Encyclopedia of Chemical Processing and Design: Volume 22 - Fire Extinguishing Chemicals to Fluid Flow: Slurry Systems and Pipelines (Chemical Processing and Design Encyclopedia) [1 ed.] 0824724720, 9780824724726

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Encyclopedia of Chemical Processing and Design

22

EXECUTIVE EDITOR

JOHN J. McKETTA The University of Texas Austin. Texas

ASSOCIATE EDITOR

WILLIAM A. CUNNINGHAM The University of Texas Austin, Texas EDITORIAL ADVISORY BOARD

LYLE F. ALBRIGHT Purdue University Lafayette, Indiana

JAMES R. FAIR Professor of Chemical Engineering The University of Texas Austin, Texas

JOHN HAPPEL Columbia University New York, New York

ERNEST E. LUDWIG Ludwig Consulting Engineers, Inc. Baton Rouge, Louisiana

Encyclopedia of Chemical Processing and Design EXECUTIVE EDITOR ASSOCIATE EDITOR

22

John J. McKetta William A. Cunningham

Fire Extinguishing Chemicals to Fluid Flow, Slurry Systems and Pipelines

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

First Published 1985 by MARCEL DEKKER, INC. Published 2021 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 1985 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government works ISBN 13: 978-0-8247-2472-6 (hbk) ISBN 13: 978-1-00-320981-2 (ebk) DOI: 10.1201/9781003209812 This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please accesswww.copyright.com(http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. ( CCC), 222 Rosewood Drive, Danvers, MA 0 1923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Library of Congress Cataloging in Publication Data Main entry under title:

(Revised)

Encyclopedia of chemical processing and design. Includes bibliographical references. 1. Chemical engineering -Dictionaries. 2. Technical-Dictionaries. I. McKetta, John J. II. Cunningham, William Aaron. TP9.E66 660.2'8'003 ISBN 0-8247-2451-8 (v. 1)

Chemistry, 75 40646

I nternational Advisory Board

RAY C. ADAM

LUCIANO BENINCAMPI

FRANK CHRENCIK

Chairman of the Board N L Industries, Inc. New York, New York

Manager of Public Relations CTIP-Compagnia Tecnica Industrie Rome, Italy

Vulcan Materials Co. Birmingham, Alabama

M. A. ALLAWALA

LLOYD BERG

Managing Director National Refinery Ltd. Karachi, Pakistan

Profe~sor

Department of Chemical Engineering Montana State University

Bozeman. Montana

HAMED H. AMER

R. G. ANTHONY Professor, Department of Chemical Engineering Texas A & M University College Station. Texas

H. J. AROYAN

Research Associate Mobil Research and Corp. Dallas, Texas

Professor of Engineering Engineering Center Arizona State University Tempe. Arizona

D. J. BlICKWEDE Vice President and Director Research Bethlehem Steel Corp

or

M. J. P. BOGART Fluor Engineers and Constructors, Inc. Santa Ana. California

F. SID ASKARI

Con­

AZARPOOR AZARNOOSH Bechtel National. Inc. San Francisco, California

DONALD l. BAEDER Executive Vice President- -Science and Technology Occidental Petroleum Corporation Los Angeles, California

Wm. A. BAILEY, Jr. Former Director, MTM Process Re­ search and Development Lab Shell Development Company Houston, Texas

TRAVIS W. BAlN Vice President National Sales, Inc. Jackson, Mississippi

GAREN BALEKJlAN C. F. Braun Arcadia, California

CESAR BAPTISTE

Development

JAMES R. COUPER

Bethlehem, Penn,'>yIY

2.2

£

2.0

00

rS

+

8

Q)

-'"

0 ()

6

100

FIG. 6.

Yield of coke and light ends.

0.85,...--------.--------,-------,--------.------,------, Parameter: correlating factor, " o ~

0.80

~

.c

g 0.75 00

rS + Q)

~ 0.70 ~

8

0.65

F-----------.2!902-~~_ _ _ _~

0.604·'="0------:'=------=---------!=------~1------!:.:-------::'1 00

FIG. 7.

TABLE 2

Relationship of coke and light ends.

Composition of Light Ends

Component Hydrogen Methane Ethylene Ethane Total

Based on Average, wt. %

Based on Fresh Feed (wt.%)

l.7 41.3 23.0 34.0

0.05 1.21 0.67 0.99 2.92

100.0

237

Fluid Catalytic Cracking, Predicting Yield 0.6 r - - - , - - - - - , - - - - - , - - - - , - - - - , - - - - - - - ,

0

.~

0.5

if!~

ui '0 QJ

~

0.4

(/)

I u:::I

e

'0

c..

0.3

0.2 .L:---~--::'::_--_==_--_+__--'--_:!:,___-~

m

~

100

Correlating factor. "

FIG. 8.

Yield of hydrogen sulfide.

Step 17. The gravity of the C 5-400°F gasoline is 58.5 API, based on Fig. 9. This is equivalent to a specific gravity ofO. 7447. Then the gasoline wt. % is the vol. % multiplied by the ratio of the specific gravities for gasoline and fresh feed, or (62.9)(0.7447/0.8996) = 52.07 wt.%. Step 18. The yield of total cycle oil is 100 minus the gas oil conversion, or 100 - 82 = 18 vol.%. Then the volume yield is related to a weight yield by adding 1.15, based on Fig. 10. Thus the weight yield is 18 + 1.15 = 19.15 wt.%.

62.---r----,----,----,---.---.

60

c::

; 30, and for large diameters Eq. (17) holds. These equations are also valid

in boundary layers. The constants K and B are found from experiment to be K = 0.4 and B = 5.5. To get the average velocity we can assume that Eq. (17) applies across the entire pipe and integrate to get Vave/U* =

1.75 + 2.5 In (Rpu*//l)

(18)

or (19) An empirical fit to this equation for N Re between 4000 and 10 5 is the Blasius equation (20) which corresponds to a velocity profile v = vmax (1 - r/R)I/7

(21)

These equations apply for smooth walls, but roughness will change the flow near the wall if the height of the wall irregularities extend beyond y+ = 5. Then there is an additional dimensionless group, the ratio of roughness height to diameter, E/D, to consider. Commercial pipe friction factors can be calculated from the empirical Colebrook formula which includes the roughness:

j -1 / 2

I [- E 1.26 _ - -4.0 oglO l3.7D + Pl2NRe

j

(22)

The values of E, the surface roughness, are given in handbooks or by manufacturers of pipes. The design value of E for commercial steel pipe is 0.0002 ft (0.06 mm), and for copper tube E is 5 X 10-6 ft (1.5 X 10-3 mm). A formula to calculate the friction factor explicitly has been proposed by Churchill:

! = 2[(8/NRe)12 + l/(A + B)15]1/12 A = {-2.457 In [7/N Re )09 + 0.27E/D]}16 B

=

(37530/N Re )16

(23)

This formula includes the laminar and turbulent regimes. It is also possible to solve Eq. (22) for! using two iterations starting with an initial guess. If the initial guess is! = .0075,

246

Fluid Flow

1-1/2 --

-4.

lE

(E

0I 5.02 14.5)J og 3.7D - N Re log 3.7D + N Re

(24)

Equation (24) is adequate for rapid computer calculations and can be used to replace Reynolds number vs friction factor charts. Some information regarding the optimum size of pipe can be obtained from the friction factor analysis. A more detailed analysis can be found in Perry's Handbook. We assume the cost of pipe per year including capitol, depreciation, interest, and maintenance can be expressed as KaDa L and the power cost can be expressed as KbQIL1pl, where Q is the flow rate ~D2vave. The total cost C T is then CT

=

Ka DaL + KbQlL1pl

=

KaDaL + 2fpv~veQL/D

=

Ka DaL + 32fpQ 3L/D s1C2

(25)

The optimum pipe diameter holding Q, f, and L constant is found when

dCT/dD =0: D oP!

=

kQ3/(s+a)

(26)

where, for liquids, we can absorb p into the constant k. For a completely rough pipe, f is constant, but even for a smooth pipe the variation off with diameter is small compared with D S• Typically, a is between 1 and 2 so the optimum pipe diameter is roughly proportional to Q1/2. Then substituting for Q in terms of D 2 vave , we find the velocity at the optimum diameter is a constant. This appears to be true in practice where average velocities of 38 ft/s (1-2.5 m/s) are typical for pipe flows ofliquids. For noncircular cross sections, the previous results can be used for turbulent flow if an equivalent hydraulic diameter is defined, DH =

4(volume) wetted surface 4( cross-sectional area) wetted perimeter

(27)

The straight horizontal pipe itself may be only a small part of a complete piping problem. In addition, gases cannot always be considered incompressi­ ble. It is necessary to develop macroscopic momentum and energy balances to treat these problems.

III. Momentum and Energy Balances The differential balances are developed in most elementary textbooks on fluid mechanics or transport phenomena. In a rectangular coordinate system,

247

Fluid Flow

the balance oflinear momentum in the x direction for a Newtonian fluid is

av x av x av x apx -a t + PV.r-ax + pV.v-ay + pV z-az -1 2

=

The terms in Eq. (28) are: 1. The rate of accumulation oflinear momentum per unit volume in the x direction 2. The linear momentum in the x direction convected with the flow 3. The input oflinear momentum due to molecular motions 4. The gravitational force (gx is the x component) 5. The component of all other body forces in the x direction, Fx 6. The pressure force

A differential energy equation may be obtained in a similar manner for (29)

where E is the sum of internal energy per unit mass U, and the kinetic energy per unit mass, 1V 2• Then

aE aE aE aE P -a t +pv·< -a x + pVy -a y + pv -- -a z = (30)

+ work done by viscous forces The terms here are: 1. 2. 3. 4. 5.

The rate of accumulation of energy per unit volume The energy convected with the flow The energy input due to molecular motion The gravitational work The pressure work

An equation for the conservation of kinetic energy can be found from the scalar product of the velocity vector and the vector form ofEq. (28):

248

Fluid Flow a(~V2)

a(~V2)

P ----at + PV x ax + PV y

a(~V2)

---a:Y + PV

a(~V2)

z

iiZ =

ap ap ap ( ). k' d'" -vx -a x - Vv. -a y - Vz -a z + P v· g + VISCOUS wor + VISCOUS IssipatlOn

(31)

Equation (31) is often combined with Eq. (30) and simplified for many special cases of engineering interest to form the differential energy equation which is used for nonisothermal problems. In pipe flow problems it is common to take a macroscopic balance over a much larger volume than a small differential element. This balance can be obtained from the previous equations by integrating over the volume. Many assumptions can be used to simplify the result. These include: a. Steady-state b. The fluid properties do not vary across the cross sections of the inlet or outlet pipes c. The molecular transfer of momentum and energy can be neglected at the inlets and outlets d. The mean velocity is parallel to the pipe walls Then the macroscopic continuity equation is

L

inputs

p;(v;) S; =

L

outputs

p;(v;) S;

(32)

where S; is the cross-sectional area of pipe i and the average (v) is

(v)

=

!If S

s

vds

=

r

3..

R

R jo

(33)

vrdr

for a circular pipe. The macroscopic momentum equation is

L {P;(v/) Sin; + P;S;n;} + 'i.E = L {P;(v/)S;n; + PS;n;} + EdT• g - mtg (34)

inputs

outputs

where n; is the unit outward normal vector from the input or output surface. The sum of the external forces on the fluid F, the net drag force E dT• g, and the weight of the fluid (the total mass m t multiplied by the gravitational accelera­ tion g) also appear in this equation. The macroscopic energy balance is

p}

..

~ 1 (v 3) L { U + 2: -() + ¢ + - p(v)S + Q - W = mputs

V

P

ou~uts

{

0 + 4~:! + ¢ + ~

}

p(v)S (35)

249

Fluid Flow

where ¢ is the potential energy per unit mass, Qis the rate of heat transfer through the walls, and W is the rate of work done by the system by pressure and viscous forces. An additional equation can be obtained by integration of the mechanical energy balance, a special form of the energy equation. The result for a system with one inlet and one outlet and without chemical reaction is called the engineering Bernoulli equation: (36)

where g is the gravitational acceleration, z is the height above a reference, W is the work per unit mass, ~ represents out minus in, and Ev is the energy loss by viscous dissipation. This energy loss appears as an increase in internal energy in Eq. (35). The integral must be evaluated over some representative streamline in the system considering all of the thermodynamic states between the input and output. For flow in piping systems there are three limiting cases of interest: incompressible flow, isothermal flow, and adiabatic flow.

IV. Incompressible Flow The engineering Bernoulli equation becomes (37)

In the straight pipe considered previously (38)

Another example that can be solved exactly is a sudden expansion from a smaller pipe with cross-sectional area A I to a larger pipe with area A 2• The continuity, momentum, and Bernoulli equations are as follows: (39)

SI(VI) = S2(V2) PISI + P(VI2)SI + F

~ (V2 3 ) 2 (V2)

_

=

P(V22)S2 + P 2S 2 + Fd

~ (VI 3) + P 2 - PI + E 2 (VI)

P

_0

v-

(40)

(41)

Here F = -PI(SI - S2) when we neglect the frictional drag and consider only the force of the fluid pressure on the expansion surface. Then let

250

Fluid Flow

so from Eq. (40)

and from Eq. (41)

When (Xi = PI = 1, we find (42)

The result is in the form E, = K~V2. Extension to all forms of fittings and valves leads to the total friction loss: (43)

V. Compressible Pipe Flow When the downstream pressure Pl is less than 90% of the upstream pressure for the flow of gases in pipelines, compressible flow calculations are recom­ mended. The continuity and Bernoulli equations are needed in differential form to develop the design equations. Therefore we place Planes 1 and 2 very close together to obtain for one input and one output with (X = P= 1:

dp

p

d(v) _ 0

+ (v) -

dp (v)d(v) + - + dE\' = 0 A

P

(44)

(45)

Since E,. = J(v)2z /2D in terms of the differential length dz,

E =[(v?dz "

2D

(46)

A. Isothermal Ideal Gas

Equation (45) can be integrated when G = p(v) is constant andp = Mp/RTto gIVe

251

Fluid Flow

(47) where R is the gas constant, M is the molecular weight, and T is the absolute temperature. This equation has a maximum mass flow rate which can be found by equating the derivative of G 2 with respect to (P 21P I )2 to zero. This maximum corresponds to the exit velocity V 2max = (P 2Ip2)1/2, the speed of sound in an isothermal ideal gas. The pressure at the exit cannot fall below the value corresponding to V 2 max, and the velocity cannot exceed this value in a straight pipe of constant diameter. For pipe flow in long small pipes or where the velocities are less than 30% of the speed of sound at the exit, the isothermal equation is valid. The adiabatic flow results are closely approximated by the isothermal results under these conditions also. Usually the second term containing the In P21 PI is small compared to the other terms, and Eq. (47) without this term can be solved directly for G. Equation (47) applies to the flow of a compressible gas from Point 1 to Point 2 in a horizontal pipe. If there is also a flow through a nozzle from a reservoir at pressure Po preceeding PI, the velocity at 1 is related to the pressure ratio by

where

is the square of the isothermal Mach number, and 1.

