ X c~ 1 c~ c--5 - + - ~ = r/~ = -- ~ r c~ KI/.t ~ ~: ~ In reduced coordinates,

(16)

THE CLASSIC DIFFERENTIAL EQUATIONS

02~ ~+ cT 2

X c~ c ~ - roZrl~ =~ r & K / p 3t 3l

99

1~

(17)

where

f-rlro t t--

x

roZr / - - ~

=

#

K/p

The value for "r0" may be treated as the upper limit of r, or conceivably may be used as a correlating coefficient (or constant). For the record, a method of solution will be rudimentarily reviewed in outline based on Bessel's equation. Using the principle for the separation of variables, let the solution for 9 be of the form = Z *. 0. + constant m

where ~. is a function of t Therefore,

alone and each 0~ is a function of r alone.

(~ d2 0. n

n d'~2

d,n)

O. dO. [

~

-o

d~

A solution exists if, for each n,

dZO. -"d72

§

qk. dO. r d? _

O.

dO,,) d[

-0

Hence 1 d20n

O. d7 2

1 dO. ~0. cl?

1 dqk.

Since each side of the last-cited equation must be independent of the other, the equality must also be equal to some constant. Let that constant be -a~ 2 , for each n. Then

100

UNSTEADY-STATE FLUID FLOW

1 d~

2

or

~,, - C e x p ( - a , 2 [) where C is a constant introduced via the integration. The remaining equation is

F9 2 ~d2~" +r_ d O , , + O a . 2 F2 d-r 2 dF

-0

m

If ~ r = R., Bessel's equation of order zero is obtained:

R2 d20n ~+R dF 2

dO,, +o~ R,,2 - o dR,,

The solution, as a matter of record, is

0,, - A,, ,]o(R,,)+B,,

Yo(R,,)

where the An and B. are constants and X2

X4

X6

Jo (x) - 1 - - T + 2 2~(2T) ~

2~(3T) ~

1

Yo(x) - Jo l n x + 2 J2 - ~ J 4

+

1 1 +-~J6 - ~ J s

1 } +5Jlo -...

The Jn in tum are solutions to the Bessel equation of order n,

xE d+y + xdY +(x2 _ n 2 ) y _ O dx 2 dr and for n positive are

THE CLASSIC DIFFERENTIAL EQUATIONS

Xn

J,

(

X, 2

X

+

"--" ~2" nT I 1

2 2 (n + 1)

101

4

2 T24 (n + 1)(n + 2)

3T2 6 (n

+ 1)(n + 2)(n + 3)

The derivatives are related by dJo - _j] dr

dCx"J,,) = x"J,,_l dr and therefore

d(x J, ) dx

=xJ o

where it is understood that here the Jn are functions of x as the variable. Furthermore, another property of the Bessel fimctions is that J , - (-1)" J_~

or

J.. - (-1)" J ,

If the boundary conditions can be so met, it is permissible to drop the Y0(x) terms. Therefore,

*-CZexp(-ct.2i){A.Jo(a.?)+*y} m

where 9 = O f if t ~ o~. The constant "C" may altemately be incorporated into the A.. If the botmdary conditions are such that 9 = ~0 at t = 0, then

O-CZA,,Jo(a,,F)+~

s -dO o

This is satisfied by the requirement that

1-~A~do(a,~)

and

C-~o-OI

A properly behaved fimction f(x) can be fit to a series of zero order Bessel functions,

}

102

UNSTEADY-STATE FLUID FLOW

by the device of multiplying by x Jo (o~kx)dx and integrating between the limits 0 and 1. Performing the operation, 1

1

o

o

1

... + A,, IxJo(a.x)Jo(akx) dx o

The integrals are evaluated as follows. Consider the general form I x Jo (c~

(fir)dr

-

V

(ax)Jo (aX)Jo (flx)d(coc)

o

o

where either a ~ 13 or c~ = ]3. Note that the Bessel function variable is now represented as x = (ax) and/or as x --- (13x). Since, from the afore-stated derivative relationship between J1 and J0 whereby

a{(ax)J,

(ax)}} - ( a x ) J o(ax)a(ax)

the above integral transforms to the following, upon integrating by parts and utilizing the derivative relationship between Jo and J~: 1

12 I J~176 a

-

o

a2

pi x J, (ax)J, (1~)dr

o+-a

o

Furthermore, on juxtaposing ~t and 13 prior to the integration, 1

I x Jo (c~ o

1

(fix)dr - ~

1

I (fir)Jo (a:r)Jo (flx)d(/~) o

THE CLASSIC DIFFERENTIAL EQUATIONS

103

and it follows that 1

1

flz I (flr) J~ (ax) d{ (flx)J' (flx)}

-

o

/~ O~ 1 p: J,(~)Jo(~)l'o +--~ IxJ,

(flx)J, (ax) dx

o

On subtracting and clearing fractions, 1

OC o

(aX)Jo (/~)J,

Therefore, 1

XJo (aX)Jo (px)ax -

a~ o(fl)J, (a) - fll o(a)J, (/3) 02 _ #2

o

If a and [3 are both roots o f the equation Jo(x) = O, then, when a v 13, 1

f~Jo(~)Jo(~)~- o o

The first ten roots are: xl = 2.4048

x6 = 18.0711

x2 = 5.5201

x7 = 21.2116

x3 = 8.6537

x8 = 24.3525

x4 = 11.7915

x9 = 27.4935

x5 = 14.9309

Xlo = 30.6346

104

UNSTEADY-STATE FLUID FLOW

On the other hand, when 13 ~ a where Jo(ct) = 0 , the derivative may be considered. Therefore, at constant J~, 1

I x J~ (ax)J~ (flx)dx - (z], (a)J, (or) = J~ (a) 2a 2 0 since (dJo)/(dx) = -J1. The functions J0 (otx) and J0 (13x) are said to be orthogonal under these circumstances, when a and 13 are roots of the equation Jo(x) = 0 .

Thus it follows that 1

21 x f (x)J o( a . x ) d x 0

A?./ ---

J((a.) where the o~ are roots of the equation Jo(x) = 0 . For the problem under consideration, f(x) = 1. Therefore, An

m

a,,Jl(a.)

Hence

1- f(x)-

2Jo(a,,x) a,,J,(a.)

Finally,

oo - oo y = 2 Z J~ (a"F) ~ o - OOy

2

e x p ( - a , , i)

(18)

a ,,J, (a ,, )

where the o~ are the roots of Jo(x) = 0. The value (I)f is the value of 9 at the wellbore. It may also be designated as ~ , -- that is, the value of 9 at r~, the wellbore or sandface radius.

Depletion Rate At any point "r" the mass rate of flow Q is given by

O - -x (2xr)( ft

)p

THE CLASSIC DIFFERENTIAL EQUATIONS

105

where h or Ah is the thickness. If

p - A e ~P or p = l l n c and

p A

9 = p , then Q _ ~K(2zcr)(Ah) _ /~

1 1 c~ c~~ (19)

K 1 lc~ = - - (2zcF)(Ah)~ /~ c~~

The partial derivative with respect to "r" is c ~ = 2 Z - J ' ( a . F ) dF e x p ( _ a , 2 [)

tT~

anJl (an ) dr

m

where d r/dr = l/r0. Furthermore, the value of r0 can be regarded as arbitrary. If r0 --~ oc then the partial is equal to zero. For finite values of r0, than it is required that r0 take on such values that, at the closed outer producing radius rb the partial derivative will be equal to zero. That is, O - 2 E - J ' (a~Fb) 1 e x p ( _ a 2 {)

a . J , (a. ) r o or

2 [)

0 - Y~ - J ' ( a ~ . r b / r o ) e x p ( a

a~J,(a,,)

(21)

n

In this fashion, in principle, the behavior of r0 can be regarded as some function of t or t . More than this, for each value of t assumed, there would be multiple roots of ro, which can be designated as (ro)i or as r j ( r o ) i . For which ever choice of root, the continuity should be sustained as t or t is allowed to vary. There is a further qualification, however, for the flow rate at the wellbore or sandface radius r,. Here, r = ra = rJro The flow rate is to be designated as a constant, to be specified as a boundary condition. It will accordingly be required that

106

UNSTEADY-STATE FLUID FLOW

Q(at r~ ) - ---K( 2 ~ ) ( A h ) l /u

2 2 cr o

-J, (a.~ ) a.J,(a,,)

e x p ( - a n 2 t-)

(22)

It may be noted that the J1 have both positive and negative values, as do the J0. From the above, the behavior of the root (ro)i so chosen should also provide a reasonable behavior for the flow rate Q at r,. In effect, an extremely complicated calculational procedure has been builtup in order to utilize the *-solution for an infinite radial system, in order that there exist a finite closed outer boundary. Moreover, there will result a production rate at the well bore or sandface that is required to vary, negating the premise of a constant production rate. Or conversely, a constant production rate would negate the premise of a closed outer producing boundary. Brought to mind is that there may be an over-subscrilStion of the number of degrees of freedom, a matter to be discussed in Chapter 11. For the record, a schematic juxtaposition for r,, rb and r0 is provided in Figure 5.1. The relative position of r0 is entirely arbitrary, depending upon the root behavior chosen, and will vary with time.

f

/

/

/

Q-o

\

\

\

/

\\

/ I I

\ r, rb

I I r~

or

\

// \

/ \

\

/ \

/

/

Figure 5.1 Schematic of relationship between r,, rb and ro.

THE CLASSIC DIFFERENTIAL EQUATIONS

107

Constant Ffflux In the preceding derivations, the value of 9 (or pressure or density) at the inner radius remains constant, and the efflux varies with time. By the principle of superposition, to be discussed in the next section, the pressure may be varied in increments to give an essentially average flow rate---through r0 may be expected to vary. This is a more comprehensive solution, since values for rates and for radii are prefixed, whereas in the method previously described, the pressure is fixed at the inner radius (for successive values, corresponding to an average efflux rate) and a corresponding value found for the outer radius. However, a generalized solution is presented by Carslaw and Jaeger (4) by which the rate of efflux or influx at both radii may be fixed. Constant values may be selected, presumably including zero. In any case, the solutions are very complicated. The implicitness that occurs does not allow a ready overview of the effect of one variable or parameter on another. Add to these considerations the fact that the artifices and the solutions, if any, would apply only to flow systems composed of compressible liquids in the entirety. Moreover, the outer reservoir boundary is closed, with the expectation that flow would only be of short duration in any event. Otherwise, the flow of liquids per se is reduced by a gas drive, either from a gas cap or from the exsolution of gases as the pressure is reduced, and/or reflux from an aquifer.

Superposition Principle In the course of production, the pressure at the well bore (or bottom-hole pressure) may be varied at a succession of levels (Pf)i -- giving a corresponding ((I)f)i or (Pf)i. Each of these successive values of pressure -- or (~f)~ --denotes a succession of solutions 9 (or ~0, or

-(OI)0 where (Of)0 corresponds to the original producing pressure at the wellbore. Each applies to a time interval At~ or A ti. In the limit, the (~f)~ become functions (or step functions) of time, denotable say as Of = ~f (t). In fact, the behavior of Of(t) may be that function or functions whereby the mass flow rate Q at r, is varied in a prescribed manner -- continuously or incrementally -- or held constant (over an interval or increment of time).

108

UNSTEADY-STATE FLUID FLOW

In this fashion, in principle, it is possible to speak of a solution determined by successive production rates. The foregoing may be said to be based on the superposition principle, whereby the separation of variables can be written as

where each ((I)f)i applies only to a time interval Ati. In other words, ((I)f)i is a step constant taking on successive and different values for each time interval Ati. The values are treated as constants for the differentiation.

5.4 COMPUTER SOLUTIONS An early published major attempt at a computer solution to the linear and radial flow problem is represented by the work of Bruce, Peaceman, Rachford and Rice (6). Later treatments include that presented in Fundamentals of Numerical Reservoir Simulation by Peaceman (7), and in Petroleum Reservoir Engineering, by Aziz and Settari (8). More on the subject of unsteady-state solutions is presented in References 1, 9, 10, and 11, with additional references provided. Reference 12 is specifically aimed at computer solutions of reservoir engineering problems. References 13, 14, and 15 are directed more at the mathematical complexities which can be introduced in representing reservoir behavior. The subject will be further evaluated in Chapter 9 and succeeding chapters. In the work by Bruce et al., the unsteady state flow equation for gaseous flow V 2 p 2 = 2 ~X ~ aT KIp~ was computer-resolved for prescribed boundary conditions in linear and radial flow for production from a reservoir of known dimensions at a constant depletion rate. Experimental results for the depletion of a linear gaseous reservoir (embodied as a pipe) were obtained and compared with the computer-calculated results. The generalized computer-calculated results for radial flow (for a specific outer/inner radius ratio) are presented in graphical form in the same paper. This information will be utilized in a subsequent chapter (Chapter 10). Interestingly,

THE CLASSIC DIFFERENTIAL EQUATIONS

109

the calculated results at any given time tend to correlate against and agree with the steady-state profiles, as will be subsequently demonstrated.

Applications to Production It goes without saying that there are a number of major oil and gas producing companies and also service companies, for instance, which have developed computer routines for numerically solving -- if this is the word -- the flow equation(s) subject to various constraints and conditions. Add to this the advent of the personal computer, with off-the-shelf s o l . a r e available. These programs may apply comprehensively to multiple wells and well-spacings, various drives (e.g., water, solution, gas), and injection methods (e.g., water-flood, CO2flood, polymer-flood, fire-flood, etc.). Though in principle generalized to a wider variety of circumstances, each becomes a special study. Moreover, the aim is generally to predict only the reservoir behavior~given the special conditions and the known or assumed extent of the particular reservoir. The objective is therefore not in the mare to predict the extent of reserves, if given the behavior of limited pressure/production performance data versus time (as in well testing). Finally, there is the thought that the solution to an equation can say no more -- if as much -- than the original equation itself, albeit the solution may be in a more useful or convenient form. Which in tum gets back to the question of the validity of the original equation or equations in the first place, and the a priori assumptions made in the formulation. These will be matters for further consideration, and have bearing on the entire edifice of mathematical physics.

On the Mathematical Representation of System Behavior It is well to keep in mind that the solution to an equation is at best but a tautology, and can say no more -- if as much --than the original equation, as previously noted. It is, hopefully, merely a more convenient or useful way to state the information revolved. In other words, everything rests on the original assumptions -- not to mention intermediate assumptions. And it may be added that in any sequence of steps in a proof or derivation, each must constitute a tautology or else error is introduced. And the more steps, the greater the opportunity for error, and cumulative error, as in the parlor game of "telephone" or pass it on." (In the number-crunching computer world, this all is manifested by the welltraveled acronym GIGO, for "garbage m, garbage out.")

110

UNSTEADY-STATE FLUID FLOW

And it may be added that the explanations of science are also suspect. For science, strictly speaking, merely signifies that the experiments or observations are reproducible, and the data or results are reproducible -- by which we are able to formulate laws. All the rest -- the theories, hypotheses, and conjectures -- lies in the aeries of the imagination. The nearest example of pure science is embedded in Kepler's laws of planetary motion. These laws reproduce planetary motion, but do not explain. (An example of the branch of philosophy called Positivism.) Such concepts as force and gravity but follow from the motion. At the most abstract level, irregardless of any assumptions made, the concept of energy arises as but another way to represent the results of experiment, as developed in Reference 16. In other words, if there are k variables involved, then k-1 differential forms can be written for the variation of each variable in terms of the kth variable. In tum, the integration or solution of these k-1 simultaneous differential forms in principle yields k-1 energy forms or functions (with "energy" in each case first introduced as a constant of integration). Assigning values to the energy functions for some particular set of boundary conditions or statements, will in theory, by successive simultaneous solutions, permit each of the k-1 variables to be presented algebraicly in terms of the kth variable, treated as the independent variable. Such a representation does not require any sort of model or mechanism, but is based entirely on an array of experimental results or observations. The mathematical manipulations based on Kepler's laws are the prime example (16). Kepler's first two laws consist of the statements that the planetary planar orbits are ellipses with the sun at one focus, and that the motion of each planet cuts off equal sectors in equal times. These two statements can be expressed mathematically, and in rum can be transformed and partially integrated to yield two independent equations, which may be denoted the energy forms. These energy forms involve the two polar coordinates of each orbital plane, and time -- and include the temporal derivatives as the velocity components. (The first equation is the customary form for kinetic and potential energy terms (the latter term, in the absolute form, is negative), with the second equation a repetition of Kepler's second law. (Kepler's third law is a boundary condition or boundary statement, linking the orbits one to another.) A further integration operation yields the trajectory, consisting of two equations which relate the two polar position coordinates to time. Inversion or circularity once again produces Kepler's laws. Other mathematical manipulations may be performed to yield the Langrangian and Hamiltonian forms, or equations. The entirety constitutes a tautology, being but a different way or ways to represent the same exact thing, namely the motion of the planets.

THE CLASSIC DIFFERENTIAL EQUATIONS

111

As Henri Pomcare once stated in Science and Hypothesis, the concept of energy is tautological (17) -- provided of course that the mathematics is itself tautological. In the same volume, Pomcar6 exaggerated that if a phenomenon has at least one mechanical explanation, then there is an unlimited number of other explanations. Even systems of logic are suspect, in that a logical step revolves a discrete change, and can take off in any direction. If not, only a tautology exists. Thus it has been known from almost the very beginning that in Aristolelian logic, the concludion is buried in the premise (18). And there is G6del's proof or incompleteness theorem(s), the tortuous exercise which demonstrates that in any system of logic diverse enough to be of interest, there will always be inconsistencies or contradictions. A further recognition of this inadequacy lies in the idea of the antmomy, whereby two different conclusions can be reached, each of which appears logically correct.

REFERENCES 1. Katz, D.L., D. Comell, R. Kobayashi, F. H. Poettmann, J. A. Vary, J. A. Elenbaas and C. F. Wemaug, Handbook of Natural Gas Engineering, McGraw-Hill, New York, 1959. 2. Scheidegger, A. E., The Physics qf Flow Through Porous Media, University of Toronto Press, 1957, 1963. 3. Hoffinan, E. J., The Concept qf Energy." An Inquiry into Origins and Appfications, Ann Arbor Science, Ann Arbor, MI, 1977. 4. Carslaw, H. S. and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press (1959). 5. Van Everdingen, A. F. and W. Hurst, "The Application of the Laplace Transformation to Flow Problems in Reservoirs," J. Petroleum Technology, 1,305-326 (1949). 6. Bruce, G. H., D. W. Peaceman, H. H. Rachford, Jr. and J. D. Rice, "Calculations of Unsteady-State Gas Flow Through Porous Media," J. Petroleum Technology, 5, 79-92 (1953). 7. Peaceman, D. W., Fundamentals Of Numerical Reservoir 3~mulation, Elsevier, New York, 1977. 8. Aziz, K. and A. Settari, Petroleum Reservoir Engineering, Applied Science Publishers, London, 1979. 9. Dake, L. P., Fundamentals qfReservoir Engineering, Elsevier, Amsterdam, 1978. 10. Smith, R. V., Practical Natural Gas Engineering, PennWell, Tulsa OK, 1983.

112

UNSTEADY-STATE FLUID FLOW

11. Katz, D. L. and R. L. Lee, Natural Gas Engineering: Production and Storage, McGraw-Hill, New York, 1990. 12. Nobles, M. A., U~ing Computers to Solve Reservoir Engineering Problems, Gulf, Houston, 1974, 1984. 13. Mathematics in oil production, Sir Sam Edwards and P. R. King Eds., Clarendon Press, Oxford, 1988. 14. The Mathematics of Oil Recovery, P. R. King Ed., Clarendon Press, Oxford, 1992. 15. Mitlin, Vladimir S., Nonliner Dynamics of Reservoir Mixtures, CRC Press, Boca Raton FL, 1993. 16. Hoffman, E. J., Analytic Thermodynamics: Origins, Methods, Limits, and Validity, Taylor & Francis, New York, 1991. 17. Pomeare, Henri, Science and Hypothesis, Dover, New York, 1952. p. 132, 222. Preface by J. Larmor. 18. Durant, Will, The Story of Philosophy." The Lives and Opinions of the Great Philosophers, Second Edition, Simon and Schuster, New York, 1961, p. 50

Chapter 6 INTEGRAL FORMS FOR DESCRIBING UNSTEADY-STATE FLOW

For any number of reasons a petroleum reservoir is more than likely not ideal~that is, is not representable as a cylindrical annulus in radial flow. Thus the formation may have recline, be irregularly and asymmetrically shaped, may revolve linear flow or spherical flow moreso than radial flow, and may exhibit water reflux and/or a gas cap or solution drive, etc. Therefore, a more generic approach may be indicated for representing unsteady-state behavior. This can be based on a material balance phrased in terms of volume and surface integrals over an arbitrary element of volume (the reservoir). The formulation and practical application, however, will require reductions to more regular geometries -- i.e., to linear or radial flow. The subject has been previously developed and detailed in Reference 1 and particularly in Reference 2, and the derivations and arguments are upgraded and restated in this chapter.

6.1 VOLUME AND SURFACE INTEGRALS The general unsteady-state fluid flow problem may altemately be formulated in terms of volume and surface integrals, which serve to relate mass density to mass flux in terms of the system geometry. The change in mass in an as yet unspecified element of volume over an interval of time is xz Y2(x)z2(x,y)

m-I I I

- xp)

dz

Xl yl(x) zl(x,y)

where the order of integration depends upon the manner in which the volume geometry is to be specified. Most usually the porosity X is a constant or at the most a function of position rather than time.

114

UNSTEADY-STATE FLUID FLOW

If the difference term above is positive, then there is a net outflow or effiux and Am will be defined as a positive number. In general, for any set of coordinates, at constant porosity, and utilizing the Jacobean determinant,

lcg(x,y,z) du

xII I

dv dw

(2)

where the limits of integration define the element of volume. The most basic circumstance would be coordinate systems adaptable variously to a parallelepiped, a cylinder, and a sphere. In tum, the mass of fluid Am flowing at or across the surface enclosing the element of volume may be represented by the surface integral t

; ;;Gl,,(u,v) [du

(3)

dv dt = Am

0

where G is a flux function defined within the above equation, and where

["("'v)l - I

~x,y)

2

O(x, z)

2

c~y,z)

2tlj2

The surface is defined by the functions z = z (u, v )

y - y (u, v) x = x (u, v) This is the parametric form for the surface. Integration is around and over the surface, and may encompass both regions of effiux and reflux. If the total efflux exceeds the total reflux, then the integral will be defined as positive. If the surface area is also defined positive, then the behavior of G will be positive for a region of effiux, negative for a region of reflux. The convention for integration around and over a surface is analogous to the convention for integration of a line integral. Thus consider a function G (u, v) of segments G1 (u. v) and G2 (u, v), where the coordinate system (u, v) is

INTEGRAL FORMS

115

imbedded in the surface. Integration of the fimction G "around" the surface is defined by u2

v2(u)

U1

WI(U)

u2

v2(u)

U1

Vl (bt)

I ~i,,~l~v~u§

II~,,~,,~v-I

I ~l,,21~v ~u (4)

or

u2

v 2 (u)

ul

V(u)

- I I ~..i,,~l~v ~u_ I I ~t,,~l~ ~u ul

Vl(U)

u2

Vl(U)

The limits are for the projection in the "u-v" plane. The former sequence of integration will be adopted here, where both segments of the surface are integrated in the same directions. The above considerations will be first adopted to a parallelepiped, a limiting case since the enclosing surface is not both continuous and differentiable at all points (the edges and comers of the parallelepiped).

Parallelepiped Consider an arbitrary surface defined by z = z (x, y) with the x-y coordinates imbedded in the surface. Here, in what will be designated Equation 5,

- xfff(;0-

;) ~ .

~

where z2 (x, y) is the upper surface z~ (x, y) is the lower surface The above expression may be differentiated in any order.

~

116

UNSTEADY-STATE FLUID FLOW Differentiating, holding z~ constant,

2 ~Y2

+

+ 1 - X c?(,o0-,o)

Representation may altemately be made in terms of x and z to yield

or in terms of y and z to yield 2

_____"-__'-

+ 1 - X cq(p~ - p ) .

All are equivalent. The choice of equations and variables for the surface have merely been switched. However, the surfaces which were held constant are z~ (x, y)

yl (x, z)

x,(y, z) These are common only at their intersection. The surfaces which were allowed to vary are z~. (x, y)

y2 (x, z) x2 (y, z) These are also only common at their intersection. Accordingly, an equivalence may be set up based on this latter common point of intersection. In this fashion a relationship may be established between the several partial derivatives, each of which becomes equivalent to the others, such that any one of the independent spatial variables can be used for the representation, say the variable x .

INTEGRAL FORMS

117

At any such common point, 2

+1

&

+

+1

2

+1

For a parallelepiped oriented along the x-y-z axes, the partial derivations of the surface are zero. Hence in the limit, at the point of intersection of the planes, c ~ z______A-_=) c~] ~_,_____-~, ~ ~C~Jx2

(6)

In general, we can speak of a function G = G (x, y, z) which reduces to

(-]z~ - G [x,Y, Z2 (X,Y)] if z = z2 (x, y). Altemately, there are also the expressions

In general terms, there results the equivalencies representable as

(7)

which denotes the limiting condition for the point of intersection of the plane surfaces of a "nest" of parallelepipeds.

