Processes in GeoMedia―Volume III 3030690393, 9783030690397

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Springer Geology

Tatiana Chaplina   Editor

Processes in GeoMedia— Volume III

Springer Geology Series Editors Yuri Litvin, Institute of Experimental Mineralogy, Moscow, Russia Abigail Jiménez-Franco, La Magdalena Contreras, Mexico City, Estado de México, Mexico Soumyajit Mukherjee, Earth Sciences, IIT Bombay, Mumbai, Maharashtra, India Tatiana Chaplina, Institute of Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia

The book series Springer Geology comprises a broad portfolio of scientific books, aiming at researchers, students, and everyone interested in geology. The series includes peer-reviewed monographs, edited volumes, textbooks, and conference proceedings. It covers the entire research area of geology including, but not limited to, economic geology, mineral resources, historical geology, quantitative geology, structural geology, geomorphology, paleontology, and sedimentology.

More information about this series at http://www.springer.com/series/10172

Tatiana Chaplina Editor

Processes in GeoMedia—Volume III

Editor Tatiana Chaplina Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences (IPMech RAS) Moscow, Russia

ISSN 2197-9545 ISSN 2197-9553 (electronic) Springer Geology ISBN 978-3-030-69039-7 ISBN 978-3-030-69040-3 (eBook) https://doi.org/10.1007/978-3-030-69040-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

Scale Invariant Structure of Lithosphere Earthquake Source . . . . . . . . . . I. R. Stakhovsky

1

Permeability Determination by the Fracture Pressure Drop Curve in Laboratory Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Helen Novikova and Mariia Trimonova

13

Geology and Petrogeochemical Features of the Kumak Ore Field Carbonaceous Shales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. V. Kolomoets and A. V. Snachev

25

Hydraulic Fracturing in Conditions of Non-linear Soil Behavior . . . . . . . M. A. Trimonova and I. O. Faskheev

37

The Exact Equation for the Waveform of Potential Stationary Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. V. Kistovich

45

Identification of the Initial Spot Location of Pollution in the Kerch Strait on the Basis of Adjoint Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kochergin Vladimir Sergeevich and Kochergin Sergey Vladimirovich

55

North Atlantic Oscillation and Interannual Mixed-Layer Heat Balance Variability in the North Atlantic . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. B. Polonsky and P. A. Sukhonos

63

Marangoni Convection in a Cylindrical Container in the Absence of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. V. Kistovich

71

Filtration in Gassy Coal Seams Taking into Account the Dependence of Permeability on Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . V. I. Karev and T. O. Chaplina

83

v

vi

Contents

Development and Sampling of the Device for Collecting Liquid Hydrocarbons from the Water Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. O. Chaplina and E. V. Stepanova

95

Physical Modeling of Deformation, Fracture, and Filtration Processes in Low Permeability Reservoirs of Achimov Deposits . . . . . . . . 107 V. I. Karev, Yu. F. Kovalenko, and Yu. V. Sidorin Features of Accumulation and Spatial Distribution of Microelements in Bottom Sediments of the Crimea Coastal Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 K. I. Gurov, Yu. S. Kurinnaya, and E. A. Kotelyanets The Effect of Perturbations in Marine Environment on Optical Parameters of Near-Water Atmospheric Layer . . . . . . . . . . . . . . . . . . . . . . . 131 V. N. Nosov, S. G. Ivanov, S. B. Kaledin, and A. S. Savin Analytical Representation of a Group Structure Sea Surface Waves . . . . 139 A. S. Zapevalov Methane Fluid Flow from Seafloor: Data from Laspi Bay Seepage Area Compared to Other Gas Emission Regions . . . . . . . . . . . . . . . . . . . . . . 147 T. V. Malakhova, A. A. Budnikov, I. N. Ivanova, and A. I. Murashova Split Between Madagascar and India . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 A. A. Schreider, A. E. Sazhneva, M. S. Kluyev, and A. L. Brekhovskikh The Influence of Geomechanical Factors on the Oil Well Productivity and the Bottom-Hole Zone Permeability of Reservoir Exposed by Slotted Perforation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 S. E. Chernyshov and S. N. Popov Laboratory Research and the Development of Analytical Models of the Elastic-Strength Properties Changes of Cement Materials Used for Casing Wells, Depending on the Hardening Time and the Impact of Clay Acid Reagent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 S. N. Popov and I. Y. Korobov Prediction of Groins Impact on a Sandy Beach . . . . . . . . . . . . . . . . . . . . . . . 195 I. O. Leont’yev and T. M. Akivis Egyptian Pyramids as Indicators of Angular Displacement of the African Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 S. Yu. Kasyanov and V. A. Samsonov Formation of System of Intense Vortices in the Mantle When a Large Temporary Earth Satellite is Immersed . . . . . . . . . . . . . . . . . . . . . . 217 S. Yu. Kasyanov

Contents

vii

Variability the Black Sea Hydrometeorological Characteristics Interconnection with the NAO-Index Extreme Values and Position of Troposphere Frontal Zone Over the European Region . . . . . . . . . . . . . . 233 A. A. Sizov and V. L. Pososhkov Geomonitoring Features with Reduced Load-Bearing Capacity of the Subsoil-Rock Mass in Megalopolis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 D. L. Neguritsa and A. A. Tereshin Modeling of Seepage into a Well Accounting Anisotropy of Rock Strength Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 V. I. Karev, Yu F. Kovalenko, Yu V. Sidorin, E. V. Stepanova, and K. B. Ustinov Analysis of Extreme Daily Precipitation in the Black Sea Region Based on the Max-Spectrum Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 V. L. Pososhkov Hydrodynamic Analysis of Pyrolytic Studies for the Kerogen-Containing Rocks of Romashkinskoye Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 M. N. Kravchenko, N. N. Dieva, and G. A. Fatykhov Assessment of the Pollution by Organic Substances of Water and Sea Bottom Sediments of the Kerch Strait and the Adjacent Azov-Black Sea Water Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 E. A. Tikhonova, O. V. Soloveva, and N. V. Burdiyan On the Issue of Applying Foreign Experience in Environmental Management in the Coastal Territories of the Russian Federation . . . . . . 295 A. Mosalev and A. Sanin

Scale Invariant Structure of Lithosphere Earthquake Source I. R. Stakhovsky

Abstract The formation of the earthquake source is a non-linear and nonequilibrium process of microcracks fractal cluster development in the material of the lithosphere, due to changes in the thermodynamic entropy of the seismogenerating system. The basic structural factor of the source is its scale invariance (multifractality). Due to scale invariant cluster structure, the microcracks coalescence causes macroruptures of any scale, being perceived on the surface of the Earth as earthquakes. The paper outlines the results of computer modeling of the structural organization of microcrack clusters and discusses the possible identification of the earthquake source in the categories of fractal geometry. Keywords Earthquake source · Microcracks · Scale invariance · Multifractals

1 Introduction The existing models of the earthquake source [considered, for example, in the monograph (Rebetskiy 2007)], despite their formal diversity, are united by general conceptual approach to the subject of modeling: the original is interpreted as a material system in which the laws of classical physics are fair, and the metric are expressed in the categories of Euclidean geometry. Meanwhile, experimental studies of recent decades have revealed a new fundamental property of rocks, which in principle does not find an explanation within the framework of classical (equilibrium) physics— scale invariance (self-similarity, fractality) of disjunctive structures in rocks. This property is inherent only in non-equilibrium, dissipative systems that evolve far beyond the applicability of classical physics laws (Keilis-Borok 1990; Turcotte et al. 2003). The destruction of rocks in the Earth’s interior occurs in a strongly nonequilibrium conditions (in equilibrium, destruction is impossible), so any model of the earthquake source have to be based on the fact that an earthquake incident itself is a consequence of a highly non-equilibrium medium state. I. R. Stakhovsky (B) Schmidt Institute of Physics of the Earth, Russian Academy of Sciences (IPE RAS), Bolshaya Gruzinskaya str. 10-1, 123242 Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Chaplina (ed.), Processes in GeoMedia—Volume III, Springer Geology, https://doi.org/10.1007/978-3-030-69040-3_1

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I. R. Stakhovsky

Today, the geometry of rock destruction is studied practically in the entire scale range in which the destruction is realized under lithosphere conditions. Thus, by means of direct measurements by X-ray and electron scanning microscopy methods it is shown that microcracks form sets of fractal structures already on the scale level of the crystal lattice (crack length of 10−9 to 10−7 m) (Radlinski et al. 2000; Jouini et al. 2011). In studies of fracture formation in samples of rocks loaded on laboratory presses (crack lengths of 10−6 to 10−3 m), by methods of optical microscopy and acoustic emission, it is determined that spatial distributions of cracks are almost always scale invariant (Johansen and Sornette 2000). Using computer-aided photo processing methods, the fractal nature of crack distributions is also detected in such objects as fault-gouge, slickensides on mountain outcrops, etc. (crack lengths of 10−2 to 102 m) (Badri et al. 1994). The distributions of seismic active faults on the Earth’s surface mapped by aerial photography (fault lengths of 102 to 106 m) are also subjected to multifractal statistics (Stakhovsky 1996; Ouillon et al. 1996). Thus, the fractal organization of crack sets in rocks is typical for all mesoscales from micro to macro level of destruction, i.e., from the crystal lattice scale level to the scale of earthquakes and seismic active faults, which is a consequence of non-equilibrium character of the destruction process in rocks. The magistral rupture in brittle rocks is always preceded by appearance of cracks, and the spatial distribution of cracks is always fractal. Logic requires to recognize that macrofractures in the Earth’s interior are formed in fractal sets of cracks due to their coalescence. The notion of an empirical criterion of crack coalescence is introduced more than 30 years ago (Zhurkov et al. 1977); at the same time, it was shown that the criterion of crack coalescence has scale invariant nature. The effect of the scale invariant coalescence criterion in a scale invariant ensemble of cracks is considered in Stakhovsky (2017), wherein a principal conclusion is output: for the formation of macrofracture it is sufficient that the fractal set (cluster) consists of only germinal microcracks which have size of several crystal lattice parameters. The mechanisms of appearance of such microcracks are discussed in Zhurkov (1965), Gilman and Tong (1971), where the appearance of microcracks is associated with phonon or quantum fluctuations of the crystal lattice of solids. An avalanche-like coalescence of cracks into macrorupture occurs when the set of cracks reaches the critical value of fractal dimension, i.e., in the form of critical transition. In this paper, the process of brittle rock destruction is considered from the standpoint of multifractal theory (Schertzer and Lovejoy 1987; Mandelbrot 1989), which is used in studies of nonequilibrium systems. The purpose of the study is to generalize modern knowledge about the process of rock destruction in a new definition of the earthquake source concept, which would directly reflect the non-equilibrium state of the medium and would take into account the scale invariant structure of crack sets as a system-forming factor. Such a generalization allows to consider the earthquake source as an element of a natural system without making unreasonable assumptions in the definition of the source and without involving unobvious physical mechanisms in the process of earthquake preparation.