VI

is the average velocity at

B. Adiabatic Flow

There is a maximum velocity for adiabatic flow, the sonic velocity a, where (48) for an ideal gas with isentropic path pp-Y = constant. Here y is the ratio of specific heats epiC. We define the Mach number N Ma as the ratio of the velocity to the speed of sound from Eq. (48). Then integration of Eq. (45) for an ideal gas with adiabatic flow from Point 1 to Point 2 in a horizontal pipe gIves (49)

where

252

Fluid Flow

A

y-1 NMa.l 2 ) N 2Ma ,2 ( 1 + 2 = ---r------:---~

y-1 N 2Ma,2 ) N 2Ma,1 ( 1 + -2-

andfisan average friction factor (f; + f2)/2. The average friction factor can be used because the variation of viscosity with temperature for a gas is small. The use of Eq. (49) to solve for the flow in a pipe is trial and error. An additional factor is the limiting condition of sonic velocity at the exit. This limits the mass flow rate for a given length of pipe or the length for a given mass flow rate. The actual pressure at the exit may be greater than the pressure of the downstream reservoir. This comes from the limiting pressure corresponding to sonic velocity. When the upstream reservoir is at pressure Po, adiabatic flow into the pipe at PI for a rounded entrance gives (SO) (Sl)

Similar relations between 1 and 2 in the pipe are 1 + [(y - 1)/2]N~a,2 1 + [(y - 1)/2]N~a.l

(S2)

and TI Tz

-=

1 + [(y - 1)/2]N~a,2 2 1 + [(y - 1)/2]NMa.l

(S3)

The relationship between the maximum length and the flow rate can be found from Eq. (49) when N Ma ,2 = 1. Lapple's method of solution using charts for three different valves ofy has been reproduced in Perry's Handbook. Present­ day computers and hand calculators make an iterative solution of the equations about as easy when PI does not differ significantly from Po. Both Lapple's graphs and the equations presented are limited to one pipe diameter, ideal gases, and constant specific heat ratio. In industrial designs these assumptions may be relaxed when the follow­ ing approximations are used. 1. More than one pipe diameter-The most accurate calculations are made by computing each portion separately. An approximation can be made by setting (fbLb/Db) = (faLa/Da)(Db/Dat 2. Inclusion offittings-The equivalent L/ D of the fittings is used unless the equivalent L/ D for any fitting is greater than 100. 3. Nonideal gases-When the gas does not condense to form two phases, the temperature T is replaced by zT; that is, the gas equation becomes Mp = zpRT. If z and y change with pressure, calculations over incremen-

253

Fluid Flow

tal lengths of pipe can be used to increase accuracy. Similarly, when two phases are present, it would be appropriate to divide the total length into several segments. 4. Exit pressure-There is usually a pressure recovery at the pipe exit. About 10% of the kinetic energy is converted to pressure energy at an exit to atmospheric pressure. Except for a limited set of conditions, the exit pressure can be assumed to be atmospheric. These conditions are P 2/P l < 0.75 and 4jL/D < 2. Then use (54) which is solved by trial and error for P 2• Here W is the mass flow rate in lb/h when T is in oR, d is the pipe diameter in inches, and all pressures are in Ib/in. 2abs. When W is in units of kg/h, T in oK, and d in meters, the constant factor in Eq. (54) becomes 5.2 x 10-5 for pressures in Pascals.

VI. Special Topics A. Drag Reduction

High molecular weight polymers when added to liquids in parts per million by weight proportions can often cause large reductions in pressure drop at constant flow rate. Although these additives are expensive, there can be significant savings in capital costs if the equipment can be scaled down to take advantage of drag reduction. Some examples of the use of drag-reducing additives are in the 48-in. Trans-Alaska Pipeline as a replacement for uncon­ structed pump stations, and in an old storm sewer system as a substitute for an enlarged system. Unfortunately, there is not a sufficient understanding at present to predict the effect of adding these long-chain molecules to a liquid from laboratory data in small pipes. For the Alaska Pipeline a scale-up was made from a 14-in. experimental study to the full-sized 48-in. pipeline by Burger, Chou, and Perkins of Arco Oil and Gas Co. Their method, based on the technique originally developed by Savins and Seyer, will be described here. Additional references on drag reduction theory are listed in the Bibliography. Let (55) where I1P p is the pressure drop for the solution containing the polymer additive and l1Po is the pressure drop for the pure solvent and both pressure drops are for the same flow rate. The fractional drag reduction is usually defined as 1 - (J. Additional parameters in the model are

a= (v~

=

(~r2

Op = aqi(Yo(JY

254

Fluid Flow

and

where u* = (-I1PoDI4pL)I/2, ¢ is the polymer concentration, and a, b, and c are constants. All of the quantities are in terms of the solvent system. Then

a[l + a(23/2 log a + 1.454YoaOp - 0.8809)F = 1

(56)

From experiments on small pipes for different concentrations ¢, in parts per million by weight, the constants a, b, and c can be determined. These constants are unique for a polymer-solvent system and must be determined from experiments. Once these constants are found, Eq. (56) can be used to find the expected drag reduction in the larger pipes. For example, data from 1, 2, 14, and 48 in. pipes for a Conoco drag reducing additive in Sadlerochit crude gave a = 0.0516, b = 0.489, and c = -0.579. To find the expected pressure drop in a 48-in. pipe with 20 ppm of this polymer in Sadlerochit crude at a Reynolds number of 3 x lOs and a temperature giving a kinematic viscosity of9 cSt, we proceed as follows. The friction factor is 0.0037 at the Reynolds number of 3 x lOs and the velocity is 7.42 ft/s. Then

a

=

([/2)1/2 = 0.043

Yo = f(v?12v = 1.05 x 10 3 S-l 0= (0.0516)(20)0.489(Yoato. s79 = 3.98 x 1O-3a-o.s79 and Eq. (56) becomes

a[0.9621 + 0.1216 log a + 0.2613aD.421F = 1 Solving by trial and error, a is 0.725, or with the 20 ppm polymer additive the pressure drop would be 27.5% less than with the crude oil alone. This is equivalent to the elimination of one of every four pumps. Some polymer additives degrade readily in pipelines and must be continu­ ously added along the pipeline. Scale-up from laboratory pipe data with inside diameters ofless than 1 in. is not recommended.

B. Flow of Heavy Crudes

Another example of an industrial fluid flow problem where extensive labora­ tory investigation is necessary to design the facility is the flow of waxy crudes. For these crudes the pour point is between 60 and 115°F and the fluids must often be pumped at temperatures below the pour point. At temperatures below approximately 20°F above the pour point, the crudes have non­ Newtonian viscosities. The wax can crystallize as the crude is cooled to form a

255

Fluid Flow

gel or a partial gel. Under static conditions a rigid gel is formed, but if the crude is cooled while in motion, the apparent viscosity will increase but the material remains fluid. Therefore, the rheological properties are functions of temperature, shear rate, shear stress, and past history. Problems in pumping these crudes will occur if the temperature drops and the fluid becomes non­ Newtonian and if gel formation occurs after a shutdown. The pipeline facility must be designed to recover from these problems or prevent them. First of all, laboratory work is required to find the properties of the crude. Measurements must include pour point, density, specific heat, wax content, distillation, nonorganic content, and experimental flow data. The flow data are necessary to find the apparent viscosity vs rate of shear. Several other flow tests are desirable including yield stress under static and dynamic conditions of cooldown. Operation of a heavy crude pipeline involves start-up, shutdown, and continuous operation. The start-up process will require pump pressures to overcome the yield stress of the gel formed under static conditions. This start­ up pressure may be higher than the maximum pressure at the maximum flow rate for continuous operation. From the experimental yield stress Y, the restart pressure is p= YLCjA

where L is the pipe length, C is the circumference, and A is the cross-sectional area. When the fluid cools and the flow rate is reduced, the pressure is at first reduced. Lowering the flow rate will lower the required pump pressure until a minimum pressure occurs. Further lowering of the flow rate will generate a pressure increase as the fluid becomes non-Newtonian. Therefore, high pressure at low flows may be found in the cool-down process. For normal operation the waxy crudes are Newtonian, but temperature gradients may lead to a buildup of wax on the walls. Some design procedures to avoid problems in the pumping of heavy crudes have been suggested by Smith. All possibilities of avoiding gel forma­ tion as a result of an unforeseen shutdown cannot be built into a facility design, so standby pumps, heaters, and injection equipment should be available. Example 1: Laminar Flow. A I-mm diameter capillary tube is to be used to monitor small flows to a laboratory reactor. If pressure taps are 2 m apart and the maximum pressure drop is to be 7 x 104 Pa, find the maximum flow rate when p = 1000 kg/m 3 and,u = I cP q

(-llP)nD 4

= vav.,A = 128,uL = =

(7

x 104 )n( 10- 3)4

(128)(10

3?

51.54 cm 3/min

= 8.59 x 10

-7

3

m /s

256

Fluid Flow Check to see if the flow is laminar:

Example 2: Turbulent Flow. A fluid is to pump 200 km through a 0.7366 m i.d. cast iron pipeline at an average velocity of 1 m/s. Find the pressure drop when f.1 = 2 cp, p = 103 kg/m 3, and E = 0.46 mm. Churchill's equation is appropriate here, but only the term involving A is different from zero.

N Re = (0.7366)(l)(W)/(2 x 10-3)

f

=

=

3.683 x 105

2[1/A 1.5r/12

A = [-2.457 In ([(7/3.683 x 105) + (0.27)(0.46)(736.6)]]16 =

(20.6376)16

f

=

2/(20.6376f = 0.00469

f

=

(-IlP)2D 2Lpvave

-IlP = 2.55 x 106 Pa (368 Ib/in. 2)

Example 3. What flow rate can be obtained in a 4-in. schedule 40 pipe (0.10226 m i.d.) for a pressure drop of 2 x 105 Pa. Let p = 10 3, f.1 = 2 cp, and E = 0.06 mm, and the length is 1000 m. When pressure drop, diameter, and length are known, the right-hand side ofEq. (22) is all known.

D P = 3 6558 X 103 11/2N Re = [-IlPD]1/2 4pL f.1 •

1-1/2 -

I [0.06 X 10-3 1.26] -4.0 og (3.7)(0.10226) + 3.6558 x W

-IlPD

1 = 0.005745 = ~L pVave Solving for the one unknown, Vave =

1.334 m/s

Example 4. A hydrocarbon product must be pumped from a chemical plant to the loading dock. An existing 3 in. schedule 40 line (77 .93 mm i.d.) and a pump capable of producing a 100 Ib/in. 2 (6.925 x 105 Pa) pressure difference between inlet and outlet are available for this project. If the fluid density is 860 kg/m 3 and the viscosity is 3 cp, find the maximum flow rate that can be supplied. The line is 1000 m long and has eight 90° elbows and 4 gate valves. Here

257

Fluid Flow

does not contain the unknown average velocity, and the friction factor can be found directly from Eq. (22). The result isf = 0.0062. Then Eqs. (38) and (43) can be used to find . A 90 elbow has an L/D equivalent of 30 and an open gate valve of 15. The total L/D is 0

(L/D)totJil

=

8(30) + 4(15) + (0.1000)/0.07793

=

13,130

then

v

2

(-I1P)

= :;:--i=--=':-

2pj(L/D)totJil

and

v = 2.23 m/s A check is necessary to confirm turbulent flow N Re = (2.23)(0.07793)(860)/0.003 ~

50,000

Example 5: Compressible Isothermal Flow. A 2-in. schedule 40 (52.5 mm i.d.) air line runs 2000 m from a tank at a pressure of 106 Pa to another tank at a pressure of 5 x 105 Pa. The temperature is 300 K and the process is assumed to be isothermal. Find the flow rate. We assume 1/ $= -410g(e/3.7D)togetf= 5.07 x 10-3• Then we neglecttheln term in Eq. (47).

2G 2(5.07 x 10- 3)(2000) _ 29[10 12 - 0.25 x 10 12 ] (0.0525) - 2(8.314 x W)(300) G = 106.24 kg/m 2 • s The densityisPI = PM/RT = 11.63 kg/m 3 and VI = 9.14 m/s. Thus V2 = 18.27 m/s.A check of the assumptions shows that less than a 1% change in V2 is obtained if a correction to fis made and the In term is included. We have also assumed that Po = PI. This can be checked:

Vrnax =

MI

=

PI = Poe

1& VAT = 293.3 m/s

9.14/293.3 _M2/2 I

=

0.0312

= 0.9995Po = Po

Example 6. Air (y = 1.4) is exhausted from a vessel at a pressure of7 x 105 Pa to the atmosphere through 120 m of2 in. schedule L/D steel pipe (52.5 mm i.d.). The pipe contains fittings with an equivalent L/D of 300. The temperature in the vessel is

258

Fluid Flow 310 K. Find the flow rate. (None of the individual fittings has a L/ D greater than 100.) First we must estimate the friction factor by assuming only the roughness is important. f= [-1/(4 log E/3.7D)F =

[-1/4 log (0.06)/(3.7)(52.5)]"

=

0.005

Then the total 4jL/ D is 4jL/D pipe 4jL/ D fittings For entrance add 0.5 4}L/D

We assume Po

=

=

45.7 6 0.5 52.2

PI and solve for N'Ma.! from Eq. (49) when N Ma .2 = 1. Let N Ma .!

52.2

=

/ I

0.714~X2 - [ - 1.2 In

x

0/=

=

X.