118

UNSTEADY-STATE FLUID FLOW

6.2 THE DEPLETION PROBLEM It is possible to develop heuristic functions which will satisfy the equivalencies and the boundary conditions. The technique is somewhat similar to that for the separation of variables or variates (groups of variables) used to obtain solutions for partial differential equations. Let the surface function G be represented by G = ye ~

(8)

where 0 = a function of position = a fimction of time T = a function of time At the region of effiux, it will be required that 0 ~ 0 such that the effiux is equal to T, a function of time. When t = 0, flow has not yet started and 0 ~ -oc. The depletion time is tf where to = 0 is the startup time. At the region of effiux, therefore, as t --~ tf then 9 -+ 0 The distraction is also to be made that at exactly t = tf, then p = 0 everywhere. Differentiating the heuristic flux function G,

re

o, ao

p0-p)

x

(9)

At the region of effiux, and at t = te, c~p o - p)

-~0

and

~ 0

(10)

If G is integrated with respect to time, an integration by parts will yield the following expression, which can be adapted to fit the boundary conditions. Indeed, this is the purpose of the exercise, to yield an expression which can be curve-fit so to speak. Thus

INTEGRAL FORMS

'

I0 G dft

'

-

119

[! ]l'

~e ~ d t - e ~162

0

gdt

o

t

- 0~ [Tdtle ~ d~ dt dt 0 The boundary conditions are: when t = tf, ~) = 0 t=0,~-oo Therefore, 'j"

''[! ]

Gdt - ~ gdt - O~

o

o

7tit e~162 dqk dt

o

dt

If the substitution is made that

[~~~t]--~-- nr d#

(11)

where "n" is an arbitrary constant, then tf

tf

tf

0

0

0

Since

tf

tf

0

0

then it follows that ,f

tf

~ Gdt-~l ~ )dt o l+nO o

(12)

120

UNSTEADY-STATE FLUID FLOW

The fluid depleted up to time tf is therefore a fimction of position only. Furthermore, d~b- t

= n din

E! 1 ?fit

0

Integrating between limits, t

,

~7"dt -

n l n ;~

's

= n In V

f ydt 0

where ~a = 0. Thus t

f yat 0

- nln 9

= n In V

(13)

I yat 0

where ~g is defined by the substitution t

0 --

(14)

tf

0

The relation

d;

~o X4po-p)

integrates with respect to t to yield

fre o~Oat- X(Po-P) c~i ~ t

0

(15)

I

~N o~

II

i

0

c~ j

C;Q

I

I

II

9 .

ct~ C~

~D

I_

~r

I

,I

o t . . m , ~ ...

II I

I

I

II

,I

0

~D

0

II

r

-e-

I

r

II

I

I

C/)

I

I

I

+

......

.........

o q t g . , ~ ..~

I

I

I

I~

II

I

!

,.j

O'Q

~D

122

UNSTEADY-STATE FLUID FLOW

Therefore,

' {E! }

t

ye"+~t

- 1 + nO

0

7ttt e~

"E! ] o}

- 1 + nO

7ttt e

or

i ~~

-

o

(1 + n 0 ) 2

+-,}

eO,,,,{l' + "O n

Since ~ = n In ~g, then

and

t t

(1 + 110) 2

(n) - 1 +

n

n In ~g

0

It follows that t

f rat, 0

(l

+

,.,o) ~- ~

"~ (n)[-I +(1 + nO In I//] -

X(po-p) o39/&

(17)

X(po-;) ooic~

X(po-p) 30/~ At t = tf, tit = 1 and there is obtained

n

(l+nO) ~

tf

!

7dt-

~ o

o39 / &

.X;o

~0

o39/~

(18)

INTEGRAL FORMS

123

Since 0 is independent of time, a solution for the above is

Xpo[(X - x, ) + ( y - y , ) + (z-

Zl)])

tf

l+nO

(19)

f y~ 0

where it is stipulated that at the point (x~, y~, z~), 0 ~ oc and G ~ 0 . that ~ is negative in the expression G = ye~ .) A material balance requires that XP0

(Note

1

(~, x,)(y,

tf

-

- y, )(:+

-

:,)

(20)

I aydt 0

where "or" is the area of effiux and (x2, Y2, Z2) are the coordinates of the other surfaces. The result is that ty

! yJt (~2 - +,)(y, - y,)(:, -++,) I + nO-

tf

[x- x, + y - y, + z - z,]

(21)

I aydt 0

The solution for the flux function may be placed in the form

G - 7'~ "e

(22)

where the functions have been determined in terms of x, y, z, and t. For the density function, dividing the solution at t by the solution at tf gives

Po -P P0

nO[-1 + (1 + nO)ln ~ ] Iif

q ! ,lI-,l

124

UNSTEADY-STATE FLUID FLOW

Since t

f rat 0

(14)

tf

f 0

then Po - P

In V nO+l

= V"~

Po (23)

-- f i l l - lnfl]

where 13 is defined by the substitution nO+l

(24)

If y is a function of t as follows,

7 - atb

(25)

where a and b are constants, then t

~ atbdt 0

(26)

tf

~ atbdt 0

where b + 1 = c , and where here "c" is a correlating coefficient or constant as distinguished from the liquid compressibility. Therefore,

?.--~.l r

]3- V "~ or

I t

I n f l - c(nO+ l ) I n t - c ( n O +

(27)

l)Intf

(28)

INTEGRAL FORMS

125

1.0

.d ,,///J

o

,'//4'1

, , / / ' / " I'1 // I/t /// Ill O/ J tx I I l l II/ I.' '/ /

/

/

_/I I

0ol I

I

/

I

/ i/

/i/, I i

/~ /

/ A/ / / / / / 1//"////tll x / i I , I / I t[ t~ i/':/ I / iJf I I I I

f

/

/ / /

0,01

/

'

0 /

/n

/'

/

/

/

/

/

/

i

/ //

I

/I

,, /

/

x/

/

0.001

ii /

I

/

/I

I 9

!

I

/

/

I" .

I

.

.

.

.

.

.

.

I00

PO

t (minutes)

Fig. 6.1 Plot of 13 vs time for linear data of Bruce et al. (3).

I000

126

UNSTEADY-STATE FLUID FLOW

Since 0 is a function of position only, a plot of In 13vs In t should produce a straight line at any position. This is shown to be very nearly the case using the data of Bruce et al. (3), as shown in Figure 6.1. The depletion time also satisfies the material balance, as will be shown subsequently in Example 6.1.

Cylindrwal Annulus It may be expected, by analogy, that in cylindrical coordinates tf

l+nO-

o

'I

[2er(r-r.)+

x - xl]

(29)

I aydt 0

and tf

j"

,Oo

-

-

(30)

0

The above would apply to the depletion of a cylindrical reservoir by radial flow.

Behavior of fl A plot of (po - p)/p vs 13 =

t/tf

is shown is Figure 6.2 for the relation

,Oo-,o_ _lnp ] Po At lower values of 13 =

t/tf,

the curve may be approximated by

P0-_____PP_ 3.0771(fl) 0.86977 Po This plot may be used to determine the corresponding 13 value from experimental values of (po - p)/po.. Then, in turn, I~ may be plotted versus In t to establish and predict

0.1

1.0

0.01

0.1

Z

Po-P Po

0.001

0.0001

0.001

0.00001

0.0001

0.001

0.01

0.1

Fig. 6.2 Plot of density difference ratio vs 13. t~

128

UNSTEADY-STATE FLUID FLOW

reservoir performance --that is, to extrapolate to tf at j3 = 1 for a given flow rate, from which the material originally in place can be calculated. At least two values are needed in order to extrapolate the straight line. Altemately, only one value is needed if correlations for the constants are developed in terms of reservoir properties (permeability, etc.). Since at the region of efflux, 0 = 0 , it is only necessary to correlate the constant (or coefficient) "c".

Geometry o f Flow System The preceding derivations appear independent of the flow geometry as far as the form of the density function is concerned. The methodology is applicable to any arbitrary shape, and is independent of whether flow is linear, radial, or spherical. The value of the coefficient "c", however, may depend upon the system geometry as well as the reservoir and fluid parameters. We are here only concerned that "c" behave essentially as a constant, to be evaluated from experiment. The fact that "c" is constant for a given system is born out by the several examples at the end of the chapter.

Generalization It may be noted that the solution may be generalized by the introduction of a coefficient "m" such that

P o - P = f l [ 1 - m l n fl ] Po This will provide for an extra degree of correlation.

6.3 PERMEABILITY FORM Drawdown and buildup curves are sometimes used to calculate permeability (4). The calculation is in part based on the premise that a plot of log p2 vs log time is a straight line--at least for reasonable intervals of time. The previously derived formula for the permeability at the region of efflux (0 = 0) may also be used to calculate the permeability for the initial conditions.

INTEGRAL FORMS

129

Permeability Conversion It is convenient to be able to convert between permeability units of darcies or millidarcies and the English units of ff3/hr2. The conversion is as follows. A permeability of one darcy is given by a velocity of one foot per second under a pressure drop of one atm/cm for a fluid of one centipoise viscosity. Therefore, in cgs units, let Kt

G'=

~t

- -p-' p ' -&- '

where G' = mass flux in gm/sec-cm 2 bt' = viscosity in centipoises p' = density in gm/cc P' = pressure in atm x' = distance in cm (positive in the direction of flow) K' = permeability in darcies

In English units, let K aT G----p-p 3x where the coresponding variables or parameters are in the following units' G = lb/l~ tx = lb/ff-hr p = lb/ff 3 p = lb/ft 2 x = ff (m direction of flow) K = ff3/hr2 Dividing, one to another,

G' G

K' p ,d cP'/&' Ki.t'pcT/~

In terms of the conversion of units, the above expression may be expressed numerically as

130

UNSTEADY-STATE FLUID FLOW 453

r

1

{

w

~I

K' 1 [1 / ~2.53)I,12)J - ~2.42-3600 [(2.53)12] 2 K 62.4 14.7(144) Therefore, performing the operations, K (m l~

2) = K' (m darcies) times (0.00442)

This statement provides the conversion.

Transformation to Permeability In terms of time, the density function has been written as Po - P _

t

1 - In

Po

\tl)

where the mass flux function becomes

G-y Furthermore, for a parallelepiped, 1 +nO-~

V

1

a x - x~ + y - y~ + , ~ - .."~

where V is the volume. Differentiating the density function with respect to x,

cg(po - P) / Po _

I-t/l' a(nO+l) (-)[ln

t](cn)

c~

.(,,o+1) (~~__i)~(.o~,)[ln t)]t(sc n

: - ( jt_

)"("~

t~

c(nO + l)(cn) 09

c~

INTEGRAL FORMS

131

where c90 ?l

'

1

V

' -

c,3c

a ( x - x l + y - yl + z - z 1 )

z

The discussion of the derivative will be confined to the point (or region or area) of efflux: x = x2 y = Y2 Z = z2 where most usually, for a lengthy parallelepiped, it can be assumed that X2-

X l >>

Y2 --

>>

y~

Z2 -- Zl

At the point of effiux, 0 = 0 such that G = 7 and

1 a

V Y2 - Yl + Z'2 - - Z1

X 2 -- X 1 +

Since it has been established that

1

XPo(X-X

~ +y-y~

1 + nO

+z-z~)

t~ 0

then at the point of effiux (where 0 = 0), with the sustained flux or effiux denoted

by -

Y =

XP~

2

-x,+y2

--

tf

where

tf

- f rat o

Furthermore, Z

m

Xp o V

._.,m_~_=_~

tf

a

Yl +

Z2

-Zl)

(31)

132

UNSTEADY-STATE FLUID FLOW

or

_

Xpo

Z -

ty

( - ) ( x 2 - x , + Y2 - Yl + zz - gl )2 l l - -

Substituting for n(00/c'~x), at 0 = 0,

-~_

~, _

X

(

Xz - X, + y 2 - y , + z2 - z ,

tj.

[In

t

or

--

--

G -7 =

0t9

(32)

(,+1

X (X 2 - - f l + Y 2 - - Y '

t].

)2

9

]2

+Z2 __11)2 +

fl[lnfl] z

c3c

(33)

This result may be transformed in several ways, depending upon the behavior of the density function. To continue, introducing the density function p into the expression for the mass flux, 2

G -- -

X

( x z - x 1 + Y z - Y l + zz - Zl

ts

P Op

fl[ln fl]z P

(34)

c3x

If, for a compressible gas, p

MP _

_

zRT

then there will be obtained 2

--Xl q-Y2 _ Yl + z 2 _ Z1) If

M

bT'

fl[ln fl]2 (p / P o )Po (35)

~ m K- - p - c7' ,u &

INTEGRAL FORMS

133

where K/~ is the substitution in braces. Thus the same form as Darcy's law may be achieved (for flow in the x-direction). The numerical behavior of K will depend upon the behavior of the [3-fimctions, which will be subsequently discussed. Thus maxima will be found to occur, denoting that K is a variable in unsteady-state flow, which will be pointed out in Example 6.3, for instance. It should be emphasized again that the expression for G , above, pertains to the point or area of effiux (where 0 = 0). Thus the permeability so derived would pertain to that point. Other positions in the reservoir would no doubt yield other expressions and values, as also examined in Example 6.3. For small changes in P or p it is difficult to distinguish between, say,

3p

07~ M P--~Po

61'

and

p

_

c~ c?x _

- - -

M 1 vT 2 zRT 2 &

M P + Po 6( P - Po) zRT 2 &

~...

8P Po &

That is, the value of p or P outside of the integral is essentially constant for small changes, and G would behave linearly either with cT

M

cT

p0-~ = zR---7P0 ~ [ at P"Po or

cT P0 & =

M

16T 2

~RT2 ~ l at P - V 0

The foregoing suggests that the gradient, even at steady-state, is not necessarily clearcut as to the nature of the potential function to be used. The issue is complicated by the relation between density and pressure. Thus, consider the other two forms relating density and pressure. If, for instance, l/a

or

then

OP = abP,,_ ~ c7'_ a "~ ~ -7

p

cT ,~

134

UNSTEADY-STATE FLUID FLOW

If a = 1 (perfect gas), then obtained. Otherwise,

- f

G -

--

X

p = bP = M P / R T , and the previous relation is

( x 2 - x I -'l- Y2 - Y l + Z2 -- Z1

fl[ln ~]2

If

'21-e p a

c 7~

(36)

K aT ~ p ~ p If for a compressible liquid,

p - Ae ~P then

~cT ~ =

Ace

cgx

cgx cT = Cp~

cgx Therefore,

+z

-- Z1) 2

I aT cp& (37)

m m

K~

p

cT

~

As a special case of the preceding, if

/9 ~ Po [1 - c ( p o -/9)] - / 9 0 [1 - cPo ) + cP

= po[B + cP] then

Of,

c,T

-~- po~ ~,

INTEGRAL FORMS

135

and

g-

I

x (x~ - x, + y~ - yl + z~ - z,) ty fl[lnfl] 2

2 1

.p0 c 19

(38) K

c~

- - - - P j~

For the radial flow of a compressible liquid, by analogy with Equation 38, it follows that the equation form for Darcy's law is again produced:

G - - {t-~ [2rc(rb-ra)+(z2-zl)]2fl[ln fl] 2

p9o C - ~ z

P (39)

K

-

cT PaY

For the radial flow of a compressible gas, by analogy with Equation 35,

_ _{ x [ 2 z ( q - ro ) + (z2 - z,)]~ tf

fl[ln fl]2

(p / 190)Po

M1}

zRT 2rc Pc)"

(40) K -

c3o Pa3"

Note that in general G signifies a mass flux term, not a total mass flow term Q , whereby in radial flow Q = 27rr(Ah)G, where &h = z2- Zl. The foregoing radial expressions, therefore, apply to trait thickness, and would be multiplied by 2nr to obtain the mass flow rate Q at a radius position r. In effect, the transformation is x = 2nr. The sustained mass flow or mass effiux rate at the wellbore or san&ace, denotable as Q or G , respectively, would be for r = r,. Differences in the coordinate "y" do not have bearing, being incorporated in the radial coordinate transformations. Furthermore, note that for radial flow the term

136

UNSTEADY-STATE FLUID FLOW

TABLE 6.1 Behavior of 13-Functions

INTEGRAL FORMS (

X~. - X~ + Yz

-

Yl

~

+ ~2

137

- - %1

)z

is replaced by the term [2rc(r b

-

r~)

+ "~2

_ ~ 1 ]z

2re

Moreover, 2n(rb-r,) >> z2-z~ . In fact, rb >> r,. In steady-state flow, the permeability is ordinarily regarded as independent of position (for an isotropic medium). In unsteady-state flow, however, there is the implication that the permeability would vary with position (and time), and relate to the extent of the reservoir and the depletion time (or rate of depletion) -or vice versa.

Reservoir Dimensions Altemately, if the reservoir permeability K is known, along with the porosity X , it is possible to estimate the dimension of the reservoir from tf.

Behavior of the fl-Functions The ratio 13 = (t/tf)r is related to the density function at 0 = 0 by the expression Po-P_ Po

[l-In

t

] - f l [ 1 - lnfl] \ tr J

In tum, the behavior of p/p0 is of interest, as is 13[ln 13]2 and the product

13

[3] (p/p0)

The numerical detennmations are shown in Table 6.1. It may be observed that a maximum occurs in the behavior of each of the latter two functions tabulated. In Figure 6.3 the behavior of 13 [In 13]2 versus [1 - (p/p0)] is plotted for lower values of 13. For values of 13 less than circa 0.01, a straight line is produced on the log-log plot. This may be approximated by the curve fit 13 [In 13]2 = 2.64052 (1 - p/p0) ~

1.0.

I--'--'1

I: i i 1 t#

d~ 0.01

~7

0.001 O.

0.001

0.01

0.1

po" P

P. Fig. 6.3 Curve for detenummg permeability fit.

I.O

INTEGRAL FORMS

139

The foregoing relationship would apply to only smaller values for time. As a point of reference, note that t/tf 0.0001 (for C = 1) would correspond to one day out of 10,000 -- or one day out of 27.4 years. =

Maxima. That a maximum occurs in the behavior of 13 [In 13]2 is readily shown by taking the derivative and setting it equal to zero. Performing the operation,

0 = ,13 (2) [In 13] (]113) + [In 13]= and solving for In [3, ln13 = - 2

and

13 =0.135,335

therefore, at the maximum, 13 [In 13]2= 0.54134 The other combinations follow in tum, and may be evaluated in sequence as follows. For the product

13 [ha 131~ (plpo) = 13 [In 13]2 (]- 13 [~ 13]2) taking the derivative, setting it equal to zero, and collecting terms and simplifying, will yield 0 = 2 ]3 [In 13]2 + (1 + [3) In [3 + 2(1 -13) Solving for 13by trial-and-error, 13 = 0.069,674

or

-- 0.07

and it follows that 13 [In 13]2 (P/po) - 0.368,221 at the maximum. These maximum values may be assessed as the more proper values to use in detemammg a number for the permeability in unsteady-state flow. That is, the

140

UNSTEADY-STATE FLUID FLOW

result would more closely correspond to the steady-state determination of permeability. In support of this assumption, it can be recognized that during the initial period of depletion, or drawdown, there will be a lower pressure gradient across the reservoir -- for a given, constant production or efflux rate. During the latter stages of depletion, or long-term depletion, there will be a lessening of the pressure gradient~for the same given, constant production rate. Both situations will require that the permeability show an increase. At the intermediate between the two situations above, or the transition, the pressure gradient will show its largest value---for the same given, constant production rate. This in tum will require that the permeability have its lowest value. Which implies that the 13-functions as previously derived and calculated will exhibit maxima, since they occur in the denominator of the permeability term. The particular fimctions used will depend upon whether gaseohs or liquid flow is involved. The above observations are bom out in an analysis of the experimental data of Bruce et al. (3) for a linear gaseous reservoir, as shown in Examples 6.3 and 6.4, following. As an aside it may be commented that modem conservation practices limit the production of gas wells to a nominal 25% to 33-1/3% of the open flow rate. Therefore, based on the back-pressure equation

(,2- C ( _ A p , ), _ C(Po

- p,.o" ),

where P0 = P,b is the initial formation pressure or shut-in pressure, then for open flow, and where the bottom-hole flowing pressure or sandface pressure P,, is assigned a value of zero (or thererabouts), it follows that

Qo,o - C(eo )" and

p2). Q

=1_

Qo~.

r~

Po2

where Pr. is the bottom-hole flowing pressure, as previously utilized. If, say, n = 1 and Q/Qopo~= 0.333, then

P,, / Po

= 0.8165

(41)

INTEGRAL FORMS

141

Whereas, it may be noted from Table 6.1, that the value of the density ratio at the maximum for gaseous flow is p/p0 = P/P0 = 0.74472 Thus gas wells are produced at rates markedly similar to the rate whereby the unsteady-state permeability is at a minimum, and the pressure gradient (or gradient of the pressure squared) is at a maximum. That is, for a specified ot given flow rate, the permeability varies reversely with the density ratio, and vice versa. This will correspond to the transition between drawdown and long-term depletion. Furthermore, the maintenance of this condition will depend upon whether a water-drive exists, and the system is open rather than closed, as will be commented upon in Section 10.3 of Chapter 10.

6.4 PRODUCTION PERIOD If a reservoir is produced at some given constant flow rate, there will come a time when this flow rate cannot be sustained. This will occur when the density (pressure) at the point of effiux is at its lowest reachable value. That is, if the reservoir flared directly to the atmosphere, as in the experiments of Bruce et al., the minimum pressure would be -- 14.7 psia. In producing from an oil or gas formation, the minimum pressure at the bottom of the well bore (at the sand face) would be dictated by the first stage separator pressure and the pressure drop in the well tubing (and valves or chokes). The first stage separator pressure, etc., may of course be reduced to an allowable limit in order to maintain production. In any event, some limiting pressure may be assigned for the bottom-hole pressure. This in tum will give a corresponding value for time, which establishes the flow or production period for the specified production rate. Accordingly, this time value is located from the 13vs t plot as follows. Since Po - / 9 ~ Po

f i l l - lnfl]

Po-P Po

lnfl]

or, for a gas, ~~fl[1-

142

UNSTEADY-STATE FLUID FLOW

the limiting value of p (or P) will establish a limiting value for 13. In tum, from the plot of 13 vs t , the corresponding limiting value for time is determined. This value will be less than the theoretical depletion time tf

6.5 PREDICTION OF PRODUCTION It is preferable that at least two experimental points be used in predicting the production behavior for In 13 vs In t . A straight line can be projected as long as production remains constant and pressure is maintained, as will be shown in the examples at the end of this chapter. However, it is possible to provide estimates from just one point, and even to anticipate the point if the reservoir parameters are provided. It has previously been derived that, for gaseous flow, in consistent units, as per the previously-determined Equation 40 -- here designated as Equation 40a -the mobility can be estimated:

K /t

X ts

9

2z

M

fl[lnfl] (p/po)Po

ZRT

2

(40a)

Knowing the other variables or parameters, it is possible to determine a value for tf. This will provide the termination point for a plot of In J3 vs c In t . (Note again that "c" as used here is an introduced correlational constant or coefficient, as distinguished from the liquid-phase compressibility.) If the slope "c" is known, or can be estimated from the behavior of similar systems, then the plot is completed. Otherwise, another point will have to be determined. From the steady-state relationship for radial flow, for a gas, in consistent units, Q -

K

Ah

,u

vv/-P/ zR T

lnrb q

where Pb is also the initial pressure of the reservoir (P0). This may also be interpreted as Q-C(p

_ p

INTEGRAL FORMS

143

where "n" is introduced as a correlating exponent to distinguish, say, between laminar and turbulent flows. For a liquid,

K Pb-P Q - 2er-- Ah~ " kt

In rb r~

m

where p is the mean density. Other forms may be obtamed depending upon the behavior of the density function and the flow regime, etc. These may be designated as the "backpressure" equations. Using a back-pressure equation for the appropriate flow system, and knowing the parameters, it is possible to obtain a bottom-hole flowing pressure from the Q-rate assigned. Unfortunately, the exact value for the time at the point cannot be anticipated. Some sort of experimental backup reformation is needed. From the initial bottom-hole flowing pressure, it is possible to evaluate

Po-P

Po - P

po

1'o

In tum, Po - t 9

~=fl[1-1nfl] P0 rom which a value of 13 is obtained, where

/3-

t J

At the area of efflux, 0 = 0 and c

fl_

t r

If the value of "c" were known, then the initial value of t can be calculated to correspond to the initial back-pressure point. This would complete the plot of In 13vs l n t .

144

UNSTEADY-STATE FLUID FLOW

At this writing, it is suspected that "c" may also depend on the state or states of the fluid. It may also depend on flow rate and reservoir characteristics. Alternately, a value for time may be estimated from the unsteady-state solutions of Section 5.3.

Estimation of Initial Time Value From the radial solution described in Section 5.3, O-~r

=2Z

9o - O f

J~

e x p ( - a . . 2/-)

(42)

a . J 1( a . )

is a function of pressure or density, or may in fact be pressure or where density, and

F-rlr o

t-

X

?'~11 ......

Kip

The value of ro is as yet an arbitrary value. However, at some point (r0)~ it is required that

-J, (a.. r 0- Z

t~ e x p ( - a n 2i)

(43)

a,J.(a,.)

where rb is the outer radius. The particular values of (ro)~ or r~/(r0)i which satisfy the equation are the roots. Any root may be chosen for r0 such that the solution is physically reasonable. At the region of efflux, r a = rdro where here r~ is the wellbore radius, and where 9 = ~ , , . The solution of the two equations above is trial-and-error for t and (r0)i. Thus given a bottom-hole flowing pressure, it is possible to calculate a corresponding value for t. This may be used to obtain a time value t for the back-pressure used. The calculation is more hypothetical than practical.

INTEGRAL FORMS

145

A plot of w"

l

at

r =r.

~o -~jwould be most useful for these purposes, as would a plot vs t at r = r~.

Production Fall-Off A plot of P0 - P

vs f l [ 1 - In,B]

Po shows that P0 - P

>P

P0 and that as 13 falls off, so does (p0 - p)/p0. This is indicated in Figure 6.1. In a performance plot, the value of 13 [as calculated from (p0 - p)/p0] is plotted versus time in log-log coordinates. At a uniform or constant production rate, a straight line is produced. Therefore, in a performance plot of 13 versus time t , if 13 falls off, so does (p0 - p)/p0 9This indicates that p maintains a higher value than expected. The reference, therefore, is that the production rate falls off, since the foregoing derivations are predicted on a constant production rate. Similarly, if 13 should climb above the straight line performance, the production rate would have been increased.

Change in Phase The foregoing derivations utilize the density p . It is generally more convenient to use the pressure P , provided P is directly proportioned to p . As a system changes phase and the gas/oil ratio changes, then p and P are of course not directly proportional. Therefore, under these circumstances, pressure readings should be converted to density based on the gas/oil ratio, if known. As a limiting condition, as the gas/oil ratio approaches zero, the system enters the liquid phase and the depletion equations no longer apply since there would be no pressure drive -- for an incompressible fluid at least.

146

UNSTEADY-STATE FLUID FLOW

The formation must be pumped and the recovery cannot be complete. Thus the situation is encountered where on the average only some 30% or so of petroleum liquids may be recovered.