Scale Invariant Structure of Lithosphere Earthquake Source

3

Onwards the term “earthquakes” will refer to lithospheric tectonic earthquakes occurring in brittle rocks of the lithosphere. The structure of deep-focus earthquake sources will not be discussed, we know very little about it so far.

2 Multiplicative Cascade For accordance with the results of the experiment, the physical definition of the earthquake source must reflect the topological properties of the crack distributions known to us by direct measurements. Thus, the source model should include the basic phenomenology of dissipative structures formation in non-equilibrium systems leading to their scale invariant organization. The equations of classical mechanics (partial differential equations) generally have no scale invariant solutions, so for modeling a realistic picture of the microcracks distribution in the zone of macrorupture let’s use such a mathematical procedure as the multiplicative cascade, proposed in Kolmogorov (1962), Obukhov (1962) for description of turbulence. Procedure can be defined as follows. A single D-dimensional segment is subjected to λ D -adic partition (λ is an integer, λ ≥ 2). At the first iteration a positive multiplier m j is associated to each i-th element of the partition, so all of m j form a finite assembly of initial multipliers meeting the conditions: 0 ≤ m 1 ≤ m 2 ≤ · · · ≤ m λ D ≤ 1, λ D j=1

m j = 1.

(1) (2)

In the second iteration, each element of the partition is again subjected to λ D -adic partition and each newly formed element is again associated to one of the initial multipliers. The results of the multiplicative procedure at the second iteration are the products of multipliers of “parent” and “child” elements. The procedure is repeated in the following iterations. Let k denote the number of iterations. At the k-th iteration,  kϕ the i-th element is associated with the product m j j , where ϕj are the relative frequencies with which the multipliers mj are presented in these products. Since λ   kD

kϕ j

mj

= 1, (k → ∞)

(3)

i=1

those quantities pi =



kϕ j

mj

(4)

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I. R. Stakhovsky

can be considered as fractions of a certain multinomial measure P, restricted to the i-th elements of the partition (in the i-th scale grid boxes). Each subsequent iteration modulates the distribution of the measure inherited from the previous scale level, increasing the intermittency of the distribution. The singular measure P in the limit k → ∞ is a multifractal, also for arbitrary shuffles of multipliers of “child” boxes within the “parent” box (multipliers “shuffling”). If the multipliers mj in the cascade procedure keep constant values, the result will be a self-similar, strictly renormalizable measure or so-called “geometric” multifractal. However, the measure P will keep statistical self-similarity also in case if the  D multipliers mj are random numbers for which the condition λj=1 m j = 1 is satisfied only as a mean over the field. In this case, the singularities of the multifractal field will not be localized. When the scale levels change, they will perform random walk over the field [such multifractals are called “stochastic” or “universal” (Schertzer and Lovejoy 1987)]. In the following calculations geometrical multifractals will be used as adequately simulating spatial distributions of microdestructions in solids. If the geometric 2D-cascade contains n non-recurring multipliers related only to the conditions of measure generation (1)–(2), then each iteration creates λ2k boxes of a scale grid, which represent a superposition of (n, k) subsets of boxes with the common value of the restricted to them fractions of the measure pi . These subsets are fractal or, more precisely, become such at k → ∞. At the first iteration, obviously: (n, k) = n, (k = 1)

(5)

From the binomial measure theory, it is known that: (n, k) = k + 1, (n = 2)

(6)

Values (n, k) at arbitrary n and k can be obtained using the recurrence formula (Stakhovsky 2017): (n, k) = (n, k − 1) + (n − 1, k), (n > 2, k > 1)

(7)

The theory of multifractals describes the topological properties of self-similar measures in terms of singularity indices ai and singularity spectra f (a) [more explicit description of the theory is given in particular in Mandelbrot (1989)]. Singularity spectra can be obtained from the results of the experiment by the help of Legendre transform of the so-called cumulant-generating function τ (q). Singularity indices are determined by fractions of measure (4) in grid boxes: 

 ln( pi ) , ai = lim r →0 ln r

(8)

where r—the dimension of the box (scale). For the multiplicative cascade with constant multiplier values, the cumulant-generating function is expressed through

Scale Invariant Structure of Lithosphere Earthquake Source

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the multiplier values as follows:

τ (q) =

ln

 2 λ

q



j=1 m j

.

ln λ

(9)

After performing the Legendre transform, the singularity spectrum expressions are:  λ2

q

m j ln m j , λ2 q ln λ j=1 m j j=1

(10)

f (a(q)) = τ (q) + aq.

(11)

a(q) =

Expressions (10) and (11) allow to construct f (a)-spectrum of the generated scale invariant measure with given values of multipliers of a multiplicative cascade.

3 Mathematical Modeling of Realistic Crack Ensembles in Rocks While investigating crack formation in rocks, self-similar measures emerge as the functions of sets of cracks. The multiplicative cascade provides an analogous opportunity thereby allowing the “visualization” of the structure of self-similar sets. The multiplicative procedure is not limited to any dimension of the embedded space for the generated set, but the graphical representation of 3D-sets is quite difficult, so let’s focus on 2D-distributions. For simplicity of physical interpretation of the results of multiplicative procedure let’s characterize non-empty grid boxes with integer numbers that simulate the “number of microcracks” in boxes at iteration. It should be noted that in experimental studies of 2D crack distributions in rocks the values of fractal dimensions are measured up

  to d f ≈ 1.6−1.7,  while the values N (r ) /ln(1/r ), where N (r )—number of d f ≥ 1.8 do not occur (d f = lim ln r →0

non-empty boxes in renormalizable scale grid). Apparently, critical values of fractal dimensions of crack sets in rocks (i.e. values d f corresponding to an avalanche-like coalescence) are within the range 1.7 ≤ d f ≤ 1.8 and probably depend on the rock type. Therefore, let’s use for the purpose of modeling such a multiplicative generator, which will allow to obtain a set similar in structure to the set of microcracks ready to coalesce in macrorupture, i.e. set with fractal dimension 1.7 ≤ d f ≤ 1.8. The multiplicative generator for this purpose may be as follows: 1

1 1

1

1 1 (continued)

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I. R. Stakhovsky

(continued) 1 1

1

1

1

1

1

1

1

1

1

1

1 1

This generator contains a grid with 5×5 (λ = 5) resolution and the units in the grid boxes (“elements of partition of 2D-segment”) indicate the locations of 17 non-empty boxes, with all non-empty boxes being “equivalent”. Mathematically it means that all the cascade multipliers are equal to m j = 1/17 = 0.0588… In physical interpretation it may mean that as a result of iteration we get a grid with one “microcrack” in the non-empty boxes. The multiplicative procedure will create in this case a set of nonempty boxes (a “set of cracks”), which (after an infinite number of iterations) can be considered as a monofractal with fractal dimensions d f = ln17/ln5 = 1.760…, that corresponds to the above mentioned physical preconditions of modeling. The format of a book illustration can represent the results of only second iteration of such procedure. Strict adherence to the order of multipliers in “child” boxes within “parent” boxes leads to set structure, shown in Fig. 1a (the grid itself is not shown, the points indicate the locations of non-empty boxes). Each point of the generated set is an indicator of the location of exactly the same set of smaller scale (each element of which is also the same set of even smaller scale, etc.). Of course, the probability of finding out such a systematic distribution of microcracks in rocks tends to zero. A more realistic distribution can be obtained by adding to the described procedure the multipliers “shuffling” with the help of random number generator. One of the implementations of the set which can be obtained by such a procedure is shown in Fig. 1b. Despite the obvious difference in the configurations of the sets in Fig. 1a, b, both of them represent strictly renormalizable monofractal sets with the same value of fractal dimension. However, the real ensembles of cracks in rocks have a self-similarity of general form, so to model them it is necessary to use generators with unequal values of multipliers. For example, at the first iteration the generator may be like this: 1 1 1

1

1

1

2

1

2

3

2

1

2

1

1

1 1

In this generator, the maximum and minimum values of the multipliers are equal to m min = 1/23 = 0.043 . . . , m max = 3/23 = 0.130 . . . . Withstanding the strict order of multipliers at iterating process we obtain after the second iteration the set, shown in Fig. 1c. In this case, the generated set shows the rarefaction and condensation of elements.

Scale Invariant Structure of Lithosphere Earthquake Source

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Fig. 1 Spatial distributions of microcracks generated by multiplicative cascade: a and b monofractal distributions, c and d multifractal spatial distributions; left column—with strict uphold of the multipliers order during iteration, right column—with multipliers “shuffling” by the random number generator

Adding to the computer procedure the multipliers “shuffling” we can obtain the set, which can be considered as a “realistic” spatial distribution of microcracks in the seismo-generating system. One of the implementations of the set obtained by multipliers “shuffling” is shown in Fig. 1d. The set in Fig. 1d preserves signs of the algorithmic and program solution of the problem (the influence of grid structure and grid borders can be distinguished) but the topology of the set and the random character of the distribution of its elements correspond to the real microcracks ensembles in rocks. The sets in Fig. 1c, d induce measures with the same spectrum of singularities. Constructed using expressions (10)–(11), the f (a)-spectrum is shown in Fig. 2. The spectrum extremum is f (a)max = 1.760 . . . (as d f in the monofractal case).