I +0.2x 2 ) 1.2x 2

0.[[2

From Eq. (52) we now find N Ma .2 for N Ma .! = 0.[12. Lety = N Ma.2• 7 x 105 Y l'I+0.2y211/2 1.0[8 x 10 5 = 0.112 1.0025 J y

=

0.733

We go back to Eq. (49) to get an improved value of x until the two unknowns x and y converge. Here x = O. [[2 does not change upon another iteration. To find the mass flow rate

and at Point I: 10 5 )"(29)]1 /2 2 [ (1.4)(7 X 103)(310) G=0.[[2(8.3[4X =311.2kg/m·s

W = 3[1.2(3600{~)c0.0525f = 2425 kg/h Several checks need to be made here. First, from Eq. (50) Po PI

=

[

y - I , ]YI(;' - I) I + -2- N Ma.! = 1.0088

and the assumption that Po = PI is verified. The final temperature T2 = 0.905 TI from Eq. (53). The friction factor at Position I is! = 0.0054; and at Position 2,/ = 0.005. The average f= 0.0052 is slightly greater than the initial assumption. Only a small

259

Fluid Flow

change in G is expected from this change. Finally, P 2 within the pipe is actually slightly less than atmospheric pressure. The estimate from Eq. (54) gives _ (5.2 x 10- 5)(2425)"(0.905)(310) (29)(P 2)0.0525)4

P _P 2-

=

atm

9.78

X

10 4 Pa (96% of atmospheric)

Again the change is small. For this problem Lapple's charts are easy to use. P 2/P o = 0.145 and 4J/LD = 52.2. The ratio G/G o, = 0.22 and the temperature ratio T2/To = 0.9 are read directly from the chart. Go,

=

then G = 313 kg/s·m 2 which is the same answer using the equations.

Symbols Ai a CT C

D DH

E

E,

J

gi Fi G k

Lo L M M, N Re N Ma

n,

Pi Po PL Q

Q r

r

P o[M/2.718RTo

area at location i sonic velocity total cost circumference pipe diameter hydraulic diameter energy lost energy by viscous dissipation Fanning friction factor gravitational acceleration component i component of force mass velocity, p thermal conductivity (Eq. 30) entrance length characteristic length or pipe length molecular weight total mass Reynolds number Mach number unit outward normal vector pressure at location i pressure at upstream reservoir pressure at downstream location flow rate heat transfer rate radial coordinate

/2

260

Fluid Flow R S,

T

u* u'v' U U+

v Vave V max

v,

0

W

W W y

y y+

z

y E

v f1 P r

TJi'

cp

pipe radius or gas constant cross-sectional area at location i temperature shear velocity turbulent stress term velocity dimensionless velocity local velocity average velocity maximum velocity i component of velocity internal energy per unit mass mass flow rate work per unit mass rate of work yield stress distance from the wall dimensionless distance from wall vertical coordinate or compressibility factor ratio of specific heats roughness height kinematic viscosity, f1/P viscosity density shear stress wall shear stress potential energy per unit mass

Bibliography The derivation of the laminar flow equations can be found in most unit operations or fluid mechanics texts. For example: Bird, R. B., Stewart, W E., and Lightfoot, E. N., Transport Phenomena, Wiley, New York,1960. Denn, M. M., Process Fluid Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 1980. The derivation of the turbulent flow profile is found in Millikan, C. B., "A Critical Discussion ofTurbulent Rows in Channels and Circular Tubes," in Proceedings ofthe Fifth International Congress on Applied Mechanics, Wiley, New York, 1939, p. 386. and also Tennekes, H., and Lumley, J. L., A First Course in Turbulence, MIT Press, Cambridge, Massachusetts, 1972.

Fluid Flow, Crude Oils

261

The working formulas are based on the following additional sources: Churchill, S. w., "Friction Factor Equation Spans All Fluid-Flow Regimes," Chem. Eng., 84, 91 (November 7,1977). Colebrook, C. F., and White, D., "Turbulent Flow in Pipes with Particular Refer­ ence to the Transition Region between Smooth and Rough Pipe Laws," J. Inst. Civil Eng., II, 133 (1938-1939). Shacham, M., "Comment on 'An Explicit Equation for Friction Factor in Pipe,'" Ind. Eng. Chem., Fundam., 19, 228 (1980). The optimum pipe size analysis is presented in Perry, R. H., and Chilton, C. H. (eds.), Chemical Engineers Handbook, 5th ed., McGraw-Hill, New York, 1973. Peters, M. S., and Timmerhaus, K. D., Plant Design and Economics for Chemical Engineers, 3rd ed., McGraw-Hill, New York, 1980. The derivations in Sections III, IV, and V are based on the equations in Bird, Stewart, and Lightfoot and McCabe, W. L., and Smith, J. c., Unit Operations ofChemical Engineering, 3rd ed., McGraw-Hill, New York, 1976. Additional sources on drag reduction are Berman, N. S., "Drag Reduction by Polymers," Annu. Rev. Fluid Mech., 10, 47 (1978). Burger, E. D., Chorn, L. G., and Perkins, T. K., "Studies of Drag Reduction Conducted over a Broad Range of Pipeline Conditions When Flowing Prudhow Bay Crude Oil," J. Rheol., 24, 603 (1980). Virk, P. S., "Drag Reduction Fundamentals," AIChE J., 21,625 (1975). Discussions on pumping heavy crudes are given in Smith, B., "Pumping Heavy Crudes," Oil Gas J., p. 111 (May 28, 1979); p. 148 (June 4, 1979); p. 110 (June 18, 1979); p. 105 (July 2, 1979); p. 69 (July 16, 1979). Withers, V. R., and Mowill, R. T. L., "How to Predict Flow ofViscous Crudes," Pipe Line Ind., p. 45 (July 1982). NEIL S. BERMAN

Fluid Flow, Crude Oils

(see also Fluid Flow)

Many pipelines around the world are now successfully transporting waxy crudes under conditions where the ambient temperature is lower than the pour point of the liquid. DOI: 10.1201/9781003209812-16

262

Fluid Flow, Crude Oils

In each case the most economical method had to be determined for transporting the particular crude. An economic evaluation must consider the energy required to pump the waxy crude. Prime considerations are the pipeline diameter and the tempera­ ture of the crude. These factors determine the pump horsepower and govern the selection of the pump. Flow characteristics must be determined in the early stages of pump selection and design. Determining the flow characteristics requires a look at the complete system and at alternate methods of transporting high-pour­ point crude. There are many design alternates which are "industry-accepted" methods in pipelining high-pour point oil. Some of these are: 1. 2. 3. 4. 5. 6. 7. 8. 9.

Preheating the crude to a higher inlet temperature to allow it to reach the destination of intermediate station before cooling to below its pour point. The pipeline mayor may not be insulated. Pumping the crude at a temperature below the pour point. Adding a hydrocarbon diluent such as a less-waxy crude or light distillate. Injecting water to form a layer between the pipe wall and the crude. Mixing water with the crude to form an emulsion. Processing the crude before pipelining to change the wax-crystal struc­ ture and reduce the pour point and viscosity. Heating both the crude and the pipeline by some method such as steam tracing or electrical heating. Injection of paraffin inhibitors. Combinations of these methods. Before deciding which design alternate to use, several design parameters must be investigated: Physical properties of crude heat transfer, restart after shutdown, facilities design.

High-Pour-Point Crude Properties

High-pour-point crudes req uire higher than normal temperatures before their pour points are reached (normally between 60 and IIYF). Additionally, high­ pour-point crudes exhibit non-Newtonian viscosity behavior at temperatures below about 20°F above the pour point. This means the effective viscosity is not a function of temperature alone, but is also a function of the effective rate of shear in the pipeline. Shear stress and rate of shear must be determined to predict the pressure required to deliver specified production volumes. With each waxy crude discovery, extensive laboratory tests should be made to determine its exact behavior under temperature variation. These tests determine the crude's rheology. In determining the rheology of liquids, anyone of five basic behavior patterns (fluid types) may be found upon agitation of the liquid at constant temperature (Table 1).

263

Fluid Flow, Crude Oils TABLE 1

Basic Fluids

Fluid Type Newtonian Dilatant Plastic (Bingham) Pseudoplastic Thixotropic

Fluid Characteristics Unaffected by magnitude and kind of motion to which they are subjected Viscosity will increase as agitation is increased Have a definite yield value which must be exceeded before flow starts. After flow starts, viscosity decreases with increase in agitation Do not have a yield value, but do have decreasing viscosity with increase in agitation Viscosity will normally decrease upon increased agitation, but this depends upon duration of agitation and viscosity of fluid and rate of motion before agitation

The crude may exhibit pseudoplastic or thixotropic behavior and/or act as a Bingham plastic in the transition phase between the onset of wax crystalla­ tion and the fully gelled state. With pseudoplastic behavior the fluid displays increasing viscosity and decreasing shear rate. This means there is a nonlinear relationship between shear stress and shear rate. Also, pseudoplastics are time dependent. A thixotropic liquid's viscosity varies as a function of time and shear rate. A waxy crude may exhibit Bingham plastic characteristics after gelling. The behavior of this type of waxy crude varies from that of Newtonian fluids only in that its linear relationship between shear stress and shear rate does not go through the origin. A finite shear stress is required to initiate flow. Further, the viscosity or pumpability of a Bingham plastic is rate-of-shear and time dependent. When waxy crude is allowed to cool below its pour point under static conditions in a pipeline (no flow in the pipeline), the paraffins will crystallize, causing the entire mass of crude oil to gel. Because of this gelling effect upon cool-down, most operators' initial reaction is to specify that the line be operated at temperatures above the crude's pour point. While this mayor may not be a valid operating criterion, there are no particular problems in pumping waxy crude below its pour point, provided the fluid is kept in motion. More pressure is required to pump in the non­ Newtonian range, but there is no sudden change in fluid characteristics at the pour point. It must be kept in mind that, if a waxy-crude pipeline being pumped below its pour point is shut down for any reason, the resulting gelled state will require, upon restart, substantially more pressure to put it in motion. However, this additional restart pressure will be substantially less than if the crude oil had been pumped above the pour point, then allowed to cool down statically.

264

Fluid Flow, Crude Oils Heat Transfer

In order to determine the required pumping equipment in a pipeline system, pressure-loss calculations must be made. With conventional crude oils, this determination is made using conventional hydraulic equations. These calculations become more complex with non-Newtonian crude since the viscosity is both temperature and shear-rate dependent. Pressure­ loss calculations for non-Newtonian liquids usually are made by dividing the pipeline into segments and analyzing each segment separately. This is best achieved by computers. To analyze a segment effectively, its inlet temperature and the tempera­ ture loss must be calculated. This involves an examination of the heat transfer from the oil to the surrounding environment. Defined as the transmission of energy from one region to another as a result of a temperature difference between them, heat transfer has three distinct modes of heat transmission: 1. Conduction is the process by which heat flows from a region of higher temperature to a region of lower temperature within a medium (solid, liquid, or gas) or between different mediums in direct physical contact. 2. Radiation is the process by which heat flows from a high-temperature body to a lower temperature one when the bodies are separated in space. 3. Convection is the process by which heat flows by fluid motion between regions of unequal density caused by nonuniform heating. Convection is the mechanism of energy transfer between a solid and a liquid or a gas. A pipeline's surrounding environment is critical in determining its poten­ tial heat transfer. Table 2 lists possible environments. The resulting range of overall heat-transfer coefficients (U) is shown to indicate the critical nature of this variable. Once a U value has been ascertained, calculation of heat loss in the pipeline can be determined by the Eq. (1) which takes into account heat lost to the environment as well as heat gained from friction between the oil and pipe wall:

Q=

UA(To - T) - (1.0381)PM

(1)

where Q = heat flow (Btu/h/ft) U = overall heat-transfer coefficient (Btu/h/ ft 2tF) A = outside surface area (ff/ft) T = outside temperature of air ground, or water CF) To = bulk oil temperature CF) P = pressure loss due to friction in the length being considered (lb/in.2/ft) M = flow rate (bbl/h) 1.0381 = conversion factor for dimensional consistency.

Fluid Flow, Crude Oils

265

TABLE 2

Environment vs U Value" Environment Above-ground pipeline, exposed to atmosphere: Uninsulated 2-in. thick insulation Buried on shore pipeline dry soil (desert 2-ft cover): Uninsulated 2-in. thick insulation Buried onshore pipeline, moist to wet soil (2-ft cover): Un insulated 2-in. thick insulation Offshore, unburied pipeline lying on bottom exposed to water currents: Uninsulated 2-in. thick insulation Offshore uninsulated pipeline, suspended, unburied, exposed to water currents Offshore buried pipeline: Uninsulated Insulated

U value, (Btu/h· of· ff)

1.5-0.7 0.1-0.21 0.15-0.65 0.05-0.15

0.3-0.8 0.1-0.2

8-12 0.1-0.2 15-100 0.5-0.7 0.1-0.2

aOverall heat-transfer coefficient U is based on pipe-surface area in square feet. Values are from various sources and experimental data.