6.6 REPRESSURIZATION In the production of liquids -- i.e., oil -- any accompanying gas may be injected back into the formation at appropriate points in order to maintain flowing pressures. While natural gas production would serve the purpose, the value of the gas -- if it is lost -- becomes overriding. Carbon dioxide is an altemative. It not only provides pressurization but dissolves to a limited extent, lowering the viscosity of the oil. While the phase behavior of pure gaseous carbon dioxide would limits it use (P~ = 1071 psia, Tr = 87.9~ in mixtures with natural gas components the criticals of the mixture will necessarily be higher. Supercritical CO2, at pressures circa 2000-3000 psi, is more effective. While some formations do contain major amounts of carbon dioxide, the buildup of a synthetic fuels industry could in theory provide more. (For instance, in the conversion of coal to synfuels, half or more of the coal is converted to carbon dioxide in one way or another. This is aside form, or in addition to, the carbon dioxide produced from the final combustion of the fuel.) In the production of so-called condensate or distillate wells from formations which are at high pressure and in the single phase region outside the phase envelope, it is desirable to maintain formation pressure. For if the formation pressure is reduced into the two-phase region, there will be a separation of the liquid phase in the formation, which may in part be subsequently lost to production. If this separation occurs in the retrograde region, a maximum separation of liquid will occur since the L/V ratio increases as the pressure is dropped. There is the remaining possibility, however, that if the pressure is further reduced, the liquid will be eventually re-vaporized into the gas-phase region. As such, it would be in part retrievable provided the remaining formation pressure is sufficient to drive well production. This will depend on the well depth and the bottom-hole pressure vs the formation pressures. In general, repressurization pertains to the injection of both gases and liquids (e.g., water flooding). Flow and pressure gradients will be set up, and the definitive analysis of multiphase flow behavior will be complicated indeed.

INTEGRAL FORMS

147

EXAMPLE 6.1 The experiments of Bruce, Peaceman, Rachford and Rice (3) have previously been mentioned. The plot of In J3 vs t (Figure 6.1) indicates that the correlating coefficient "c" is very close to unity, since the slope for the point of efflux is nearly unity. The point tf obtained from the plot very closely satisfies the material balance, as follows: Pertinent data are"

Length Diameter Porosity Permeability Flow Rate Temperature Initial Pressure Gas used Viscosity

20.58 ft. 1.049 inches 0.367 13.2 millidarcies 0.000101 lb-mole hr 78~ 108.1 psia Nitrogen (0.018 cp)

The initial molar density calculates to 108.1 10.73(460 + 78)

= 0.018725 lb - mole / ft 3

The depletion time calculates to 1.049)2 (0.367)(0.018725) ( 2 0 5 8 ) 4 (--12--

tf= =8.4044 hr

0.000101 or

504.3 min

This value is very close to the extrapolated value at 13= 1. The initial material present thus can be calculated from the flow rate and depletion time: 0.000101 (8.4044) = 0.000848884 lb-mole In calculations for actual reservoirs, the data are never so clear-cut.

148

UNSTEADY-STATE FLUID FLOW

EXAMPLE 6.2 From the previously given data for the experiments of Bruce et al. (Example 6.1), substitution in the formula for K/~t for linear gaseous flow yields

K P

0.367

(20.58) 2

28

8.40 0.3682(0.018725)(28) 1543(460 + 78)

- 0.00323 3

where tx = 0.018 (6.72x104) = 0.121x10"* lb/ft-sec = 0.018 (2.42) = 0.0455 lb/ft-hr and K = (0.003233)(0.0435) = 0.000141 ft3/hr = (0.000141)/(0.00442) = 0.0318 darcies or 31.8 md The experimental steady-state determination was 13.2 md as compared to the above calculated value of 31.8 md..

EXAMPLE 6.3 The linear performance data of Bruce et al. (3) is shown in Figure 9.1 (Chapter 9). This data may be used to give some indication of the behavior of the permeability during tmsteady-state flow, as will be indicated in Tables 6.2 and 6.3. TABLE 6.2 Reduced Pressure vs Reduced Distance [From Results of Bruce et al. (3)] t 0 4 10 2O 4O 90 135

x=0 1.00 0.93 0.87 0.79 0.68 045 0.25

x = 0.125 1.00 0.98 0.95 0.90 0.805 0.63 0.505

x=0.3 1.00 0 995 0.98 0 96 0.89 0.75 0 64

x=0.5 1.00 1.00 1.00 0.09 0.97 0.885 0.795

x = 0.75 1.00 1.00 1.00 1.00 0.958 0.94 0.88

1.0 1.0 1.00 1.00 1.00 0.995 0.96 0.91

INTEGRAL FORMS

149

TABLE 6.3 Square of the Reduced Pressure vs Reduced Distance [From Results of Bruce et al. (3)] t

0 4 10 20 40 90 135

x=0 1.00 0 865 0.770 0 624 0.462 0.203 0.063

x = 0.125 1 00 0 960 0 990 0810 0 648 0 397 0.255

x=0.3 1.00 0.990 0.960 0.922 0.792 0.563 0.410

x=0.5 1.00 1.00 1.00 0.980 0.941 0.783 0.632

x = 0.75 1 00 1 00 1 00 1 00 0 970 0 884 0 774

x=l.0 1.00 1.00 1.00 1.00 0.990 0.922 0.828

In Table 6.2 values of the reduced pressure are tabulated vs reduced distance as obtained from the plot made by Bruce et al. for parameters of time. The squares are in Table 6.3 with a plot of the squares vs distance in Figure 6.4. TABLE 6.4 Comparison of the Gradient of the Square of the Pressure [From Results of Bruce et al. (3)] Between x = 0 and x = 0.125 Ap2/Axbar Ap2/Ax t(mm) /~2 (psi2/ff) 0 0.095 0.76 431.5 4 013 1.04 590.5 10 0 186 1.488 844.9 20 0 186 1.488 844.9 40 0 194 1.552 881.2 90 135 0 192 1.536 872.2

Between x = 0.75 and 1.0 Ap2 AP2/Axbar AP2/Ax (psiE/ff)

0.010 0.020 0.030

0.040 0.080 0.120

22.7 45.4 68.1

With allowances for discrepancies in reading the curves, nevertheless the first two points give an indication of the derivation at the closed end. A comparison of the differences in the squares of the reduced pressures is given in Table 6.4. The derivative as calculated first shows an increase and then levels off with time. For a constant mass efflux rate, this could suggest that the permeability would be higher initially, then would decrease and level off. Though changes in

150

UNSTEADY-STATE FLUID FLOW

gas viscosity with pressure would have some effect, the effect is small. Moreover, gas viscosity increases with pressure, which would require an even greater increase in K for the initial stages. Interestingly, at the closed end, the derivative hypothetically should be zero, based on potential theory (e.g., the unsteady-state Darcy relation), whereas experimental values other than zero are indicated as the depletion proceeds. Perhaps, as will be indicated in Chapter 11, the behavior of the pressure gradient at the closed end is dependent, and cannot be preassigned as a boundary condition.

~.0

.s .4

.3 .2

0

.1

.2

.3

.S

X red.

1.0

Fig. 6.4 Plot of the square of the reduced pressure versus reduced distance. From the results of Bruce et al. (3).

INTEGRAL FORMS

151

EXAMPLE 6.4 As a check against the pressure gradient from the data of Bruce et al. (Example 6.1), the linear relation in moles is 1K

Q~-

1

C'~ 2

A~

(where P increases with x)

2 ju z R T

where Qm - 0.000101 lb - moles / hr

/t-0.018

cp = 0.04356

A - (1049)2 rc _ 0.00600 144 4

lb ft - hr ft 2

R - 1543 T = 460 + 78 = 53 8 ~ R

The pressure above is to be in psf. Hence, using a value of ~792

psf 2 (in

CT 2 ) -

ft

c~

c3P2/c3x ~

in psi ft ) times (144 )2 - 8 7 0 ( 1 4 4 ) ' -

-

870 psi2/fc,

18.0x10

-6

Therefore, K

m

0.000101(2 )(0.04356)I 53(53 8) = 67.7xl 0 -6 ft 3 hr 2 (0.00600)18.0x10-6

In millidarcies, K -

67.7x10 -6 0.00442

(1000) - 15.3 m d

The steady-state permeability was measured as 13.2 md with a variation of about 10% from one end to the other. At shorter transient times, the derivative is smaller, which would cause the permeability to increase above this value.

152

UNSTEADY-STATE FLUID FLOW

EXAMPLE 6.5 Drawdown and buildup curves are sometimes used to estimate reservoir size or fluid in place (4). Pressure vs time is obtained for an essentially constant withdrawal rate, over a significant period of time. A plot of p2 vs log time is made and used in the calculation (4). The semi-log plot is assumed to be a straight line. The subject will be further discussed in Chapter 9. Interestingly, if the data of Bruce et al (3) are plotted on log-log coordinates, a curve is obtained as shown in Figure 6.5 using the tabulations of Table 6.5. The initial portions, however, may be approximated by a straight line over a sufficiently short interval. This sort of limited approximation is characteristic of the shape and sweep of logarithmic or exponential data.

TABLE 6.5 Data of Bruce et al. (3) at Point of Effiux t (min) 0 4 10 20 40 90 135

P(dimensionless)

Po- P/Po

1

0.00 0.07 0.13 0.21 0.32 0.55 0.75

0.93 087 079 068 0.45 0 25

p2 1.000 0.865 0.757 0.624 0.462 0.203 0.063

p2

GO

10

I00

t (,,,I,,.)

Fig. 6.5 Plot of pZ versus t from the data of Bruce et al. (3).

154

UNSTEADY-STATE FLUID FLOW REFERENCES

1. Hoffman, E. J., The Concept of Energy." An Inquiry into Origins and Applicanons, Ann Arbor Science, Ann Arbor, MI 1977. 2. Hoffman, E. J., Phase and Flow Behavior in Petroleum Production, Energon, Laramie, WY, 1981. 3. Bruce, G. H., D. W. Peaceman, H. H. Rachford and J. D. Rice, "Calculations of Unsteady-State Gas Flow Through Porous Media," J. Petroleum Technology, 5, 79-92 ( 1953). 4. Katz, D. L., D. Comell, R. Kobayashi, F. H. Poettmann, J. R. Elenbaas and C. F. Wemaug, Handbook of Natural Gas Engineering, McGraw-Hill, New York, 1959.

Chapter 7 TWO PHASE AND MULTIPHASE FLOW: GAS, OIL, AND WATER

The two extremes of two-phase or multiphase flow behavior are (a) displacement of one fluid by another and (b) the concurrent or cocurrent flow of the phases. In petroleum reservoirs the situation is compounded by the presence of the original or connate water (or brine) interspersed or dispersed with the hydrocarbon fluids (gases and/or liquids) in the reservoir rock interstices or pores. Moreover, this water may either form a distract phase or emulsify. In other instances, stratified layers of the separated phases may coexist, with the more dense aqueous phase constituting the bottom layer, and such free or undissolved gases as are present forming a gas cap. Ordinarily, there is a connection with the aquifer, designating an open system, signaled by the formation pressure corresponding to the hydraulic head. Otherwise, the formation is closed and can be. called geopressured or abnormally-pressured, at either a higher or lower pressure than the hydraulic head. Thus, generally speaking, a formation will exhibit a different effective permeability to gas, oil, and water--and each permeability may be subject to change. This difference is reflected in the concept of relative permeability. Stratification or holdup may also affect the relative flows. The situation starts to get out of hand even without the additional complications of water encroachment into the reservoir formation and such other problems as may be present. For these reasons, and other reasons no doubt, a simplified overall generalization would be most welcome if it can be demonstrated to reproduce and predict production and depletion behavior. Furthermore, as the hydrocarbon system originally exists in the petroleum reservoir, it may be totally in the single-phase region as a less-dense single-phase (gas) or a more dense single phase (liquid), as indicated in the following Figure 7.1 on a pressure-temperature or P-T diagram. In other words, the mixture may be outside the two-phase region, and in the region near or beyond the critical point pressure can be designated as a supercritical fluid. Or the mixture may exist originally in the two-phase region, with the less-dense gas phase occurring as a gas cap. Hence the term gas-cap drive or gas drive, albeit a gaseous phase may exsolve during flow.

156

UNSTEADY-STATE FLUID FLOW RETROGRADE REGION'~

Liquid

~ / /

/ / /

P

/

/

/

/

/

/

RETROGRADE REGION

/

//

~ ~

Critical

~/"

J

/

Single Phase Region

I

/ /

/

/',,I /

/,,.

/ /

~

o~

Vapor (LESS DENSE SINGLE PHASE)

Fig. 7.1 Phase diagram in P-T coordinates for a complex hydrocarbon mixture.

For as the pressure is reduced during flow, there is the tendency for the gas to evolve or exsolve from the liquid -- referred to as so-called dissolved gas. Thus the term solution-gas drive. The action is most pronounced for so-called condensate or distillate wells, where the hydrocarbon mixture may exist originally in the formation as a moredense single phase mixture at high pressures, circa the critical. Thus there can be flow by displacement due to the expansion of a gas cap and/or by the encroachment of water from the aquifer. At the same time, or in lieu of, there may be the simultaneous flow of two or more phases, due to the exsolution of dissolved gases and the presence of connate water. And/or there may be revolved the channeling or mixing of the entrained gas or water phase. Beyond this, a gas or water phase may be injected. The possibilities lead to complexity, with one and the other occurring simultaneously in varying degree. It is associated with the concepts of a waterdrive, gas-drive or solution-gas drive, and which may occur naturally and/or be artificial. The former or natural occurrence is reduced by the reduction of pressure at the wellbore. The latter or artificial occurrence is more associated with an increase in or maintenance of pressure at some point or points in the outer producing regions by the injection of a gaseous or aqueous phase via an injection well or wells.

TWO-PHASE AND MULTIPHASE FLOW

157

Moreover, there is the possibility that more than one hydrocarbon-derived liquid phase may exist. That is, liquid-liquid heterogeneity can occur in hydrocarbon-rich phases, particularly if oxygenated compounds are present--e.g., as in asphaltic or heavy crude mixtures (1). Even solid-like materials may tend to precipitate, such as waxes or resins. This phase separation action in the liquid region is in fact used as the basis for the solvent refining of crude oils, for instance to produce more desirable lubricating fractions. A number of selective solvents can be used, including the lighter paraffinic hydrocarbons such as propane (1, 2, 3). Phase separation effects may also occur in the liquid region or regions, and have been observed, e.g., during the injection of liquid or supercritical carbon dioxide for the purposes of enhanced oil recovery (1). In general it may be said that two-phase or multiphase flow may involve either or both the parallel, concurrent or simultaneous flow of phases, or the series flow or displacing flow of phases. Moreover, phase changes may accompany flow, as in the exsolution of gases from the liquid phase. Also of significant mention is two-phase or multiphase flow up the well bore, whereby the presence of, or exsolution of, a less-dense phase (e.g., gaseous phase) will contribute to flow. The effect may be described as one of buoyancy. There will resuk a smaller pressure increase from the wellhead to the bottom of the hole. Thus the bottom-hole pressure will show a decrease for a given wellhead pressure. As the pressure at the wellhead decreases, or is too low, it becomes necessary to reduce flow by pumping and/or compression at the bottom of the hole or at the surface. The subject of artificial lifts is treated exhaustively in Reference 4. The subject of displacement will be dear with in a subsequent chapter based on unsteady state long-term behavior. Relationships more applicable to the concurrent flow of phases are presented and discussed in the next section.

7.1 CONCURRENT TWO-PHASE FLOW The relationships for concurrent two-phase or multiphase flow are developable from the dissipative or lost work effects for flow through porous media, which are equivalently a statement of Darcy's law for each contributing phase. This in tum leads to the idea of relative permeability. Such correlations as are usually developed provide a means to calculate or correlate the two-phase pressure-drop in terms of the respective flow rates of the two phases and their properties, and in terms of the single-phase pressure-drops as

158

UNSTEADY-STATE FLUID FLOW

if the entire flow system were one or the other of the phases. Or in other words, the two-phase pressure-drop is prorated in terms of a single-phase pressure-drop. A particular representation will be derived here based on Darcy's law for each of the respective phases. As previously developed in Chapter 4, for laminar or viscous flow it may be written (Darcy's taw) that Q

_KA p z dtw ,tt dL

(1)

K dP Q ---Ap~ p dL

(la)

-

or for horizontal flow,

-

where here L is the distance in the direction flow, and where A is the crosssectional area normal to flow. Consider two phases "V" and "L" where V is the less-dense phase and L the more dense phase. For convenience, let Qv = mass flow rate of phase V QL = mass flow rate of phase L Also QT = Qv + Qt.

(2)

so that it may be written that here,

I_x_Q~,,

= mass fracton of V

(3)

x_QL -~

= mass fraction of L

(3a)

QT

QT

(The symbol X is also used for porosity, as is the symbol ~. It is usual problem of reconciling symbols that are customary in different disciplines.) For steady-state flow for two phases V and L it may be written that

TWO-PHASE AND MULTIPHASE FLOW

159

Qv = - Kt---LA P t dPr tu v dL

(4)

K QL - - --__LA P L dL l.tL

(4a)

where the subscripts V and L denote the respective properties, and the subscript T denotes the two-phase pressure-drop~that is, the common pressure-drop for the total two-phase flow system. Furthermore, the permeabilities Kv and KL are the effective permeabilities based on the total crossectional area A. It may altemately be written that x Qr - - ~ Av P v Pr dL

(5)

QL - - ~,uL AL PL dL

(5a)

where K is an absolute permeability for the medium, and Av and AL are the effective or prorated crossectional areas for each phase. It is evident that

Qv

Kv Pv PL

QL

KL P L l'tv

Qr

Ar Pv Jut.

QL

AL P r. P z

(6)

or

(7)

and that

Also,

Kv

Av

KL

AL

(8)

160

UNSTEADY-STATE FLUID FLOW Kr --Pv ktz,

=

Qr Kr ----PL QL PL

(9)

or

QL Kr

----Pv Qr PL

Kv

(9a)

=-pt. /aL

These expressions interrelate mass velocities and flow properties,

Velocity and Mass Flow For a two-phase mixture in fully-developed steady-state flow, both phases will move at the same actual linear velocity. This may be based on the total crossectional velocity. This common velocity will be

Qv / pv + Qr. / p l = superficial vel. A

(lO)

Therefore, if the crossectional area assigned to V is Av, then

Qv / pv + QL / P~. = Qr / Pv A

(1 1)

Ar

and for L,

Qr / pv + QL / P L

Qr / pL

A

AL

(1 la)

whereby

Qr lpr..

QL IPL

A

Ar~

and

Av/A-

AL/A-

Qr i pv Qv l p v +QL / p L

QL /PL Qr /p~, +QL /PL

(12)

(13)

(13a)

TWO-PHASE AND MULTIPHASE FLOW

161

from which mV ___jr

A

mL

(14)

-1

A

Thus A-A

v +A L

(15)

or

A

Ar

AL

AL

+I

or

A - I + ~m L Av Av

These expressions prorate the assigned crossections.

Relative Permeabilities

Note also that, ideally, Kv

Av

KAv - K v A

or

~ = -K A

(16)

KA L - K LA

or

KL AL -- = K A

(16a)

and

The quantities Kv/K and KIJK would be the relative permeabilities of phase V and phase L. Furthermore,

Kv = Av KL

(17)

Ar

Since Qr / PL __ QL / Pz, = vel. Ar AL

(18)

then, altemately, for the relative permeability of phase V to phase L (Kv/KL),

162

UNSTEADY-STATE FLUID FLOW Av = (Jr.- PL _ Kf___2-_ Ar QL Pv KL

(19)

k follows in tum that

A K - Kv A/. = x v (1 + Av - K,.. (1 + ~ f ) - K v + K L

(20)

such that a

g V

K

g L

+ ~

K

(21)

from which K Kr =~+1 KL KL

(22)

or

K = 1 + ~g L Kv Kv

(22a)

These expressions relate phase permeabilities to absolute permeability,

Comparison to Single-Phase Flow If the total flow system was wholly constituted of one or the other of the phases, then it may be written that K Qr = - ~ A p v t.t v

dPrT dL

(23)

or

Qr - _ K___KA_ p v dPrr dL t.tL

(23a)

where dPvr/dL and dPLT/dL would be the respective single-phase pressure-drop. The value K is the absolute permeability.

TWO-PHASE AND MULTIPHASEFLOW

163

Therefore,

Pv Dp.

p~ dP~

Pv

PL dL

dL

(24)

Substituting for the permeability terms,

.K•?

--Pv fir"

ae~ dL

Kv --Pv Pr

4

d~7

QL Kv

(-dL)

Qv v pv

KL Pv Pv

Qv KL ~~Pv QL ItL

+

dPLT

KL (pL Pv ~)(_ ItL PL PL

dP~,,r Qv Qv +QL (- dL )+

QL

(_ dPLT )

Q,, + Q~

dP~

aL )

dL

dP~

- ( 1 - X ) ( - dL )+ X ( - 'dL )

(25)

Similar results may be obtained by other means. Other such derivations, correlations and references for two-phase flow and the accompanying pressuredrop are supplied for instance in Reference 1.

Conversions to Volume Fractions or Saturations The respective volume fractions for each phase may be interpreted as the fraction or percentage saturation. Thus, for phase L , the liquid volume fraction (VLF) or saturation SL would be

LVF- Sr =

X / PL X / PL + ( 1 - X) / Pv

(26)

164

UNSTEADY-STATE FLUID FLOW

and for phase V, the vapor volume fraction (VVF) or saturation Sv would be

V V F - 1- L V F - S v =

X/Pv X/PL

+ (1- X)/Pv

(26a)

The above are the respective saturations, where Sv + SL = 1. If it is so desired, the figures may be adjusted to the bulk volume of the porous medium. And strictly speaking, the saturation should refer to the fractions or percentages of the fluids at rest, or as originally occurring in the reservoir.

Cumulative Pressure Change The cumulative pressure-drop is given by L2

(- aL.,)aL

(27)

L1

If pv or pt is a function of pressure, then this can be so entered into the determination. Such may require numerical procedures for the integration or solution. If a phase change occurs, then the integrated pressure-drop may be represented by

x2 dL _APr_ i (_dP~) ,dr dL dX X1

(28)

where d ~ d L denotes the degree of phase change per unit distance.

Degree of Dispersion If one phase is dispersed (as droplets) in the other phase, then the degree of dispersion and droplet size can have an effect on the two-phase flow behavior. Generally speaking, the two-phase flow can be based more upon the continuous phase, and utilizing the viscosity of the continuous phase. An interesting problem in two-phase flow, therefore, is whether the liquid phase, say, will form a continuum with the gas phase fully dispersed, or whether the gas phase will form the continuum with the liquid phase fully dispersed. Such can have a marked effect on frictional losses and pumping or compression

TWO-PHASE AND MULTIPHASE FLOW

165

requirements. It can be of significant consequence in vertical flow, where the liquid-phase hydraulic head may or may not exist, depending upon whether the liquid or gas forms the continuum. Thus there can be more to two-phase flow behavior than mere correlations against only the liquid/gas ratio and the respective phase properties.

7.2 MULTIPHASE FLOW The representation may be extended to more than two phases by analogy. In other words, the results may be generalized to three or more phases -- that is, to multiphase flow -- by prorating the pressure-drops in single-phase flow on the basis of the respective mass fractions. The phases existing in petroleum reservoirs will most likely consist of a gas or vapor phase, an oil or hydrocarbon-rich phase, plus water.

7.3 IMMISCIBLE AND (PARTIALLY) MISCIBLE DRIVES Generally speaking, in a petroleum reservoir, the existing phases will not move in true concurrent flow. There will be holdup and displacement of one phase with respect to another, and channeling may occur -- attributable in part to surface and interracial effects. Displacement in particular requires special consideration, and is allied with gas and water drives, natural or induced. The subject is further expanded upon in Chapter 15. There is also the subject of miscibility or partial miscibility such as occurs in enhanced oil recovery, e.g., the injection of hydrocarbons or carbon dioxide. Here there may be mixing or solvation, even emulsification, in addition to or in lieu of displacement. The most common type of drive is no doubt that of water-drive, due to the presence of or connection to an aquifer. The oil in the formation is (partially) displaced by encroaching water under the hydraulic head. This is essentially a problem of a moving boundary at a constant outer boundary pressure. It shares a kinship with water-flooding. In lieu of a water-drive, or in addition to water-drive, there may be a gas cap which propels the liquid phase. This may be viewed as gaseous encroachment at varying pressure, as the gas cap expands. Finally, there may be dissolved gases, as in condensate or distillate liquids. Or in other words, there is a more dense, liquid phase which will evolve gas -- causing the system to enter, or further enter, the two phase region when the

166

UNSTEADY-STATE FLUID FLOW

pressure is reduced. Thus as flow proceeds under a pressure gradient, there will be a simultaneous evolution of gas. The liquid may be variously viewed as flowing simultaneously or concurrently with water and/or gas, with constant or varying phase proportions, or as being displaced -- or both. The representation in any event will be complex, with a different permeability and viscosity for each phase. It is therefore much simpler to use a composite representation, assuming average or mean reservoir and fluid properties, particularly for two-phase or multiphase flow systems. As to immiscible displacement, as in a water-drive, in the limit the situation can be viewed as steady-state flow in which the outer producing radius is varying. That is, in terms of steady-state radial liquid flow, Q _ K (2n.)z~p &P /~

(29)

In rb r~

where AP and r, are constant but rb is varying by the material balance

Q(t - 01 - zc{[(rb )0 ,. - ra 2 )] -[ro E - r a 2

(3o)

where (rb)0 is the initial outer producing radius or radius of drainage. A more detailed representation will be provided in Chapter 15. The foregoing presentations may be perceived as a more "primitive" viewpoint for expressing the relationships involved in two-phase flow and multiphase flow. For more complicated or sophisticated representations, referral may be made to the volume Nonlinear Dynamics of Reservoir Mixtures by Vladimir S. Mitlm (5), and to the appropriate chapters and authors in Mathematics in oil production, edited by Sir Sam Edwards and P.R. King (6), and The Mathematics of Oil Recovery, edited by P.R. King (7). The subject can be made very complex indeed.

7.4 ENHANCED OIL RECOVERY Multiphase flow is generally involved in enhanced oil recovery in one way or another. There is the injection of a phase which may be immiscible or partly miscible with the oil in the formation such as to cause additional recovery. In effect, the mobility is improved.