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Fig. 2 The singularities spectrum for multifractal measures induced by the sets shown in Fig. 1c, d

Maximum and minimum values of singularity indices can be determined using expression (8): amin = lnm max /ln(1/λ) = 1.265 . . . , amax = lnm min /ln(1/λ) = 1.948 . . . . At the considered structure of the multiplicative generator the measure f (a)-spectrum is asymmetric. By changing the structure of the cascade and the values of multipliers we can obtain an infinite variety of scale invariant sets, topologically equivalent to real sets of microcracks in rocks. We can see that realistic distributions arise as a result of mathematical procedure with a high level of nonlinearity and randomness, i.e. a procedure that cannot be reduced to an analytically integrated system of differential equations. In nature the situation is analogous, i.e. self-organization and formation of scale invariant dissipative structures are observed in strongly nonlinear, nonequilibrium, open and non-integrable systems consisting of statistically large number of metastable subsystems. Such a system is the source of a forthcoming earthquake. The scale invariance of crack spatial distributions in rocks is explained not by the stress field features in the medium or its texture, but by the properties of nonequilibrium multicomponent systems adequately represented by a multiplicative cascade procedure. The multiplicative cascade allows modeling the structure of real ensembles of cracks observed experimentally and, consequently, the structure of ensembles of cracks in earthquake sources.

Scale Invariant Structure of Lithosphere Earthquake Source

9

4 Earthquake Source Definition and Conclusions The generalization of the results of experimental and theoretical studies of rock fracture leads to the definition of the crustal earthquake source in the categories of fractal geometry. Indeed, the strong instability and disequilibrium of the earthquake preparation process destroy the deterministic understanding of the source. The nonequilibrium state of the lithosphere material and the scale invariance of microcrack sets formed in the lithosphere material prove to be the key factors of macrofracture in the Earth’s interior (earthquakes). The process of earthquake preparation is a process of microcracks accumulation up to achievement of critical value of fractal dimension in sets of microcracks that leads to avalanche-like coalescence of microcracks in macrorupture. In this case mechanical stresses are not the cause, but a catalyst for fracture, i.e. a catalyst for dissociation of molecular bonds in the material (Zhurkov 1965; Gilman and Tong 1971). The system in which the earthquake source is formed: (i) is open, (ii) is nonlinear, (iii) is non-equilibrium, (iv) is self-organized, (v) has a scale invariant structure. Thus, the earthquake source should be defined as a fractal cluster (set) of microcracks formed by changes in the thermodynamic entropy of the seismogenerating system and accumulating microcracks up to avalanche-like coalescence of cracks (in consequence of achievement of critical value of fractal dimension). It should be noted that this phenomenological scheme has no any assumptions— all stages of the macrofracture preparation process are experimentally confirmed. However, because of the fundamental impossibility of direct verification, this definition (like any other) can only be established as a hypothesis or a model. Nevertheless, such a model does not contradict any of the known facts and therefore allows some conclusions. 1.

2.

The source of a forthcoming earthquake in the form of a fractal cluster of microcracks has no specific size, volume, envelope or characteristic scale in the Euclidean sense of these concepts. As can be seen from Fig. 1d, even the formation of a magistral rupture in the fractal cluster of microcracks most likely occurs not in the form of the propagation of a single crack from a certain point, but in the form of a coalescence of microcracks simultaneously in many points when the fractal dimension of crack set reaches a critical value. Due to scale invariance of microcrack set, earthquakes (large-scale rupture of rocks in the lithosphere massif) are caused by changes in thermodynamic entropy, i.e. the cause of the fracture emanates from the non-equilibrium state of the crystal lattice itself. External factors can only accelerate or decelerate the process of accumulation of microcracks. The movement of seismic fault banks, generation of seismic waves, etc. are macroscopic consequences of nonequilibrium thermodynamic processes occurring at the crystal lattice level—thermodynamic fluctuations, destruction of molecular bonds, accumulation of microcracks, spatial organization of microcracks in the form of self-similar sets, critical transitions in their structure (avalanche-like microcracks coalescence), etc.

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3.

For earthquake genesis no intermediate physical mechanisms (like fluid diffusion as “trigger” effect) are required on the mesoscales between microcracks caused by thermodynamic fluctuations and the macrorupture. It is enough that the process of microcracks accumulation has a critical transition when the corresponding value of fractal dimension is reached in the set of microcracks. The model of earthquake source in the form of fractal cluster of microcracks is fair not only in conditions of the Earth, but also, for example, in conditions of the Moon (where there is seismicity, but there can be no any unbound fluids). The processes leading to earthquakes occur in any fragment (piece) of crystal rock regardless of its size, inasmuch as the cause of fracture is a non-equilibrium, fluctuating state of the crystal lattice itself, giving rise to the appearance of microcracks in rocks. Natural fracture begins to be perceived as seismicity, when the “rock fragment” becomes very large, for example, the size of a lithospheric plate, and the fracture, due to its scale invariance, reaches “seismic” scales. The very fact of scale invariance of seismic structures, apparently, means that the description of the seismo-generating system cannot be reduced to an analytically integrated system of differential equations. The basic physicochemical phenomenon inherent in the preparation process of any crustal earthquake is self-organization of the seismo-generating system caused by its non-equilibrium state.

4.

5.

These properties make earthquake to be perceived as a result of non-equilibrium evolution of physicochemical system. In terms of the kinetic concept of strength (Zhurkov 1965) earthquake should be considered as nonequilibrium solid state dissociation reaction in the lithosphere material. More radical opinions are also expressed in literature, for example, analogies between earthquakes and “chain chemical explosions” are drawn, whereas the lithosphere is proposed to be considered as a “macroreactor” (Buchachenko 2014). The definition of the earthquake source in terms of fractal geometry is its description at the level of topology. However, as Fig. 1d shows, a realistic distribution of microcracks occurs only as one of the realizations in the mathematical procedure that generates the distribution of probability of microcracks appearance, i.e. probability multifractal field (note that unlike the above examples, the real multifractal field is three-dimensional). The distribution of probability is a more general physical category than one of the specific sets generated by this distribution from an infinite number of possible variants. Thus, the singularity spectrum of a multifractal field similar to that shown in Fig. 2 (i.e. the singularity spectrum of the probability distribution defined in an infinite scale range) can be considered as a universal structure characteristic of the earthquake source defining the properties of any fractal cluster of microcracks that may be engendered by this multifractal field. For the practical construction of the f (a)-spectrum of multifractal probability field, which forms the earthquake source, it is indispensable to have information on the spatial distribution of microcracks in the source. At the same time, the process of microcrack formation occurring in lithosphere massif at depths of tens of kilometers on the scale level of the crystal lattice of minerals cannot be directly controlled.

Scale Invariant Structure of Lithosphere Earthquake Source

11

However, due to the scale invariance of brittle fracture, seismicity allows extrapolating the observed phenomena into the inner space of the lithosphere inaccessible for observation. Microcrack formation and small-scale seismicity can be defined as different scale subranges of a single scale invariant fracture process, thus the topological characteristics of small-scale seismicity repeat the topological characteristics of microcrack sets. The singularity spectrum of the spatial distribution of hypocenters of small-scale events in the focal region of a strong forthcoming earthquake is analogous to the singularity spectrum of the spatial distribution of microcracks in the source. Thus, the singularity spectrum of the spatial distribution of hypocenters of small-scale events in the focal region of a strong earthquake can be interpreted (at least in principle) as the singularity spectrum of the three-dimensional multifractal probability field, which engendered the earthquake source (i.e. fractal cluster of microcracks). The technical difficulties arising on the way of such interpretation are quite obvious: lack of seismic data, errors in determining the coordinates of hypocenters, etc. However, the quality of seismic data is continuously improving, and technical problems tend to be solved with time. The possibility to have an objective structure characteristic of the earthquake source makes consider the construction of f (a)-spectrum of multifractal (probability) field engendering the earthquake source as a perspective problem.

References Badri A, Touchard G, Velde B et al (1994) Image processing software for fractal analysis of fractures in rocks. Acta Stereol 13(1):183–188 Buchachenko AL (2014) Magnetoplasticity and the physics of earthquakes. Can a catastrophe be prevented? Phys Usp 57(1):92–98 Gilman JJ, Tong HC (1971) Quantum tunneling as elementary fracture process. J Appl Phys 42(9):3479–3486 Johansen A, Sornette D (2000) Critical ruptures. Eur Phys J B18:163–181 Jouini MS, Vega S, Mokhtar EA (2011) Multiscale characterization of pore spaces using multifractals analysis of scanning electronic microscopy images of carbonates. Nonlinear Process Geophys 18(6):941–953 Keilis-Borok VI (1990) The lithosphere of the Earth as a nonlinear system with implications for earthquake prediction. Rev Geophys 28(1):19–34 Kolmogorov AN (1962) A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J Fluid Mech 13:82–85 Mandelbrot B (1989) Multifractal measures, especially for geophysicist. Pageoph 131(1–2):5–42 Obukhov A (1962) Some specific features of atmospheric turbulence. J Geophys Res 67:3011–3014 Ouillon G, Castaing C, Sornette D (1996) Hierarchical geometry of faulting. J Geophys Res 101(B3):5477–5487 Radlinski AP, Radlinska EZ, Agamalian M (2000) The fractal microstructure of ancient sedimentary rocks. J Appl Crystallogr 33(1):860–862 Rebetskiy YuL (2007) Tectonic stresses and strength of natural massifs. Academkniga Research Center, Moscow, 406 p (in Russian) Schertzer D, Lovejoy S (1987) Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes. J Geoph Res D 92(8):9693–9714

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Stakhovsky IR (1996) Multifractal analysis of fault structures in basement rocks. In: Oncken O, Janssen C (eds) Basement tectonics 11. Europe and other regions. Kluwer Academic Publishers, Dordrecht, Boston, London, pp 101–110 Stakhovsky IR (2017) Scale invariance of shallow seismicity and the prognostic signs of earthquakes. Phys Usp 60(5):472–489 Turcotte DL, Newman WL, Shcherbakov R (2003) Micro and macroscopic models of rock fracture. Geophys J Int 152(3):718–728 Zhurkov SN (1965) Kinetic concept of strength of solids. Int J Fracture Mech 1:311–323 Zhurkov SN, Kuksenko VS, Petrov VA et al (1977) To the rock fracture prediction. Izv USSR Acad Sci Phys Earth 6:11–18 (in Russian)