If the pipeline is insulated or coated, the heat loss can be described by (2)

where In = logarithm base e = pipe inside radius (ft) = pipe outside radius (ft) = pipe center to insulation surface (ft) hi = inside heat transfer (film) coefficient (Btu/h/ft2tF) ho = outer surface heat transfer coefficient (Btu/h/fetF) kl = thermal conductivity of pipe (Btu/h/fttF) k2 = thermal conductivity of insulation (Btu/h/fttF)

'I

'2 '3

A further effect that must be considered is that a film of wax or coating formed on the inside of the pipeline that shields the hot oil from direct contact

266

Fluid Flow, Crude Oils

with the pipe wall. While the net effect on the overall heat-transfer coefficient U is not extreme, it should be considered. Wax deposits from the crude oil would have an insulating effect similar to that of an internal lining. Since it cannot be assumed that wax will automatically form on pipe walls just because wax is present in the crude, laboratory tests should be performed to determine at what conditions such formations could start. Wax will not deposit at high temperatures, particularly in turbulent flow, so part of a pipeline may have a thin wax film and part may be clean. When a hot pipeline is buried, it heats the soil surrounding it. Heat moves from the pipe in a parabolic shape, with the widest area of the parabola at the ground surface. The term "shape factor" (S;) expresses the amount of heat flow through this parabolic-shaped field. The magnitude of the shape factor depends primarily on the burial depth and pipe diameter, as shown by (3)

or Sf =

2n cosh - ,2Dn Di

----=-=-

(4)

Heat flow from the pipeline then can be calculated by (5)

where Sf = shape factor Q = soil thermal conductivity (Btu/h/fttF) Ks = soil thermal conductivity (Btu/h/fttF) Ti = pipe surface temperature ("F) Ts = ambient temperature (soil) ("F) Dn = depth of burial, soil surface to pipe center (ft) Di = pipe outside diameter (ft) To determine heat flow (Q) by the above equation, the soil thermal conductivity must be measured. Methods of direct measurement of onshore soil thermal conductivity are available and well documented. However, these methods are not yet adapted to direct measurements offshore. The same basic concepts could be used, but equipment modifications would be required. Restart after Shutdown

As waxy oil congeals under stationary conditions, a finite pressure is required to initiate flow. This starting pressure gradient is related to the rheological property known as yield stress. Yield stress is the shear stress that must be developed in the oil to initiate

267

Fluid Flow, Crude Oils

flow. Yield stress for waxy crude is an inverse function of temperature, the stress increasing with decreasing of temperature. It is also a function of the rate of shear when cool-down occurs and depends upon whether or not the cool-down occurs during static or dynamic conditions. Since start-up pressure depends to a large extent on whether the oil is cooled under static or dynamic conditions, rheological studies should be made for static and dynamic cool-down rates so that the exact start-up pressures can be determined for use in designing pumping equipment. Experiments have shown that restart pressures can be 5 to 10 times higher for a statically cooled pipeline (the fluid was above the pour point while flowing and allowed to cool while shutdown) than for one that has been dynamically cooled (the fluid was already below the pour-point temperature before shutdown). Facilities Design

In addition to restart problems and their effect on operation and pumping equipment, consideration also must be given to the requirements of facilities appurtenant to the pipeline, such as storage tanks, scraper traps, test equip­ ment, auxiliary lines, and spare parts for equipment repair. When dealing with waxy crudes, pump selection for the mainline units requires a thorough understanding of the pipeline system, as well as the crude characteristics. However, once the design parameters are determined, the same approach as for Newtonian fluids is used. As in the normal pump design, a flow vs head requirement must be calculated. Figure 1 shows a typical waxy crude flow vs pressure curve.

I I I I I

Newtonian flow I Non-Newtonian flow •••••••••••• Pressure vs flow for nonwaxy crude

/

I\~on-Newtonian

flow

-

- Newtonianflow

'I'

F Min,"

'\

.r

j~

.'\. ........

.... .. '

r-...

... A..

,...I-""

,/

~

Flow

FIG, 1,

Flow vs pressure for waxy crude,

V

V

oJ

/ I""p F

~ Max

Max_

268

Fluid Flow, Crude Oils

It can be shown that, as flow decreases from some optimum point A, the pressure requirement will increase; because the oil is in the line longer, it cools more, and its viscosity increases. Additionally, as the flow decreases, the velocity decreases, which decreases the rate of shear. As the rate of shear decreases in non-Newtonian flow, the viscosity increases (Fig. 2). When the flow starts to increase past point A, Newtonian characteristics take effect and pressure again increases-this time because increased pressure is required to offset increased friction losses. With increased flow, resulting incoming temperatures will start to increase after the pipeline and its surrounding soil have adjusted to the new temperature profile. In choosing a main pump for the system curve outlined in Fig. 1, the operating point P rna" F max must be met. If the point P 2 , F min cannot be met by the mainline pumps because of temperature restrictions at the low flow rate,

10 000

9 8

7

6

--

s = 1.5

-

~ -

s=J

,000

S=6 10,

...

~= S

;= ~

+"'

'2 .~

54

>

(J

';;

co

2

=" "'-

S=6~~

"-

--

~

--

"'"\.\\.

"\.\

~ 'f\

~

3

V>

a .~

~v,

"~

~ l\\

s= 100 ' \

\

..... 1'\ ~\\ 100 '\ S=3~ 9,_

E 8 OJ

~- S

.~ 6 :,,: 5

"'_\.

500'\

'\.

3

\\

\\

10

8 7 6 5

-'~

"-

" '" i'.

I'

3

~

r\

'\ .\

2

,

-"'.

"-

i'.. I'-..

~

r---..

Non-Newtonian range

Newtonian range

2

1

60

70

80

90 Temperature,

FIG. 2.

100 o

110

120

F

Temperature vs viscosity (experimental data) for various shear rates S.

Fluid Flow, Crude Oils

269

or because P2 might be greater than P max> either special start-up/restart pumps, such as positive-displacement (high pressure, low flow) pumps, or the other possible solutions for restart presented elsewhere might be considered. An example of transporting high-pour-point crude oil is a field producing oil with a pour point of 80a R After comprehensive rheology tests were performed in the laboratory, heat-loss and hydraulic calculations resulted in the following recommended design premises and operating philosophy: 1. Design pumping facilities to handle maximum expected volumes. Pumps should be selected to allow a parallel/series arrangement which could transport early production volumes at slower rates and higher pressures when necessary. The piping could be manifolded so that parallel arrange­ ment would be accommodated by repositioning of valves to handle higher flow rates. 2. Use pour-point depressants. 3. Design production separators for higher than normal outlet pressures to allow as much gas and light hydrocarbons as possible to remain in solution. 4. Schedule pipeline start-up during period of warmest ambient tempera­ tures. 5. Establish minimum flow rate to be maintained during initial start-up and for at least 2 weeks thereafter. 6. Include redundant provisions for emergency and planned shutdowns. Systems to consider would include crossover connection with a parallel gas line, standby pumps for displacing the crude oil in the pipeline with water, and pour-point-depressant injection facilities. Once the pipeline system is defined in terms of fluid rheology, hydraulic head requirement, and operating philosophy, pump selection may begin.

Part A: How to Determine Properties of Crude Oil Several design parameters must be investigated before a determination is made of how to pipeline a high-pour-point crude. These parameters are: 1. 2. 3. 4.

Physical properties of crude Heat transfer Restart after shutdown Facilities design This section will describe the steps needed to determine the first parame­

ter. To determine the physical properties of a high-pour-point crude, samples of the crude must be tested in the laboratory.

270

Fluid Flow, Crude Oils

As an example of this complicated process, consider a project which involved the feasibility of transporting a waxy crude, designated R-82, from an oil field in North Africa to a coastal sales terminal. Since the production from the R field was only 10,000 bbljd, installation ofa separate pipeline was precluded. The principal question to be answered was whether it would be technically feasible to inject the R-82 into an existing pipeline handling C-65 crude. The work involved collection of field oil samples, examination of the subject crude oils through laboratory analysis, evaluation of existing C-65 operations, prediction of the total system response to blended oil transport, estimation of new and/or additional facility requirements, and consideration of the effect of blended oil on saleability. Before conclusions could be drawn, detailed laboratory work was needed. Sample Information

The C-65 crude oil samples were taken at a sampling station in the oil field. During the sampling period, the pipeline pressure was 30.24 kg/cm 2 (430 Ib/in.2gauge). The temperature was 57.8°C (136°F) and the production rate was 165,000 barrels of oil per day (bo/d). The R-82 crude-oil samples were taken downstream of the producing separator at a new well location remote to the producing oil field. During sampling the separator pressure was 15 kg/cm 2 (213.3 Ib/in.2 gauge) and the temperature was 5SOC (l31°F). At the laboratory, the following tests were performed on the samples: Pour point, density, specific heat, wax content, and water and sediment. Also, tests were run to determine salt content, distillation, rate of shear, and yield stress. Pour Point

The test for the pour point ofR-82 crude and a blend of the C-65 and R-82 crudes (80%/20% by weight, respectively) was conducted according to ASTM Method 097-66. This is accomplished by heating the sample, then cooling it at a specified rate and examining it for flow characteristics at each interval of 3°C (5°F) in temperature drop. The lowest temperature at which movement of the oil is observed is recorded as the pour point. The results of three separate experiments were: Crude oil R-82: Mean pour point 47°C (llrF). Crude oil C/65/R-82 (80%/20%): Mean pour point 33°C (91°F). Density

The densities ofR-82 and the blend C65/R82 crudes were determined with a DMA 50 digital precision densitometer. The determination is based on the

271

Fluid Flow, Crude Oils

measurement of the vibration frequency of a thermostated U-shaped tube filled with a sample. The precision of the measurements with this apparatus is ±l x 10-4 g/mL. Two series of duplicate measurements were performed. The first series started at 5SOC (l31 F) and ended at lOT (50°F). The second series started at lOT and ended at 5SOc. The heating and cooling rate was about 2T/min. The equilibration time at the measuring temperature was about 10 min. The results of the two series are given in Table 3. The observed differences probably were caused by slow crystallization effects. 0

Specific Heats

The specific-heat measurements were made with an adiabatic calorimeter. The sample energy in the measuring vessel is dissipated by means of a calibrated electrical heating coil. The temperature of the sample is recorded as a function of the energy input. The energy input in the sample and measuring vessel amounted to 6 cal/min. Of this, about 50% was used for the oil and 50% for the vessel with accessories. Heat input into the oil was about 0.12 cal/g/min. In order to avoid slow crystallization effects, the measurements were performed at rising temperature after equilibration at about O°c. The mean specific heat was calculated in intervals of SOc. The temperatures given in Table 4 are the mean temperatures at the chosen intervals. Wax Content

The wax contents of crudes R-82 and C-65 were determined by crystallizing the wax in methylene chloride at ~2YF (~32°C) according to BP Method

TABLE 3

Density Test Results

Temperature

Crude R82 Density (g/mL)

Crude Blend R82/C65 Density (g/mL)

CC)

Cooled

Heated

Cooled

Heated

10 15 20 25 30 35 40 45 50 55

0.8514 0.8458 0.8397 0.8330 0.8255 0.8175 0.8099 0.8057 0.8013 0.7972

0.8514 0.8461 0.8403 0.8338 0.8266 0.8186 0.8124 0.8077 0.8022 0.7977

0.8543 0.8496 0.8444 0.8390 0.8337 0.8389 0.8250 0.8212 0.8175 0.8140

0.8543 0.8496 0.8444 0.8391 0.8340 0.8298 0.8255 0.8216 0.8177 0.8141

272

Fluid Flow, Crude Oils TABLE 4

Specific Heats Specific Heat (caljg°C) Temperature

CC)

Crude R82

Crude Blend R82/C65

5 10 15 20 25 30 35 40 45 50 55

0.68 0.71 0.75 0.81 0.90 0.89 0.80 0.87 0.79 0.73 0.63

0.60 0.63 0.63 0.66 0.68 0.60 0.54 0.57 0.57 0.55 0.54

237/55. Asphaltic materials were removed by first swirling a solution of the oil in petroleum ether with concentrated sulfuric acid. The solution was then centrifuged and filtered through a layer of a filler aid (Celite). After this, the wax was crystallized from methylene chloride and isolated by filtration. The wax was dissolved again in petroleum ether and filtered through a glass filter. The results by weight were: Crude (%) Experiment 1 Experiment 2 Experiment 3 Mean

R-82 39 38.2 38.2 38.5

C-65 17.6 18.2 17.0 17.6

Water and Sediment

The water and sediment contents of crudes R-82 and C-65 were determined according to ASTM Method 096-63. The samples were diluted with an equal volume of "water-saturated" toluene and centrifuged until no droplets or particles were observed microscopically in the oil solution. The volume of sediment was then measured. The results were: Crude (vol. %) Experiment 1 Experiment 2

C-65

R-82

0.0 0.0

0.2 0.2

The amount of water condensed was 0.1-0.2% for the C-65 oil and negligible for R-82 oil.

273

Fluid Flow, Crude Oils Salt Content

The salt content was determined by careful extraction of a solution of 100 mL of crude oil in petroleum ether with three 50-mL portions of warm distilled water. The chloride content of the extract was determined by adding O.1N of silver nitrate. The obtained turbidity was compared with the turbidity of standard solutions of salt with the same amount of silver nitrate. The results were: Crude C-65 3 x 10\ then Cd = 0.62. Other possibilities exist for orifice geometry and tap locations. These are standardized, and complete information on discharge coefficients are given in other publications. The sharp-edged orifice plate is typically used with the following tap locations: Flange Taps-l in. upstream and 1 in. downstream of the orifice Vena Contracta Taps-One pipe diameter upstream and at the minimum pressure location downstream Radius Taps-One pipe diameter upstream and one-half pipe diameter down­ stream

291

Fluid Flow, Measurement

Pipe Taps-2.5 pipe diameters upstream and 8 pipe diameters downstream An orifice meter has the following advantages: 1. Construction is simple with no moving parts and the cost is low 2. Standards for construction and calibration are readily available 3. The method can be used with most fluids

and disadvantages: 1. The accuracy of the orifice and pressure measurement instrumentation is usually ±1% with calibration and less ifuncalibrated 2. A large proportion (50-80%) of the pressure drop is not recovered 3. A long, straight run of pipe is necessary to avoid effects of upstream conditions 4. The pressure drop flow rate relationship is not linear so that the range is limited to a 4: 1 ratio between maximum flow and minimum flow 5. Highly viscous fluids and fluids containing particulates are not suitable

If pressure taps are used, there are some limitations on their size so that errors are minimized. A rule of thumb is to have the length of a pressure tap at least 2.5 times the diameter, and the diameter should be less than I/S of the pipe diameter. The taps should not be placed on the bottom of the pipe where dirt can collect and cause plugging. In order to minimize the permanent pressure loss across the meter, constrictions which avoid separated flow on the downstream side of the constriction are available. However, the cost is considerably higher than the simple orifice plate. The Venturi meter has a conical entry with a cone angle about 21° and an exit cone angle of about r. Taps are located at the throat and upstream of the entrace cone. For pipe Reynolds numbers greater than 2 x lOS, the discharge coefficient is 0.984 and the permanent pressure loss is less than 15% of the pressure differential, PI - Pz. Another common constric­ tion is the flow nozzle. It has a flared approach section leading to a short cylinder. Typical permanent pressure losses are 60% of the pressure drop. The differential pressure meters can also be used for compressible flow. We need to integrate the differential form of Eq. (2), assuming an adiabatic expansion from PI to Pz and subsonic flow. If (XI = (Xz = 1, we have 1 1 1 pdp + "AdA + (v)d(v)

0

(7)

~dp + d( (1Z) + d(1.) = 0

(8)

=

and

292

Fluid Flow, Measurement

For Venturi meters, flow nozzles, and orifice plates, we can obtain a working equation for the average velocity in the form of Eq. (6) with an additional factor Y on the right to account for expansion. In terms of the mass flow rate which is constant, p,(v,;A, = P2(V2;A 2= ril, (9)

(10)

For orifice plates, empirical fits to experimental data are used for flange taps, vena contracta taps, or radius taps. Y = 1 - 1 - r(OAI + 0.35/]4) Y

(11)

Y = 1 - 1 - r [0.333 + 1.145(13 2 + 0.713 5 + 121313)] Y

(12)

and for pipe taps

For a Venturi or flow nozzle the maximum flow through the constriction is limited to sonic velocity at the minimum area. For a sharp-edged orifice the area of the vena contracta increases as downstream pressure decreases. All cases can be treated similarly if the differences are taken into account by changing the discharge coefficient. For rc = P 2IP, at the point of maximum velocity and assuming a thermo­ dynamically reversible adiabatic process in an ideal gas,

rc

(1 - Y)/i

+

Y - 1134 2/y _ Y + 1 - 0 2 rc 2-

(13)

When 13 < 0.2, the second term can be neglected and (14)

Row nozzles are often used as gas meters when operated at the maximum mass flow rate. Instead of varying the pressure differential across a fixed sized constric­ tion as a function of flow rate, a meter can be constructed to vary the flow area of the constriction while the pressure drop is constant. The rotameter shown in Fig. 2 is an example of such an area meter. A float is free to move up

293

Fluid Flow, Measurement

SCALE

FIG. 2.