TWO-PHASE AND MULTIPHASE FLOW

167

The degree of miscibility may serve to reduce viscosity (and decrease density) as is the case for supercritical or liquid carbon dioxide injection. There may be accompanying thermal effects as in the injection of steam or the injection and combustion of combustibles. There may be simultaneous flow or displacement, or more likely a combination of both, with the eventual breakthrough of the injected phase. The use of the relationships for simultaneous flow are warranted -without or with a change in phase, i.e. phase contraction or expansion may occur with regard to any one or more of the phases present. The problem, however, starts to get out of hand in view of the several permeabilities and viscosities which enter into the representation. The representation of displacement also has its problems in that the outer reservoir boundary is no longer govemed by the production material balance, as previously explained, and which will be further developed in Chapter 15. A miscellany on the expanding subject of enhanced oil recovery for heavy crudes in its many aspects, including heavy crudes, is introduced via References 815. Additional and more current reformation is found in other volumes, publications, papers, conferences, and symposia such as exist. The subject is ongoing -- e.g., the Society of Petroleum Engineers, Dallas -- and interest waxes and wanes with imports and oil prices, but is of primary concem here only in terms of the more elementary flow relationships.

REFERENCES

.

Hoffinan, E. J., Phase and Flow Behavior in Petroleum Production, Energon, Laramie, WY, 1981, pp. 341 ff. Gruse, W. A. and D. R. Stevens, The Chemical Technology of Petroleum, Second Edition, McGraw-Hill, New York, 1942, p. 301,337 ft., 476. Nelson, W. L., Petroleum Refinery Engineering, Fourth Edition, McGrawHill, New York, 1958, p.347ff. Brown, K. E., The Technology of Artificial Lift Methods, in six volumes, PennWell, Tulsa OK, 1977-1984. Mitlm, Vladimir S., Nonlinear Dynamics of Reservoir Mixtures, CRC Press, Boca Raton FL, 1993. Mathematics in oil production, Sir Sam Edwards and P. R. King Eds, Clarendon Press, Oxford. 1988. The Mathematics of Oil Recovery, P. R. King Ed., Clarendon Press, Oxford, 1992.

168

UNSTEADY-STATE FLUID FLOW

8. Haynes, H. J., Enhanced Oil Recovery, EOR: An Analysis of the Potential for Enhanced Oil Recovery from Known t~)elds in the United States, 1876 to 2000, National Petroleum Council, Washington D. C., 1976. 9. Enhanced Oil Recovery: Secondary and Tertiary Methods, H. H. Sehumaeher Ed., Noyes Data Corp., Park Ridge, NJ, 1978. 10. The Future of Heavy Crude and Tar Sands, R. F. Meyer and C. T. Steele Eds., J. C. Olson Asst. Ed., United Nations Institute for Training and Research (UNITAR), Alberta Oil Sands Technology and Research Authority (AOSTRA), and United States Department of Energy, Edmonton, June, 1979. Mining Information Services, McGraw-Hill, New York. 11. Van Poollen, H. K. & Associates, Fundamentals of Enhanced Oil Recovery, PennWell, Tulsa, OK, 1980. 12. Heavy Oil and Tar Sands Recovery and Upgrading (International Technology), M. M. Sehumaeher Ed., Noyes Data Corp., Park Ridge NJ, 1982. Based on research by Roebuck Associates and Booz-Allen and Hamilton, Inc. 13. Heavy Crude Oil Recovery, E. Okandan Ed., Martinus Nijhoff Publisher, The Hague, 1984. 14. Enhanced Oil Recovery, Donaldson, E. C., G. V. Chilingarian and T. F. Yen Eds., Elsevier, Amsterdam, 1985. 15. Careoana, Aurel, Applied Enhanced Oil Recovery, Prentice Hall, Englewood Cliffs NJ, 1992.

Chapter 8 STEADY-STATE: PRODUCTIVITY TESTS

The flow-testing of naturally-flowing wells is ideally conducted at stabilized conditions of flow and pressure (and temperature) at the wellhead or at bottom-hole. If two or more phases occur, the flow rate of each phase is to remain constant. Such a stabilized condition can be referred to as a pseudo or quasi steadystate condition. Or for practical purposes, this condition may be designated simply as "steady-state." Or, as will be developed in a subsequent chapter, may be more properly considered as the transition between initial drawdown and longterm depletion. Such a steady-state condition may be attained to a degree in the singlephase flow of either a compressible liquid or a compressible gas. The pressure effects are more pronounced if a gas phase is involved, due to its greater compressibility and its lower density. The latter contributes to lift (or a reduction in head) at the wellbore. Thus in the presence of a gaseous phase, there will be a gas-drive whereby a gas/oil ratio can be established at the wellhead for different separator conditions (pressure and temperature and flow rates). The determination is especially appropriate for condensate or distillate wells. As a means of rating oil and gas wells, or oil/gas wells, the concept of productive capacity is used. The productivity or productive capacity of an oil well to produce is called the well's potential (1), and is generally measured in barrels per hour or barrels per day. It has been characterized as a measure of the formation to deliver fluid into the wellbore, but not of the reserves in the reservoir (1). As will be subsequently indicated, such tests can be used to estimate an outer producing radius or radius of drainage, whereby a figure can be obtained for the fluid in place. The most common methods for determining the production capacity of oil wells are as follows (1):

Flowing Wells: The well is produced at different rates using different chokes or valve settings. The corresponding bottom-hole flowing pressures are measured. Ideally, a plot of bottom-hole pressure versus

174

UNSTEADY-STATE FLUID FLOW production rate will yield a straight line. The reformation can be extrapolated down to say zero pressure to yield a hypothetical open-flow rate.

Pumping Wells: The height of the fluid column in the casing is measured while the well is pumped at different speeds. A steady-state condition is reached at each speed when a sonic meter indicates the fluid level ceases to fall. The production capacity of gas wells may be determined directly or by extrapolation as follows (1):

Open-Flow Method: The gas well is opened up to maximum capacity and the volumetric flow is measured using a pitot tube. The method is simple and direct but wasteful of large volumes of gas. Moreover, the formation and well fittings may be damaged, and interpolation to actual operating conditions at lesser flows is suspect. Back-Pressure Method: The well is produced at different rates, with the corresponding wellhead pressures measured and adjusted to bottom-hole conditions. A log-log plot of the difference between the squares of the initial and flowing pressures versus the production rate will tend to yield a straight line. This plot can be extrapolated to zero bottom-hole pressure to yield a hypothetical open-flow rate. Ideally the slope of the log-log plot is unity, but in general may have other values. Ideally, the back-pressure method will yield a straight line with a slope of unity on log-log plot, which would mean that a linear plot could also suffice. More usually, however, the log-log plot will have a slope other than unity, for reasons to be further discussed. Thus in the testing of either oil or gas wells, particularly, and at different stabilized flow rates, a pattem emerges in the comparison of pressure versus rate. These test results can be projected to provide a hypothetical open-flow rate, the flow rate if the bottom-hole pressure was zero, or at least atmospheric pressure. While hypothetical, it provides a means to compare the performance potential of one well (or field) against another. The further utilization of the above reformation is as follows. Of first interest here will be its projection to estimate reservoir size and reserves, based upon the producing radius or radius of drainage, which signifies the outer reservoir boundary. a

STEADY STATE: PRODUCTIVITY TESTS

175

8.1 DETERMINATION OF PRODUCING RADIUS A reiteration for the steady-state relationship is as follow, from which an estimate can be made for the outer producing radius or radius of drainage.

Liquid Flow The integration of the differential form at constant flow rate (steady-state) requires that the density be some know fimction of pressure. For near-incompressible liquids, the density is a constant by definition. For slightly compressible fluids (liquids), the relationship is often used that p - poe-C(Po-P) = poe-~Po e ~P _ Ao e~P or

p-11n_ P c Ao where c is the fluid compressibility and the substitution A or Ao becomes another correlating coefficient. As previously noted, the above is perhaps more appropriate for compressible liquids, much less so for gases. Therefore, integrating between limits, at constant temperature, will yield the following forms for Darcy's law, as previously indicated in Chapter 5: Linear Flow (m -x direction)

K

P -P,

O or Qm = - - A P o X2

-- X 1

Radial tqow Q-

K (2rc)Ahpo /2

In the above form for linear flow, the symbol crossectional area normal to flow. It may be noted that

0 Q"=G=--A

A

represents

the

176

UNSTEADY-STATE FLUID FLOW

where Q" or G is the mass flux. The above form for the radial flow of a liquid can be written conveniently as

Q = P I (Pb- P a) or

Q = PI (P2- P0 where ideally the productivity index PI is given by the substitution

101_~K2 7 r ~ P o

1 r~

where the units are required to be consistent, and p ~ p0 is assumed essentially constant. In this fashion, knowing the productivity index PI and the permeability K in consistent units, then the outer producing radius rb can be calculated from the other properties and parameters. Thus rb

- r,, exp{ (K

/ P)2rcAhp~

}

where r, denotes the sandface or wellbore radius. In tum the reservoir volume V is

v-.(#: -#:) from which the original volume of oil in place is VX (which can be converted to barrels, e.g., one bbl = 42 gal = 6.614 cu if), and the original mass of oil in place is mo = VX p0 where p0 is the initial density, and traits are to be consistent. Note that if VX is in cubic feet, then the conversion to barrels is

1

V X i n v a r r e l s --

WXincubicfee t

(7.48)~ 42

If po is in pounds per cu It, then mo = (VX)~. ~ po will be in pounds.

STEADY STATE: PRODUCTIVITY TESTS

177

Gaseous Flow

For gases it is more appropriate to utilize the form of the gas law whereby MP zRT

where M here is the molecular weight and z is the compressibility factor. For an ideal gas, z = 1. It is also convenient (if not necessary) to assume a constant or mean value of z for the purposes of integration. Therefore, assuming the form for the gas law, integration between limits for the steady-state will yield the following, at constant temperature and constant z, as also indicated in Chapter 5: Linear Flow (m - x direction) 1 K

M

Pz 2 - 1"12

2 t,t

zRT

xz - x1

where "2" and "1" denote the reservoir limits.

Radial I~7ow

~

zRT

In G r~

where "b" and "a" denote the outer and tuner reservoir limits.

Application to Practice.

The radial integrated form (or the linear form) may be

written as ~

2 - /,,2 )

or

where Pb or P2 is the initial formation pressure and P, or P~ is the flowing pressure at the sandface or wellbore at the bottom of the well.

178

UNSTEADY-STATE FLUID FLOW

K has been found that this form correlates very well with reduced changes

in the flow rate versus bottom-hole pressures, during stabilized production following the transient period of drawdown -- as distinguished from the long-tema depletion performance of the well. These short-term tests, referred to as "backpressure" tests, are relatively expeditious tests conducted upon the completion of a gas well, as stability ensues following drawdown, for each prescribed flow rate. That is, a log-log plot of the difference in the squares of the bottom-hole shut-in pressure and flowing pressure versus the mass flow rate (or its volumetric equivalent in SCF) will tend to yield a straight line. From the intercept of this line, a value for C can be determined. It is sometimes also required to adjust the value of the squared exponent, or rather to apply an exponent to the difference of the squares. The open-flow rate is defined by a mathematical extrapolation, whereby the absolute pressure at the sandface is by definition zero. Since in theory the value of C may be ascribed as K M C = rc~ Ah--

,u

1

zRT In r~ ro

then if the reservoir permeability, gas viscosity, reservoir thickness, gas molecular weight and temperature are known, the ratio rb/r, can be calculated. In tum, from the radius r, at the sand face or well bore, a value for rb can be calculated, which is called the radms of drainage. The calculation is, however, very sensitive to the value of the permeability and other parameters used, since logarithmic (or exponential) behavior is revolved. Moreover, the reserves vary with rb2 It is therefore preferable that the outer radius rb be determinable independently, rather than based on r~. For rb will establish the reservoir extent, and should not be susceptible to assumptions about r~. An independent, first appraisal of rb is in fact possible via the tmsteady-state solutions of Bruce et al. (2), from which r, can then be estimated from the steady-state relationship, as will be further developed in Chapter 10. Thus in one way or another it is possible to estimate the dimensions of the reservoir and hence its volume V or pore volume VX : v -

-

from which the number of initial moles of gas in place is

tl 0

=

~

.

zoR?o

STEADY STATE: PRODUCTIVITY TESTS

179

Since the number of standard cubic feet of gas in place is SCF = 379 no then, in millions of standard cubic feet (MMSCF), MMSCF = 379 no (10 "6) In the experiments by Bruce et al. (2) using a linear reservoir closed at one end and depleted at a constant mass flow rate at the other, the pressure profile at successive times tended to follow Darcy's law as previously shown in Example 6.3. Though the permeability or slope showed variation, it was not extensive. The experimental results are shown in Figure 9.1 of Chapter 9. And while eventually the pressure at the closed end fell off with time, in the initial period of production, the pressure at the closed end tended to remain at the initial shut-in pressure. Interestingly, however, in the calculations and correlations of Bruce et al. for unsteady-state gaseous flow, at any given time the pressure-distance profile tended to approximate that for steady-state flow. This occurred for both the linear and radial unsteady-state calculations, as will be demonstrated in Chapter 10.

Other Observations As long as the pressure at the closed end remains at the initial value, then "drawdown" is assumed to be occurring. With further or long-term production the pressure at the closed end will then tend to fall off. The point or interval of transition is that point perceived as most applicable to representation by steady-state flow. It can therefore be concluded that the steady-state integrated formula is a fair approximation for determining the radius of drainage, or outer radius of a reservoir, at the start of production -- that is, from a back-pressure test. And by definition the back-pressure test pertains to that transition period or interval between drawdown and long-term depletion.

Water Influx If the outer boundary, say, of the reservoir is connected to an aquifer, then as the pressure at the outer boundary tends to fall, there will be an reflux or encroachment of water (or brine).

180

UNSTEADY-STATE FLUID FLOW

....~

I

.... t

k

I

2eeo

i Shul.... t '~. . . . . L.... Ir.....L:2J . . . . L ]

I\-I ,

. . . . . . . . .

.

.

.

.

.

.

,4 0 . . . . .

:

.,:~o,,

~ "

-,

.~j ! LI I-!II I! ~221-I i--\] ' 'I 'I 'I"'
P a . These are the customary forms for linear and radial steady-state flow,

rb > r >

except that here the limits P2, P , and P~ are for the corresponding values of x2, x , and x~, and Pb, P , and P, for the corresponding values of rb, r, and r, _-or In rb, In r , and In r~ Furthermore, these pressure values or limits are not constants but are in general functions of time -- that is, depend upon time as a parameter.

Uniform Total Mass Flow Rate The above results assume that the total mass flow rate q or Q remains

constant, with respect to both time and position. Whereas this will produce no consternation in true steady-state flow for open systems, it seems at odds with normal expectations, otherwise. Nevertheless, the results of the unsteady-state calculations and experiments of Bruce et al. (5) indicate that unsteady-state behavior can at least be approximated by a succession of steady-state profiles. The situation, therefore, will bear further examination, for both liquids and gases.

CRITIQUE

323

For a liquid, the foregoing conclusion does not seem too far awry. For example, consider a long trough filled with water. If one end is partially opened so that flow ensues, the linear flow rate expectedly will be the same, or nearly so, all along the trough, even though one end remains closed -- barring frictional effects. (There are of course vertical changes as the water level drops, but these changes are not of consequence if purely linear flow is to be controlling, and the only consideration. The vertical changes merely serve to sustain the flow.) Similarly for a circular bucket filled with water. If a plug or valve in the center is partially opened, say, then conceivably the total mass rate of flow will be uniform in the radial direction, even though the outer radius is closed. The water level merely drops. (We hereby exclude considerations of a vortex being set up, the so-called Coriolis effect. That is, looking downward, the rotation would be counterclockwise in the northem hemisphere, and clockwise in the southem hemisphere. Only at the equator would tree radial flow be expected.) Since a liquid is more or less incompressible, or only slightly compressible, and if a continuity exists, then the total mass flow rate will have to be the same at every point x in linear flow, and at every point r (or In r) in radial flow, even if flow is through a porous medium. For a gas, the situation is more circumspect, since its flow behavior under compression and decompression is not so readily observable or analyzable. The effect of compressibility, however, has already been taken into account in the flow equations. And here we are talking strictly of total mass flow rate rather than of total volumetric flow rate. It has been the conventional wisdom that for flow through porous media, in the flow direction normal to and away from a closed surface, the pressure gradient at or contiguous to the closed surface or boundary will be, or will approach in the limit, zero. This, however, is not bom out by the linear flow experiments of Brace et al. (5), nor in the calculated results, nor for that matter in the profiles cited in Reference 20. In matter of fact, a pressure gradient will exist at and away from the wall surface in the direction of flow, and will further increase in the direction of flow. Furthermore, as an analysis of the results of Bruce et al. for gas flow indicates, that at any given time, this gradient -- that is, the gradient of the pressure squared - will tend to calculate the same rate of total mass flow as at other isochronal points along the line of flow. Admittedly, for a given system, if the mass rate of flow is constant, the volumetric rate of flow will vary, but on the evidence the mass rate of flow does not vary throughout the system if a constant mass rate of efflux is incurred. The situation for varying mass efflux or depletion was not examined.

324

UNSTEADY-STATE FLUID FLOW

Consider then a compressed gas in a long tubular pressure vessel with a valve at one end, and with no mtemal restrictions. If the valve is partially opened, and the gas is slowly discharged, the pressure throughout will be uniform, and there will be no pressure gradient the length of the vessel. In other words, wall effects or gas viscosity are not to enter into the consideration. Furthermore, under these conditions, the mass flow rate will be tmiform along the linear dimension. If not, a contradiction will be produced -- that is, there will be holdup within the cylinder, which by definition is not allowable according to the conditions stipulated. This will pertain to a constant efflux rate, certainly, and will also apply to a varying efflux rate, if in both cases the discharge rate is relatively slow enough. We are thus speaking of the decompression or expansion of a gas from a horizontal cylinder, with no intemal velocity effects (or frictional effects). If the gas in the cylinder exists within a porous medium, or if other restrictions or surface effects occur, and the valve at one end is opened, then expectedly a pressure gradient will be set up in the linear flow direction. The mass rate of flow must be uniform with direction, however, otherwise it can be argued that the gas in the pressure vessel would never be depleted. Similarly for radial flow. In can be argued in tum that, in the limit, if the mass rate of flow was not uniform throughout the contacting vessel or reservoir, whether flow was through a porous medium or not, complete depletion could never be consummated at a finite depletion rate. A contradiction or anomaly would therefore be encountered.

Density or Pressure as a Function o f Time

The formal statement must in fact be questioned or dropped from consideration that, say for a gas, 0 - Op

or

0

c~P

For the above-cited statement to hold true, it would be required that there be a point in unidirectional unReady-state flow where density or pressure be independent of time, and would be a function of position only -- i.e., at a closed surface. This, however, is contrary to the experimental evidence for the behavior at the closed surface of a reservoir, as per the results of Bruce et al. (5). The overall material balance, however, requires that density or pressure can otherwise be made a function of time, as will be developed in Chapter 14 using Ready-state profiles which change with time. That is, when an overall

CRITIQUE

325

material balance is introduced, then the above-cited statement would be in excess, and would produce a contradiction. There would in fact be too many boundary statements. In another way of looking at the situation, what occurs for the depletion of a gaseous closed reservoir can be perceived as the expansion of a gas through a porous medium. In the limit, with no intemal restrictions, there is merely the expansion of a gas. If velocity effects should be incurred in the direction of flow, enter then the concepts of elongational flow, and of elongational viscosity or Trouton viscosity (21). Whereas the concept of viscosity normally encountered and used is that of shear viscosity, whereto velocity effects occur across, or normal to, surfaces parallel to the direction of flow, in elongational flow the velocity effects occur in the direction of flow. In other words, although the terminal rate of flow is made constant, there will be a decreasing flow rate upstream from the direction of flow. This, in substance, is what is assumed by the conventional wisdom for the flow of a gas through a porous medium where the outer boundary is closed. In a way of speaking, the flow of gas would be a manifestation of elongational flow, and elongational viscosity effects would come into play. The concept of elongational flow, however, pertains to elastic or viscoelastic materials which display cohesion -- what are ordinarily called solids or semi-solids (21). Thus a specimen of asphalt say, when subjected to a tensile force at one end, will display elongational flow. Furthermore, the linear velocity within the specimen will vary, being at its maximum rate at the end where the force is applied, and becoming zero at the stationary end. Plainly, gases and liquids are not cohesive in this usage of the word. There is also the distraction to be considered of whether shear viscosity should be viewed and measured in terms of mass linear velocity changes normal to flow, or in terms of volumetric linear velocities. Most times there would be no distraction, other than in units, since a point condition is assumed, which may be regarded as steady-state. Thus the use of a constant shear viscosity in say the Darcy relationship in itself refers steady-state behavior. The experimental and calculated results of Bruce et al. for gaseous flow at a constant mass depletion rate tend to confirm, however, that the mass velocity would be the preferential criterion rather than volumetric velocity.

326

UNSTEADY-STATE FLUID FLOW REFERENCES

1. Carslaw, H. S. and J. C. Jaeger, Conduction of Heat in Solids, Second Edition, Cambridge University Press, 1959. 2. Sneddon, I. N., Elements of Partial Differential Equations, McGraw-Hill, New York, 1957. 3. Churchill, R. V., Fourier Series and Boundary Value Problems, Second Edition, McGraw-Hill, New York, 1963. 4. Van Everdingen, A. F. and W. Hurst, "The Applications of the Laplace Transform to Flow Problems in Reservoirs," Trans. AIME, 186, 305 (1949). 5. Bruce, G. H., D. W. Peaceman, H. H. Rachford and J. D. Rice, "Calculations of Unsteady-State Gas Flow Through Porous Media," Trans. AIME, 198, 79 (1953). 6. Hoffman, E. J., The Concept of Energy: An Inquiry into Origins and Applications, Ann Arbor Science, Ann Arbor, MI, 1977, pp. 348-352, 361362. 7. Churchill, R. V., Modern Operational Mathematics in Engineering, McGraw-Hill, New York, 1959. 8. Katz, D. L., D. L. Comell, R. Kobayashi, F. H. Poettmann, J. A. Vary, J. R. Elenbaas and C. F. Wemaug, Handbook of Natural Gas Engineering, McGraw-Hill, New York, 1959. 9. Hoffinan, E. J., Azeotropic and Extractive Distillation, Wiley, New York, 1964, pp. 208-211. Revised Edition, Robert E. Krieger, Huntington NY, 1959. 10. du Noiiy, Lecomte, Human Destiny, Longmans, Green and Co., New York, 1947, p.18, 28. 11. Bell, E. T., Men of Mathematics, Simon and Schuster, New York, 1937, Chapters 25,27 (p. 521), 29, and Chapter 28 (p.538). 12. Kline, M., Mathematics: The Loss of Certainty, Oxford University Press, New York, 1980. 13. Berlinski, David, Black Mischief The Mechanics of Modern Science, Morrow, New York, 1986, p. 219-220. 14. Hof~an, E. J., Heat Transfer Rate Analysis, PennWell, Tulsa, OK, 1981. 15. McAdams, W. H., Heat Transmission, Third Edition, McGraw-Hill, New York, 1954, p.414. 16. Gleick, J., "New Images of chaos that are stirring a science revolution," Smithsonian, 18, 9), 122-134 (December, 1987). 17. Gleick, J. Chaos: Making a New Science, Viking Penguin, New York, 1987. 18. Prigogine, I. and I. Stengers, Order Out of Chaos: Man's New Dialogue with Nature, Bantam, New York, 1984. Foreword by Alvin Toffler.

CRITIQUE

327

19. Davis, P. J. and R. Hersh, Descartes'Dream: The World According to Mathematics, Harcourt Brace Jovanovich, San Diego, 1986, pp. 159-164. 20. Katz, D. L. and R. L. Lee, Natural Gas Engineering: Production and Storage, McGraw-Hill, New York, 1990, p.329.

Chapter 12 THE RESULTS OF BRUCE ET AL. IN TERMS OF INTEGRAL FORMS

Altemate approaches (and simplifications) based on Darcy's law have been applied in order to provide solutions for unsteady-state radial flow, as presented in Chapter 5 and Chapters 9 and 10. Not to be over looked is the pressure dependency of the viscosity and the compressibility factor for gaseous systems, and which can alter the composing of the flow equation. Joule-Thomson expansion effects can at the same time affect the flowing temperature. For most purposes, however, mean or constant values suffice over an interval, and the temperature can be assumed constant. In other developments in tmsteady-state flow, the flow behavior has been described in terms of volume and surface integrals rather than the classical approach of partial differential equations. This integral approach is described at length in Chapter 6, along with applications to reservoir behavior and to the depletion problem and the estimation of reserves. The methodology will be further applied in this chapter, based on the results of Bruce et al. (1), and utilizing actual reservoir performance data.

12.1 REVIEW OF THE DERIVED RELATIONSI~P AND CORRELATIONS As developed in Chapter 6, the statement for the depletion of an element of volume in Euclidean 3-space, with constant porosity X , is the volume integral

Am-

fff

-

where the limits of integration define the element of volume. The mass of fluid Am flowing at or across the surface(s) enclosing the element of volume is in turn represented by the surface integral

f GIn(,,.v dvd.dt- Am where G is a flux function defined on the surface and

330

UNSTEADY-STATE FLUID FLOW

O(X,Z) 2 + O(Y'Z) 2

C)(x,y)

I,,(.. v)l - {

v)

v)

+

The coordinates (u, v) represent the coordinate system imbedded in the surface. For a parallelepiped, for instance, (u, v) may take on the altemate coordinate systems (u,v)-(x,y) -

(x,z)

- (y,z)

On denoting the upper and lower surfaces of an enclosure for a given physical embodiment (e.g., a parallelepiped), and the corresponding flux functions, and by equating the volume and surface integrals so-defied, performing the operation of successive differentiation will yield a set of differential forms. In the case of a parallelepiped these forms are constrained at the intersecting surfaces, such that c~ OG OG d (p o - p ) ~~~~~--'X

ox

c~

~

c~

A flux fimction G may be represented by the heuristic form G - )e "~ - y exp(O~b) where 0 = a function of position only r = a function of time only 7 = a function of time only The boundary conditions are to be such that: At the region of efflux, 0 ~ 0 and the efflux is a function of time or a constant. When t = 0, flow has not started, and r ~ - oo. For a depletion time tf (where to = 0) as t ~ tf then p ~ 0 everywhere. Without going through the mechanics of the further derivation -- which is presented in Chapter 6 -- a solution is obtained for a parallelepiped whereby

RESULTS OF BRUCE ET AL. AS INTEGRAL FORMS

331

Po - P ~=/311-lnfl] Po where 13 is of the form

At the area of effiux (the sandface), 0 = 0 and

Correlation of this formula has been made with the linear depletion data of Bruce et al. (1). The correlation is excellent, as per the plot of In 13 versus In t which is shown in Figure 6.1 of Chapter 6. The values for 13 are obtained from the solution of the equation

and where for a gas,

P0 - P = f l [ 1 - l n f l ] P0

Po-P

1~ - P

po

Po

Straight lines are produced for each pressure point, and which tend to converge at tf ~ 500 minutes, and which agrees with the experimental value calculated from the flow rate and initial pressure (density), as indicated in Example 6.1 of Chapter 6. Moreover, at the point of efflux, where 0 = 0 , it is found that for linear flow c ~ 1.0. (For actual gas reservoirs, it is expected that c may be less than unity.)