Permeability Determination by the Fracture Pressure Drop Curve in Laboratory Experiments Helen Novikova

and Mariia Trimonova

Abstract This paper considers one of the known methods for reservoir permeability determination in the time interval after the closure of the fracture that appears during the hydraulic fracturing process. The described technique was applied to pressure dependency curves obtained during a series of laboratory experiments on formation and propagation of a fracture. In this study, the permeability values obtained by using some approximations, such as steady pseudoradial regime in the reservoir after the hydraulic fracturing crack is closed and the possibility to neglect the not fully closed crack in the reservoir, are considered. The values obtained during the analysis by this permeability determination method are compared with the real values of permeability which are measured in preliminary experiments. Keywords Hydraulic fracturing · Pressure curve · Permeability of the laboratory sample

1 Introduction The data obtained during a series of laboratory fracturing experiments is analyzed in this work. The hydraulic fracturing (Fig. 1) method is the most effective method of oil production intensification in a real field, and consists of a mechanical effecting on a productive oil or gas reservoir by injecting high pressure fluid into it through a well (Turuntaev and Kocharyan 2007; Economides et al. 2002). The majority of currently known methods for determination of permeability and other reservoir characteristics require additional procedures at the real field, in some cases demands even stopping field development for a long period of time (Mangaseev et al. 2004; Kremenetsky and Ipatov 2008; Hayrullin et al. 2012). Also the results H. Novikova (B) · M. Trimonova Sadovsky Institute of Geospheres Dynamics, Russian Academy of Sciences, Leninsky ave., 38-1, 119334 Moscow, Russia e-mail: [email protected] M. Trimonova e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Chaplina (ed.), Processes in GeoMedia—Volume III, Springer Geology, https://doi.org/10.1007/978-3-030-69040-3_2

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H. Novikova and M. Trimonova

Fig. 1 Fluid injection and crack formation: qi —rupture fluid flow, which is supplied to the well under pressure

obtained by the mentioned methods highly depend on the accuracy of supplementary measurements during the additional stages of field processing, which requires large financial expenses, especially when developing low permeability formations. Consequently the search for a method of miniFrac data analysis has become relevant, which will make possible to reveal some reservoir characteristics (permeability, coefficient of fluid leakage into the reservoir and initial pressure in the reservoir) before the start of the full field development process (Soliman 1986; Baree et al. 2007; Castillo 1987). The principal purpose of this work is to determine the permeability of the laboratory sample by analyzing the data of pressure variation with time obtained during a series of laboratory experiments on the formation and propagation of fractures during hydraulic fracturing.

2 Method Description The method for reservoir permeability determination during hydraulic fracturing, applied in this study, is constructed on the theory for the formation with a single well in which the radial flow regime exists proposed by Horner (1951). According to this theory, the relation of pressure in the well on time can be recorded as follows: pw = pi +

  qμ tc + θ , ln 4π kh θ

(1)

Permeability Determination by the Fracture Pressure Drop Curve …

15

where pw is well pressure, pi —initial pressure in a model sample, θ = t − tc time after the crack has been closed, t—actual time, tc —crack closure time, q— fracturing fluid injection rate, μ—fluid viscosity, k—model sample permeability, h—height of model specimen. However, paying attention to the specification of laboratory experiments, which are relatively close to the real conditions of hydraulic fracturing, it is impossible to consider an infinite continuous formation with a single well. The applied method to determine the permeability values for reservoir is implemented to the data of pressure drop in the period of time after the hydraulic fracture formation unto its closure. The developed theory for the case of a present closed crack with the radial fluid flow regime, according to Nolte et al. (1997), and from which there are minor fluid leaks into the reservoir with a constant rate, leads to modification of expression (1) into the following form pw = pi +

  χ tc qμ 1 , ln 1 + 4π kh χ t − tc

(2)

where χ tc —crack closure time, which is equivalent to the time of liquid leakage, χ = 16/π2 . In the right part of the equation there is some function depending on the time of crack closure and the current moment of time in the experiment, so expression (2) can be simplified by introducing a new dimensionless function of time: Fr (t, tc ) =

  χ tc 1 ln 1 + 4 t − tc

(3)

Then Eq. (2) becomes linear: pw − pi = Mr Fr (t, tc ),

(4)

where Mr = πqμ/16hk—the slope factor of the pressure-dependence curve on the dimensionless time function Fr (t, tc ). Knowing from laboratory experiments the dependence of pressure on time, permeability can be expressed from the formula (4). Since this theory considers the formation in the time interval when the crack has already closed, it is necessary to determine the value of the time of the fracture closure. For this the G-function method is used, based on the works of Nordgren (1972) and Nolte (1979). The given method is directed on construction of dependences of the first derivative of pressure and a semi-logarithmic derivative of pressure, and also pressure itself from dimensionless G-function, depending on time. The characteristic behavior of these curves is used to determine the time of crack closure. The pressure dependence on the G-function is derived from the continuity equation written for fluid injected into the reservoir and propagating in the fracture and formation

16

H. Novikova and M. Trimonova

−

∂ A(z, t) ∂ Q(z, t) = λ(z, t) + , ∂z ∂z 2C H p λ= √ t − τ (z) h/2

A=

w dz = −h/2

π π H2 Wh = p 4 2E

(5) (6)

(7)

where Q—fluid flow rate, λ—fluid losses per unit of crack length, A—crack cross-section area, C—fluid loss factor, H p —height of crack at which leaks into the formation emerge, τ —time of crack propagation  start in point z, H—crack height,  W —maximum crack width, E  = E/ 1 − ν 2 , where E—Young modulus and ν— Poisson’s coefficient. By integrating the resulting ratio along the entire length of the crack, the necessary pressure dependence on the dimensionless function of time is obtained P(δ0 , δ) = δ=

√ C H p E  t0 G(δ0 , δ), H 2 βz

(8)

t t − t0 = , t0 t0

(9)

P(δ0 , δ) = P(δ0 ) − P(δ), G(δ0 , δ) =

 16  3/2 (1 + δ)3/2 − δ 3/2 − (1 + δ0 )3/2 + δ0 , 3π

(10) (11)

where t0 —pumping time before closing, δ—dimensionless time dependent function, βs—ratio of average pressure to the downhole pressure during a fluid injection stop.

3 Description of the Experiment and the Experimental Facility Experimental facility for modeling the formation and propagation of hydraulic fracturing process is located in the Institute of Geosphere Dynamics of the Russian Academy of Sciences (IDG RAS). Its design consists of two steel caps, one of which has a groove for pouring material that simulates the formation in which hydraulic fracturing occurs. The diameter of the working chamber is 43 cm and its height is 6.6 cm. The model reservoir sample is separated from the top cover by a rubber membrane. There is some sealed airtight space between the cover and the membrane,

Permeability Determination by the Fracture Pressure Drop Curve …

17

Fig. 2 Schematic illustration of the facility

which is filled with water to simulate the lithostatic pressure on the sample. This pressure is provided and maintained by a separating cylinder, the upper part of which is filled with compressed nitrogen at the required pressure magnitude and the lower part is filled with water. Horizontal loading of the model sample is attained by injecting gas or liquid into the copper chambers located on the sidewalls (Fig. 2). The experiment is conducted in several stages. Initially, a sample which will be used for hydraulic fracturing is prepared from a mixture of gypsum with the addition of cement. The mixture is then poured into the facility groove and, after it has hardened, saturated with water. After some time the fracturing fluid, i.e. oil, is injected at a constant rate to create a hydraulic fracture. Prior to the fracturing experiments, the permeability of the sample material is determined experimentally. For this purpose, cylindrical specimens with a diameter of 105 mm and a height of 65 mm with a 15 mm hole in the center are manufactured. These samples are then placed in a special facility between parallel plates and rubber gaskets, where they are compressed by a screw-down press (Fig. 3).

Fig. 3 Experimental facility for permeability determination of reservoir model samples

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H. Novikova and M. Trimonova

Fig. 4 Example of the relation of the water flow rate through a cylindrical sample on the pressure drop obtained in a laboratory experiment

Canals with an outlet into the center hole of the sample are made in the plates. Through this canals the liquid is fed. Permeability measurements are carried out at different flow rates. Relation between flow rate and pressure drop is found based on the results of the experiments (Fig. 4). During the laboratory experiments the permeability of the model material is determined to equal to 2 mD. Permeability is calculated according to the Dupuy formula Q=

2π kh( pc − pe ) μ ln Re /Rc

(12)

where k is the sample permeability, h—the height of the sample, pc —pressure in the center hole, pe —pressure on the external surface, μ—the viscosity of the fluid, Re —the outer radius of the sample, Rc —the radius of the center hole. After a series of laboratory experiments on the formation and propagation of fractures were obtained pressure dependences on time (Fig. 5a–c), recorded with a sensor located inside the well. As a result of laboratory experiments, the vertical crack that came out to the model sample surface (Fig. 5a), horizontal crack that came out to the model sample surface (Fig. 5b) and horizontal crack that did not come out to the model sample surface (Fig. 5c) are formed.

4 Results of Methods Application The pressure data obtained during laboratory experiments are initially smoothed out, then the G-function method is applied to the obtained dependencies in order to find the time of crack closure, which in turn is necessary for further calculations of permeability of model samples used in laboratory experiments.

Permeability Determination by the Fracture Pressure Drop Curve …

19

Fig. 5 Relation of pressure to time in experiments: a–c first, second and third respectively

According to the G-function method, the crack closure time is determined by the characteristic cumulative behavior of three curves: the pressure curve on the G-function, the first derivative of pressure on the G-function curve, and the semilogarithmic derivative of pressure on the G-function. Since this technique is applied to the data recorded during the time interval after stopping the injection of the fracture fluid into the sample, it is assumed that the fluid leaks into the reservoir (sample) at a constant rate. In this case, the semi-logarithmic derivative behaves like a straight line passing through the origin until the crack is closed. Therefore, the goal of the technique is to determine the moment of time when the semi-logarithmic pressure derivative deviates from a straight line passing through the origin. The correctness of the time value found is checked by the fact that at the same time the first pressure derivative has the behavior of a horizontal line (Castillo 1987). This technique is applied to data from three laboratory experiments on the formation and propagation of fractures. Since all pressure dependency curves showed sharp decrease in pressure immediately after stopping the injection of fracturing fluid into the sample, this pressure jump is not considered in further application of the G-function technique. The results of data processing by G-function method are presented in graphs (Fig. 6). Thus, the crack closure times are determined in each of three experiments mentioned above. The next step is to determine the permeability of formation models samples used in laboratory experiments. Graphs of pressure dependence on dimensionless function F r (t, t c ) are plotted. In the initial period of time, when, according to the applied method, the pseudoradial mode of flow is set inside the model sample, the curve behaves like a straight line, which has a characteristic slope necessary to calculate the permeability of the model formation sample (Fig. 7).