Schematic diagram ofa rotameter.

or down in a tapered tube which has a linear scale inscribed on it. The tubes are usually glass or plastic so the float can be seen. Metal tubes with a magnetic readout can also be used. The meter operates by letting fluid (liquid or gas) enter the bottom of the tube and the flow through the annulus between the float and the tube leads to a pressure drop which balances the other forces on the float. The theoretical analysis of the rotameter must use a force balance in addition to the mass and energy balances. We place Plane I at the bottom and Plane 2 at the top of the float and assume that the tube area A r is constant between the two planes. Thus for an incompressible fluid (VI) = (V2) = q/A r , where q is the flow rate. At Plane I the velocity is considered uniform across the plane and at Plane 2 let (15) (16)

then the balances become 1()2 -Q2 ()2 - P 2+g( h 1 - h 2)-lv=O V2 + PI P 2VI m[(vI) - P2u

~ 0.83 u u. u. W W

0.82

-' 0

z

« 0.81 I Z

«

c..

0.80

0.79

0.78 300

15

10 400

500

m 3!d (millions) 600

20 700

FLOW - RATE (MMSCF/D)

FIG. 5.

Panhandle plots ofEq. (I) plus transmission factors.

25 800

900

312

Fluid Flow. Natural Gas I

I

20.0

r

.....

I.

t

a: 19.5

o

IU

~ LL

z o

19.0

(f) (f)

~

18.5

(f)

z

~

a:

I- 18.0

'.

17.5

15 m 3 !d (millions)

10 300 12

400 14

16

20

25 800

500 FLOW _ RAT~OO (MMSCF!D) 700 18

I

20

I

22

I

24

,

26

I

28

30

32

900 34

REYNOLDS NUMBER (MILLIONS)

FIG. 6.

Calculated transmission factor versus Reynolds number.

plot should be regarded as the most reliable indication of dry-gas behavior. It can be seen that use of "efficiencies" in the region of 90% brings the Panhandle plots into reasonable agreement with theAGA plot (assuming the use of that Panhandle equation which yields the lowest transmission factor for any given Reynolds number). As expected, the Weymouth equation has no relevance to partially turbu­ lent flow. In the fully turbulent flow region, Weymouth's equation is consis­ tently conservative by about 5~% relative to the AGA plot. Comparison of the operating data plot with the various gas equations reveals several interesting points. It appears that the North Sea gas pipeline operates in the fully turbulent flow region at rates above about 450 MMSCFjd (12.7 x 10 6 m 3 jd). In fully turbulent flow, the presence of condensates in the gas stream effectively reduces the transmission factor (predicted by AGA) by about 4!%. Hence the Weymouth equation closely follows the operating data at flow rates above 450 MMSCFjd. The overall behavior of the pipeline bears little resemblance to either Panhandle equation. The Panhandle "efficiency" factor could be varied to enable the Panhandle equations to characterize the pipeline's performances but it is more logical to compare the line's performance with that predicted by theAGA (dry-gas) design method. Comparison of the operating data plot to the AGA plot is interesting for two principal reasons. . First, the apparent fully turbulent transmission factbr is approximately 5% below that suggested by the AGA dry-gas method (assuming a pipe

313

Fluid Flow, Natural Gas

24

23

a:

~

;t

? PS-lrlP-NDLE 22

NlODlr lED

z

o Ul Ul

:2; 21 Ul

z

u z

70

w U

u.. u..

w

«

60

w ...J

0

Z

« J:

50

Z

«

a..

40 (Pipe Diameter 30

15 inches) (381 mm)

3.0 m3 !d (millions) 4.0

2.0

5.0

6.0

7.0

20~----~----~~----~----~--~~~-----T~----~~~-r--~~

25

50

75

100

125

150

175

200

225

250

FLOW· RATE (MMSCF!D)

FIG. 8.

FPC flow versus efficiency curve.

One further point of interest which arises from this analysis is the abrupt transition which occurs between partially and fully turbulent flow in this pipeline. Although relatively few of the data points are in the partially turbulent region, the analysis appears to favor theAGA abrupt flow transition rather than the broad Colebrook transition.

Transient Behavior and Sphering Regarding the data analysis presented here, it is noteworthy that "steady­ state" conditions are unlikely to be attained in this pipeline due to the frequent changes in flow rate dictated by varying demands for gas by the purchaser.

316

Fluid Flow, Natural Gas

Data points were eliminated from the pipeline pressure surveys if the measured flow rate had altered by more than 10 MMSCF/d(280,000 m 3/d) in the previous 2 hours. This does not eliminate transients since a large jump in flow rate may cause pressure oscillation for several hours. One small and predictable source of transients is provided by the practice of sphering wet-gas lines. This practice is very common since the use of spheres can both increase the "efficiency" of the line and prevent large random liquid slugs appearing at the pipeline outlet. At the time the pipeline pressure surveys were carried out, spheres were launched at about 6-h intervals when the flow rate was below 600 MMSCF/d (17.0 x 106 m 3/d). Sphering was unnecessary at flow rates above 600 MMSCF/d. This sphering policy had been formed from operating experience with the pipeline. When the line was not sphered, large slugs occasionally appeared at the shore terminal at rates below about 550 MMSCF/d (15.6 x 106 m 3/d). The slugs caused handling problems at the terminal and large pressure drops in the pipeline.

100

- --- -Hours between spheres

90

i--

80

>U

5 20 40 80

.......

....... Il}fil} . Ity_

70

, "" , .... ""...., , ,,

Z

w

u

60

w w

50

.........

u.. u..

Z

" , " , ,

:J w

a. a.

'I.

40

en Y> X). The vertical broken line represents the Reynolds number which will occur in the pipeline at maxi­ mum flow rate. The maximum flow rate will be dictated by the maximum operating pressure of the pipeline and hence by the pipe wall thickness. In the situation illustrated in Fig. 10, the pipeline will operate in fully turbulent flow (at maximum flow rate) if there is less than Z bbl/MMSCF condensate in the line. At all liquid loadings greater than Z bbl/MMSCF, fully turbulent flow will not occur within the operating range of the pipeline. It is apparent from Fig. 10 that a change of pipe diameter not only affects the ranges of Reynolds number and transmission factor which may occur, but also determines the flow behavior in the pipeline. At the present time insufficient pipeline data have been published to enable the presentation of a complete design method. However, the authors

318

Fluid Flow. Natural Gas

Specified:

I I Maximum

Pipe piamefer Wall Thickness I

I

I

I

Gas Flowrate

I

~

u

0) or injection (q < 0) rate of the well f.l = fluid viscosity k = permeability h = formation thickness

0.02. Ei(-x) can be calculated from the equation shown with an error ofless than 0.6% for smaller values of x.

r for a noninfinite reservoir containing multiple wells that produce with a time-varying rate. The solution can be adapted to gases and to a multiphase reservoir. Lee's textbook [16] has a discussion of the Ei function and the other analytical solutions for unsteady-state flow. For those desiring to calculate unsteady-state behavior with Eq. (23), we suggest the following: Use the basic data in Example 3, except assume that the reservoir is infinite with Pi = 6000 Ib/in. 2 gauge and that the well starts producing at 250 bbljd at time t = O. Assume that 1> = 0.20 (fraction) and c = 1 X 10- 5 (lb/in. 2tl, which is a typical value of compressibility for a reservoir

340

Fluid Flow, Porous Media

containing crude above its bubble point. [Hint: Plot p vs log r for times of 1, 10, and 100 h after the well starts producing. Use a scale of5500 to 6000 Ib/in. 2 gauge on the pressure axis, and a scale of 0.25 to 4000 ft on the distance axis.]

III. Two-Phase Flow The native fluids in a specific reservoir will be oil and water, or gas and water, or oil, gas, and water. Water is ubiquitous in petroleum reservoirs-it is in the same microscopic pores as the hydrocarbon. (It is common practice to use the term "water" to denote the aqueous phase in a reservoir. Actually, the aqueous phase contains dissolved salts; salinities will range from several hundred to over 200,000 ppm total dissolved solids.) At some point during depletion of practically all reservoirs, there will be two-phase and/or three­ phase flow of the native fluids. We limit this discussion of two-phase flow to the oil-water system. First, we discuss effective permeability and relative permeability, which extend Darcy's law to each phase in a porous medium that is not saturated with a single phase. Next, we introduce capillary pressure, which provides the link between Darcy's law for each phase. Finally, we present the Buckley-Leverett equation, which allows unsteady-state two-phase behavior to be calculated for special conditions.

Effective and Relative Permeability

Effective permeability is the permeability of a porous medium to a fluid when the medium contains more than one fluid. Practice is to write effective permeability as k with a subscript identifying the fluid (e.g., ko and kw for effective permeability to oil and water, respectively). Relative permeability is defined as the effective permeability divided by the absolute permeability. Thus, (26a) (26b) define the relative permeability to oil and water, respectively. Note, however, that a denominator other than absolute permeability may be used in defining relative permeability. The effective permeability to oil at the irreducible minimum water saturation, (ko)cw, is often used as the denominator for the oil-water system. Darcy's law (Eq. 3) can be extended to a porous medium containing more than one phase with either effective permeability or relative permeability.

341

Fluid Flow, Porous Media

Using relative permeability, we can write Darcy's law for the oil and water phases as

qo A

=

v =_C]krok dpo 0

l1 0

dx

(27a)

and

qw = v =_c]krwk dpw A

w

I1w

dx

(27b)

where the subscripts 0 and w refer to oil and water, respectively. Notice that a different pressure, Po and Pw, in the oil and water phases is implied in Eqs. (27a) and (27b). These phase pressures are related to each other by the capillary pressure, which is discussed later. Relative permeabilities must be determined experimentally; several methods are available [6, 8, 11]. For a specific fluid system, relative perme­ ability is a function of (1) saturation, (2) saturation history, (3) rock charac­ teristics, and (4) wettability. We will illustrate the effect of saturation, satura­ tion history, and rock characteristics on relative permeability for a completely water-wet rock with the aid of Fig. 5. The effect ofwettability is discussed in Section IV. Oil-water relative permeabilities are depicted by plotting kro and k rw versus watersaturation,* Sw, as shown in Fig. 5(a). When absolute permeabil­ ity is used as the reference as in Eqs. (26a) and (26b), relative permeabilities have these upper bounds: k ro ::;;; 1, k rw ::;;; 1and, kro + k rw ::;;; 1. When (ko)cw is the reference permeability as in Fig. 5(a), these upper bounds are k/(ko)cw instead of 1. In effect, relative permeability is an empirical way of accounting for the interference between multiple phases flowing in a porous medium. Notice in Fig. 5(a) that k rw becomes zero at a certain Sw; so does kyo. The saturation, S;w, at which k yw = 0 is called the irreducible minimum water saturation (often called connate water saturation). Siw is the lowest water saturation that can be obtained. Similarly, the saturation at which kyo = 0 defines the residual oil saturation, SO" the lowest oil saturation that can be obtained in a normal water-oil system. Special processes-i.e., improved or enhanced oil recovery methods-are capable of obtaining a lower oil satura­ tion [12-15]. Different relative permeability curves are obtained depending on whether water saturation is decreasing (called drainage relative permeability curves) or increasing (imbibition relative permeability curves). This dependence on saturation history is shown in Fig. 5(b) for a core initially saturated with water. Multiple values of relative permeability at a given saturation occur because the fluids occupy different regions ofthe pore space depending on the *Saturation ofa phase is the fraction of the pore volume occupied by that phase. For a porous medium containing water and oil: Sw + So = 1. In figures and text, saturations are often expressed as percent pore volume (% PV).

342

Fluid Flow, Porous Media

z 1.0

z 1.0 i=

0

0

i=

u

u

--

:J 0.6 iii

:J 0.6

f-

f-

CD

0.2

i=

-' UJ

-'

0

UJ

0

20

40

60

80

100

~

[

0.8

>--

0.7

aJ t hl • Obtain (Sw)2 from the (dJ:JdS w) vs Sw relation (Step 2), and (fwh from the!w vs Sw relation (Step 1). Calculate the average water saturation, (SItL, for the specific time t using Eq. (34). The increase in the average water volume in the porous medium over its initial value must equal the volume of oil produced from the porous medium at the specific time t: (37)

where Qo == total volume of oil produced. Step 7. Repeat Step 6 for other values of time to obtain the production

351

Fluid Flow, Porous Media

curve, the volume of oil produced versus the volume of water injected. Usually, volumes are normalized to the pore volume of the porous medium and the production curve is plotted as pore volume of oil produced versus pore volume of water injected. Example 5. A linear horizontal core is 2.54 cm diameter and 40 cm long with a fractional porosity of 0.20. Initially, the core is saturated with a 20-cP oil (S wi = 0). The core is flooded with water of l-cP viscosity at a constant rate of 0.5 cc/min. Relative permeability values are listed in Table 2. Calculate the saturation distribution in the core after 15 min and at water breakthrough, and the pore volume of oil produced vs pore volumes of water injected after breakthrough.