Extrapolation of Production History In practice it is preferable that the production history (pressure vs. time for a constant flow rate) be known and available over a sufficiently long period of time. The extrapolation of log beta (log 13 or In 13) versus log time (log t or In t) will then establish tf at 13 = 1.0 . The reservoir size and initial reserves then follow. However, a single point for the flowing pressure and the corresponding flow rate obtained from a back-pressure test can also be used to estimate reserves and other properties when the permeability is not otherwise known, as will be demonstrated subsequently.

332

UNSTEADY-STATE FLUID FLOW

Permeability It is also found that, for linear flow in the (-x) direction and as an approximation for gaseous flow, the mean value for the flux function can be obtained as follows:

--

X

( x 2 - x l + Y2 - Y l + z2 - z l ) 2

fl(ln fl)z. p

G - - {tf

M

oT

zRT}P (1)

_

{K}p

where the function fl(ln fl)2 p _ fl(ln fl)2 ( p / Po )Po is determined from 13 ; which, to be rigorous, requires a solution of P0 - P = f l [ 1 - In/3] P0 where 13 = (t/tf) ~ . The overall calculation is very sensitive to the value used for the density ratio:

Po - P= I Po

P = l - Z~176 Po zTPo

Hence the permeability (or mobility) can be related directly to the depletion behavior. A check on this formula using the data of Bruce et al. (1) was made in Example 6.2 of Chapter 6. The above formula gave a value of 26.5 millidarcies (assuming a gas viscosity of 0.018 cp) versus the steady-state reported value of 13.2 md. Density simplifications were used, however, which affect the results. It was also shown in Example 6.3 and 6.4 of Chapter 6 that the unsteadystate permeability will vary, based on the data of Bruce et al. (1). In fact, it may be considerably larger in the initial depletion period (twice as high as for the later period--with the latter corresponding more closely to the steady-state value). The temperature T under flowing conditions may differ somewhat from the temperature To at shut-in, which also affects the compressibility factor z at flowing conditions. This can be traced to Joule-Thomson expansion effects. Heat

RESULTS OF BRUCE ET AL. AS INTEGRAL FORMS

333

transfer from the porous medium and surrounding rock formations will expectedly minimize thermal changes, however.

Radial t, low The same equation forms for depletion behavior can be used for radial flow (or even spherical flow) as for linear flow: P0-P

= f i l l - Infl]

P0 where at the region of effiux

and where for a gas,

Po - P = I

P

1 - z~ T~P~

Po

Po

zTPo

and as a simplification, Po-P=I

- P

Po

Po

The expression involving permeability is different, however. Thus for radial flow, in cylindrical coordinates, the coordinate transformations utilized are x = 2~r (y is not applicable) Z--7.

and it will follow that

where

K

X[2rc(rb - r ~ ) + z 2-z~]2 M

kt

tl

{fl(ln,/3) z . p

z 2 - Zl - A h , t h e r e s e r v o i r

thickness,

P

zoToP

Po

zTPo =

P

Po

if

zRT

1 2rc

and where

Z "~ Z 0

and

T- To

(2)

334

UNSTEADY-STATE FLUID FLOW

From a material balance it follows that

Q t l - Po XAhrc(rb 2 - r~

(3)

or

Qt s

(3a)

Po Xkhlr where r, is the bottom-hole radius, and here can be neglected. And for all practical purposes, ~ [ 2 r c ( r b - r. ) + z 2 - z, 12 ~ 2rcrbz 2er

(4)

since (rb- r~) >> (z2- zl ) and rb >> r,. Therefore, K

2Q

M

r

fl(lnfl)Z.ppoAh

zRT

(5)

This relates K/~t to the flow rate Q and flowing pressure P after the well performance levels-off and stabilizes after the initial drawdown.

12.2 RELATION TO THE RESULTS OF BRUCE ET AL. The foregoing relationships resulting from correlations based on the unsteady-state solutions of Bruce et al. (1) for radial flow may in tum be related to or referenced to solutions or correlations as were developed in Chapter 6. The latter pertain to a body of theory for unsteady-state flow based on an initial representation in terms of volume and surface integrals. As previously noted, the density function is expressed as P0 - P

P0 - P

Po

Po

- f l [ 1 - In fl]

whereby at the area of efflux t

r

RESULTS OF BRUCE ET AL. AS INTEGRAL FORMS

335

TABLE 12.1 Behavior of 13 for Q = 0.1 m

i

QO (o) 0.00001 0.001 0.001 0.01 0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.5

1- P/Po

P/Po

(1.o)

(o)

0 96 0.93 0.90 0.87 085 0.83 081 0.765 0.64 0.515 0.36 0.16

0.04 0.07 010 0.13 0.15 0.17 0.19 0.235 0.36 0.485 0.64 0.84

Solution for 13 0.007 0.013 0.020 0.028 0.035 0.040 0 045 0.06 0.115 0.18 0.29 0.50

TABLE 12.2 Behavior of 13 for Q = 0.01 g

QO (o) O.00001 0.001 0.01 0.02 0.04 0.06 0.1 0.2 0.3 0.4 0.5 0.6 0.7 08

P/Po

1- P/Po

(1.0) 0.99 0.98 0.97 0.96 0.945 0.925 0.885 0.795 0.68 0.58 0.475 0.37 0.26 013

(o) 001 002 003 0 O4 0 055 0.075 0 115 0 205 032 042 0 525 063 0 74 087

0013 003 .005 .007 0096 .014 024 .052 .095 .14 .20 .28 .37 .53

340

UNSTEADY-STATE FLUID FLOW

1.0

A

A A

Long-term

0.8

depletion

A

A

0.6 Slope r

A

0.4

0.2

0

0 0

0

0

0 Drawdown

0.0 O. 001

I O. 01

I O. 1

1.0

Fig. 12.3 Slope c versus log Q.

Square of the Pressure Doubtless, correlations could be developed in terms of the squares of the pressures in lieu of the pressures (or densities) to the first power. This is suggested by the relationships stated and developed in Section 9.2 of Chapter 9 for gaseous unsteady-Rate flow, and based on the material in Reference 2. Interestingly, the relationship can be generalized as 1 - ( P / 19o )" - f i l l - In fl]

where 13 = ( Q 0 ) b a n d where P/P0 = 0 and 13- 1 at Q0 = 1. And such is the nature of logarithmic behavior that different (and higher) values of n an be determined such that a log-log plot of either

1- ( P / Po )" ---~fl vs Q O

(with slope b)

RESULTS OF BRUCE ET AL. AS INTEGRAL FORMS

341

or simply

] - (P \ Po)"'

vs

0-o

will yield straight lines for parameters of Q over the entire range of Q 0 . Thus reservoir behavior in its entirety, encompassing both drawdown and long-term depletion, can be correlated with a single equation. The value of the coefficient n or n' can be adjusted in order to more closely approximate a straight-line relationship.

iii_

'

F I 'x !' ,I

o~o ~_,oo .... ~,o~. ~3~-'~o;n%,%o~:''w-~oi~,i' 60 ._

80

100

120

140

160

Durotion of flow, rain

180

200

220

1 jjJ !

440

"o

~o 420 o

240

260

I

-

~._~9AO. pl

i 5401 0

20

40

60

80

100

120

140

160

Durotion of flow, hr

180

~

~

200

220

4o~B0g 240

260

Fig. 12.4 Drawdown of flowing wellhead pressure on high- and low-permeability wells (from Reference 2, re the Federal Power Commission).

338

UNSTEADY-STATE FLUID FLOW

where tf is the theoretical depletion time and c is a constant (or coefficient) to be determined from the production behavior. Thus a plot of In 13 versus In t should at least produce a curve with slope or tangent c and terminating at tf. And since

-Q O-

qzRT 2

t - Bt

rc r~ X A h P o a plot of In 13 versus Q0 should also produce a curve with tangent c , terminating at Q0 = 1. The visually inspected solutions from Bruce et al. in Tables 10.1 and 10.2 for Q = 0.1 and Q = 0.01 (for rb/r, = 200) are listed in accompanying Tables 12.1 and 12.2 to first obtain the behavior of the function 13. (Figure 12.1 restates the graphical relation between 1 - P/P0 and [3 ). In tum, [3 is plotted versus Q0 on log-log plots in accompanying Figure 12.2 for the two cases. The result in each case is at least a curve terminating at t = t f for Q0 = 1.0, within the limits of the vicissitudes of visual inspection of the graphical representations for the numerical solutions (Figure 10.1). Significantly, however, the initial portion of the curves, up to about Q0 = 0.01, tends to be a straight line. This can be characterized as drawdown. Furthermore, there is a transition between about Q0 = 0.01 and Q0 = 0.1, after which the curve also tends to straighten out. The latter would represent the long-term behavior of a system whose outer boundary is closed. The behavior, above, is consistent with the results of 13 vs t for actual reservoir and field performance as illustrated in the examples which are provided. For the purposes of correlation, the initial and final slopes of In [3 vs In Q0 have been estimated for the several values of Q . This information is tabulated in Tables 12.3 and 12.4, and shown graphically in Figure 12.3. The difficulty of reading the graphical solutions of Figure 10.1 makes the trend of the lower slope questionable. For the upper slope, there is an increase as Q decreases, and which presumably levels off to c -~ 1.0 as Q --~ 0. From the foregoing, given a value for

--

2qpzRT

Q-

rcPgXh

by virtue of the known parameters, a value of c can be obtained for drawdown and for long-term behavior. In turn, the density function or pressure behavior can be predicted in terms of _ qzRT QOt-Bt rcr~XhPo which establishes P (or P/P0 or 1 - P/P0) as a function of t .

RESULTS OF BRUCE ET AL. AS INTEGRAL FORMS

339

TABLE 12.3 Determination of Slope "c" during Drawdown

Q =0.2 Q =0.1 -.,,,.._

Q =0.05 Q =0.02 Q =O.O1 Q =0.005

P/Po

1 P/Po

0.935 0.74 0.96 0.87 0.975 0.935 0.985 0.96 0.99 0.97 0.995 0.98

0.065 0.26 0.04 013 0 025 0 065 0.015 0 04 001 0 03 0 005 002

-

Q 0 0.00001 0.01 0.00001 0.01 0.0001 0.01 0.00001 0.01 0.00001 0.01 0.00001 0.01

c (loglog slope) 0.201 0.171 0 138 0 142 0 159 0 201

TABLE 12.4 Determination of Slope "c" during Long-Term Depletion (Closed Outer Boundary)

Q =0.2 Q =0.1 Q =0.05 Q =0.02 -.--.,__

Q -0.01 - ! , _ _

0=0.005

P/Po

1 - P/P0

o 0.60 0 0.765 0. 0.835 0 0.875 0 0.885 0 0.89

1 0.40 1 0.235 l 0.165 1 0.125 1 0.115 1 0.11

Q 0 10 0.1 1.0 0.1 1.0 0 1 1.0 0 1 1.0 0 1 10 0 1

c (loglog slope) 0.398 0.629 0,783 0 903 0 939 0.959

0.1_

1.0

0.01

0.1

P,o-P Po

0.001

0.01

0.0001

0.001 0.00001

0.0001

0.001

0.01

Fig. 12.1 Plot of (po- p)/po vs [3 (duplicates Fig. 6.2).

0.1

1.0

1.0

,

' .....

'

'

O A O 0

0.1

0 Q

0

0

0

In 13

0

A

A ,..]

A

C~

A

C3 t'n t'n -]

A

m

Q=

0.1

A

0

>

A

0.01

A A A Q = 0.01

0.001 0.00001

|

O. 0 0 0 1

,,,

|

|

0.001

,

0.01

ii

0.1

1.0

In q o

Fig. 12.2 13 vs QO (for rdr. = 200).

"..4

342

UNSTEADY-STATE FLUID FLOW

EXAMPLE 12.1 A performance curve for a low-permeability gas well (and another, high permeability well) is shown in Figure 12.4 from Reference 2. (Duplicates Figure 8.7) While the flow rate falls off somewhat, for the purposes here a constant or average value of about 1, 850 MSCF/D can be assumed over the period of the test. Data taken from the curve is shown in Table 12.5, assuming density linear with pressure.

TABLE 12.5 Drawdown Date for Stookey No. 1 t 0 hr 1 8 25 100 220

P

(P0- P)/Po

13

426 psia 395 380 367 352 342

0 0.0728 0.10798 0.13849 0.17370 0.19718

0 0.014 0.023 0.031 0.041 0.050

The plot of 13 vs t depletion time of

appears as in Figure 12.5 giving an extrapolated

tf = 25 x 107 hr = 1.04 x 10 6 days = 2.85 x 103 years The value of c is c = 0.236. Assuming the flow rate to be 1850 MSCF/D, the estimated reserves in place for the field would be 1850 (1.04

x

1924 x 106 MSCF = 1.924 x 106 MMSCF = 1.924 trillion SCF

10 6 ) =

This figure is within the range of conjecture.

1.0"

9

0.1

Z

J

o-" .01 0.1

1'o

loo

t (hrs)

Fig. 12.5 13 vs t for StookeyNo. 1.

1o~)o

~o,ooo

4~

r

1700

i

: 2

K'a.~,~,.~'~_

S ': \ \ = : ::~ ..... ! .~ ~ "~- . . . . i \ t~" Q-

.--o--- Ca.lcu,~'edpressuresm' wes~rn bounc/Qry oF the F/eld I Observed averaae f/eldpressure ~ ~ Cplculaled~.r boundary pre~su..'esassumZ'r,auniForml i -.

"

~60o

,

/h.o~,ahou~ whole produc//o,~/,,:~orr Ca/c,,/,./ebl western boundary p r e . o r e . . . . . . . . inq u t ~ , ~ r . ~./,~lreservo/rw,thd~owalro~e ofSISOOOba:relxperdoy.

i

1 J

i

~. o

Ib00 o

--. If4oo

L-0~

:

I lO0 170o

'

I

1

...... I

'

I

I

~

I --

;

I

i

!

i

4

3

-i300

~L. ~>

1200

2_o l'~OO

i--

~

......! .

.

.

.

L_J

!

_ _

.

II00

1200

I,,~176176 800 ~

.,3O0

-

" .....

. ...... ..-.

600

:.~

400

.._= o

2oo >.

0

~,1 '.' g.' [J A J o ~J A J O' ~ 1931 1932

' llJ!,l~ AJ 01JAJ 0 I 1933 1934

i ii, ! i1!i, A J 0 iJ A J 0 ~--A J 0 r 1935 1936 193"/

A J O I938

A J 0

1939

]J

AIg~IO0

lJ A O 0

1941

A ~40

( A J 0 IJ A J 0 194S 1946

Fig. 12.6 The pressure and production history of the East Texas field. (From Copyright 1949 by McGraw-Hill. Used with the permission of McGraw-Hill Book Company.)

Physical Principles of Oil Production by M. Muskat.

0

~

d~ -]

RESULTS OF BRUCE ET AL. AS INTEGRAL FORMS

345

EXAMPLE 12.2 The same techniques may be applied to other types of pressurized formations, under other circumstances. The East Texas Field is an example of a water-drive reservoir (3). The pressure and production history is shown in Figure 12.6. Data taken from the curve is shown in Table 12.6 assuming density to be linear with pressure. TABLE 12.6 Pressure Data from the East Texas Field t 0yr 1 2 3 4 5 8 10

P

(Po- P)/Po

13

1615 psia 1450 1380 1270 1235 1200 1100 1050

0 0.I0216 0.14551 0.21362 0.23529 0.25696 0.31888 0.34985

0 0 022 0.033 0.055 0.062 0.070 0 095 0 107

A plot of 13 vs t is shown in the following Figure 12.7, giving an extrapolated value oftf = 270 yrs at - 1 and a value of c = 0.687. The daily production rate is about 600 x 103 barrels. Therefore, the total oil in place would be 600 x 103 (365) (270) = 5.9 x 10 l~ bbls = 59 billion barrels If 30% is recoverable, this comes to 17.7 billion barrels. This is in the range of other estimates. If the final depletion pressure or hydraulic head is 900 psi, and if this figure is subtracted from the pressure readings, the [3-plot will yield tf ~ 40 years. The oil originally in place calculates to between 8 and 9 billion barrels, compared to a commonly-accepted nominal figure of 7 billion barrels for the recoverable reserves. This is more in line with the high recoveries reported for water drive and water flooding.

1.0-

-d

0.1

/f 0/

~ 0.01

, 1

f lO

lOO

t (y,.~)

Fig. 12.7 13 vs t for the East Texas Field.

1000

RESULTS OF BRUCE ET AL. AS INTEGRAL FORMS

347

Therefore, at least in correlating and predicting the performance of waterdrive reservoirs, the use of well--head pressures is indicated rather than bottomhole or formation pressures. If the compressibility form for density is used, then

Po-P= l_(P/po)~ Po where c is the compressibility. This may also be used to adjust the 13-plot. Decreasing values of c produce increasing values of tf.

28

\

A

,,

24

N

v

~20 E

5 E

C 0

e

,\

P,,)~

~r/ o %8_ ~ 2

4

}o

t~

.

.

o 0

~

~

_n

o'5

Production rate i=,========~= ===q '

I I

I

-

.

I

~.

I.

.

L

3

~ .~

L

....

~

J

L

t.Oos-oIl rafiO

~--

/

]

i

4

8

12

16 20 24 28 Cumulaf;ve production(101bbrs.)

~2

36

40

Fig. 12.8 The pressure and production history of the Wilcox Sand reservoir of the Oklahoma City Field. (From Physical Principles of Oil Production by M. Muskat. Copyright 1949 by McGraw-Hill. Used with permission of McGrawHill Book Company.)

348

UNSTEADY-STATE FLUID FLOW

EXAMPLE 12.3 Pressure and production for the Wilcox Sand reservoir of the Oklahoma City field are shown in the preceding Figure 12.8 (3). This is an example of a gasdrive reservoir, with the pressure diminishing to a low level. The pressure history is shown in Table 12.7 with the plot of 13vs t shown in Figure 12.9. TABLE 12.7 Pressure Date from the Wilcox Sand t

P

(P0- P)/Po

13

0 yr 1 2 3 4 5 6 7

2700 psia 2100 1500 900 500 230 170 100 50

0 0 2222 0.4444 0 6666 0.8148 0.9148 0.93703 0.96290 0.98148

0 0.058 0.150 0.30 0.50 0.70 0.74 0.83 0.90

8

While the initial portion of the plot is a straight line, there is a "tailing off" at the upper regions. A number of explanations can be offered, including the fact that a two-phase system is revolved, and the loss of pressure is due to gas losses (the gas/oil ration drops), with hydrocarbons liquids retained in the formation~ which can be pumped.

-]

O~

1.0"

O OO0

t~ r

J3

O -]

0.1 -

G

0.01

t (yrs)

Fig. 12.9 13 vs t for the Wilcox Sand, Oklahoma City field.

350

UNSTEADY-STATE FLUID FLOW

EXAMPLE 12.4 The production, rock and fluid data for an initially undersaturated Wilcox reservoir has been presented by Guerrero and Stewart (4). Four points spanning the production history are given in Table 12.8.

TABLE 12.8 Pressure Data from the Wilcox Reservoir t 0 days 547 1003 1460

P

(Po- P)/Po

13

3793 psia 3613 3408 3277

0 0.0475 0.1015 0.1360

0 0.0080 0.021 0.301

The plot of 13 vs t in Figure 12.10 yields a value of tf = 15,000 days. The cumulative production was 3,175,948 bbl over 1460 days, for an average of 2175 bbl/day. Using this figure, the original oil in place calculates to 32,625,000 bbl. Guerrero and Stewart determined a value of 24,700,000 bbl based on other estimates of the reservoir size and properties.

1.0

0.1

B

0.01

0.001 10

~o

T

T

1000

10,000

| -days

Fig. 12.10 13 vs t, Wilcox reservoir.

352

UNSTEADY-STATE FLUID FLOW

EXAMPLE 12.5 Data for production of the Wilcox reservoir has been presented by Guerreroand Stewart (4)" 0.209 275 md 100 cp

Porosity Permeability Viscosity of oil at reservoir conditions Compressibility Radius of reservoir

6.8 x 10-6 per psia to 15.6 7,100 it

The value calculated previously for tf was 15,000 days or 360,000 hours. The calculated oil in place was 32,625,000 bbl. The estimate from reservoir size was 24,000,000 bbl. The adjusted reservoir size is r~

32,625,000

(7000) 2

24,000,000

r b - 8,278 fi The formula to be used is m

G--{

aT

x 9p o c } p

tl 2.83311-0.8438 p ]Po Po

K

o3o

p

c3c

c,3c

Substituting in the brace term and assuming p0 ~ P, K

0.209

(8,2782)

~

p

6.8

X l 0 "6

9

360,000 2.83311-0.8438]

144

The viscosity is 100 x 2.42 = 242 lb/ft-hr. Therefore,

= 4.24

x 1 0 "6

RESULTS OF BRUCE ET AL. AS INTEGRAL FORMS

353

K - 4.243 x 10 .6 (242) - 1030 x 10 .6 fi 3 / hr 2

=

1030 x 10 .6

= 0.233 darcies

0.00442 = 233 m d All things considered, this value is in substantial agreement with the reported value of 275 md.

EXAMPLE 12.6 The data from Table 9.4 of Chapter 9 for a low-permeability producing gas well may be converted to [3 by the afore-used relationship

Po-P

Po - P

Po

Po

= fl[1-1nfl]

The conversion and results are shown in Table 12.9. TABLE 12.9 Calculation 13 and c t (hrs) 0.25 0.5 1.0 3 6 10 16 21 24

1 - P/P0

13

c

0 063 0 113 0119 0 127 0 132 0.139 0.145 0.147 0.147

0.0238 0.0255 0.0277 0.0291 0.0311 0.0328 0.0334 0.0334

0.0995 0.0753 0.0753 0.1301 0.1130 0.0667

Values of the constant c for successive intervals are also entered into the previous tabulation. The performance of c is somewhat erratic, due in part no

354

UNSTEADY-STATE FLUID FLOW

doubt to inaccuracies in the visual readout of the pressure-time data. In general, however, the value of c should fall-off due to the incursions of the water drive. If 13 = (t/tf) r then for any interval between points "1" and "2",

fll

tl

or

ln(fl 2 / f l , )

C=

ln(t 2 / t , ) where c becomes the shape of the log-log plot of 13 vs t . From the initial values of c , and which pertain to drawdown, it may be referred that c ~ 0 . 0 7 5 , and 13 ~ (t/tf) ~176 . At say t = 3 and 13 = 0 . 0 2 7 7 , it follows that tl

t 3 fl1/0.075 (0.0277)133 = 1.556 x 1021

This result is clearly out of line. For any interval "1" and "2", taking the difference yields 1 _

_

[t2 ~ - t ,

~]

For the values, say, of 13 = 0 . 0 2 5 5 13 = 0.0277

t=l.O

t=3.0 it will follow that r

tf

(3) ~ - 1 ~ 0.0277 - 0.0255

For c = 0 . 0 7 5 , ty

-

1.0859- 1

=39.0389

0.0277 - 0.0255 t I - 39.03891/~176 - 1.153 x 10 ~7

RESULTS OF BRUCE ET AL. AS INTEGRAL FORMS

353

Similarly, other anomalous results can be obtained using other combinations of points. On the other hand, if it was inferred that c = 1 and [3 -- t/tf, then over the range of values t=0.5 t = 24

13 =0.0238 13 = 0.0334

it will follow that, by taking the difference, !

0.0334 - 0.0238 - --:---(24 - 0.5)

tf

and tf = 2448 hr. This is in line with other results. The difficulties can be reconciled by an inspection of Figure 12.2 and 12.3. The plotted results show a characteristically low value for the exponent c during drawdown followed by a transition to near-unity during long-term depletion. Thus the extrapolation to yield tf for a particular value of c obtained during drawdown is not permissible. Only after long-term depletion is recurred, where c will be near-unity, can extrapolation be made to yield tf. It is furthermore required that the outer reservoir-boundary be closed -e.g., a water drive or intrusion does not exist.

12.3 GENERALIZED INTEGRAL FORM If the density or pressure solution is generalized by the introduction of a coefficient m , then Po - P = ~ 1 Po

m In fl]

This form can still meet the boundary conditions whereby Po - P

~ - ~ 0 Po /9 0 - / 9

~ - ~ 1 Po where at the efflux,

p=

when

fl-~0

when

fl--~l

356

UNSTEADY-STATE FLUID FLOW

It is advisable, however, that c behave as a constant, and preferably that c = 1 . In this way the density or pressure relationship can be used to extrapolate or predict. This will require the determination of a suitable or best value for m .