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H. Novikova and M. Trimonova

Fig. 6 Dependencies of pressure on G-function (purple solid line), dependence of pressure derivative on G-function (thin dotted line), dependence of semi-logarithmic pressure derivative on Gfunction (blue dashed line). The moment of crack closure is marked by a vertical dash-dotted line

Permeability Determination by the Fracture Pressure Drop Curve …

21

Fig. 7 Dependence of the pressure on the dimensionless function of time in the period after the crack closure in the experiments: a–c first, second and third respectively. Blue solid line shows pressure dependence on time. Red dash-dotted line shows the characteristic slope equal to M r

The found values of crack closure time and slope of the pressure dependence curve on the time function F r (t, t c ) allowed to calculate permeability for each laboratory experiment. The data from three laboratory experiments on formation and propagation of hydraulic fracturing cracks are processed in the presented paper. The obtained values of some characteristics calculated during the data processing are presented in Table 1. By comparing the calculated permeability values of the model samples with their real permeability determined experimentally and equal to 2 mD, the inconsistency is found. This discrepancy shows that under the conditions of these experiments the steady pseudoradial flow regime does not exist in the sample after the crack closure, i.e. it is assumed that the crack has not totally closed. This assumption is confirmed at further opening of the samples from the experimental facility. It is found that oil is flowing out of the crack in the model sample, hence the crack have not actually closed firmly enough that the leaks could be neglected. Table 1 Characteristics obtained through the processing of pressure–time data Experiment No

Crack closure time, s

Permeability, mD

1

235.2

169.6

2

72.8

23.2

3

221.8

196.0

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H. Novikova and M. Trimonova

5 Conclusions The study is resulted in processing data from three hydraulic fracturing experiments. The data are analyzed using a special permeability technique based on the study of pressure drop curves obtained during hydraulic fracturing. This made possible the determination of the model sample permeability and further comparison of the obtained values with the real value of permeability known from the conditions of laboratory experiments. In order to apply this technique, the data of hydraulic fracturing pressure dependence on time are processed using the G-function method in order to determine the necessary value of fracture closure time. Significant differences in experimental and calculated permeability values are explained by the presence of an unclosed fracture. Thus, the study shows that when processing hydraulic fracturing pressure curves with standard methods, an unclosed fractures in the reservoir can significantly affect the flow regime established after fracture is closed, resulting in the permeability value. Acknowledgements The research was supported by RFBR, project No. 20-35-80018 and state task 0146-2019-0007.

References Baree RD, Baree VL, Craig DP (2007) Holistic fracture diagnostic. Society of Petroleum Engineers, USA, 13, SPE-107877-MS Castillo JL (1987) Modified fracture pressure decline analysis including pressure-dependent leakoff. Society of Petroleum Engineers, Denver, pp 273–281, SPE-16417-MS Economides M, Olini R, Valko P (2002) Unified design of hydraulic fracturing. Institute of Computer Science, Orsa Press, Izhevsk, Moscow, p 236 (in Russian) Hayrullin MH, Khisamov RS, Shamsiev MN, Badertdinova ER (2012) Hydrodynamic methods for investigation of vertical wells with hydraulic fracturing crack. Institute of Computer Science, Izhevsk, Moscow, p 84 (in Russian) Horner DR (1951) Pressure build-up in wells. World Petroleum Congress, Netherlands, pp 25–43, WPC-4135 Kremenetsky MI, Ipatov AI (2008) Hydrodynamic and field-technological studies of wells. Gubkin Russian State University of Oil and Gas, Moscow, p 476 (in Russian) Mangaseev PV, Pankov MV, Kulagina TE, Kamartdinov MR, Deyeva TA (2004) Hydrodynamic well research. Center for Professional Retraining of Oil and Gas Specialists at Tomsk Polytechnic University, Tomsk, p 340 (in Russian) Nolte KG (1979) Determination of fracture parameters from fracturing pressure decline. Society of Petroleum Engineers, Nevada, p 16, SPE-8341-MS Nolte KG, Maniere JL, Owens KA (1997) After-closure analysis of fracture calibration tests. Society of Petroleum Engineers, Texas, pp 333–348, SPE-38676-MS

Permeability Determination by the Fracture Pressure Drop Curve …

23

Nordgren RP (1972) Propagation of a vertical hydraulic fracture. Society of Petroleum Engineers, Houston, pp 306–314, SPE-3009-PA Soliman MY (1986) Analysis of buildup tests with short producing time. Halliburton Services Research Center, Society of Petroleum Engineers, New Orleans, pp 363–371, SPE-11083-PA Turuntaev SB, Kocharyan GG (2007) Introduction to geophysics of hydrocarbon fields. Moscow Institute of Physics and Technology (MIPT), Moscow, pp 177–335 (in Russian)

Geology and Petrogeochemical Features of the Kumak Ore Field Carbonaceous Shales A. V. Kolomoets

and A. V. Snachev

Abstract The geological structure of the Kumak ore field is concidered. Special attention is paid to the carbonaceous shales of the Bredy formation (C1 bd), which compose elongated submeridional blocks of the Anikhov Graben. Rocks are saturated with carbonaceous matter to varying degrees. According to chemical and thermogravimetric analyses, the content of Corg in them reaches 11.1%, which allows to classify the considered rocks as carbonaceous. It is shown that black shales fall into the fields of terrigenous-carbon and siliceous-carbon formations. Carbonaceous matter is represented by two types: weakly metamorphosed sapropel sedimentationdiagenetic and metamorphic graphite. Terrigenous high-alumina planting material underwent minimal transfer and was formed mainly due to the destruction of rocks of protophile composition, as well as the products of erosion of acidic volcanites at the Bredy formation section base. Sedimentation experienced a high degree of weathering that is characteristic for a humid climate with oxidative and partially suboxidative conditions. The deposition of material occurred in a transition from riftogenic to collisional geodynamic environment. Black-shale deposits of the Kumak ore field according to the parameter Na2 O/K2 O = 0.62 belong to the normal potassium formation type, which is typical for deposits with mainly gold-sulfide mineralization. Keywords South Ural · East Ural uplift · Anikhov Graben · Bredy formation · Carbonaceous shale · Kumak ore field · Gold

A. V. Kolomoets (B) Orenburg State University, 13 Pobedy ave., Orenburg 460018, Russia e-mail: [email protected] A. V. Snachev Institute of Geology, UFRC RAS, 16/2 Karl Marx street, Ufa 450077, Russia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Chaplina (ed.), Processes in GeoMedia—Volume III, Springer Geology, https://doi.org/10.1007/978-3-030-69040-3_3

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A. V. Kolomoets and A. V. Snachev

1 Introduction Gold manifestations of the vein-included type are known in the black shale strata in the Orenburg region. One of the significant ore-transmitting structures containing such objects is the Anikhov Graben, one of the fragments of a major tectonic disturbance (the Chelyabinsk deep fault) that can be traced along the East-Ural Rise. In the central part of the folding zone, in the interval of ten kilometers along submeridional extension, there are a number of gold ore manifestations of the Kumak ore field (Kumak, Kumak-Yuzhny, Zabaikalskoe, Centralnoe, etc.) (Fig. 1). The industrial mineralization has confinedness to the carbonaceous-terrigenecarbonate sediments of the Lower Carboniferous Age. The native gold is found both in the shale part of ore bodies and in quartz veins saturating the shale. Interest in the black shale strata has increased significantly while the primary sources of gold in the formation of gold-bearing rocks are found there, and sometimes the black shale strata contain proven reserves and manifestations with industrial concentrations of noble and rare metals (Arifulov 2005; Loshhinin and Pankrat’ev 2006; Sazonov et al. 2011; Snachev et al. 2013; Khanchuk et al. 2013; Cherepanov et al. 2017; Kolomoets et al. 2019).

2 Research Methods Silicate analysis of carbonaceous shales is carried out by standard methods at the Institute of Geology of the UFSC RAS (Ufa, analyst S.A. Yagudina). The rock sections are described using the polarization-optical microscope AxioObserver with the digital video camera AxioCam HRc (1300 × 1030) at the CUC “Spectrum” (Institute for Physics of Molecules and Crystals of the UFSC RAS, Ufa).

3 Geological Structure Carbonaceous terrigenous-sedimentary formations form the lower part of the Lower Carboniferous section and are represented by the Bredy Formation, which is widely developed in the Kumak ore field area. The formation was isolated by A.A. Petrenko in 1946 and named after the village of Bredy in the Chelyabinsk region. Its section is dominated by carbonaceous terrigenous-sedimentary formations: siltstones, carbonaceous clayey shales and sandstones, as well as rare interlayers of limestone and coal. At the base of the section there are effusives of dacite and andesite porphyrites and their tuffs. Mineral paragenesis of rocks corresponds to green shale facies. The washed Beredinsk Formation is located in the Bereznyakov Stratum (D3 C1 bz) and is overlapped with by the washed Bergildin formation (C1 br). Its capacity is 350–700 m. The age of the deposits is determined by foraminifera detection in

Geology and Petrogeochemical Features of the Kumak …

27

Fig. 1 Geological map of the Kumak ore field (Southern Urals) (by Ljadskij et al. (2018), simplified by authors). Symbolic notation: 1—Birgildinsk stratum (conglomerates, sandstones, limestone, carbonaceous shale), 2—Bredy formation (carbonaceous shale, sandstone, siltstones, conglomerates), 3—Bereznyakov stratum (tuffs of basic and acidic composition, less often lava, layers of siltstones and carbonaceous shale), 4—Kokpektinskaya stratum (lava and tuffs of basalt, subvolcanic bodies of gabbrodolerites, rhyolites), 5—Jabyk-Sanar granite-leukogranite complex, 6—Kumak diorite-plagi granite complex, 7—gold manifestations and deposits: 1—East Tykashinskoye, 2— Commercheskoe, 3—Milya, 4—Tamara, 5—Zabaikalskoe, 6—Baikal, 7—Centralnoe, 8—Kumak, 9—Kumak-Yuzhny

the limestone interlayers and by the traces of microfauna and spores of ancient ferns, calamites and other plants (Ljadskij et al. 2018). Near the zones of tectonic disturbances shown along the East and West Anichov deep faults, there is developed adjoining folding with intensive folding and metasomatic elaboration (quartzing, sericization, chloritization, pyritization and tourmalization) (Novgorodova et al. 1981). The composition and facial variability of the studied sections of the Bredy