Solution: We will use Eq. (31) until water breakthrough at the outflow face and the Welge method afterwards, as described in Steps 1-7 above. First, we obtain!w using Eq. (32) and the relative permeability data in Table 2, and we evaluate (d!w/dSw) vs Sw using cubic spline interpolation. Computed!w and (d!w/dS w) values are listed in the fourth and fifth columns ofTable 2. For an injection time ofl5 min (assumed to be before breakthrough), we calculate the distance x various Sw values have moved using Eq. (31) and the data in Table 2. Results are plotted in Fig. 8. Notice the false multiple values of Sw between 18 and 29.5 cm. To get the true Sw vs x curve, we must find the value of XI that satisfies Eq. (35). This value is xf= ll.5 cm, from which we obtain the saturation at the front, Swf= 0.565. The trueS w vsx curve at 15 min hasa discontinuity in Sw atx = ll.5 cm­ this is the Buckley-Leverett shock front. From Eq. (36) we calculate the time of water breakthrough at the outflow face (tbl = 52.2 min) and generate the corresponding Sw vs x plot as shown in Fig. 8. After water breakthrough, we use the Welge method to calculate pore volume of oil produced vs pore volume of water injected. Results are in Table 3 and Fig. 9. Note that at water breakthrough the pore volume of oil produced is equal to the pore volume of water injected. Also note that the pore volume of oil produced after breakthrough is equal to the increase in (Sw)av over the initial water saturation. Since the latter is zero for this example, the pore volume of oil produced is identically equal to (Sw)av, TABLE 2

Relative Permeability Data for Example 5a k rw

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80

0.690 0.611 0.535 0.457 0.378 0.294 0.210 0.120 0.050

O.otl

o

0.004 0.009 0.016 0.032 0.055 0.086 0.129 0.184 0.250 0.327 0.403

(d!./dS.) 0.1039 0.2276 0.3743 0.5834 0.7442 0.8540 0.9247 0.9684 0.9901 0.9983 1.0000

2.455 2.513 3.718 3.963 2.625 1.773 1.114 0.635 0.270 0.078

o

aRelative permeability data are adapted from Richardson (Ref. 8, p. 16-57, Table 16.5) with permission of McGraw-Hill;!w is calculated from Eq. (31); (d!wldS w ) is obtained from a cubic spline interpolation with (d!./dS w) forced to zero at Sw = 0.8.

352

Fluid Flow, Porous Media

.9 .8

> a.. c

.7 tbt = 52.2 min.

0 ';:0 t.l

....~ Z· 0 I-

«

a:

::>

t = 15 min •

.6

o

.5

o o

.4

o

I-

«

a..

.5

6

UJ

u

::> 0 0

.4

a:

a..

-l

.3

0

.2

.1

0

2

0

FIG. 9.

345

WATER INJECTED, PV

6

Production curve calculated by the Welge method (Example 5).

7

353

Fluid Flow, Porous Media TABLE 3

{(min.) 52.2 60 80 100 150 200 300 400 500

Use of the Welge Method in Example 5 Water Injected (PV)

(djw/dSw)2

(Swh

ifw)2

(Sw).v

0.644 0.740 0.987 1.233 1.850 2.467 3.700 4.933 6.167

1.351 1.013 0.811 0.540 0.405 0.270 0.203 0.162

0.580 0.609 0.629 0.663 0.682 0.700 0.710 0.718

0.900 0.934 0.953 0.976 0.984 0.990 0.993 0.994

0.644 0.654 0.674 0.687 0.708 0.721 0.737 0.746 0.754

Example 5 mimics a common laboratory test, called a waterflood dis­ placement. By measuring pressure drop across the core in addition to the volume of oil produced versus time, and by conducting the flood at a rate high enough to minimize capillary forces, relative permeability curves can be calculated from the data. Although based on a linear homogeneous laboratory model, Example 5 plus Figs. 8 and 9 also illustrate the basic behavior of waterflooding. (Also see Fig. 10 and the accompanying text.) Reservoir waterflooding, however, is considerably more complex; Craig's monograph [11] should be consulted for an in-depth treatment of the engineering aspects of this recovery process.

100

....--------------4

100 a.

~

g so >-

II:

o>

60

60

cP OIL

151 cP OIL

w

>

40

0

u

40

w

II: ...J

>-"

1.S

II:

w

6

so

0 oR.

w

u

6

II:

20

...J

o L -_ _ _.L-_ _ _...l....._ _ _....L......j o 2 3

6

COMPLETELY WATER WET

20 0

10

0

0.5

1.0

1.5

2.0

2.5

WATER INJECTED, PORE VOLUMES

WATER INJECTED, PORE VOLUMES

(0) Effect of wettability [after Raza et ai, Reference 29].

(b) Effect of oil viscosity [after Richardson, Reference 30; @ 1957 SPE-AIMEJ.

FIG. 10.

Effect ofwettability and oil viscosity on waterflood displacement behavior. Lab flood data are plotted as oil recovery (% of original oil-in-place) vs pore volumes of water injected. (a) Noticeably different behavior is obtained when flooding a low viscosity oil from completely water-wet and completely oil-wet sandstone cores. (b) Noticea­ bly different behavior is also obtained when flooding 1.8 and 151 cP oils from completely water-wet sandpacks; flooding behavior of the 151 cP oil could be erroneously interpreted as indicating a system that was not completely water wet if the effects of viscosity were not considered.

354

Fluid Flow, Porous Media

IV. Wettability Wettability of reservoir rock describes its affinity for water in the presence of oil, and vice versa. Even though the concept has been around for over 50 years, there is no generally accepted way of quantitatively measuring rock wettability, although several methods have been proposed. The only generally accepted way of characterizing wettability is qualitative, often using three categories such as completely water wet, completely oil wet, and intermediate wettability. Even here, various authors employ different terms and/or use more than three categories. For example, instead of the terms "completely water wet" and "completely oil wet," some use "strongly water wet" and "strongly oil wet," respectively. Others will subdivide the intermediate wetta­ bility category using descriptive names such as "slightly water wet," "slightly oil wet," and "neutral wettability." One author might classify a rock as being oil wet, while another would call the same rock preferentially oil wet. Thus, in reading the literature on rock wettability, one has to be careful to understand the qualitative definitions used by each writer. Nevertheless, wettability is an important characteristic of reservoir rock. This fact is generally accepted. Wettability affects the very nature of the capillary pressure and relative permeability curves and, hence, the basic mechanics of fluid flow and displacement behavior in reservoirs. The effect ofwettability on the waterflood displacement of a low-viscosity oil in linear sandstone cores is illustrated in Fig. 1O(a). For a completely water­ wet core, a large percentage of the original oil in the core is recovered before water breakthrough and little additional oil is recovered after breakthrough. For a completely oil-wet core, the oil recovery prior to breakthrough is much less but considerable amounts of oil are produced afterwards. Unfortunately, reliance on a waterflood displacement test to characterize wettability can be misleading if all factors are not taken into account. This dilemma is illustrated in Fig. 1O(b) for displacement of 151 and 1.8-cP oils in 1.09-0 sandpacks that were completely water wet. Flooding behavior of the 151-cP oil could be erroneously interpreted as indicating a system that was not completely water wet if the effect of oil viscosity were not realized. The literature dealing with rock wettability is too vast to summarize here. As an introduction to the subject, we limit the following treatment to two topics: an expanded discussion on the classification of rock wettability and a description of methods for ascertaining it.

Classification of Rock Wettability

We prefer to use the terms completely water wet, completely oil wet, and intermediate wettability to qualitatively describe rock wettability. We will use the spontaneous imbibition behavior of a rock as an aid in defining these states. A simple version of this test consists of immersing a core, initially containing a low water saturation and a high oil saturation, into a beaker of water (or immersing a core with low So and high Sw in oil) and observing the

Fluid Flow, Porous Media

355

amount, if any, of oil (or water) expelled from the rock by spontaneous imbibition of the other fluid. Completely Water Wet. A completely water-wet rock has a continuous film of water over the entire pore surface at any water saturation. Oil does not contact the rock surface; it exists in continuous filaments in the pore channels and as isolated droplets in certain pores. A completely water-wet rock will spontaneously imbibe water, but not oil. Completely Oil Wet. A completely oil-wet rock is the exact opposite of a com­ pletely water-wet one. Oil exists in a continuous film on the pore surface. Water exists in continuous filaments in the pore channels and as isolated droplets in certain pores. Water does not contact the rock surface. A completely oil-wet rock will spontaneously imbibe oil, but not water. Intermediate Wettability. It is generally thought that a rock in an intermediate wettability state has water in intimate contact with the pore surface in some places and oil in other places. These water-wet and oil-wet places are comingled on the micro­ scopic-not macroscopic-scale. This model* of the intermediate state has been called heterogeneous or "dalmatian" or "spotted." A rock in the intermediate wettability state will spontaneously imbibe both water and oil although the amounts imbibed may be very small and, in some cases, undetectable. A special subcase of intermediate wettability merits mention. Salathiel [31] postulated that in some instances the oil-wet places are somehow joined together to form a continuous but tortuous surface network which allows oil to continue to flow through a rock even at very low oil saturations. Starting with completely water-wet cores, he was able to effect this peculiar intermediate wettability state in the labora­ tory. Salathiel called this subcase "mixed wettability,"* and used it to explain the very low residual oil saturations «10% PV) found in portions of the East Texas Field that had been under waterflood for several decades.

At one time, reservoir engineers believed all reservoirs were completely water wet. The basis for this belief was the supposed mechanism by which petroleum or its precursors migrated to and were trapped in a reservoir. Before this migration the reservoir's pore space was filled with water and the rock was presumed to be completely water wet. During migration the less dense organic phase displaced the water by buoyancy forces (primary drain­ age). However, the rock surface was always protected from direct contact with the organic by the intervening water film. Thus, the reservoir maintained its completely water-wet state even after it trapped the organic phase. Evidence collected over the past 30-40 years has overwhelmingly proved that this is not true: many reservoirs are in an intermediate wettability state, some are completely water wet, and a few are completely oil wet. There are several plausible explanations for other than a completely water-wet state; we mention two. First, crude oils contain small amounts of polar organic compounds such as carboxylic acids. Over time, these polar molecules *An alternate model for intermediate wettability invokes a homogeneous state for the pore surface. The contact angie, if one could be measured on the rough mineralogically heterogeneous rock surface, would be at some intermediate value and everywhere the same. This depiction is no longer accepted by most. *The term "mixed wettability" is used by some authors as a synonym for the intermediate wettability state-not to indicate the state postulated by Salathiel.

356

Fluid Flow, Porous Media

diffused across the intervening water film and adsorbed on the rock surface, creating oil-wet regions. Second, all or just portions of the rock surface were coated by organic molecules (such as waxes or asphaltenes) that phase separated from the original crude oil during the pressure and temperature changes that many reservoirs experience over geologic time. Methods of Assessing Wettability

There is no way of measuring the wettability of reservoir rock in situ. These measurements have to be made in the laboratory (or at the well site) on cores from the reservoir. Whether or not core samples retain the in-situ wettability of the reservoir as well as the best methods for obtaining, transporting, and handling cores for minimum disturbance of their wettability have received extensive attention over the years (e.g., Refs. 32-34). It is beyond the scope of this article to delve into this aspect ofwettability. Once a core is in the laboratory, however, 10 methods are available for assessing its wettability.: 1. Characteristics of capillary pressure and/or relative permeability curves, (e.g., Ref. 11) 2. Spontaneous imbibition tests (e.g., Refs. 33 and 35) 3. Amott's method [36] 4. DOE centrifuge method [37] 5. Contact angle measurements (e.g., Ref. 38) 6. Threshold pressure measurements [39] 7. Dye adsorption method [40] 8. Capillary rise method [41] 9. Nuclear magnetic resonance relaxation time measurements [42] 10. Flotation method [43]

The above numerical ordering has only the following significance: Methods 1-5 have enjoyed the most use, while Methods 6-10 have had little use since they were proposed. (The flotation method (No. 10) was proposed most recently-in 1980-and has not had the exposure time of the other methods.) Methods 1, 2, and 10 give a qualitative or relative assessment, while Methods 3-9 were designed to yield a quantitative measure-i.e., some type of "wetta­ bility number." As mentioned, none of these methods has been accepted as a standard by the petroleum industry. It is instructive, however, to examine the principles behind the most used ones, Methods 1-5. Method 9 will also be discussed because of some recent research results.

Capillary Pressure/Relative Permeability Curves This method differs from the rest in two aspects: it has no formal proposal date, evolving instead over the years, and it is strictly qualitative. In this

357

Fluid Flow, Porous Media - 48 Ol

I

LL

- 40

TENSLEEP CORE TS-2

0

k = 48.1 rnD

E

u

w

- 32

(( ~

(/)