Determination o f the Coefficient m

At any two points "1" and "2" it may be written that P 0 -- P l

= fl, - m In fl,

P0 P0 m P2

- f12 - m In f12

Po Taking the difference, Pl ~ P2

~2

Po

Pl

If e = l , Pl - P2

Po

tz - tI = ~ - m If

t2 ln~ tI

or

Pl - P2 + m In t-A-2- t2 - tl Po

t1

If

For points "2" and "3", it follows that P2

-

P3

In t-2-3- t3

.~_ m

Po

t2

-

t2

tf

Dividing the above and solving for m , Pl m P2

+

Po

m-

12

In--

1'1

and

12 -

tl P2

13 -

tt

12 - t~ In

13

13 - t 2

2

Pl m P2 Po

Po

-

t 2 -t

if-

-

I

t2

+ m ln--

tl

P3

RESULTS OF BRUCE ET AL. AS INTEGRAL FORMS

357

These determinations are illustrated by example as follows.

EXAMPLE 12.7 u

Q = 0.1, consider the following tabulation, as per Table 10.1 of

For Chapter 10:

u

m

QO = t

P/Po

1)

O.O0001

0.96

2)

0.0001

0.93

3)

0.001

0.90

4)

0.01

0.87

5)

0.1

0.765

P1 - P2/Po

0.03 0.03 0.03 0.105

Note that the data above is in terms of the reduced time t that on extrapolating to time tf it follows that tf = 1.0. Utilizing the first three points,

-(0.o3) + mIn

0.0001

-

0.00001

0.0009 0.0009 0.00009

where t = t/tf

such

(0.03) In

0.0009

0.001 - ~ 0.0001

_- -0.03 + 0.003 __ - 0 . 0 1 3 0 2.3026 - 0.2303

and

tf

-

0.0001 - 0.00001 0.03 + (--0.0130) In

0.001

=

0.00009 0.000067

= 1.355

0.0001 In principle, tf should be equal to 1.0. The deviation can be viewed in part as due to inaccuracies in the graphical inspection of the calculated results of Bruce et al.

358

UNSTEADY-STATE FLUID FLOW Utilizing points 2, 3, and 4,

0.0009

-(o.03) + m 0.0001 I n - 0.00001

0.0009 0.00009

(0.03) 0.001

In

0.0009

_- - 0 . 0 3 + 0.003 -- - 0 . 0 1 3 0 2 . 3 0 2 6 - 0.2303

0.0001

and 0.001 - 0.0001

il

0.001

0.03 + ( - 0 . 0 1 3 0 ) In

0.0009

- 13.55

0.000067

0.0001 The above suggests a pronounced error or sensitivity in the graphical interpolation of the data results of Bruce et al. among these three points -- or conceivably errors in the results so utilized. Note that, for points 3 and 4, if

10 4 --- 10 3

-- 0.02

Po m

then m = -0.0135 and tf = -0.830 (which is not allowable). If, however, 10 4 m 10 3

~- 0.04

Po then m = -0.0125 and tf 0 . 7 3 9 . Thus the foregoing indicates a high degree of sensitivity to the trend of the data. Utilizing the last three points, =

0.009 -(0.03) + ~ (0.105) m 0.09 _ - 0 . 0 3 + 0.0105 - - 0 . 0 6 5 6 0.01 0.009 0.1 2 . 3 0 2 6 - 0.2303 In - In 0.001 0.09 0.01

and

0.009 ~f

=

m

0.03 + ( - 0 . 0 6 5 6 ) l n ~

0.01 0.001

as compared to the theoretical value of tf

=

1.0.

0.009 0.01490

= 0.604

RESULTS OF BRUCE ET AL. AS INTEGRAL FORMS

359

Utilizing the first, third, and fifth points, 0.001 - 0.00001 m

( 0 . 9 0 - 0.765)

-0.06

0.01 - 0.001

_

0.01 In

.

.

.

0.001 - 0.00001 .

0.001

0.1 ln--

0.1 - 0 . 0 0 1

+ 0.0 l(O.135)

4.60517 - 0.04605

= -0.01286

0.01

and

0.001

m

tf=

- 0.00001

0 . 7 6 - 0.90 + (-0.01286)In where as noted the theoretical value is

tf =

0.001

=

0.00099 0.0007775

= 1.273

0.0001 1.0.

EXAMPLE 12.8 For Q = 0.01, consider the following, taken from Table 10.2' D

Q0 = t 1)

P/Po

0.00001

Pl-P2/Po

0.99 0.01

2)

0.001

0.98 0.01

3)

0.01

0.97

4)

0.1

0.885

0.085

Utilizing points, 1, 2, and 3,

-(0.01) + m

B

0.01

In ~ 0.0001

-

0.00099 0.0090 0.00099 00.0090

(0.01) In

0.01

-0.01 + 0.0011 4.60517 - 0.25328

= -0.002045

0.001

and b

0.001 - 0.00001

0.00099

0.001 0.01 + (-0.002045) I n 0.00001

0.0O0582

= 1.70

360

UNSTEADY-STATE FLUID FLOW

1.0 , but there are difficulties and Again the theoretical value should be tf discrepancies in reading the graphical results of Bruce et al. Nevertheless, with the appropriately accurate data, the initial drawdown behavior can be used to extrapolate to tf. =

EXAMPLE 12.9 The data from the previously-described producing gas well can be similarly analyzed using the drawdown data from Table 9.4 of Chapter 9:

1) 2) 3) 4) 5) 6) 7) 8)

t (hrs)

P/P0

0.5 1.0 3 6 10 16 21 24

0.887 0.881 0..873 0.868 0.861 0.868 0.853 0.853

Utilizing the first three points, 0.5 -(0.006) + -2- (0.008) m~

In

1.0 0.5

-0.006 + 0.002

0.5

3 In - 2 1.0

0.693147-0.274653

= -0.009558

and

tI =

1.0 - 0.5 0.006 + (-0.009558) In

1.0

=

0.5 -0.0006251

= - 8 0 0 hrs.

0.5

A negative number for tf is not allowable, hence the data is presumably not consistent. Choosing points 2, 3, 4 gives the following calculations. Solving for m , 2 -(0.008) + 3 (0.005) m-

3 2 In~--In1.0 3

6 3

-0.008 + 0.00333 =

].0986]-0.462]0

= -0.007337

R E S U L T S O F B R U C E E T AL. AS I N T E G R A L F O R M S

and 3 - 1.0

tf-

2.0

0.008 + ( - 0 . 0 0 7 3 3 7) In

= - 3 3 , 0 4 8 hrs.

-0.0000605

3

1.0 Again, a negative value for tf is obtained, which is not allowable. Utilizing points 2, 4, and 6,

5

-0.013 + - - (0.013) 10

m-

=

6 5 16 In . . . . . . l n - 1 10 6

-0.013 + 0.0065

= -0.0049948

1.79176 - 0.49041

and

tf-

6 - 1.0

5.0

6 0.013 + ( - 0 . 0 0 4 9 9 4 8 ) In - - 1.0

-0.0040505

= 1234 hrs.

Utilizing points 2, 4, and 7, 5

m _

-(0.013) + - - ( 0 . 0 1 5 ) 15 In

6

5

1.0

15

In

21

-0.013 + 0.005 1.79176 - 0.41759

= -0.00582

6

and 5.0

6 - 1.0 t f ~-

0.013 + ( - 0 . 0 0 5 8 2 ) In

6

0.00257

= 1944 hrs.

1.0

Utilizing points 2, 4, and 8, 5

-(o.o 13) + = : (o.o 1 s) m _

In

6

1.0

1~

-0.013 + 0.004167

5

1.79176 - 0.38508

24 In ~ 18 6

= -0.006280

and

tf=

6 - 1.0

6 0.013 + ( - 0 . 0 0 6 2 8 0 ) I n - - 1.0

= 2860 hrs.

361

362

UNSTEADY-STATE FLUID FLOW

The value for tf calculated by other means in Exam_pie 10.1 of Chapter 10 is 3,333 hours. The foregoing indicates that the stabilized value for the pressure should be used for the third point in the evaluation of m and tf. The calculation is also sensitive to the initial pressure values, and care should be taken in measuring the initial points of the drawdown data.

REFERENCES 1. Bruce, G. H., D. W. Peaceman, H. H. Rachford and J. D. Rice, "Calculations of Unsteady-State Gas Flow Through Porous Media," J. Petroleum Technology, 5, 7 9 - 92 (1953). 2. Katz, D. L., D. Comell, R. Kobayashi, F. H. Poettmanr~, J. A. Vary, J. R. Elenbaas and C. F. Wemaug, Handbook of Natural Gas Engineering, McGraw-Hill, New York, 1959. 3. Muskat, Morris, Physical Principles of Oil Production, McGraw-Hill, New York, 1949. Reprinted as a second edition by Human Resources Development Corporation, Boston, 1981. 4. Guerrero, E. T. and F. M. Stewart, Applied Reservoir Engineering, Vol. 2, Petroleum Publishing Company, Tulsa, 1962.

Chapter 13 THE COMPUTATION OF RESERVES AND PERMEABILITY FROM STABILIZED FLOW-TEST INFORMATION (Back-Pressure Tests)

The reservoir extent or reserves and the future performance of an oil reservoir can be estimated from the productivity index (PI) as previously outlined in Chapter 8. The determination is straightforward. The calculations for gaseous flow are not so straightforward, however, and will be further developed as follows. It will be assumed that the idealized Ready-state flow equation can be used to estimate the outer producing radius or radius of drainage. The test information is further assumed to be from stabilized flow behavior approximating the transition between drawdown and long-term depletion (or the onset of waterdrive). The above requires that the permeability be known or estimated. An estimation can be made based on the solutions provided by the use of integral forms to describe unsteady-state flow. A systematic adaptation will be provided in computer notation based on the initial formation pressure and open flow rate, as per the listing in the U. S. Bureau of Mines databank "Analyses of Natural Gases." This is perhaps the minimal flow test information which can be utilized to estimate gaseous reserves.

13.1 RESERVES AND PERMEABILITY CALCULATIONS The integral relationships may be utilized to estimate reservoir size and reserves, and the permeability, as previously derived. A systematic recapitulation is in order. If the bottom-hole shut-in or reservoir pressure is given, and if the flow rate and corresponding bottom-hole flowing pressure is known when the well performance levels-off after the initial drawdown, then the reservoir size may be estimated as follows. Basic to the determination is the assumption that the pressure gradient has reached the outer producing radius, which defines the transition between drawdown and long-term depletion.

364

UNSTEADY-STATE FLUID FLOW

After the well performance levels off after the initial drawdown, a flow test will yield a coefficient C 9

C=

Q po2 _ p 2

For extrapolation to open flow, as a special case, C = QoF

1,o The above permits the determination of C if only the open flow rate is given -that is projected from the data at two or more flow rates. It is required that beta (13) be evaluated from the relation Po - P ~ = P0

1 ~ 1 - lnfl]

where

Po - P= 1 - z~176 Po zTPo

and

The exponent c is not evaluated at this point, however, but rather after the solution is obtained. It will next be assumed that for gaseous flow K

2Q

M

~

,

,u

, ,

f l ( l n f l ) 2 - p p o A h zRT

from which (K/~t) is calculated. Since a condition of steady-state flow is approximated after the well performance stabilizes (after drawdown), then ideally M

c-

1

zRT In rb r~

and

M rb = exp { r~

p

zRT C

COMPUTATION OF RESERVES AND PERMEABILITY

365

Given r,, the well-bore radius at the sandface, and the reservoir thickness Ah or h , the outer reservoir radius or radius of drainage rb can be calculated. Therefore, from the material balance, the initial reserves in place are

O t,. - p o X ~ ~ ( r ~

- r))

in say pounds mass. In SCF, or billion SCF, Reserves - ~Mtf (379)x10 -9

BSCF

In tum, a reservoir permeability K is calculable from K/~t'

ic- (-K)~ P in consistent units, where ~ is in lb/ft-hr: ~( in lb/ft-hr) = 2.42 ~ (m cp). In turn, K will be in ft3/hr2' The conversion is K' (in darcys) = and

1

0.00442

K (in ft 3 / hr 2 )

K' (in md) = 1000K' (in darcys) Thus a figure for the reservoir permeability may be established, which may or may not agree with the laboratory value from a core sample. The theoretical depletion time tf for a constant flow rate Q is

tf=

poX~~(~

- ~) )

Q Reserves (in lbs)

Q If of interest, the exponent "c" can be calculated. Since P0 - P = / 3 1 1 - In fl] P0 solving for 13 at the flowing condition will yield

366

UNSTEADY-STATE FLUID FLOW

where t is the time at which the flow test was made. Thus r

lnfl

ln(t/t I) where tf has been previously estimated. The foregoing does not take into account the nonideal behavior of a backpressure flow test, where the exponent n is introduced: Q = C(po2-p2)~ . In other words, as an expediency it is assumed that n = 1.

E X A M P L E 13.1 Data from p. 444 of Reference 1 (re the preceding Chapter 9) will illustrate and confirm the technique(s). The initial drawdown curve may be approximated as shown in Table 13.1 (as taken from Table 9.4). The producing rate was essentially constant during drawdown, varying only from 1.5 to 1.43 MMSCF/D, for an average of 1.465 MMSCF/D.

TABLE 13.1 Drawdown

Roughly after about 24 hours, or one day, the well performance levels off to a flowing pressure of about 405 psia.

COMPUTATION OF RESERVES AND PERMEABILITY

367

The reservoir and fluid properties are as follows (re Table 9.3): Depth Temp Gas Grav MW Porosity Permeability Thickness Comp factor Viscosity Shut-in pressure Av. Flow rate

1,190 ft 84~ or 544~ 0.62 18 0.205 74 md 17 0.92 0.0113 474.4 psia 1.465 MMSCF/D

In lb/hr, the flow rate of 1.465 million standard cubic feet per day calculates to 1.465x106

18

24

379

- 2899 lb / hr

with a flowing pressure of abut 405 psia after drawdown. The initial density po is A4P Po = ~ zRT

18(474.7) 0.92(10.73)544

= 1.591 lb / ft 3

The density at flowing pressure is. p

=

18(405)

= 1.3571b/ft 3

0.92(10.73)544 The viscosity in English units is 0.0113 (2.42) = 0.0273 lb / ft - hr . The density ratio is first assumed to be as follows (neglecting compressibility factor adjustments): Po-P=I_~= P Po Po

1

405 474.7

The relevant calculations are as follows.

= 0.14683

368

UNSTEADY-STATE FLUID FLOW

Calculation of C: C =

2899 [(474.7) 2 - (405) 2 ](144) 2

= 0.000,002,28

Note that open flow extrapolates to C(474.7)z(144) z = 10,654 lb/hr.

Calculation of Beta (13): Po - P

= 0.14683 - / 3 1 1 - In fl]

Po Solving for 13 by trial-and-error yields 13 = 0.03337 . Furthermore, it follows that 13 (In 13)2= 0.38578.

Calculation of K/lu (m consistent units)' K

2(2899)

18

fl(ln fl)2 (1.357)(1.591)(17) 0.92(154)544

P

= 0.00954

Calculation of rv%" zc(0.00954)17 r b = exp{ r~

18 0.92(1

544)544.} _ exp{5.2053604} - 182.2

0.000,002,28

If r, = 1 if, then Rb = 182 ft.

Calculation of Reserves" 1.591 (0.205)(17)zc[(182)' ] = 0.578 x 10 ~ lb 0.578X106 18

(379)(10 -9 ) - 0.0122 B S C F

Calculation of Permeability: K-

/( ('-t-")/.t - 0.00954(0.0273) - 0.000260 ft 3 / hr 2 P

COMPUTATION OF RESERVES AND PERMEABILITY or

0.000260 0.00442

369

- 0.059 darcies = 59 md

This result is as compared to the reported value of 74 md.

Calculation of Theoretical Depletion Time: tf =

Reserves (in lb)

0.578 x 106

Q

2899

= 199 hr = 8.3 days

The foregoing calculation assumes that the prescribed constant production flow rate could be maintained.

Calculation of the Exponent c: lnfl

c-

= In

ln(t / t I )

ln(0.03337)

= 0.97

ln(24 / 797)

This is as compared to an ideal value of unity.

Note. The foregoing calculations are extremely sensitive to the ratio

Po - P = 1 - P--P-= 1 - z~176 Po Po zTPo and to the density p , both of which affect the mobility (K/~t). Whereas Zo is evaluated at the initial shut-in condition (at To and P0), the compressibility factor z at the bottom-hole flowing condition is evaluated at T and P . And whereas P is the bottom-hole flowing pressure at the sandface, and is known, the bottom-hole flowing temperature T may be different than To, for instance, due to the Joule-Thomson expansion effect. (At ordinary or subcritical conditions, there is a cooling effect. At supercritical conditions there could be a heating effect.) Thus for example if it is assumed that zT ~ (0.89)(533.4), then 1 - z~ T~

zTP

- 1-

0.92(544)(405) = 0.1005 0.89(533.4)(474.7)

Also, p-

18(405) 0.89(10.73)(533.4)

= 1.431

370

UNSTEADY-STATE FLUID FLOW

Hence 0.1005 = 13 [1- In 13] and solving for 13 by trial-and-error, check), whereby it follows that

13 = 0.0206 (which gives 0.10058 as a

fl(ln f l ) ' - 0.3105 Therefore, 2(2899)

K /a

18

=0.01184

0.3105(1.431)1.591(17) 0.89(1544)533.4

and K = 0.01184(0.0273) = 0.000323 fl:3/hr2 or 0.000323

- 0.073 darcies = 73 m d

0.00442

which agrees closely with the reported value of 74 md. Furthermore, r ~ = exp {6.4617} - 640 G and rb = 640 ft. The reserves calculate to 1.591(0.205)(17)~ (640) z = 7.135 7.135

x 10 6

(379)(10 -9 ) - 0.15 B S C F

18 Also,

7.135 tf

--

x 10 6

2899

- 2,461 hrs -

Since /9 0 - p

-~=0.1005 Po and 13 = 0.0206, then r --

1n(0.0206) -

ln(24 / 2461) as compared to unity.

0.838

103 days

x 10 6

lb or

COMPUTATION OF RESERVES AND PERMEABILITY

371

The value of c possibly can depend upon the eccentricity of the well with respect to the center of the reservoir, or on other factors not yet understood.

Effect of Permeability The calculation for reserves is extremely sensitive to the value used for the permeability, as indicated by the following tabulation:

The results are affected markedly by even slight discrepancies in the permeability value used. (And which is affected by other phases present.) The value of r, assumed will also have an appreciable geometric effect. Thus halving r, will reduce the reserves by a factor of four. Or doubling r, will increase the reserves by a factor of four. It is apparent therefore that the value of r, could also serve as a correlating parameter. As a point of reference, the independent reserve estimates for the particular reservoir were 1.1 to 1.26 BCF. The reservoir was never produced. On addition to the citation in Reference 1, p.444, reformation was supplied by K. W. Robertson, Illinois Power Company, Decatur IL.) The above illustrates the difficulty of estimating reserves from well tests. The use of acreage estimates is therefore viewed as more reliable. (Agreement would be most fortuitous.) Perhaps best of all is the extrapolation of pressure decline versus production. This, however, requires a long-term production history -- and again there may not be agreement with the other methods. In sum, the prediction of reserves in place is fraught with hazards, and is imperfect. There is disagreement, and consensus may be the exception. As the saying goes, it also depends upon "who is selling and who is buying." Needless to say the subject can stand further investigation, including comparison and correlation against known system behavior.

372

UNSTEADY-STATE FLUID FLOW

Note. Additional reformation is provided in the report "Preliminary Engineering Study of the Gas Storage Potential, Cypress Gas Reservoir, Dubois Field, Washington County, Illinois," prepared for the Illinois Power Company by James A. Lewis Engineering, Inc., August 25, 1955. Some eight wells were drilled, and flow tests were variously performed on five of the wells. The particular flow data from the above report do not match with the data presented in Reference 1, which may be a composite. There are also other factors which can be introduced, namely water saturation and its effects. An isopachous mapping (same thickness) was used to estimate the acreage involved, and further using the estimated sand thicknesses, the reservoir sand or formation volume was assigned a value of 5567 acre-feet, or 242.5 x 10 6 ft 3. Adjusting for porosity, etc., this translates to a gas storage capacity of about 1.17 BCF or BSCF, at 475 psia. Further correcting for the original water present--say about 50% saturation--gives for the original gas in place about 0.5 BCF, a value in the realm of that previously estimated from the flow test data in Reference 1. The wells tested gave widely varying open flow rates, which may indicate a number of things, including different asymmetries to the well location, and different effective water saturations, which affect both the porosity to the gas phase and the relative to gaseous flow.

EXAMPLE 13.2 Another example for utilizing the steady-state relationship is as follows. The data pertains to the Entrada Sand Reservoir, Harley Dome Field, Grand County, Utah (information supplied by D. Gililland, Grand Jtmction, CO): K = 150md ~t~ 0.014 cp Ah = 38.5 ft X = 0.22 P0 = 172 psia M ~ 16 mol. wt. T = 75 ~ F or 535~ Open flow = 2 MMSCF/D Water sat. = 55% Therefore, 2,000,000 24(379)

(16) - C [(172) 2 - 0](144) 2

COMPUTATION OF RESERVES AND PERMEABILITY

373

and C - 5.734 x 10 .6

0.150(0.00442) 0.014(2.42)

(re)(3 8.5)

16

1

1543(53 5) lnr b / ro

Solving for the logarithm, In rb/r, = 7.9995 and rb/r, = 2979. Assuming a value r, = 1 it, then it follows that rb = 2979 it. The reserves calculate to rc(2979) 2 (38.5)

172(144)

(0.22)(1 - 0.55) - 3.188xl 06 lb - mole - 1.2 BCF

1543(535) which closely compares with the figure of 1.3 BCF arrived at from estimates of the acreage. The calculation is also extremely sensitive to the viscosity value used, and to other fluid and reservoir properties, as previously noted for the permeability, etc. Moreover, the reservoir water saturation can enter into the estimation, as shown above.

Comparison of Open Flow Rates. A fact not commonly appreciated is that, all other things being equal, the higher the open flow rate, the smaller the reservoir, and the lower the open flow rate, the larger the reservoir. That is, at steady-state or near steady-state conditions, whereby stabilized operations signal the transition between drawdown and long-term depletion. Non-Idealities. Also to be acknowledged is the possibility that what a producing well "sees" -- defined as the radius of drainage -- may differ from the actual confining or enclosing surface boundaries of the reservoir. The problem is exacerbated by multiple producing wells from the same reservoir, and is further compounded by other nonidealities in reservoir and fluid properties, and the eccentricity of the well relative to the reservoir.

13.2 COMPUTER APPLICATIONS The following derivations for gas flow may be applied systematically to the U. S. Bureau of Mines (USBM) data base (2), or to other routine field tests. The enumerations provide the basic relationship and simplifications needed.

374

UNSTEADY-STATE FLUID FLOW

T A B L E 13.2

Designators for U S B M Data F ile (2) RESERVOIR DESCRIPTORS STATE COUNTY FIELD FORMATION GEOPROV GEOAGE WELLS SAMPLED INDEX SAMPLE NO WELLNAME LOCATION OWNER COMPLETED SAMPLED PUBLISttED DEPTH, FT THICK, FT ELEVKB, FT WELLHEADP, PSIG OPENFLOW, MSCFD C1, PCT C2 C3 NC4 !C4 NC5 IC5 CC5 C6PLUS N2 02 tMR It2 H2S CO2 HE BTU SPGR, G/L REMARKS SAMPLED BY MS HEMS TABLETAG AIRFREE TOTALPCT FILLER

AV

(ETC.)

COMPUTATION OF RESERVES AND PERMEABILITY

375

In addition to the methodology described hereto, there are the many other computational techniques described in the literature, as already utilized from References 1 and 3, and as may be further cited in the brief listing that comprises References 4-7, and as previously cited in Section 5.4 and Chapter 9 for instance. This is aside from the proprietary computer programs developed mtemaUy within petroleum and engineering companies, in progress and continuing. A listing of the range of information in the USBM data file (2) is shown in Table 13.2. It may be arranged in any prescribed order from the computer tape or disks supplied. By definition, a common reservoir will have the same STATE, COUNTY, FIELD and FORMATION designators. (The FIELD and FORMATION designators are controlling, however, and may be extended across the county and state lines.) The GEOPROV (geological province) and GEOAGE (geological age) designators are dependent. The designators are for the most part self-explanatory. However, ELEVKB refers to the elevation of the kelly bushing on the driUing-rig platform, and is used in the calculations as the wellhead elevation. The appropriate data from the individual wells sampled for each reservoir can be averaged, as indicated in Table 13.2, to provide a representative value for the reservoir. The information which includes REMARKS and designators which follow is not necessary to the calculations. In the absence of data points, the required reformation can be estimated or assumed.

Approach The back-pressure equation is Q = C (P02- p2)n where for the purposes here the flow rate Q is in the consistent units of lb/hr and P is in psfa. At open flow, where Qope.aowis also in lb/hr, = c(&

Therefore, the concept of the flow ratio designated FRAC may be introduced as follows" FRAC =

Q

Qopenflow

,

376

UNSTEADY-STATE FLUID FLOW

The problem reduces to determining FRAC , corresponding to a flow rate Q whereby all conditions are met. The fundamental relationship involved are: (1)

I - P / Po : ~ I - ln,B ]

(2)

Kh:

2Q

It

fl(ln fl)2 ppo

zRT

1- (P / Po)Z

2q(zRT)

(3)

Q=2

:

=

In rb / r~

(4)

M

ra~ ( Kh / p)

1 - (P / Po)2 _ 1.708~-o.s24

Q ln[2O0-(r b / r~)] 2

The last equation above, designated (4), is based on the results from Reference 3 as presented in Chapter 10, which utilizes the third equation, designated (3). By analogy with (3), above, there is referred that, for a flowing well, (5)

Q-2

[1 -

( P //90 )2 ]. l n r b / ra

=2

FRAC l n r b / ro

This relationship, designated (5), may be used to determine the ratio rb/r, and consequently the extent of the reservoir. In tum, calculating the density from the reservoir conditions will permit the determination of the reserves initially in place.