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A. V. Kolomoets and A. V. Snachev

Formation largely depend on their confinement to the main elements of the area structure. The rocks of the Bredy Formation are rich in carbonaceous matter to varying degrees (Arifulov 2005; Sazonov et al. 2011; Kolomoets et al. 2019). According to chemical and thermogravimetric analyses, the Corg content in these rocks reaches 11.1%, which allows us to classify the considered rocks as carbonaceous (Yudovich Ya and Ketris 2015). The carbonaceous substance is in a finely atomized state or as a mass cementing the rest of minerals. Its composition is dominated by sapropelic components in form of irregularly shaped fragments, but in the areas of tectonic processing and metamorphism it is represented by large vein and scaly graphite extractions. Depending on the composition and ratio of the composing components black shales of the considered area are subdivided into sericite-quartzcarbonaceous, quartz-carbonaceous-turmaline, otrelitic-carbonaceous and quartzcarbonaceous-otrelitic. The first subdivision is the most widely spread and represented rocks of greyish black, sometimes black, fine grained, with weakly expressed shale, easily splitting on shale planes with angular fracture. According to petrographic description, black shales are composed of quartz (up to 40%), sericite (5– 10%), carbonaceous matter (up to 50%), carbonates (5–10%) and sulfides (up to 5%). According to the results of microscopic study, the structure of carbonaceous rocks is due to the presence of quartz grains, tourmaline, as well as scale, leist and flake aggregates of mica. The texture is characterized by the presence of bands of mainly mica-quartz and carbonaceous-sericite composition, layers and elongated lenses of quartz, a layered accumulation of quartz from fine to coarse-grained dimensions. Capacity of layers is different. The banded texture is complexified by a series of asymmetrical sub-parallel folds, which seem to reflect cleavage (Fig. 2). Quartz is irregularly grained, mainly micro-thin grains with homogeneous extinction and conformal grain boundaries. Most commonly, in mass it is associated with mica minerals, which are developed in the interstitial space between quartz grains. In

Fig. 2 The banded rock texture, complicated by cleavage (a) and quartz-mica-turmaline veins in carbonaceous shale (b) (without evaluator, 200×)

Geology and Petrogeochemical Features of the Kumak …

29

Fig. 3 Incorporation contacts between quartz grains in layers and lenses (a) and quartz layered clusters (arrows—inclusions of carbon along crack) (b). Rock section Km-026s, without evaluator, magnification 40×

several interlayers and elongated lenses quartz is noted in sizes from 0.02 to 1.2 mm, in which it has inhomogeneous, often wavy extinction, and also incorporation boundaries between grains on account of serrated, pawed contours of grains (Fig. 3a). At large quartz in such layers and lenses is pure, without inclusions, and also practically is not associated with other minerals. Leys of muscovite are extremely rare in interstitions between quartz grains. One more quartz generation is noted, these are grains of different size (from 0.05 to 0.9 mm), of irregular, angular shape, sometimes with rugged contours, with inhomogeneous, wavy extinction, cracked, with inclusions on cracks of carbonaceous substance, less often micaceous minerals and chlorite (Fig. 3b). Micaceous scales bending is observed around such quartz grains, which indicates a later timescales of mineral extraction along with inclusions of carbonaceous matter.

4 The Research Results Carbonaceous thicknesses represent rather informative material for reconstruction of paleo-geographical and physicochemical conditions of sedimentation, as is commonly known. For the purpose of reconstruction, 27 samples of carbonaceous sediments of the Kumak Ore Field from the Bredy Formation are subjected to silicate analysis (Table 1). To determine the affiliation of the black shale formations, the A-S-C diagram is employed, which was drawn out during generalization of a large number of carbonaceous formations chemical analyses (Gorbachjov and Sozinov 1985). The analyses brought in the diagram show that the figurative points form a continuous series

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A. V. Kolomoets and A. V. Snachev

Table 1 Results of silicate analysis of carbonaceous rocks of the Kumak ore field No

Sample No

SiO2

TiO2

Al2 O3

Fe2 O3

MnO

CaO

1

KM005s

68.00

0.52

9.40

11.40

0.08

0.80

2

KM009s

50.00

1.43

31.50

1.57

0.01

0.51

3

KM011s

59.00

1.33

24.00

2.90

0.01

0.40

4

KM014s

58.00

1.26

24.00

3.50

0.01

0.45

5

KM025s

60.00

1.20

24.20

1.40

0.01

0.70

6

KM026s

52.00

1.54

27.00

6.00

0.05

2.10

7

KM031s

58.00

1.10

25.00

1.35

0.01

1.10

8

KM032s

51.00

1.89

26.60

6.40

0.06

1.40

9

KM034s

60.00

1.54

24.5

4.00

0.05

0.90

10

KM037s

49.00

1.82

27.00

2.92

0.01

2.20

11

KM039s

57.00

1.85

22.30

3.80

0.02

0.90

12

KM041s

57.00

0.70

20.00

2.20

0.01

0.90

13

KM042s

63.00

1.33

22.00

3.19

0.02

0.90

14

KM043s

52.00

0.40

15.00

15.60

0.06

0.90

15

KM044s

50.00

1.40

24.50

13.40

0.12

0.28

16

KM015g

63.50

0.82

20.50

2.40

0.01

0.50

17

KM016g

70.00

0.24

13.70

1.40

0.02

1.10

18

KM017g

52.00

1.40

27.00

6.00

0.06

0.80

19

KM018g

70.00

0.50

14.00

6.00

0.08

1.40

20

KM019g

62.00

0.75

20.00

3.75

0.01

1.20

21

KM020g

78.50

0.60

13.50

0.20

0.08

0.80

22

KM024g

71.00

0.50

12.60

5.53

0.07

0.80

23

KM025g

59.10

1.50

22.20

3.30

0.04

0.30

24

KM029g

63.80

1.22

18.60

6.00

0.04

0.50

25

KM045s

63.50

0.98

20.00

3.60

0.04

0.50

26

KM048s

70.00

0.29

14.00

1.57

0.01

0.40

27

KM049s

79.00

0.10

13.00

0.40

0.01

No

Sample No

MgO

Na2 O

K2 O

P2 O5

ppp

0.20 

1

KM005s

4.60

0.30

0.48

0.07

4.67

100.32

2

KM009s

1.00

2.70

3.70

0.03

7.93

100.37

3

KM011s

1.20

1.35

3.70

0.03

5.62

99.54

4

KM014s

1.00

1.35

3.60

0.03

6.79

100.00

5

KM025s

0.60

1.25

2.00

0.02

8.11

99.74

6

KM026s

0.60

1.08

2.50

0.07

6.95

99.99

7

KM031s

0.80

1.35

2.85

0.02

8.60

100.17

8

KM032s

2.00

1.00

2.50

0.07

6.95

99.87 (continued)

Geology and Petrogeochemical Features of the Kumak …

31

Table 1 (continued) No

Sample No

SiO2

TiO2

Al2 O3

Fe2 O3

MnO

CaO

9

KM034s

1.40

0.45

1.20

0.25

5.67

99.96

10

KM037s

0.80

2.50

2.50

0.07

11.84

100.61

11

KM039s

1.40

2.00

2.00

0.03

9.04

100.36

12

KM041s

0.80

1.05

3.50

0.02

14.33

100.51

13

KM042s

1.10

1.00

2.80

0.03

5.07

99.64

14

KM043s

6.42

0.10

0.12

0.23

9.64

100.52

15

KM044s

1.80

0.40

0.40

0.05

7.06

99.41

16

KM015g

1.60

1.25

3.50

0.04

6.06

100.18

17

KM016g

1.00

1.25

1.30

0.03

9.74

99.78

18

KM017g

1.40

1.20

2.80

0.02

7.18

99.86

19

KM018g

2.00

0.48

0.40

0.01

4.76

99.63

20

KM019g

0.80

1.35

1.50

0.02

8.94

100.32

21

KM020g

1.05

1.20

1.00

0.01

2.81

99.75

22

KM024g

1.40

0.40

0.50

0.20

6.45

99.45

23

KM025g

1.20

2.00

2.80

0.01

7.21

99.56

24

KM029g

1.40

1.35

2.50

0.01

4.34

99.76

25

KM045s

1.40

1.85

3.50

0.01

5.05

100.42

26

KM048s

1.40

0.20

6.00

0.01

5.89

99.77

27

KM049s

0.60

0.70

3.00

0.01

3.40

100.42

along the S-axis from 500 to 1200 units and belong mainly to the terrigenouscarbonaceous formation. Only a few samples fall into the left part so belong to the field of siliceous-carbonaceous formation (Fig. 4). Standard petrochemical parameters (modules) calculated from silicate analyses are used to reconstruct the composition and accumulation conditions of carbonaceous shales (Yudovich Ya and Ketris 2015). The hydrolyzate module (HM = (TiO2 + Al2 O3 + Fe2 O3 + FeO + MnO)/SiO2 ), based on the provision of five main petrogenic oxides, which is universal for most terrigenous and siliceous rocks, makes it possible to separate rocks containing either hydrolysis products (kaolinite, oxides of aluminum, iron, and manganese) or silica. The higher the value of the hydrolyzate module (HM), the stronger and deeper the weathering of the source rocks and the lower its value, the “looser” the sediment from the erosion products, i.e., the higher the rocks maturity. The considered carbonaceous sediments belong to the type of sialites and syfferlites (HM = 0.3–0.5), but some parts in their composition carry products of high alumina weathering rocks and belong to the type of hydrolyzates (HM > 0.55). This is reflected in the values of the aluminosilicon module (ASM = Al2 O3 /SiO2 ), which serves for the separation of clayey and sandy rocks, the maximum value of which