(/) - 24

w

((

0-

r

(( - 16

«

-.J -.J

~ u

-8

o L-__ o 20

~~~g.

40

__

~~-L

60

__

80

~

100

OIL SATURATION, % PV

FIG. 11.

Capillary pressure curve for a completely oil-wet rock. In this wettability state, P, is negative for all saturations (note that the abscissa is oil saturation): I is primary drainage (So decreasing), 2 is imbibition (So increasing), and 3 is drainage. Curve 3 has been sketched in based on known trends. (After Killins et al. [25].)

32

I

C>

24

LL

0

E

16

u

w

a: ~

8

(/)

(/)

w a: 0-

r

0

((

« -.J -.J

c:: « u

BEREA CORE 2-M016-1

-8

k = 184.3 rnD

-16

-24 0

20

40

60

80

100

WATER SATURATION, % PV

FIG. 12.

Capillary pressure curves for a core with intermediate wettability. In this wettability state, P, has both positive and negative values: I is the primary drainage curve (Sw decreasing), 2 is the spontaneous part of the imbibition curve (Sw increasing and Pc > 0),3 is the forced part of the imbibition curve (Sw increasing and P, < 0), and 4 is a drainage curve that also has a spontaneous portion (Sw decreasing and Pc < 0) and a forced portion (Sw decreasing and Pc > 0). Curve 4 has been sketched in based on known trends. (After Killins et al. [25].)

358

Fluid Flow, Porous Media

method, wettability is assessed from the shapes of the capillary pressure and/or relative permeability curves. (At times, electrical resistivity and/or waterflood displacement behavior are also considered; e.g., see Fig. 10 and the accompanying text, and Ref. 29.) Craig [11] presents some succinct observations on this method; we follow his approach. Wettability has a significant effect on capillary pressure curves. For a completely water-wet core (Fig. 7, especially Fig. 7b), Pc has only positive values. For a completely oil-wet core (Fig. 11), Pc has only negative values (note that the abscissa in Fig. 11 is oil saturation). For a core with intermedi­ ate wettability, Pc has both positive and negative values as shown in Fig. 12. We shall say more about the drainage and imbibition curves, which are shown on these figures, in the discussion of Spontaneous Imbibition Tests. Relative permeability curves for a completely water-wet rock (Fig. 5, especially Fig.5b) and a completely oil-wet rock (Fig. 13) are obviously different. Rules of thumb for these differences are summarized in Table 4 (after Craig, Ref. 11, © 1971, SPE-AIME). Rocks with intermediate wettabil­ ity, Craig states, "have some of the characteristics of both water-wet and oil­ wet formations." Thus, it is obvious that wettability affects capillary pressure and relative permeability curves. Ascertaining wettability from these data does require experience, however.

z 1.0

Q

IU

«

a: 0.8 u..

r-

I-

.....I

0.6

tii

« w ~ a: 0.4

w a.. w

> 0.2 t=

« .....I

w

a:

a

a

20

40

60

80

100

WATER SATURATION, % PV

FIG. 13.

Typical water-oil relative permeability curves for a completely oil-wet rock show substantially different characteristics than those for a completely water-wet rock (see Fig. Sa). For example, the irreducible minimum water saturation (Siw) is less and k m values are greater for the completely oil-wet rock. Reference permeability is (ko)cw for these relative permeability curves. (After Craig [11], © 1971, SPE-AIME.)

359

Fluid Flow, Porous Media TABLE 4

Rules of Thumb for Effect ofWettability on Relative Permeability (after Craig, Ref. 11, © 1971, SPE-AIME) Completely Water Wet

Completely Oil Wet

1. Irreducible minimum water saturation

Usually greater than 20-25% PV

2. Water saturation where k ro = k rw 3. k rw at residual oil saturation

Greater than 50%PV Generally less than 0.3

Generally less than 15% PY, frequently less than 10% Less than 50% PV

4. Effect of drainage-imbibition cycle: a. On k rw

b. On kro

Little or none Significant hysteresis

Greater than 0.5 and approaching 1.0 Significant hysteresis Little or none

Spontaneous Imbibition Tests Spontaneous imbibition tests have been and still are the most widely used method for qualitative assessment of wettability. These tests have been used on an informal basis since the 1950s, perhaps earlier. By informal, we mean immersing a core with high oil saturation in a beaker of water as mentioned earlier, or the procedure of putting a drop of water (or oil) on a core to see if it disappears. Spontaneous imbibition can be explained by following the appropriate paths on the capillary pressure curves in Figs. 7(b), 11 and 12. Increasing water saturations on the imbibition curve (Curve 2, Fig. 7b) for a completely water­ wet rock describes the path of spontaneous imbibition for this wettability state. Increasing oil saturations on the imbibition curve (Curve 2, Fig. 11) for a completely oil-wet rock traces spontaneous imbibition for this case. For intermediate wettability (Fig. 12), spontaneous water imbibition will take place along the imbibition curve (Curve 2) as long as Pc> 0, while spontane­ ous oil imbibition will occur along the drainage curve (Curve 4) as long as Pc < O. Along these paths for any wettability state, spontaneous imbibition stops whenPc=O. Bobek et al. [33] were the first to systematically use spontaneous imbibi­ tion tests to ascertain wettability. Their procedure consisted of determining whether water and/or oil would be spontaneously imbibed by a rock sample. This result was compared to a reference imbibition test on the same rock sample after it had been made completely water wet by muffling at 400°C. In a subsequent paper, Denekas et aI. [35] used initial spontaneous imbibition

360

Fluid Flow, Porous Media

rates to ascertain the relative effect of various crude oil fractions on wettabil­ ity. Care must be taken when assessing wettability by spontaneous imbibition. The initial saturation must be near (preferably at) the appropriate endpoint saturation: Siw for spontaneous water imbibition and Sor for oil imbibition. The volume of liquid expelled by spontaneous imbibition of the opposite liquid can be quite small and difficult to detect. For example, spontaneous oil imbibition for the intermediate wettability core in Fig. 12 would expel only about 8% PV water if the core were initially at Sor (e.g., only 0.8 cm 3 maximum for a core plug with 10 cm 3 pore volume).

Amott's Method Amott [36] developed a combined imbibition-displacement test, the results of which are expressed on a numerical scale covering the entire wettability range. A displacement-by-water ratio forms one-half of this scale, a displace­ ment-by-oil ratio forms the other half. To obtain the displacement-by-oil ratio, start with the core at residual oil saturation. Immerse the core in oil and measure the volume of water produced by spontaneous oil imbibition after 20 h (call this V wl ). Next, measure the volume of water produced after centrifuging the core under oil at high speeds (call this Vd. The displacement-by-oil ratio is equal to V w1 /( V wl + V w2). The displacement-by-water ratio is obtained in an analogous way. If20 h is sufficient for completion of spontaneous imbibition (it may not be!), Amott's two ratios have the following limits: For completely water-wet cores, the displacement-by-water ratio is 1 and the displacement-by-oil ratio is zero; for completely oil-wet cores, the displacement-by-oil ratio is 1 and the displacement-by-water ratio is zero; for intermediate wettability, both dis­ placement ratios will have values in the range ofzero to less than 1. In general, the magnitude of a core's preference for water or oil tracks its displacement ratio: a strong preference is indicated by a ratio approaching 1; a weak preference, by a value approaching zero. Craig [11] considers this "one of the best methods for measuring the degree of rock wettability."

DOE Centrifuge Method The U.S. Department of Energy method* [37] defines a single wettability number calculated from the areas under two capillary pressure curves meas­ ured by the centrifuge method. As indicated in Fig. 14, (38)

*Originally called the U.S. Bureau of Mines method.

361

Fluid Flow, Porous Media

where W = the DOE wettability number Al = area under Curve III A2 = absolute value of area under Curve II Interpretation of W is as follows: for water-wet cores, W >> 0, while for oil­ wet cores, W «0. For intermediate wettability, W can be positive or negative but is closer to zero. The curves in Fig. 14 are not complete Pc vs Sw curves due to intrinsic limitations of the centrifuge method: the water imbibition curve can be obtained only for Pc < 0 (Curve II) and the water drainage curve can be obtained only for Pc> 0 (Curve III). The dashed Curve I represents the path followed in the DOE method when starting with a core saturated with water. Notice that we have deviated from our terminology of "completely water wet" and "completely oil wet" in the above text and in Fig. 14, using instead the terms "water wet" and "oil wet" per Donaldson et al. [37]. The reason is this: the area A 2 will be zero for a completely water-wet core (W -----> (0), and the area A 1 will be zero for a completely oil-wet rock (W -----> - 00). Therefore, both plots in Fig. 14 are for cores in the intermediate wettability state according to our definition .

..

•

10 ,....-----"'T" •--------------~

""'/

~,/~~I.~",

w

0::

::J

en en w

0::

!l.

>-

..

10...----

w

0::

::J

en en w

g:

0 t-----

>-

0

1------1~~.......,...,.;;;;,.;:.:i~:........l

0::

0::

~

~ ...J ...J

...J ...J

!l. ~

!l.

~

U

U

-10

L -______________________- - - I

o

100 WATER SATURATION, % PV

(a) Intermediate wettability, W = 0.79 (called "water wet" in Reference 37).

FIG. 14.

-10 '------------_______

o

WATER SATURATION, % PV

(b) Intermediate wettability, W = -0.51 (called "oil wet" in Reference 37).

DOE centrifuge method for measuring wettability. The DOE wettability number is defined by: W = log [AI/A2)' where AI and A2 are the areas under the capillary pressure curves as shown. The porous media in (a) and (b) are both in the intermedi­ ate wettability state as defined in this article. The meanings of Curves I, II, and III are explained in the text. (After Donaldson et aI. [37), © 1969, SPE-AIME).

362

Fluid Flow, Porous Media

Contact Angle Measurements Treiber et al. [38] published one of the most comprehensive examples on the use of this method. Under anaerobic conditions using synthetic formation brine and degassed crude oil, they measured-at reservoir temperature-the equilibrium water-advancing contact angle on the polished surface of the dominant mineral in the reservoir from which the crude came. They made these measurements on crudes from over 50 oil reservoirs. Their results indicate the wettability of the 50+ reservoirs covers the broad spectrum from completely water wet to completely oil wet, with the majority in a "moder­ ately oil-wet" state.

NMR Relaxation Time Measurements In 1956, Brown and Fatt [42] proposed use of nuclear magnetic resonance (NMR) relaxation time measurements as a method of quantitative assess­ ment of wettability. They measured the proton spin-lattice relaxation time (TI) in packs containing known mixtures of water-wet and oil-wet sands that were saturated with water. Finding a linear correlation between l/TI and percent of oil-wet sand in the sand pack, they proposed use of this method to determine the fraction of oil-wet surface area in reservoir rock. Their proposal is appealing because the method would be simple and straightforward. But for almost two decades after Brown and Fatt's paper, there was absolutely no follow-up work on the method in the literature. Then in 1981 and 1982, Fung and McGaughy [44] and Williams and Fung [45] published two relevant papers. In brief, here's what they reported. Use of proton Tl measurements as a quantitative measure ofwettability, as proposed by Brown and Fatt, is improper: magnetic nuclei on the rock surface (e.g., Si-29, Na-23) significantly affect the measured TI values. Since the type, amount, and distribution of these magnetic nuclei will likely vary from core plug to core plug, the proton TI values will be affected in an unknown manner so that a correlation between Tl and wettability cannot be obtained. Williams and Fung showed that the deuteron rotating-frame spin­ lattice relaxation time (Tip) is quite insensitive to variations in these mag­ netic nuclei from sample to sample, and that the deuteron Tip did vary with wettability in simple systems. Further work is needed to demonstrate that deuteron Tip does correlate with reservoir rock wettability, and to reduce the method to practice. If this can be done, the method would be a valuable contribution.

Symbols Fundamental dimensions for each quantity are expressed in terms of m (mass), L (length), t (time), and T (temperature). Variables in Eqs. (1), (9), and (10) require specific units, which are specified in the text.

363

Fluid Flow, Porous Media

Cl,C2,C3,C4,Cj,C6 D

area (L2) areas under capillary pressure curves in Eq. (38) (mILt2) Klinkenberg factor (ml Lt 2 ) compressibility (Lt2Im) units conversion factors true vertical depth, positive downward (L)

Ei(-x)

- rue e-

C

j~.

ifwh

g

II du, the exponential integral or Ei function u fractional flow of water (dimensionless) fractional flow of water at the outflow face (dimensionless) acceleration due to gravity (Llt 2; 9.806650 mN, 32.17405

Jx

ftN)

h k

ka (k)av

kg kL

ko (ko)cw

k", k rw kw

In log L P P p" P, Po Pw Pwj Pc /)'P/ /),P

(/),P )a (/),P )u q q

qo qw Qo r

R

conversion factor in Newton's second law of motion; 1.0 kg· miN· S2, 32.17405 Ibm' ft/lb/" S2 formation thickness or height (L) absolute permeability or permeability of the bulk forma­ tion (L 2) permeability in the altered zone around a well (L 2) average permeability (L 2) permeability to gas (L 2) permeability to unreactive liquid (L 2) effective permeability to oil (L 2) effective permeability to oil at the irreducible minimum (connate water) saturation (L 2) relative permeability to oil (dimensionless) relative permeability to water (dimensionless) effective permeability to water (L 2) natural logarithm, base e common logarithm, base 10 length (L) pressure (ml Lt 2 ) arithmetic average pressure (ml Lt 2 ) pressure at the external radius r" (mILt2) initial reservoir pressure (mILt2) pressure in the oil phase (mILt2) pressure in the water phase (m I Lt 2 ) flowing pressure in the wellbore (mILt2) Po - Pw, capillary pressure (mILt2) entry or threshold capillary pressure (mILt2) applied pressure drop (ml Lt 2 ) additional pressure drop (ml Lt 2 ) pressure drop if permeability were uniform (mILt2) volumetric flow rate (L 3It) volumetric flow rate evaluated at p (L 3It) oil volumetric flow rate (L 3It) water volumetric flow rate (L 3It) total volume of oil produced (L 3) radius (L) radius of zone of altered permeability around a well (L) wellbore radius (L) universal gas constant (L 2It 2T); 1.9858 BtutR· Ibm' mol, 8.3143 kJ/kmol· K

364

Fluid Flow, Porous Media

t

T tbl

Ti

Tip

v

v

y

z Z ~

(J