A~mplifications

Note that, in utilizing the relationship Q - q. M -

FRAC Qopo.now 9

it will be convenient to make the following substitutions, particularly in adapting the calculations to computer notation in terms of various differences denoted as "D" and as "DD". These substitutions are as follows:

COMPUTATION OF RESERVES AND PERMEABILITY

377

D D - 1 - P / Po - f i l l - In fl] or

P/Po

- I-DD

Furthermore, let D - ( P / Po) 2 - ( 1 - DD) 2 or

2 - l-D-

1-(P/Po)

1-(1-

DD) 2

Therefore, F R A C - [ 1 - ( P / Po )2 ],, _ (1 - D)" where n is the slope of the log-log back-pressure curve, obtamed when Q is plotted as the ordinate and the difference (p02 - p2) or [1- (P/P0) 2] as the abscissa. Eliminating pressure as a variable between relationships (3) and (4) gives Q l n r b / r, - 1.7080 ~ 2

Q ln[200. ( r b / r~ )] 2

Solving for Q , Q = 0.08259 Substituting for Kh/~t from (2) into (3), where q = Q/M, there is obtained 0

--

-

(fl In fl) 2 P

In p)2 ~zRl'

(/3

Z19o2

( M

)2 p p o -

Po

Substituting for P/Po from (1), (j

(fllnfl)2 -

(1 -

fl[1

-

In ill)

7/"

Since Q = 0.08259, a trial-and-error solution for 13 yields 13 - 0.017. It follows that D D - / 3 1 1 - I n / 3 ] - 0.0863 and D - (1 - DD) 2 _ 0.83491

378

UNSTEADY-STATE FLUID FLOW

Therefore, FRAC = (1 - 0.83491) ~ = (0.16509) ~ Utilizing relationship (5), m

Q-2

FRAC lnr b/ro

whereby r b / r~ -

exp[ 2(FRAC)]_

Q

where Q = 0.08259. The higher the FRAC value, the greater the ratio rb/r,. Note that at FRAC = 1 , which corresponds to open flow, the ratio rb/r, would be required to take on the value (rb/r,)m~ = 3.2876 x 101~ At FRAC = 0, then rb/r, = 1. In turn, the quantity Kh/~t may be calculated from (2), as may K if h (or Ah) and ~t are known. The reservoir extent can be calculated from rb (e.g., assuming r, = 1). It follows that the original reserves in place will be Reserves (in lb) -

PoXh(rb

" - ra "~) ~ p o X h r b

"

or

1 Reserves (in B C F ) - Reserves (in lb) [-:-; 9 (379)(10-9)] M

The theoretical depletion time for the producing mass rate Q will be

t / ( i n years) -

Reserves (in lb) 1 Q

1

24 365

where Q = FRAC 9Qop~no~ (in lb/hr). To convert from the flow rate Q in lb/hr to MMSCF/D, multiply by 379 M A 24 )( 10-6 ) where M is the molecular weight. If the specific gravity of the gas is given relative to air, then the molecular weight is equal to the specific gravity times 29. If the specific gravity of the gas is given in grams per liter, then the molecular weight is equal to this value multiplied by 24.470.

COMPUTATION OF RESERYES AND PERMEABILITY

379

Effe c t o f n

The following correspondence is of interest:

Note: one section equals 640 acres or one square mile. In practice, state regulatory agencies generally limit production from 25 % to 33-1/3 % of open flow. Well-spacings assigned are nominally 320 acres, though 640 acres have been specified in some instances.

Bottom-Hole Pressure and Temperature

In the absence of experimental data, temperature can be estimated from the thermal roughly 0.16~ per foot. Assuming 60~ as a surface temperature, the formation temperature would be

the bottom-hole or formation gradient of the earth, which is representative surface or nearTo in degrees absolute or ~

To = 60 + 460 + (0.16) DEPTH where DEPTH is a positive term. The formation pressure P0 can be calculated from the well-head shut in pressure, assuming some mean absolute temperature T for the stationary or shutin column of fluid in the well. Thus, from an energy balance, where velocity effects do not enter, 0 - V d P + g dz gc

380

UNSTEADY-STATE FLUID FLOW

where here the variable "z" denotes elevation, and for most practical purposes g/go= 1, and V = 1/p. For a liquid column, the specific volume or density is assumed constant, and thus 191 -- P2 - Po - Pwh -/0(7-2

P(Zwh -- 7"0 ) - - P" DEPTH

- Zl ) --

where integration may be perceived as occurring from point "1" at the bottom-hole or formation conditions, also subscripted "0", to point "2" at the wellhead, also subscripted as "wh". The wellhead elevation proper is commonly assigned as the elevation of the drilling-rig platform, or more specifically as that of the kelly bushing, denoted elsewhere as ELEVKB. For a gas well and gas column, the specific volume or density varies, such that RT ~dP MP

- -dz

and where the compressibility factor (also denoted as z) can be assumed equal to unity. Integrating between limits,

e0

- exp[~

DEPTH]

Pwh

The gas constant R is required to be in consistent units, e.g., in this case R = 1543 tt-lb/~ per lb-mole. A closer approximation can be made by introducing the thermal gradient, such that T - 520 + 0.16(Zwh -- Z) -- TADJ - 0.16 z where TADJ = 520 + 0.16 Z~h. On substituting, R (TADJ - 0.16 z) = - ~ MP Separating the variables, and integrating between limits, ln(P0 / P ~ h ) or

M

1

R 0.016

In

TADJ-0.016z

0 TADJ - 0.16Zwh M

5/

TADJ - 0.016z o ) ih-o~f6-i_ (To / l'wh) R(-o-.(ii6i "Do / Pwh - ( T A D J - O.O16 Zwh

COMPUTATION OF RESERVES AND PERMEABILITY

381

where Zwh- Zo = DEPTH. The gas constant has the value R = 1543 ft-lb/~ per lb-mole, when z and DEPTH are in ft. This refinement hardly seems worth the extra trouble.

Computer Notation The afore-described calculations can be adapted to a systematized computer notation as follows, as per the USBM data compilations outlined by Table 13.2. The assumptions listed are mainly for the situations in which the required data points have not been fully provided in the USBM data file. If the open flow rate is given, the actual production rate is defined as 25% of the open flow rate. If the open flow rate is not given, a nominal open flow rate of 4 MMSCF/D is assumed, which gives a nominal production rate of 1 MMSCF/D. The assumed figure may of course be adjusted up or down. The reservoir properties include an allowance for non-combustible gases, namely nitrogen and carbon dioxide. The propane-plus fraction, or C3+, is calculated as a measure of the potential liquids content of the gas. The estimated results include an estimate of the original reserves in place -that is, the original fluid or gas in place -- based on an assumed areal extent for each well sampled of one section or 640 acres. If not given, the reservoir thickness is assumed to be 10 ft. A uniform value for the porosity, which is not given, is assumed to be 20 % or 0.20. The calculated results include the detemamation of the original reserves in place and the permeability based on the open flow rate or producing rate. Additionally, the bottom-hole flowing pressure and the wellhead flowing pressure are calculated, based on the well depth and the wellhead shut-in pressure. If this reformation is not given, a well depth of 5000 ft is assumed, and a shut-in bottomhole pressure equal to the hydraulic head is assumed. Finally, estimated and calculated depletion times may detenmned from the foregoing calculational sequence, with the necessary assumptions, as foUows..

Note. The exponent n for the slope of the logarithmic back-pressure curve is not included in the USBM data file. Accordingly, a value of 0.6 is assumed for each well, which will give a calculated reservoir extent similar to that estimated from an areal consideration of one section per well. These numbers may of course be adjusted. For instance, a commonly-used well-spacing is one-half section or 320 acres per well. In this case, the larger value was used since the wells analyzed in the USBM data were in the main selective and did not include all the wells in the field or reservoir.

382

UNSTEADY-STATE FLUID FLOW ASSUMPTIONS

DEPTHAV

= 500

if not given

THICKAV

= 10

if not given

ELEVKBAV = 0

if not given

Z=I m

.

.

.

.

TRACE = 0

in calculating average values

"NOT GIVEN" is not to be entered into the calculated averages except as above. If all values are "NOT GIVEN" then the average becomes "NOT GIVEN" WELLHEADP = 0.45 (DEPTHAV) OPENFLOWAV = 4000

if not given if not given

COMPUTATION

OF RESERVES

RESERVOIR

AND PERMEABILITY

PROPERTIES

NETYIELD = 100 - N2AV - CO2AV

ADJBTU = (BTUAV)/[NETYIELD)/(100)] -,,.-===.

MW = (SPGRAV) * (24.470)

R =

1543

E X = ( M W ) / [ ( R ) * (0.016)]

PATM = (14.696)[(1) - ([0.00000687] * [ELEVKBAV])] ,,.=..,=.,.

A = WELLHEADPAV

+

PATM

B = [([520] + [(0.016) * ( D E P T H A V ) ] ) / ( 5 2 0 ) ] Ex

RESPRESSCALC

= A *B

R E S T E M P C A L C = 60 + [(0.016) * ( D E P T H A V ) ] .

.

.

.

==

H Y D R A U L I C P = P A T M + [(0.45) * ( D E P T H A V ) ]

PRESS/HYD = (RESPRESSCALC)/(HYDRAULICP)

P O R O S I T Y E S T = 20

5"256

383

384

UNSTEADY-STATE FLUID FLOW

CALCULATION OF C3PLUS

C 3 P L U S A V = [C3AV * 27.49 + NC4AV * + IC4AV

31.50

* 32.66

+ N C 5 A V * 36.17 + IC5AV + CC5AV*

* 36.57 38.28

+ C 6 P L U S A V * 46.12 ] / (100) ADJC3PLUS = (C3PLUSAV)/[NETYIELD)/(100)]

COMPUTATION OF RESERVES AND PERMEABILITY

38~

E S T I M A T I O N OF R E S E R V E S A N D P R O D U C T I O N R A T E S

AREAEST = (1) * (NO. OF WELLS SAMPLED)

V O L U M E E S T = (AREAEST) * (THICKAV) * ([5280] 2)

T = (RESTEMPCALC) + 460

PO = (RESPRESSCALC) * (144)

RHOOM = (PO)/[(Z) * ( R ) * (T)] ,,,.. ,,..,

R H O 0 = (RI-IOOM) * (MW)

,,,,,, ,,.,,

R E S E R V E S E S T = (VOLUMEEST) * [(POROSITYEST)/(100)] * (379) * (10 9) (RHOOM) 9

F L O W l N G B H P E S T = (REPRESSCALC) * [ ( 1 ) - (0.25) 2] 0.5

DELTAP = (REPRESSCALC) - (A)

F L O W l N G W H P E S T = ( F L O W l N G B H P E S T ) - (DELTAP)

PROD/WELLEST = (0.25) * (OPENFLOWAV) * (10 3)

T O T A L P R O D R A T E E S T = (NO. OF WELLS SAMPLED) * (PROD/WELL)

DEPLETIONTIMEEST = [TOTALPRODRATEEST]

[(RESERVESEST)

*

([10 a]

/

[365])]

/

386

U N S T E A D Y - S T A T E FLUID F L O W

CALCULATION

OF RESERVES

AND PRODUCTION

Q O P E N = ( O P E N F L O W A V ) * [(10 3) / ( 3 7 9 ) ] * [ ( M W ) / ( 2 4 ) ]

N E X P = 0.6

C = ( Q O P E N ) / [(po)2] ~'~xP

B E T A = 0.017

D D = B E T A * [(1) - ( I n [ B E T A ] ) ]

D = [1 - DD] 2 . . . .

,,=

F R A C = (I -

D) NExP

P = (PO) * [(D)~

R H O M F L = (P) / [ (Z) * (R) * (T) ]

RHOFL = (RHOMFL) * (MW) .,=,.,,,,==

==

D E N O M = ( B E T A ) * [(In[BETA]) 2] * [(RHOFL)] + [RHOO]

K H M U = ([(2) * )Q)] / [ D E N O M ] ) * ([MW] / (Z) * (R) * (T)])

QBAR = 0.08259

RATES

COMPUTATION OF RESERVES AND PERMEABILITY

387

EXP = [(2) * (FRAC)] / (QBAR)

RRATIO = e Ex~ , . . . m . , . . ,

RA=I

RB = (RRATIO) * (RA)

A R E A C A L C = (x) * ([RB) / (5280)] 2)

V O L U M E C A L C = (70 * ([RB] 2) * (THICKAV) ,.

,,.,,,,,..,.,.,

RESERVESCALC

= ( V O L U M E C A L C ) * [(POROSITYEST) ( R H O O M ) * (379) * [(10) 9]

/ (100)])

GASVISC = (0.01) * (P) / (144)] / (14.696)] ~

M U = (2.42) * (GASVISC)

MOBILITY = (KHMU) / (THICK)

PERMEABILITYCALC = (MOBILITY) * (MU) * [(1000) / 0.00442)]

F L O W I N G B H P C A L C = (P) / (144)

FLOWINGWHPCALC = FLOWINGBHPCALC-

DELTAP

*

388

UNSTEADY-STATE FLUID FLOW

P R O D / W E L L C A L C = [(Q) / ( M W ) ] * ( 3 7 9 ) * [(10) "6] * ( 2 4 )

TOTALPRODRATECALC

DEPLETIONTIMECALC

= (NO. OF W E L L S ) * ( P R O D / W E L L C A L C )

= [(RESERVESCALC) ( T O T A L P R O D R A T E C A L C ) ] * [([10] 3) / (365)]

Note: Q = c [ ( p o ) 2 - (p)2]n Q O P E N = C [[(PO) 2 ]n

It follows that, if F R A C = Q / Q O P E N , then P/PO = [ 1 - F R A C ] ~ = D 0.5 1 - [ P / P O ] = 1 - D 0.5= D D

(1 - [P/pO]2) N = F R A C

COMPUTATION OF RESERVES AND PERMEABILITY

389

REFERENCES 1. Katz, D. L., D. Comell, R. Kobayashi, F. H. Poettmann, J. A. Vary, J. R. Elenbaas and C. F. Wemaug, Handbook of Natural Gas Engineering, McGraw-Hill, New York, 1959. 2. Moore, B. J. and S. Sigler, "Analyses of Natural Gases, 1917-85," U. S. Bureau of Mines Information Circular 9129, 1987. Updates include IC 9167 (1986), etc. Information is added annually or biennially. Also, Tape No. PB89-158661, etc., National Technical Information Service, available on magnetic tape or diskettes. 3. Bruce, G. A., D. W. Peaceman, H. H. Rachford and J. D. Rice, "Calculations of Unsteady-State Gas Flow Through Porous Media," Trans. AIME, 198, 7990(1953). 4. Matthews, C. S. and D. G. Russell, Pressure Buildup and Flow Tests in Wells, Society of Petroleum Engineers of AIME, Dallas, 1967. 5. Meehan, D. N. and E. L. Vogel, HP41 Reservoir Engineering Manual, PennWell, Tulsa OK, 1982. 6. McCoy, R. L., Microcomputer Programs for Petroleum Engineers. Vol. 1: Reservoir Engineering and Formation Evaluation, Gulf, Houston, 1983. 7. Sinha, M. K. and L. R. Padgett, Reservoir Engineering Techniques Using Fortran, International Human Resources Development Corp., Boston, 1985.

Chapter 14 APPROXIMATE SOLUTIONS DUR/NG DRAWDOWN AND LONG-TERM DEPLETION

In Chapter 10 it was shown that the unsteady-state numerical solutions for gaseous flow based on Darcy's law could be approximated by the steady-state distance profile at any given time, as diagrammed in Figure 10.9. This approach will be extended to depletion by both linear and radial flow, for compressible liquids as well as gases. Such will permit the expression of the drawdown and long-term depletion behavior as a function of time -- e.g., at the point or region of effiux, or producing sandface. In particular, the results for linear compressible liquid flow will be found to agree exactly with the theoretical results for flow from a semi-infinite or infinite linear reservoir. This, then, serves to further validate the general approach for utilizing steady-state profiles.

14.1 COMPRESSIBLE LIQUIDS The density function 9 for a compressible liquid is given by p = A e ~ = A exp(cP) where P = pressure c = compressibility A = proportionality constant This will be further applied to Darcy's law, as follows.

Linear 1,7ow

In unsteady-state linear flow, Darcy's law can be written, for flow in the (-x) direction, as

3 94

UNSTEADY-STATE FLUID FLOW Q " - _ _K p kt

cgx

where Q " is the mass or molar flux (e.g., mass/time-area). Substituting for p , Q,, _ K A e ~P c ~ = K 1 cg( A e ~e ) = K 1 c?p p

c3x

pc

c3x

pcc~

The foregoing will represent the behavior during both drawdown and long-term depletion. At a particular time t and assuming steady-state, Q" will be constant and the integration will revolve the ordinary differential equation

Q

t!

K ldp

/~cdx

which will represent the density/distance profile at some particular time t . The constancy of the mass flow rate at the point or region of efflux may be prescribed by a constant volumetric liquid flow rate maintained at a constant temperature, pressure, and composition. Drawdown. During drawdown the preceding differential equation represents a succession of density/distance profiles, whereby the upper density limit remains at the constant initial value p0. Thus, integrating independently of t and at constant Q" (the steadystate), the density profile is obtained:

/9 0 - p x

Q" c = ~ ( L K / /z

x)

(1)

where p0 = initial reservoir pressure L = x-intercept at p0 and where x varies from x = 0 to x = L. In tum, the upper limit L is treated as a parameter which may vary from zero to some value of the x-intercept along p = 90 9 Thus L varies during drawdown from L = 0 to some maximum value L = L~, which denotes the length or outer dimension of the reservoir. Thus the particular value of L sets the particular profile for the particular time t , and when L reaches the value Li, this will denote the transition time ta between the end of drawdown and the commencement of long-term depletion. It may be noted therefore that L increases during drawdown until it

APPROXIMATE SOLLrFIONS

3 95

reaches its maximum value L~, which also denotes the linear dimension of the reservoir, and signifies the closed end of the reservoir. That is, for flow or depletion in the (-x) direction, the linear dimension x will vary from x = 0 to x = L , with the parameter L (which is the varying upper limit for x) taking on values from L = 0 to L = Li. All during drawdown, at x = L , the density and pressure remain constant at p0 and P0, the initial condition for the reservoir. At Li. at time t~, at the end of the drawdown period, the density and pressure will have these same values, then will decrease at L~ as long-term depletion proceeds. There will eventually be a time, however, when the prescribed constant production rate cannot be sustained during long-term depletion. The limiting condition would be when P = 0 at the point of effiux, and since p = p0 exp(cP) where p0 is the reference, Q" C

P0

~L~

K / ,u

A schematic representation of successive profiles is shown in Figure 14.1.

L-

.I

Intercept

1.1" ..I"

Fig. 14.1 Successive steady-state profiles.

3 96

UNSTEADY-STATE FLUID FLOW

The solid lines represent profiles during drawdown. The dotted line represents the profile at the transition time t~. The dashed lines represent profiles during longterm depletion. Note that at the limiting condition, when P = 0 and p = A, longterm depletion can no longer be sustained at the prescribed production rate. The material balance statement for a particular profile during drawdown is L

Q" (t - O) - Xpo (L - O) - X I p d r 0

(2) L

[po

= X p o ( L - o) -

0

(,.:"0c L - x)ldr K / ,u

where X is the porosity. Thus

Xc

L2

K/lu

2

t ~

and

L - [2 K //2 ]1/2 tl/2 cX which can be utilized in the density expression that is Equation 1. Substituting for L in the density expression gives

Po - Px

_ Q"c {[2 K / tu]v2 tv2 _ x} K/----~ cX

(3)

At x = 0 ,

Po

Px-o

~

OH

[

2c X(K

t

v2

/

or PO ~ Px=o

Po

Q" [.

2c

Po X ( K / p )

If for slightly-compressible fluids,

]1/2 t 1/2

(4)

APPROXIMATE SOLUTIONS

397

p-~ po[B + cP]- po[1 + c ( P - Po] then

poc(Po - Px=o) - Q"[

2c

v2 t v2

X ( K / It) ]

or

Po - P~---o

0"

Po

2

PoPo - [cX(K / p)]v2 tl/Z

(5)

Thus density or pressure falls offwith the square root of time.

End qfDrawdown. Note in particular that the end of drawdown is marked by the transition time t~ whereby Xc L~z ti

--

K/it

2

Furthermore, the total production at the end of drawdown is

Q" (t~ - t) This value is on an areal or flux basis normal to flow.

Comparisons with Rigorous Linear Solution. The partial differential equation for a compressible liquid in linear flow may be written as follows (Sections 5.1 and

5.2):

K~ i.t 82p cX cgx2

0t9

For a constant mass efflux rate,

K10p pc& For a slightly compressible liquid, in terms of the pressure P,

K~ p O2P cX &2 and

c~ c~

398

UNSTEADY-STATE FLUID FLOW

Q,,

K

cT

-- m p 0

p

An apparently rigorous solution for heat conduction in a semi-infinite body is given by Churchill (as per Section 9.2). Rephrasing the solution in terms of fluid flow the following expression is obtained for the point or region of effiux:

2Q" PO ~ Px=o

=

K / ft

c

K/p

7( 1/2 ( c X

v2 )1/2t

or PO ~ Px=o

Q" 4~

2c

]1/2tv2

po 4Y [X(K / ~,)

Po In terms of pressure,

P O -- P x = o

--

2Q" 1 1 K / ft 1/2 X / ~ Po ~"~ ( cX )"~t

or

Q" x/2

m Px=o

Po

2

]1/2 t 1/2

PoPo 4r~ [cX( K~ It)

The rigorous solution differs from the approximate solution (Equation 5) by a factor of x/2/rc - 0.79788 . In brief comment, the fact that the steady-state representation for a finite body and the rigorous solution for a semi-infinite body tend to overlap, suggests that the profiles or gradients for the latter may be approximated by steady-state conditions -- or vice versa. This is the case only during drawdown., however, and is not the situation for the long-term depletion of a finite reservoir.

Comparison of Constants. In Chapter 9, the theoretical equation for the flow of a compressible liquid from a semi-infinite reservoir (closed on one end) was represented in the following equivalent notation: Po - &:o Po where

=

t'o - P ( O , t) Po

= m ( 2 ) ( a ) v z t 1/2

APPROXIMATE

m-

399

SOLUTIONS

158.5 Q hKP0

and Q = cubic feet of liquid per day K = millidarcies h = feet ~t = centipoises P0 = original pressure in psia and where a --

2.63(10 -4 )K /acX

The crossectional area normal to flow is h (1) = h or Ah. A darcy is manifested by a flux of one actual cc per cm 2 per second of a fluid having a viscosity of one centipoise flowing through one cm of distance under a pressure difference of one atmosphere. Therefore, in consistent units, consider the following conversions: Q (in ft 3 / day -g

- ft 2 ). (30.48)( 1 3600 1 ) - Q" (in cc 3 / s e c - cm 2 ) 24 Po

or

Q Q" - - = ~ (2834.6) h ,,90 Furthermore, since K (m darcies)xl000 = K (m millidarcies), then 0.1585,uO m

-

-

2.63(10-' )K and

a

-

hKP o

l.tcX

where K is now in darcies. Also P0 (in p s i ) - P 0 (in atm). 14.7 and 1

t (in h o u r s ) - t (in see). 3600

400

U N S T E A D Y - S T AFLUID T E FLOW

Finally, 1

c (in p s i - ' ) - c (in atm-'). ~ 14.7 Therefore, assuming no errors have been made,

Po - Px_-o 0.1585 1 Q" 2.634(10-' )(K //~) ,,2 t = (2834.6)(2)[ (c / 14.7)Xzc ] (3600)'/2 19o K~ la Po(14.7) Po 1.00 Q" ~

2

,/z

Po po 47 [~x(I< / /.,) l'~ t where the units are consistent, and K is in darcies.

Material Balance ,Statement Jbr Long-Term Depletion. The overall material balance for long term depletion may be phrased as Li

Q" (t - t~ ) - X p o (L, - O) - Q" (t, - o) - x ~ p a x 0

where the second term on the right represents the total production during drawdown. Accordingly, Li

Q" (t - o) - X p o (L, - o) - x ~ p & 0

where it is understood that t > ti in this particular equation.

Long-Term Depletion. For the long-term depletion of a compressible liquid in linear flow, the (steady-state) profiles are given by

Qtt C K / ,u or (~)tt C

(Po

-

P)-(Po

-

P L , ) - K"/ I t

(L,

-

x).

(6)

APPROXIMATE SOLUTIONS

401

where the notation is used that PLi stands for the density at Li, the closed end of the reservoir, and which varies as depletion proceeds. The material balance statement is Li

Q" (t - O) - Xpo (L, - O)- X~ pdx 0

Li : XPoL~ - X

Q"c (LZ [pr, - K /----~

L, 2 2 )

0

Q" c Li 2

= X(Po - PL, )L, -+

K/,u

2

or

Q"c Li 2 Q" t - X ( p o - Pz., )L, +

K/,u

(7)

2

The variables are t versus PLi. The quantity Li is a constant. Therefore, on solving for p0- Pti and substituting in Equation 6,

O"t t3o

-- Pri

PO

~

=

X L,

Q"

L~

(8)

X ( K / ,u) 2

and

Q" t Px:O

"- ~

-

Q" c F ~

X L~

(K / ,u)

L, (1-

I

(8a)

where, during long-term depletion, t > t~. Therefore,

Po - Px-o

O" t :

Po

-

XLip o

Q" c +

L~ ,

1

--tl-

( K / p ) po

2X

The above behavior would be linear with time. For a slightly compressible liquid, where p ~ po [B + cP}, Po - Px=o - pocPo - pocP~=o

(9)

402

UNSTEADY-STATE FLUID FLOW

and Po - Px-o 1~

Q" t

=

+

c X L, Po p o

Q"

L,

1

--(1

(K / p ) Po P o

)

(10)

- -~

The behavior will also be linear with time. The limiting condition would be at the pressure Px--o= 0.

Radial Flow In radial flow, Darcy's law reads 03o K 07' Q _ K 2 rcr h p = 2 rc h p p. c~ p 81n r where h or Ah is the reservoir thickness and Q is the mass flow rate (e.g., mass/time). Similarly, the nomenclature could be adopted that Q~ is the molar flow rate, where Pm is the molar density. Altemately, the expression can be written as b~T cT K Q,_ K 2n-rp 2rcp 81nr ,u c~ ,u .

.

.