32

A. V. Kolomoets and A. V. Snachev

Fig. 4 Position on the classification charts for points of carbonaceous shale composition of the Bredy formation. Notes a classification diagram A-S-C (Gorbachjov and Sozinov 1985). Formation fields: I—carbonate-carbonaceous, II—terrigenous-carbonaceous, III—siliceous-carbonaceous. Parameters: A = (Al2 O3 − (CaO + K2 O + Na2 O)) × 1000 and S = (SiO2 − (Al2 O3 + Fe2 O3 + FeO + CaO + MgO)) × 1000 are expressed in molecular quantities, parameter C = (CaO + MgO) in mass fraction of oxides. b Fields (Herron 1988): I—Fe shales, II—Fe sandstones, III—Shales, IV—Wackes, V—Litarenitis, VI—Sublitarenitis, VII—Arcoses, VIII—Subarcoses, IX—Quartz arsenites. c F1 –F2 (Roser and Korsch 1988), clastic material source fields: I—quartz-rich sedimentary rocks, II—magmatic rocks of the protophile composition, III—medium composition magmatic rocks, IV—acidic rocks. d DF1 –DF2 (Verma and Armstrong-Altrin 2013), fields of sedimentation: I—island-arc, II—collision, III—riftogenic

in the rocks of the Bredy Formation reaches 0.77 units (average—0.38 units), indicating that this rocks belong to the class of superalumina sediments. Stable high values of the Chemical Index of Alteration [CIA = 100 Al2 O3 /(Al2 O3 + CaO + Na2 O + K2 O)], calculated on the basis of molecular amounts of oxides and falling in the range from 70 to 94 units, indicate a high degree of weathering of sedimentary aluminosilicclastic material, typical for deposits of humide zones (Nesbitt and Young 1982).

Geology and Petrogeochemical Features of the Kumak …

33

On the classification diagram log (SiO2 /Al2 O3 ) − log(Fe2 O3gen /K2 O) (Herron 1988), reflecting the ratio of quartz, feldspars and clay minerals in rocks, the overwhelming number of figurative points of the carbonaceous shales of the Bredy Formation is concentrated in the shale field (Fig. 4b), which indicates the minimal transport of sedimentary material. According to the chemical composition of sedimentary rocks, the composition of source rocks of clastic material for these rocks can be estimated in a certain extent. A variety of different diagrams is usually implemented for this purpose, diagram F1– F2 (Roser and Korsch 1988) is of the widest use. The figurative points distribution in the composition of the considered carbonaceous shales shows that the sources of terrigenous material are mainly rocks of the protophile composition as well as products of erosion of acid volcanites at the base of the Bredy Formation section (Fig. 4c). The DF1 –DF2 diagram is proposed by C. Verm and J. Armstrong-Altrin (Verma and Armstrong-Altrin 2013). Here the same rocks form a compact swarm in the transition zone from the riftogenic geodynamic conditions to the collisional ones (Fig. 4). Taking into account features of accumulation and transport of some elements in the process of sedimentation, it is possible to reconstruct rather confidently the redox environment in ancient sedimentation basins. According to the parameters values combination V/Cr, V/(V + Ni), Mo/Mn, Ua = Utotal − Th /3 (Holodov and Naumov 1991; Jones and Manning 1994; Wignall and Myers 1988), the carbonaceous sediments of the Bredy Formation are deposited under oxidizing and partially suboxidizing conditions. According to S.G. Parada’s research (Parada 2002), there are two basic geochemical types among the carbonaceous-terrigenous rocks, which are ore-containing for gold deposits: normal potassium (average content of 348 samples: Na2 O 1.74%, K2 O 3.30%, Na2 O/K2 O = 0.51) and anomalous sodium (average content of 138 samples: Na2 O 3.31%, K2 O 3.30%, Na2 O/K2 O = 1.0). The first is typical for deposits with gold-sulfide disseminated vein ores, and the second—for gold-quartz veins. For the black shale sediments of the Kumak Ore Field, the average value of Na2 O for 27 analyses is 1.19% (Table 1), K2 O 2.33%, and Na2 O/K2 O = 0.62, which makes it possible to refer them to the normal potassium formation type, mainly typical for the deposits with gold-sulfide ore. A similar conclusion is made by Znamenskij and Znamenskaja (2009), who placed this object in the group of gold-sulfide veins or multi-formational with combined gold-sulfide and gold-quartz small-sulfide mineralization.

5 Conclusions The study of the carbonaceous shales of the Kumak Ore Field has shown the following:

34

A. V. Kolomoets and A. V. Snachev

1.

The Black shale formations of the Bredy Formation are of the carbonaceous type and fall into the fields of terrigenous-carbonaceous and siliceouscarbonaceous formations. Carbonaceous material is represented by two types: weakly metamorphosed sapropel sedimentation-diagenetic and metamorphosed graphite. Sedimentation conditions were quite specific for the sediments of the South Urals. Terrigenous high-alumina sedimentary material underwent minimal transport and was formed mainly due to the destruction of rocks of the protophile composition, as well as products of erosion of acidic volcanites of the footing of the Bredy Formation section. Sediments experienced a high degree of weathering/erosion, typical for humid climate with oxidizing and partially suboxidizing conditions. The deposition of the material took place in the transition from riftogenic to collision geodynamic conditions. The black shale sediments of the Kumak Ore Field belong to the normal potassium formation type by the parameter Na2 O/K2 O = 0.62, which is typical for deposits with gold-sulfide mineralization.

2.

3.

4.

Acknowledgements Field work was supported by the Regional Grant in the domain of scientific and technical activities in 2019 (Agreement № 23 from 14.08.2019). The analytical assignment was carried out within the framework of the State assignment, theme No. 0246-2019-0078. The authors would like to thank S. A. Yagudina for the evaluation labor and E. O. Kalistratova for the tips in the rock sections describing.

References Arifulov ChH (2005) Chernoslancevye mestorozhdenija zolota razlichnyh geologicheskih obstanovok. Rudy Met 2:9–19 (in Russian) Cherepanov AA, Berdnikov NV, Shtareva AV, Krutikova VO (2017) Formation conditions and rare-earth mineralization of Riphean carbonaceous shales of the Upper Nyatygran subformation, Russian Far East. Rus J Pacific Geol 11(4):297–307. https://doi.org/10.1134/S18197140 17040029 Gorbachjov OV, Sozinov NA (1985) Nekotorye petrohimicheskie i geohimicheskie aspekty tipizacii uglerodistyh otlozhenij dokembrija. Problemy osadochnoj geologii dokembrija 46–57 (in Russian) Herron MM (1988) Geochemical classification of terrigenous sands and shales from core or log data. J Sediment Petrol 58:820–829. https://doi.org/10.1306/212F8E77-2B24-11D7-864800010 2C1865D Holodov VN, Naumov RI (1991) O geohimicheskih kriterijah pojavlenija serovodorodnogo zarazhenija v vodah drevnih vodoemov. Izv AS USSR Ser Geol 12:74–82 (in Russian) Jones B, Manning DAC (1994) Comparison of geochemical indices used for the interpretation of paleoredoh conditions in ancient mudstones. Chem Geol 111:111–129. https://doi.org/10.1016/ 0009-2541(94)90085-H Khanchuk AI, Nevstruev VG, Berdnikov NV, Nechaev VP (2013) Petrochemical characteristics of carbonaceous shales in the eastern Bureya massif and their precious-metal mineralization. Rus Geol Geophys 54(6):627–636. https://doi.org/10.1016/j.rgg.2013.04.012

Geology and Petrogeochemical Features of the Kumak …

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Kolomoets AV, Pankrat’ev PV, Kisil’ RS, Panteleev VS (2019) Geology and gold mineralization of lower carboniferous fills of Kommercheskoye deposit (Kumakskoye ore field). Processy v geosredah 3(21):341–346 (2019) (in Russian) Ljadskij PV, Chen-Len-Son BI, Alekseeva GA, Olenica TV, Kvasnjuk LN, Manujlov NV (2018) Gosudarstvennaja geologicheskaja karta Rossijskoj Federacii. Scale 1:200000, 2nd edn. Ser South-Urals. List M-41-I (Anihovka) ref. MF FGBU «VSEGEI», Moscow (in Russian) Loshhinin VP, Pankrat’ev PV (2006) Zolotonosnost’ nizhne-srednepaleozojskih chernoslancevyh formacij Vostochnogo Orenburzh’ja. In: Mat annual sci session Mining Institute of UB RAS on res results in 2005 STRATEGIJA I PROCESSY OSVOENIJA GEORESURSOV, Perm, pp 79–82 (in Russian) Nesbitt HW, Young GM (1982) Early Proterozoic climates and plate motions inferred from major element chemistry of lutites. Nature 299:715–717. https://doi.org/10.1038/299715a0 Novgorodova MI, Jakobs EI, Shinkarenko JuG (1981) Zolotoe orudenenie i metasomatity odnogo iz rajonov Juzhnogo Urala. Voprosy petrologii i metallogenii Urala, pp 115–116 (in Russian) Parada SG (2002) O litogennoj prirode nekotoryh zolotorudnyh mestorozhdenij v uglerodistoterrigennyh tolshhah. Litologija I Pol Iskop 3:275–288 (in Russian) Roser BP, Korsch RJ (1988) Provenance signatures of sandstone–mudstone suites determined using discriminant function analysis of major-element data. Chem Geol 67:119–139. https://doi.org/ 10.1016/0009-2541(88)90010-1 Sazonov VN, Koroteev VA, Ogorodnikov VN, Polenov JuA, Ja VA (2011) Zoloto v “chernyh slancah” Urala. Litosfera 4:70–92 (in Russian) Snachev AV, Rykus MV, Snachev MV, Romanovskaya MA (2013) A model for the genesis of gold mineralization in carbonaceous schists of the Southern Urals. Moscow Univ Geol Bul 68(2):108– 117. https://doi.org/10.3103/S0145875213020105 Verma SP, Armstrong-Altrin JS (2013) New multi-dimensional diagrams for tectonic discrimination of siliciclastic sediments and their application to Precambrian basins. Chem Geol 355:117–133. https://doi.org/10.1016/jchemgeo201307014 Wignall PB, Myers KJ (1988) Interpreting the benthic oxygen levels in mudrocks: a new approach. Geology 16:452–455. https://doi.org/10.1130/0091-7613(1988)016%3c0452:IBO LIM%3e23CO;2 Yudovich YaE, Ketris MP (2015) Geohimija chernyh slancev. Direkt-Media, Moscow, Berlin. https://doi.org/10.23681/428042 (in Russian) Znamenskij SE, Znamenskaja NM (2009) Klassifikacija zolotorudnyh mestorozhdenij vostochnogo sklona Juzhnogo Urala. Geologicheskij Sbornik 8:177–186 (in Russian)