skin factor (dimensionless) irreducible minimum water saturation (also called connate water saturation), fraction pore volume, (dimensionless) oil saturation, fraction pore volume (dimensionless) residual oil saturation, fraction pore volume (dimensionless) water saturation, fraction pore volume (dimensionless) average water saturation, fraction pore volume (dimension­ less) water saturation at the Buckley-Leverett displacement front, fraction pore volume (dimensionless) initial water saturation, fraction pore volume (dimension­ less) water saturation at the outflow face, fraction pore volume (dimensionless) time (t) temperature (1) time of water breakthrough at outflow face (t) NMR spin-lattice relaxation time (t) NMR rotating-frame spin-lattice relaxation time (t) velocity (LIt) velocity evaluated at Ii (Lit) velocity along the x, y, z coordinates, respectively (Lit) volume (L 3) DOE wettability number, dimensionless x coordinate, distance along the x coordinate (L) distance Buckley-Leverett shock front has moved along the x coordinate (L) Y coordinate (L) z coordinate (L) gas law deviation factor (dimensionless) difference; e.g., ~p angle measured positive counterclockwise from horizontal (dimensionless) contact angle (dimensionless) viscosity (miLt) oil viscosity (miLt) water viscosity (miLt) density (m1L3) oil density (m1L3) water density (miL 3) interfacial tension (mlt2) porosity, void volume divided by the bulk volume, fraction (dimensionless)

Thanks to R. J. Blackwell and J. M. Maerker for their helpful comments on this article.

Fluid Flow, Porous Media

365

References 1.

2. 3.

4.

5.

6.

7.

8.

9.

10.

11.

Standard: Letter Symbols for Petroleum Reservoir Engineering, Natural Gas Engineering and Well Logging Quantities, Society of Petroleum Engineers, Dallas, 1965. (Also published in J. Pet. Technol., December 1965.) Standard: Computer Symbols for Petroleum Reservoir Engineering, Natural Gas Engineering and Well Logging Quantities, Society of Petroleum Engineers, Dallas, 1968. (Also published in Soc. Pel. Eng. J, pp. 423-442 (December 1968).) Standard: Supplements to Letter Symbols and Computer Symbols for Petroleum Reservoir Engineering, Natural Gas Engineering and Well Logging Quantities, Society of Petroleum Engineers, Dallas, 1972. (Also published in Trans. AfME, 253,556-574 (1972).) The Sf Metric System of Units and SPEs Metric Standard, Society of Petroleum Engineers, Dallas, 1982. (Also published in Trans. SPE, 273, 1022 ff(1982).) B. C. Craft, and M. F. Hawkins, Applied Petroleum Reservoir Engineering, Prentice-Hall, Englewood Cliffs, New Jersey, 1959. Although published in 1959, this well-written textbook covers the basic principles of reservoir engineering. Chapters 6 and 7 contain derivations of most of the equations in this article. J. W Amyx, D. M. Bass, and R. L. Whiting, Petroleum Reservoir Engineering­ Physical Properties, McGraw-Hill, New York, 1960. This textbook/reference book has much information on the physical properties of reservoir rocks, hydrocarbons, and oilfield brines. Chapter 1 is an introduction to petroleum, its origin, how reservoirs trap hydrocarbons, reservoir lithology, drilling operations, and reservoir performance. Chapters 2 and 3 have detailed discussions of the material in Sections I-III of this article. E. H. Timmerman, Practical Reservoir Engineering (Part f), Pennwell, Tulsa, 1982. Part I describes practical methods of obtaining the best rock parameters, PVT properties of hydrocarbons, and other reservoir engineering parameters such as subsurface temperature and static reservoir pressure. Chapter 5 is a comprehensive treatment of the performance of flowing and shut-in wells. J. G. Richardson, "Flow of Fluids Through Porous Media," Section 16 in Handbook of Fluid Dynamics (V L. Streeter, editor-in-chief), McGraw-Hill, New York, 1961. This excellent 111-page section by Richardson is an advanced treatment of the principles of single-phase, multiphase and miscible flow. Start­ ing with elementary material (similar to that in Sections I-III of this article), Richardson adroitly expands it to advanced techniques. R. E. Collins, Flow of Fluids Through Porous Materials, Reinhold, New York, 1961. Originally published in 1961, this useful and clearly written reference book was reprinted in 1976 by the Petroleum Publishing Company (Tulsa). It is an advanced work (containing a reasonable amount of introductory-level material) that treats the fundamentals of flow in porous, permeable media with emphasis on reservoir rock. D. W Peaceman, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York, 1977. Peace man treats the numerical methods used to solve the partial differential equations for multi phase multidimensional flow in reser­ voirs. The book is written for one with some skills in numerical methods. Chapter 1, however, will provide those without these skills with a nice, logical introduction to reservoir simulation (including derivations of the partial differ­ ential equations). F. F. Craig, The Reservoir Engineering Aspects of Waterflooding (SPE Mono-

366

Fluid Flow, Porous Media

12.

13.

14.

15.

16.

17.

18.

19.

graph, Vol. 3), Society of Petroleum Engineers, Dallas, 1971. Craig's monograph is an excellent summary of pertinent literature on waterflooding, the main method of recovering additional oil from a reservoir. Chapter 2 (which treats wettability, capillary pressure, relative permeability, and other basic water-oil flow properties of rock) is an adjunct to the material in Sections III and IV of this article. Appendices A-C have clear derivations of the fractional flow equation, the Buckley-Leverett equation, and Welge's method. M. Prats, Thermal Recovery (SPE Monograph, Vol. 7), Society of Petroleum Engineers, Dallas, 1982. Prats' monograph is a comprehensive treatment of all thermal recovery methods (hot-water drives, steam drives, in-situ combustion, cyclic steam injection). Included are the principles of thermal methods, basic data, operating design criteria, and evaluation methods. F. I. Stalkup, Miscible Displacement (SPE Monograph, Vol. 8), Society of Petroleum Engineers, Dallas, 1983. Stalkup's monograph treats the principles of miscibility and miscible displacement in reservoir rock, plus the various miscible displacement processes (including the CO 2-miscible process). D. O. Shah and R. S. Schechter (eds.), Improved Oil Recovery by Polymer and Surfactant Flooding, Academic, New York, 1977. This book is a compilation of papers on surfactant and polymer flooding that were presented at the AIChE Symposium in Kansas City, April 1976. As the editors state, "the symposium covered the molecular, microscopic, and macroscopic aspects of oil displace­ ment in porous media by surfactant and polymer solutions." Improved Oil Recovery, Interstate Oil Compact Commission, Oklahoma City, 1983. This reference book contains separate chapters, written by experts from the petroleum industry, on many recovery processes: waterflooding and improved waterflooding, gas injection, hydrocarbon miscible displacement, carbon diox­ ide flooding, microemulsion flooding, in-situ combustion, and steamflooding. W J. Lee, Well Testing (SPE Textbook Series, Vol. 1), Society of Petroleum Engineers, Dallas, 1982. Lee's textbook treats the technology of well testing-a technique of analyzing pressure transients to obtain individual well and reservoir characteristics such as the skin factor, bulk formation permeability, distance from a well to a fault, and static reservoir pressure. Well-testing methods are based on the principles of unsteady-state flow (e.g., the Ei-function solution in Section II of this article). M. Muskat, The Flow of Homogeneous Fluids Through Porous Media, McGraw-Hill, New York, 1937. Written in 1937, this classic work by Muskat sets forth the mathematical physics for single-phase flow in reservoirs. This book was reprinted in 1982 by the International Human Resources Development Corpo­ ration (Boston). M. Muskat, Physical Principles of Oil Production, McGraw-Hill, New York, 1949. This second classic by Muskat, written in 1949, sets forth the underlying principles of modern reservoir engineering. It is a treatise on all recovery methods known at that time. This book was reprinted in 1981 by the Interna­ tional Human Resources Development Corporation (Boston). H. Darcy, Les fontainer pubfiques de fa ville de Dijon, Victor Dalmont, Paris, 1856.

20. 21.

L. J. Klinkenberg, "Permeability of Porous Media to Liquids and Gases," in Drilling and Production Practice, American Petroleum Institute, 1941, pp. 200-

213. Recommended Practice for Determining Permeability of Porous Medium, (API RP 27), Division of Production, American Petroleum Institute, Dallas, 1956.

Fluid Flow, Porous Media

367

22. A. F. van Everdingen, "The Skin Effect and Its Influence on the Productive Capacity ofa Well," Trans. AIME, 198, 171-176 (1953). 23. Tables ofSine, Cosine and Exponential Integrals, Vol. I, Federal Works Agency, Work Projects Administration for the City of New York, 1940. 24. J. T. Morgan and D. T. Gordon, "Influence of Pore Geometry on Water-Oil Relative Permeability," J. Pet. Techno!., pp. 1199-1208 (October 1970). 25. C. R. Killins, R. F. Nielsen, and J. C. Calhoun, "Capillary Desaturation and Imbibition in Porous Rock," Producers Mon., pp. 30-39 (December 1953). 26. N. R. Morrow, "Capillary Equilibrium in Porous Materials," Soc. Pet. Eng. J., pp. 15-24 (March 1965). 27. S. E. Buckley and M. C. Leverett, "Mechanism of Fluid Displacements in Sands," Trans. AIME, 146, 107-116 (1942). 28. H. J. Welge, "A Simplified Method for Computing Oil Recovery by Gas or Water Drive," Trans. AIME, 195,91-98 (1952). 29. S. H. Raza, L. E. Treiber, and D. L. Archer, "Wettability of Reservoir Rocks and Its Evaluation," Producers Mon., pp. 2-7 (April 1968). 30. J. G. Richardson, "The Calculation of Water flood Recovery from Steady-State Relative Permeability Data," J. Pet. Techno!., pp. 64-66 (May 1957). 31. R. A. Salathiel, "Oil Recovery by Surface Film Drainage in Mixed-Wettability Rock," J. Pet. Technol., pp. 1216-1224 (October 1973). 32. J. G. Richardson, F. M. Perkins, and J. S. Osoba, "Differences in Behavior of Fresh and Aged East Texas Woodbine Cores," Trans. AIME, 204, 86-91 (1955). 33. J. E. Bobek, C. C. Mattax, and M. O. Denekas, "Reservoir Rock Wettability-Its Significance and Evaluation," Trans. AIME, 213, 155-160 (1958). 34. J. J. Rathmell, P. H. Braun, and T. K. Perkins, "ReservoirWaterflood Residual Oil Saturation from Laboratory Tests," J. Pet. Technol., pp. 175-185 (February 1973). 35. M. O. Denekas, C. C. Mattax, and G. T. Davis, "Effects of Crude Oil Com­ ponents on Rock Wettability," Trans. AIME, 216, 330-333 (1959). 36. E. Arnott, "Observations Relating to the Wettability of Porous Rock," Trans. AIME, 216, 156-162 (1959). 37. E. C. Donaldson, R. D. Thomas, and P. B. Lorenz, "Wettability Determination and Its Effect on Recovery Efficiency," Soc. Pet. Eng. J., pp. 13-20 (March 1969). 38. L. E. Treiber, D. L. Archer, and W. W. Owens, "A Laboratory Evaluation of the Wettability of Fifty Oil-Producing Reservoirs," Soc. Pet. Eng. J., pp. 531-540 (December 1972). 39. R. L. Sloblod and H. A. Blum, "Method for Determining Wettability of Reser­ voir Rocks," Trans. AIME, 195, 1-4 (1952). 40. O. C. Holbrook and G. G. Bernard, "Determination of Wettability by Dye Adsorption," Trans. AIME, 213, 261-264 (1958). 41. R. T. Johansen and H. N. Dunning, "Relative Wetting Tendencies of Crude Oils by the Capillarimetric Method," Producers Mon., pp. 20-22 (September 1959). 42. R. J. S. Brown and I. Fatt, "Measurements of Fractional Wettability of Oilfield Rocks by the Nuclear Magnetic Relaxation Method," Trans. AIME, 207, 262264 (1956). 43. M. S. Celik and P. Somasundaran, Wettability ofReservoir Minerals by Flotation and Correlation with Surfactant Adsorption (Society of Petroleum Engineers Preprint No. 9002), Presented at SPE Fifth International Symposium on Oil­ field and Geothermal Chemistry, Stanford, California, May 28-30, 1980. 44. B. M. Fung and T. W. McGaughy, "Magnetic Relaxation in Heterogeneous Systems," J. Magn. Reson., 43, 316-323 (1981).

Fluid Flow, Porous Media

368

C. E. Williams and B. M. Fung, "The Determination of Wettability by Hydrocar­ bons of Small Particles by Deuteron Tip Measurement," J. Magn. Reson., 50, 71-80 (1982).

45.

A. L. POZZI, JR.

Fluid Flow, Slurry Systems, Nomograph

In the minerals industries, insoluble mixtures of ore and liquids (slurries) are often used for transport and for processing. Examples are movement of coal­ water mixtures in pipe lines and flotation-separation of iron ore concen­ trates. Slurry calculations require knowledge of the mixture average specific gravity, percent dry solids by volume, and the amount ofliquids required. Let p sand P L = weights of dry solids and liquids, respectively; D s, D L, D M and D w = densities of solids, liquids, mixture, and water, respectively; and S, L, and M = specific gravities of solids, liquid, and mixture and water, respec­ tively. The density of an insoluble mixture is the total weight of all com­ ponents divided by the total volume: (1)

Mixture specific gravity, M = DM/D w, and S = Ds/Dw and L = DdDw Dividing both sides ofEq. (1) by Dw: (2)

Let W

=

percent dry solids by weight:

W=

1001Lps+PsP J

(3)

L

and (3a)

DOI: 10.1201/9781003209812-21

369

Fluid Flow, Slurry Systems, Nomograph

Substituting (3a) in (2) and solving for M: (4)

Let V = percent dry solids by volume:

[f;sJ

V

=

PS+PLJ

=

100[

Ps JM

PS+PL S

DwM

(5)

WM S By manipulating Eqs. (4) and (5):

M

=

VS W W

L(100 - V)

= (100 _ W) =

s+

V L + 100(S - L) (6)

100 (100 - W) L

(7)

VS W= M

VS L L+O.01V(S-L)= 100- M(100- V)

=

(8)

~L )(S~L)

= 100(M S= WM =

V

WM 100 _ M(100 _ W) L

= L(100 -V)( V

W

100 - W

=L + 100(M -L) V (9)

)

In the usual case, the liquid is water (L as shown in the nomographs.

=

1.00), and the equations become

370

Fluid Flow, Slurry Systems, Nomograph

Water at 60°F weighs 8.333 lb/gal, so for any solid of specific gravity (SG), [tons/hr][2000 lb/short ton] gpm = [60 min/h][8.333 x (SG) lb/gal] 4(tph) (SG) W

=

1001 (tphh l L(tphh + (tph)w J 100(tph)s [tons/h slurry = (tphhd

(10)

(11)

or

W)

(tph)w = (100 (tphh W

(12)

and (gpmhv = 4( 100 (tphh W (tphh (tphh (gpm)sL

(tphh

W)

100

=

TV

400 WM

(13)

(14)

(15)

and since M

(gpmh (tphh

(100 - W) + W/S 100 =

l(100 L W

W) +!sJl

(6)

(16)

where gpm = gallons per minute. Figure 1 permits a rapid, simultaneous solution of these equations for water slurries. If any two of the variables are set, the others are determined. Example: What are the other quantities if the dry solids specific gravity, S, = 2.5 and the percent dry solids by volume, V, = 40.0? Align S and V and read mixture average specific gravity, M = 1.60; percent dry solids by weight W = 62.5; gpm of slurry mixture per tph dry solid = 4.00; and that 2.4 gpm of water is required per tph dry solids.

371

Fluid Flow, Slurry Systems, Nomograph

2.1

2.0

M

= VS = (10o-V) W

1.9

V

=

=1 +

=

WM S

~(S-l) 100

(lOG-W)

10o-M(10o-W)=

a

0 200 100

100 (lOG-W) + W/S

100(M-l) =

"lS=1)

1

50 40

100 1 +S

CO~W)

30

= VS W ~ VS = 100- (lOG-V) -100 (M-l) ( S 8=1, 1 +::!.... (5-1) M M M 100

1.8

S= MW=C0o-V)(~~=l+ 100 V V lOG-W V (gpm)SL

4

a

1.7 (gpm)w

~ .s;

s

4

[CO~W)+ -§-J = CO~W

(M-l)=

20

WM WM-l00(M-l) 15

400 WM

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