.

where Q' is the mass flow rate per unit thickness. Substituting p = Ae ~ = A exp(cP) , and rearranging, K2rch O __

tu

c

3,0 31nr

This represents the behavior during both drawdown and long-term depletion, and the transition m-between. At a particular time t and assuming the steady-state, Q __

X2rch /,t

c

dp dlnr

which will connote the density/distance or density/log distance profile at the time t

403

APPROXIMATE SOLUTIONS

Drawdown. Integrating the foregoing equation between limits will yield 19o -- P r =

Qc

(K //~)2er h

lnrb r

(11)

where p0 = initial reservoir density rb = r-intercept at P0 and where r varies from r = r, to r = rb. The value of ra is a fixed constant value, the nominal radius for the wellbore or sandface. The value of rb is here treated as a parameter, however, and will vary from r, to some final value (rb)~, which denotes the outer dimension or radius of drainage for the reservoir. [In the previous derivations for radial flow, the outer radius or radius of drainage was given the value rb, which was considered a fixed value or value to be determined -- at least in estimating the reservoir extent and the reserves originally in place. Thus the former usage is now succeeded by a different usage whereby the reservoir radius or radius of drainage takes on the value (rb)i-which, incidentally, will be the value of rb recurred at transition, and for a reservoir whose outer boundary is closed, will remain at this value during the subsequent long-term depletion. The analogy is entirely with linear flow.]. It will be observed that rb reaches its maximum and sustaining value (rb)~ at the end of drawdown, at transition, signified by the subscript "i", whereby the following relationship exists: P0 - (P~)~ -

Qc In (rb)~ (K //~) 2re h r,

(11 a)

During the subsequent long-term depletion, this relationship will be modified, as will be developed forthwith. The outer radius (rb)~ will be remain a constant value. The schematic representation of successive profiles is shown in Figure 14.2. The solid lines again denote the profiles during drawdown. The dotted line represents the profile at the time t~. The dashed lines represent the profiles during long-term depletion. For convenience, the abscissa is denoted as r

x=ln-r~ which corresponds to an assumed mathematical transformation from linear flow to radial flow.

404

UNSTEADY-STATE FLUID FLOW x,

=

In ( r h } , r a

................ .iiii .......iiiiiii ..........iiiii ........................... .111

I~ ~

11

I

1111

111 ~I~ ~

.I~ ~

~I

11

~

I~ ~ 1 1 ~ 11

r X

=

hi

~ ra

Fig. 14.2 Successive steady-state profiles for radial flow.

The material balance statement for a particular profile during drawdown is rb

Q(t - o) - Xhporc(rb 2 - ra 2 ) - Xh I p 2 z r d r ra

rb

= Xhpo~r(rb ~ - ~ : , ~ ) - A ~ [ P o rr

rb

Q c X i ( _ ) r l n r dr K/

Integrating,

p~ a

-

rb

-

Oc

( K / p)2rch

In rb ]2zrdr r

(12)

APPROXIMATE SOLUTIONS t

405

cX [_r 2 In -r- + r 2 ] (b K/p 2 Pb 4 r,~

~

(13) cX [ r 2 _ r 2 K/p

q2

4

2

In rb ]

r~

This provides the relationship between rb or rb/r, and t . Rearranging, let t ' - 4 ( K / p) t - [(rb / r~ )2 _ 1 - In (rb ) 2 ]

cXra 2

(14)

r~

The behavior of this relationship is tabulated in Table 14.1 and shown in Figure 14.3. The relationship or solution may be expressed as

rb = f ( t ' ) ro where rb is a varying parameter during drawdown. Therefore at r = r~,

Po - P -

Oc

"lnf(t') (K / p ) 2rch

or

Po - P

Qc

Po

Po (K / p)2rch

lnf(t')

(15)

where, as previously stated, t'= 4 ( K / p )

cXr~

t

2

For slightly compressible liquids, where p = p0 [B+cP], it follows that t'o - P

Q

Po

190 (K / p)2rch

where t' is defined above.

lnf(t')

(16)

406

UNSTEADY-STATE FLUID FLOW

TABLE 14.1 Reduced Time vs Radius Ratio rb/r.

(rb/r~)2

1 3.16228 10 31.6228 100 316.228 103

t'

1 10 100 1,000 10,000 100,000 106

0.0 6.697414 94.3498 992.0922 9,989.790 99,987.487 999,985.18

lO s

10 7

10 6

10 5

10 4 log t 10 3

10 2

101

l

10

lO 2

lO 3

l o g r~/ro

Fig. 14.3 Logarithmic plot of reduced time versus radius ratio.

APPROXIMATE SOLUTIONS

407

At higher values of rb/r,, it is apparent from Table 14.1 that t'-~ (rb) 2 r~

or

rb = (t')'/2 ro

or

In r~ _ _1 In t' r~ 2

Therefore,

Po - P Po

Q c _1 In t' po (K / p) 2ah 2

(J7)

or for slightly compressible liquids, where p -~ p0[B+cP], Po - P = Q l l n t, Po PoPo (K //.t) 2zdt 2 or Po -

P

Q l ln 4 ( K / P) t Po P o ( K / It.)2ah 2 c X r~ z

_

19o or

Po - P Po

_ Q 1 K~ - PoPo (X / p) 2nh -~ {ln, ,., cXr~P2 t + In 4}

(18)

where In 4 may be dropped for higher values of K/p cXr~

2 --ID

whereby this controlling relationship corresponds to the tD term as previously expressed in Section 9.2. End qfDrawdown. Note in particular that the end of drawdown is marked by the transition time ti whereby from Equation 13, when r = (rb)~ or rb~, _

t~-~[

cX

rb, 2

K/p

-r~

2

4

2

r~ in rbt ] 2

r~

Furthermore, the total production at the end of drawdown is Q(t, - t)

This value is on the basis of total flow.

408

UNSTEADY-STATE FLUID FLOW

Comparison o f Constants with Rigorous Solution. A solution presented in Section 9.2 is reported to be asymptotic with the rigorous solution for higher values of tD : Po - P ( r , t ) = mP,

1"o where in this case, for radial liquid flow,

25.1 Q m_

h(X / )Po

and

1 Pt - = (In tz) + 0.8097) Z

It is interesting to note that the constant value 0.8097 as used above compares with a value of In 4 = 1.3863 as derived from Equation 18. Furthermore, in appropriate units, 2.634(10 -4 )(K / p ) tD -2 t CATr Therefore, at r = r,, Po - Pr=,-~

=

Po

2.634(10 -4 )(K / ,u) 1 t + 0.8097} -{ln cXr a2 h ( K / ju ) P o 2

25.1 Q

At higher values Of t (or to), the constant 0.8097 may be dropped. (Incidentally, 0.8097 = In 2.2472.) As in the case of linear flow, it may be noted that, in conflicting notation, (30.48) 3 Q(in ft 3 / day) 2-~6--O()) - (,)(in cc 3 / see) Q (in grams / sec)

Q (in lb / hr) 453.6

P0 (in grams / cc)

Po (in lb / fl 3) 3600

--'7

where (30.48)3/(24)(3600) - 0.3277 and course that h (in cm) h (in fl) -

30.48

1/(0.3277) - 3.0512. Note also of r a (in f t ) -

r (in cm) 30.48

409

APPROXIMATE SOLUTIONS

Thus in units made consistent, Po - P

P0

~,~

0.0251

3O.48

(2 (3.0512)~1 In{ 2.634(10-')(K

(K / ,u)h Po (14.7) Po

/ ~t)

cX( r, )2 14.7 -30.48-

2

t 3600

or

P0 - P,.. _-

Po

0.1587

Q 1 ln{(l00)(X / ~t) t}

Po ( K / l.t ) h P o 2

cXr ,

2

or

Po-P Po

rd

1.00 Q _l ln{ K / p 2 t} Po ( K / p ) h 2re P o 2 cXr ~

where the units are consistent and where here K is in darcies. This result agrees with the previously-derived formula based on steady-state profiles -- e.g., Equation 18. (Interestingly, note that 0.1587(2n) = 0.9971 ~ 1.00 .) Apparently there is a close similarity with the behavior assuming steadystate radial flow and with presumably more rigorous representations. As noted for linear flow, however, this is the situation only during drawdown and does not apply to the long-term depletion of a closed reservoir.

Material Balance Statement for Long-Term Depletion. balance for long term depletion may be phrased as

The overall material rbi

Q(t - t, ) - X h p o rc[(r~ )2 _ r 2 ] _ Q(t, - o) - X h ~ p 2rcr dr ?'t7

where the second term on the right represents the total production during drawdown. Accordingly, rbi

Q(t - o) - X h p o rc[(r~ ) 2 _ ro2]_ X h ~ p 2nr dr re7

where it is understood that t >__ti in this particular equation.

Long-Term Depletion. For the long-term depletion of a compressible liquid in radial flow, the profile is given by

p,+, - p-

Oc -

(K / It) 2~rh or

ln~

r

410

UNSTEADY-STATE FLUID FLOW

(Po

-- I3 ) -- ( P o

--/Orb, ) --

In (rb)i ( K / ,u)2rch

r

(19)

=y where y (gamma) would be a function of r -- that is, ~/= ,/(r) -- as defined by the substitution. Furthermore, the density function designated p or p, is a function of r -- that is, p = p(r) -- and here Prbi or (Prb)i denotes the behavior of the density function at r = (rb)~ or rbi, the closed end of the reservoir. The material balance statement is rbi

Q(t - O) - X h p o z [ ( r b )2 _ r 2 ] _ X h I p 2 z r dr ra

or tbi

Qc

O ( t - o ) - Xhporc[(r b )2 _ r 2 ]_ X h I [p,.~ ' _

In (t;); ] 2 a r d r

(K //.t)2rch

ra

r

(20) Integrating between limits, where 2

I x l n a x d x - x lnax 2 .

.

.

.

.

x .

2

.

will give the expression ,.

Q t - xh

(Po

- P r b , ) ~ [ ( r a ),

,. -

C) c

r

In ~ - rt (r~ ),

r~ ] + X h ( K / l.t) h [- 2

r

[ rbi '~

]

4

where integration is from r - r~ to r - ( r b ) i , also denoted respectively as r = ra and as r = rbi for convenience in the notation. Substituting the limits and simplifying,

Q t - X h ( p o - Prh, )z[(rh )2 - L , 2 1+ X

2

+t',

0._ c [(t; ), 2 - r , 2 (K / /.t) 4

In r ,

2

,9

1] + X - - - -(') - - [c (K/p)

ra2 ~(&)22 4 r. .

.

.

.

.

.

.

.

.

.

1 .

.

.

,

(rb)i ]

or

Q t - X h ( , o 0 - P,b, )nro" [(& ~'r,,

a

.

ln(r .

.

.

.

.

r

APPROXIMATE SOLUTIONS

41 1

Note that the above expression may be further simplified, since (rb)i > > r , . to yield

O t - Xh(p ..

(rb); 2 ~4 ]

o -- p,b, )rc(rb ) 2 + X Q c ; (K/B)[

Therefore.

(rb), ~ , Oc [___s

ot-x -

K/#

Po - P,.bi =

X h rc(r~ ),

2

(21)

Furthermore. from Equation 19. Po - P-

( P o - P,h; ) +

(2 c In (Q )______!_, (K / #)2xh r

which reduces to the following at r - r,

Oc Po - P,,, - (Po - P ~ , ) +

(r. ),

-

In ~

(K / #)2x h

(22)

Q

Both the latter expressions will be linear in t for t > t~, that is, for values of time greater than the transition time t~. Assuming that for a slightly compressible liquid p = po[B + cP] and dividing by the qumatity cPo will yield ()/

Po - Prbi

. _

1"

()C

Po

r (~;)i/ 2 .

,ut

K/

-

" -

. . . . . . .

4

. . . . . .

]

c1'o X h x ( r~ ),'

In t u r n , 9

Po-P

Po =

Po

----

:~.h, O t - X

Po

_

--

Oc

Xl #

[

.....4-

c P o X h rc ( r , ) , 2

2

. . . . .

]

+

(),}

(rb)

-ln___~, c Po ( K / / t ) 2 rc h r (23)

where P takes on the behavior P - P~. at r - r~, which is also linear with time The lower hypothetical limit for P at r. is of course zero.

412

UNSTEADY-STATE FLUID FLOW

14.2 COMPRESSIBLE GASES

The density function P for a gas is representable by p-

MP

zRT P =~ p M

or

zRT

where M is the molecular weight and z the compressibility factor. This will be applied to Darcy's law assuming constant z. However, z may take on constant values other than unity.

Linear blow A reiteration of Darcy's law reads

Q " -K _p

cT

where Q" is the mass flux. Substituting for p at constant z and T ,

Q

t!

.._

K M P c~

K I M

cT 2

pzRT

ta 2 z R T

c3c

cgx

Substituting for P at constant z and T ,

Q , , -K_ p zR _ _T M

t.t

Op

K l zRTc?p 2

dx

t.12 M

c~

The latter expression will serve the purposes here, such that at some particular time t , and at steady-state, K 1 zRTdP

Q

t!

__

p2M

dx

which establishes the differential form for the density profile.

Drawdown. During drawdown, the density profile can be represented as follows, obtained by integrating the preceding differential form. Thus

APPROXIMATE SOLUTIONS ,

Q"

,

Po - Px :

413

M

~ ( L - x) (K /,u) zRr

(24)

where p or px denotes the density at any point x , and where the value of the parameter L ranges from zero to L~, the value reached at the end of drawdown, that is, at transition, at the transition time ti. The material balance statement for a profile at time t during drawdown is given as follows: L

Q"(t- o)- Xpo (L- o)- x~ p dx o

- Xpo(L-

Q"

O ) - X f [po z

m

(L - x)] v2 d r

K / ,u z R T

(25)

Integrating, 2

Q" t - X p o L - X

(K / ,u) z R T O,,

-~[Po

Q"

1]/1

~ (L - x)] 3/2 [~ K / kt z R T o

2

or 2

Q" t - X p o L - X 2- ( K / p ) z R T 2 3/2 -[po 3 Q" M {[p~

-

0"

M

K / At z R T

3/2

(L)]

}

(26) This provides the relationship between L and t . It does not readily lend itself to solving for L in terms of t. If, however, this latter solution is symbolically represented by the function L = L(t), then at x = 0,

Po

2

Q".

2 - (Px =o)

=

m ~

K/ pzRT

L(t)

or 2

P0 - ( P ~ - 0 ) Po or

2

2

Q)) =

_

=

I

K / , u Po

M

2

~L(t) zRT

414

UNSTEADY-STATE FLUID FLOW

(Px=o)2

Po 2 --

Q,,

Po z

1

zRT

--L(t)

(27)

K / p Po 2 M

This would represent the drawdown curve. Obviously the above is not equal or even similar to the relation presented in Chapter 9 for gaseous flow.

End of Drawdown. From Equation 26, when L = L i , the corresponding transition time t = ti marks the end of drawdown, which is encapsulated in the following relationship 9 3;2 O" M XpoL , 2 (K / p) zRT 2 3/2 2 ._ K/pzRT(L')] } t,= Q" - X - 3 (Q")2 M {[Po ] - [ P o Whereas t~ is readily calculated from L~, the mverse is not the case.

Esnmation of L, and/or t,. If L~ is known, then ti can be estimated from the above equation. If neither are known, then they can be estimated jointly from well-test data, based on drawdown or flow tests, as presented in Chapter 13.

Long-Term Depletion. For long-term depletion, where t _> ta , the profile is represented by 2

2

PL~ -P~ =

O ...."

m

--(L, K/ pzRT

-x)

or 2

2

(Po -P~ ) - [ P o

2

-(PL),

2

Q"

m

]-K/BzRT

(L, - x)

(28)

The material balance may be stated as previously developed for the linear flow of a compressible liquid, for the domain where t _>t~ Li

Q" (t - o ) - Xpo (L, - o ) - xj" p a x 0

(29) Lt

= Xpo (L, - o ) - x

o

[(p~ ),

(2"

M (L, -- x)]l/2 dX

K/pzRT

APPROXIMATE SOLUTIONS

415

By analogy with Equation 26, the above integrates to

Q" t - XpoL , - X 2 (K / p) zRT {[(PL ) 2 ]3/2 _ [(PL )2 3 Q" M

Q"

~m

(L, )]

K/l.tzRT

(30) In principle this may be solved for (PL)i o r (PL)i 2 in terms of t to obtain the difference 2

po - (p,~),

2

(3~)

as a function of t . Substitution in the profile represented by Equation 28 , at x = 0 , will variously give 2

2 Po - (P~:o)2

Po - ( P x = o ) 2

or

2

Po 2 - ( P x - o ) 2 or

Po

(32)

Po"

as functions of t for t _>ti. One or another of the above will represent the longterm depletion curve for linear gaseous flow.

Radial Flow In radial gaseous flow, Darcy's law becomes c~P

Q - K 2nr h p -~ It which transforms to Q -

K M --zch~

/u

ca~ 2

zRT Oln r

or

Q -

K zRT 3p z ~rch /~

M

31nr

The latter form, at a particular time t and assuming the steady-state, becomes

3/2

416

UNSTEADY-STATE FLUID FLOW

Q -

K zRT dp 2 --rch~ p

M

d lnr

Integration may be pursued to accommodate the conditions for drawdown and depletion

Drawdown. Integrating between r and rb,

Po

2

- Pr

2

lnrb ~ (K / p)nh zRT r Q

=

M

--

(33)

where r varies from r = r, to r = rb. The former is a constant denoting the wellbore or sandface radius. The latter, here denoted as rb , is a parameter, varying from r, to (rb)i, where (rb)a is the closed outer boundary -- or radius of drainage. Note that during drawdown, the successive profiles terminate (or originate) at p = p0 and r = rb. The material balance during drawdown is as stated previously for a compressible liquid in radial flow, whereby for a gas it follows that

Q(t - o) - Xhpo zc(rb 2 - ro 2 ) - X h ~ p2err dr and rb

Q t - XhPorC(rb 2 - ro z) - Xh f [Po 2 ra

Q M ln rb ]2rcrdr ( K / p ) rth zR T r

(34)

Inasmuch as analytic integration is not readily discemible, the operation will not be pursued. However, the solution can be represented symbolically as -

(t)

(35)

and substituted in the profile equation, such that at r = r., 2

2

Po - P,~ = or

Q

M

r (t)

~ l n ~

(K / ,)zch zRT

ro

APPROXIMATE SOLUTIONS 2

417

2

Po - Pr~ = Q 1 M In rb (t) po 2 (K / p)rch /902 z R T ra or

Po E - Pro z _

Po

2

1 z R T In r~ (t)

Q

( K / p ) ~ v h Po E M

r~

(36)

This represents the drawdown curve -- that is, the relationship between pressure and time at r = r, .

End of Drawdown. Note that, in Equation 34, when rb takes on the value (rb)i or rbi, then the transition time ta is recurred, signifying the end of drawdown and the commencement of long-term depletion. Estimation of (rb)~ and~or t~ . The numerical integration of Equation 34 will require that the limiting condition of either (rb)~ or t~ be known. For an actual reservoir, if (rb)~ can be estimated from geologic, geophysical, or core data, then ta will follow from integration of Equation 34. If neither are known, then the estimation methods of Chapter 13 can be utilized, based on well-test data, that is, upon drawdown or back-pressure tests. Long-Term Depletion. previously,

For long-term depletion, the profile is, as suggested

(P~b)2 _ p2 _

Q M In (rb)' (K / tu)rch z R T r

or

(po

2

- p

2

)-

[po

2

-

2

]-

Q M in (rb)~ (K / kt)rch z R T r (37)

=y where y = y(r) is a function of r and is defined by the substitution. Furthermore, as before for the radial flow of a compressible liquid, p or p~ is a fimction of r, and prbi or (P~b)i represents the behavior at (rb)i or rbi. The material balance statement is rbi

Q ( t - o ) - Xhpozc[(r b ) 2 _ r 2 ] _

Xh ~ p 2 r t r d r l't2

where

(38)

418

UNSTEADY-STATE FLUID FLOW P - [(Prb)2 _

Q

M In (,rb); ]~,,2

(K / p)erh z R T

(39)

r

and where t _>ti. The nature of the analytic integration is not readily apparent. If integration of Equation 39 were possible or feasible, however, it would in principle also be possible to obtain (p~)~ or p~b~ and establish the difference 2

P o - (Prb),

2

(40)

as a function of t . Substitution in the density profile relationship, Equation 37, would then give the long-term depletion curve for t _>t . The behavior may be expressed either as the difference or as the ratio relative to p0, or else in terms of pressure in the same manner as was done for long-term depletion in linear gaseous flow.

14.3 TRANSITION (OR STABILIZATION) BETWEEN DRAWDOWN AND LONG-TERM DEPLETION As has been indicated, drawdown implies the movement of the (steadystate) transient profile to intersect the outer, closed boundary. At this point a change in the density-time curve or pressure-time curve will be evidenced at the producing boundary or region of effiux -- e.g., at the wellbore, bottom-hole, or sandface. That is, there will be a transition or reflection between drawdown behavior and long-term depletion. Thus for example in the flow of compressible liquids, the pressure drawdown curve for linear flow has been shown to be reproducible as Po - P

0"

Po

2

]l/z

(5)

Po P o c X / ( K / At)

whereas for long-term depletion,

Po - P

O" t

1~

XL ~Po p o

~ =

"-

+

Q" c

L,

(1-

( K / At) Po P o

The behavior during drawdown is proportional to depletion it is linear with t .

t 1/2

1

~__)

(10)

2X -

whereas during long-term

APPROXIMATE SOLUTIONS

419

For the radial flow of compressible liquids, the drawdown curve is a more complicated function of time. The long-term solution is also more complicated, and will behave differently than for drawdown. Similar remarks also apply to both linear and radial gaseous flow. The transition or demarcation at the end of drawdown provides the point in time, denoted as ti, at which the outer reservoir dimension can be calculated -in tum denoted as Li for linear flow and (rb)~ for radial flow. This can be most readily accomplished utilizing the relationships between Li and t , and between (rb)i and t , as previously derived.

Open or Non-Ideal Systems While the foregoing statements and derivations pertain to reservoirs closed at the outer boundary, there will also be changes occurring at the end of drawdown when considering open systems -- notably revolving water- and/or gaseous-drives. This can result in a flattening-out of the pressure-decline curve, or conceivably even an increase, depending upon the mobility and driving pressure for the particular drive in the particular porous medium (and sometimes called repressurization.) It is, moreover, complicated by matters of two-phase flow and the effects of relative saturation, and whether the drive occurs by displacement and/or by simultaneous flow of the phases. Then there are the anomalies of divergences from ideality which occur in the configuration of the reservoir itself. Etc.

14.4 ESTIMATION OF RESERVOIR EXTENT AND RESERVES Irregardless of whether the outer reservoir boundary is open or closed, the end or termination of the drawdown curve affords a means to estimate L~ or (rb)~ -- and in tum the size of the reservoir and the reserves originally in place. This estimation will be independent of the further and future behavior of the reservoir -- that is, the long-term depletion -- and it will not matter whether the outer reservoir boundary is closed or open, and whether drawdown will be followed by an extemal, intruding drive.

Chapter 15 REPRESENTATION OF WATER-DRIVES

As set forth in Chapter 7, the drive causing reservoir liquids to flow naturally may originate not only from the liquid compressibility but may originate from a gas cap, from the exsolution of gases from the liquid phase, and/or from connections with the aquifer. The latter situation is uniformly referred to as a case of water-drive, and will be of primary interest here (and may include waterinjection from external sources). Complicating the picture for the orientation of reservoir flow is that gascap drive originates from the topmost parts or the reservoir, and water-drive most likely originates from the bottom of the reservoir. That is, water or brine is a more dense phase, and very likely is revolved in an upflow of sorts, though for the purposes of analysis, may be regarded as flowing into and across the outer boundary of the reservoir proper. This idealization is especially appropriate if the reservoir is reclined. For the purposes here, water-drive will be idealized as originating from the outer reaches of the (horizontal) oil-bearing strata, and will be characterized in the mare as two-phase flow by displacement.

15.1 INFINITE RESERVOIRS (WITH DRIVE) There is the question or problem of whether a finite reservoir, with say a water-drive from an aquifer, can be treated as an infinite reservoir, at least for the purposes of calculation and estimation. Thus if the transient effects of a pressure reduction during the initial production or drawdown is instantaneously projected to the reaches of the infinite or near-infinite, then the aquifer becomes part of the physical system. That is, for mathematical purposes the reservoir can be regarded as embracing the aquifer as well -- although with different mobilities, etc. Otherwise, the approximation methods previously developed perceive the successive drawdown profiles to intercept the original pressure or density level. Using this interpretation, the effect of the aquifer wouldn't be encountered until the pressure or density profile reaches L~ or (rb)~, as the case may be.

422

UNSTEADY-STATE FLUID FLOW

L~ or (rb),

..

xorln r i".

Hydrocarbon Phase

! i

•

(b)

Water Encroachment

~j x

Fig. 15.1 Effect of water-drive on the drawdown curve.

WATER-DRIVES

423

15.2 FINITE RESERVOIRS (WITH DRIVE) For a finite reservoir with displacement by say water encroachment at the outer boundary, the situation during drawdown can be diagrammed as indicated previously in Figure 14.1. Water encroachment has not yet commenced. During drawdown, the mathematical representation is as presented before. The transition at the end of drawdown can be used to estimate the reservoir extent and reserves for both linear and radial flow, and for either liquids or gases. The behavior is schematically denoted in Figure 15.1 (a). If water encroachment then occurs from the water-drive, it can be assumed as a simplifying case that the water density or pressure at the outer reservoir boundary will be sustained at the original aquifer pressure (which is also the original shut-in reservoir pressure). There will, however, be a density or pressure drop as the water flows into and through the reservoir proper. The situation is illustrated in Figure 15.1 (b). Moreover, it can be assumed that the flow of the hydrocarbon phase (liquid or gaseous) can be represented by a steady-state condition, as previously confirmed and utilized. That is, the mass flux Q" in linear flow, or mass flow rate Q in radial flow, is independent of position and time. For the water phase, however, the mass flow rate can be assumed independent of position, but not necessarily of time, as will be dictated by the material balance. (Energy balances are not of consideration.) The following relationships are to apply during the subsequent long-term depletion.

Compressible Liquids Both the hydrocarbon phase and water phase are representable as compressible liquids, as follows, corresponding to Figure 15.1 ..

Linear Flow. The following integrated relationships will apply: Hydrocarbon Phase

Q

KlPj-P

t!

pc xs-x Aqueous Phase m

Q

tt

~

m

K lpi-P -~ ~ L~ - ~

424

UNSTEADY-STATE FLUID FLOW

where bars or overlmes are used to denote the driving phase, i.e.., water or brine. It will be noted that

O