Hydraulic Fracturing in Conditions of Non-linear Soil Behavior M. A. Trimonova

and I. O. Faskheev

Abstract Numerous laboratory and field experiments on hydraulic fracturing show that often classical elastic models are not able to accurately predict the values of breakdown pressure. In this work, computer simulation of the sample was carried out based on experiments conducted at the Institute of Geospheres Dynamics of the Russian Academy of Sciences (IDG RAS) and the Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences (IPE RAS). The geometry of the computational domain, the applied boundary conditions, and nonlinear properties were set based on the conditions of hydraulic fracturing experiments and the results of uniaxial compression tests. As a result of the simulation, it was revealed that the fractured fluid injected under pressure causes plastic deformation in the vicinity of the well, which, in turn, is the cause of the increased breakdown pressure. Keywords Hydraulic fracturing · Plasticity · Stresses · Strains · Laboratory experiments · Breakdown pressure

1 Introduction Presently the most common technology to increase oil and gas reservoir permeability is hydraulic fracturing. The extremely high outlay of this operation makes it difficult to carry out hydraulic fracturing in the amounts necessary for qualitative analysis, so laboratory and numerical experiments within modern computational packages and using models based on the continuum mechanics equations turn out to be increasingly important. With the advancement of computer technologies, numerical modeling of crack development has become one of the basic instruments for rock behavior forecast. M. A. Trimonova · I. O. Faskheev (B) Sadovsky Institute of Geospheres Dynamics, Russian Academy of Sciences, Leninsky ave., 38-1, 119334 Moscow, Russia e-mail: [email protected] M. A. Trimonova e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Chaplina (ed.), Processes in GeoMedia—Volume III, Springer Geology, https://doi.org/10.1007/978-3-030-69040-3_4

37

38

M. A. Trimonova and I. O. Faskheev

The problem of fracturing development in an elastic medium is widely and thoroughly elucidated in scientific publications, but accurate modeling of fracturing at many modern deposits often requires the use of more complex models that take into account the plastic properties of medium. This paper is devoted to study of possible influence of inelastic (plastic) properties of the rocks on fracture pressure: basic characteristic of fracture propagation during hydraulic fracturing operation. In contrast to the elastic deformation regime, the inelastic regime in hydraulic fracturing operation is much less analyzed and described in the scientific publications worldwide. Brief review of papers on the topic is brought below. Classical plasticity models in their original form were constructed to describe the various properties of metals, which, in general, have a simpler structure than soils with a complex internal system of cracks and pores. For these reasons, for an accurate description of the deformation properties of soils and rocks, special inelastic models are required that takes into account the peculiarities of the structure of the medium. The influence of plasticity on fracturing is discussed in Papanastasiou (1997). Fluid flow in the crack is modeled on the basis of the theory of lubrication. The deformation of the rock is modeled on the basis of the classical Coulomb-Mohr plastic model. Crack propagation criterion is based on softening of rocks. The corresponding nonlinear problem is solved with the use of combined finite-difference finite-element scheme. The results show that plastic deformations (which occur near the tip of a spreading fracture) affect the fracture toughness of the rock. These studies have been developed in Sarris and Papanastasiou (2011, 2013), where the model has been extended to take into account poro-elastic–plastic adhesion during hydraulic fracturing. The Drucker-Prager model is proposed in Drucker and Prager (1952), which generalizes the Coulomb-Mohr model with the associated form of the MisesSchleicher flow law, including the internal friction angle and the coefficient that takes into account the material adhesion. A variant of this model with the non-associated law of flow is proposed in Nikolayevsky (1972), Garagash and Nikolaevskiy (1989). This model allows for more accurate consideration of deformation properties of rocks. The advantages of the non-associated law are described in detail in works by Vermeer and De Borst (1984). Numerical experiments on crack formation in geomaterials for different loading options have been carried out in Stefanov (2002). Cracks near the inclusions and inside the medium have been considered. Calculations are made using the PrandtlReiss, Drucker-Prager and Nikolaevsky models. A numerical simulation of the localized deformation zone and cracks for elastic brittle-plastic materials for various types of loads was carried out and considered in Stefanov (2004). The Nikolaevsky model and the new method of mesh nodes partitioning are used for numerical modeling. The results obtained are in good agreement with the experimental data on the reaction of the geological medium under different loads. Numerical simulation of hydraulic fracturing in porous rocks showing inelastic deformations is considered in Qingdong et al. (2018). The inelastic deformation of the porous rock is described by the non-associated model of plasticity based on DruckerPrager criterion. Plastic deformation in combination with fluid pressure changes is

Hydraulic Fracturing in Conditions of Non-linear Soil Behavior

39

described by the theory of lubrication. The extended finite-element method is used to simulate crack propagation. The proposed numerical model is compared with numerical and analytical results. This publication notes the influence of the angle of friction and inelastic deformation on the hydraulic fracturing process. The works discussed above show the importance of taking into account the plastic properties of the material in the numerical and analytical modeling of hydraulic fracturing. In this piece the plasticity effect on the modification of samples subjected to hydraulic fracturing experiments is studied. The Abaqus Student Edition package is used for numerical modeling.

2 Laboratory Experiments on Hydraulic Fracturing At the Sadovsky Institute of Geospheres Dynamics of the Russian Academy of Sciences (IDG RAS) a number of experiments on hydraulic fracturing on artificial samples are conducted (Trimonova et al. 2017, 2018). The basic objectives of this work are: to develop a general methodology for conducting such experiments, to define criteria for similarity between laboratory and field conditions. The result of this work is comparison between experimentally obtained data on fracturing with real field data. In the experiments, special attention is paid to study the influence of pore pressure and contrast of stresses on the direction of crack propagation. Basing on the criteria of similarity, gypsum with the addition of cement is chosen as the model material for the specimen.

Fig. 1 Typical experimental curve of pressure relation on time for the central well

40

M. A. Trimonova and I. O. Faskheev

The graph (see Fig. 1) shows typical curve of pressure relation to the experiment time in the central well. Such a characteristic curve shape is inherent in most of the experiments performed in IDG RAS. The authors of the experimental study have noticed the fact that the values of breakdown pressure in the experiments significantly exceed the values calculated on the basis of classical “elastic” problems with precise analytical solution. For a better illustration of the above mentioned fact, the classical Kirsch problem for the stresses concentration around a circular hole (Kirsch 1898) is considered, for which biaxial compression is used as boundary conditions (two perpendicular main stresses S H , Sh , S H > Sh ) on infinitely distant boundaries and pressure on the hole Pmud [here, the medium is considered to be poroelastic, saturated by pressure Ppor (Jaeger and Cook 1979)], the expression for breakdown pressure have the following form: Pmud = 3σh − σ H + Ppor + U T S

(1)

σ H = S H − Ppor

(2)

σ H = S H − Ppor

(3)

Table 1 presents the results of the burst pressure (Pmud ) calculation according to exp the Kirsch formula and its experimental values (Pmud ) for the six experiments carried out in IDG. The experimental data differ significantly from the values estimated based on the Kirsch problem solution, as it is shown in Table 1. The significant difference in burst pressures when comparing experimental data with an elastic analytical model led the authors of this paper to study the possible effects of plastic behavior of the sample material. Table 1 Basic values in hydraulic fracture experiments exp

Item No.

S H , MPa

Sh , MPa

Ppor , MPa

UTS, MPa

Pmud , MPa

Pmud , MPa

1

0.5

0.50

1.00

0.6

0.6

2

1.0

0.55

0.50

0.8

0.95

9.0

3

1.0

0.55

1.00

1.0

0.65

13.0

4

1.0

0.78

1.25

1.0

1.09

11.0

5

1.1

0.80

1.25

0.8

0.85

7.5

6

0.4

0.35

1.25

0.9

0.30

9.0

7.2

Hydraulic Fracturing in Conditions of Non-linear Soil Behavior

41

3 Laboratory Experiments to Determine Elastoplastic Properties A series of laboratory experiments is conducted at the Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences (IPE RAS) to study the mechanical properties of samples. Cylindrical gypsum specimens are subjected to tests. Length is about 107 mm, diameter is about 54 mm, weight is about 400 g. The following tests are carried out: 1. 2. 3. 4. 5.

Determination of single axial tensile strength (TSTR); Determination of uniaxial compression strength (UCS); Creep Test (CRP); Multistage Testing at Various Comprehensive Pressures (MST); Standard Axial Arm Test (SingleStage Test).

The multistage test is designed to determine the parameters of the rock destruction criterion. For the fracture criterion of the Coulomb-Mohr these are: uniaxial compression strength (UCS) and internal friction angle (FANG). The rock sample is placed in the specified thermobaric conditions and compressed until elastic strength limit is reached. Then the sample is offloaded, comprehensive pressure (CP) is increased to the next preset point. The sample is stabilized at new CP and a cycle of sample compression towards the elastic strength limit and offloading is carried out. Thus the elastic strength limits of the sample are determined at a given set of different radial stresses. At the concluding CP the sample is loaded beyond the elastic strength limit and brought to the tensile strength. This research allowed to determine almost all basic mechanical characteristics of the model material, such as Young’s modulus, Poisson’s coefficient, limits of elastic strength, plasticity and tensile strength. In this work, curves for stress relation to axial strain in uniaxial compression tests (UCS) are mainly used. One of these curves is shown in Fig. 2. The given curve has allowed the authors to construct the elastic–plastic model in Abaqus Student Edition package, which sufficiently precise corresponds to real properties of samples used in fracturing experiments.

4 Numerical Modeling The Abaqus Student Edition package is used to simulate the plastic properties of samples in the presented study. The package allows: 1. 2.

to use classical plasticity models with Mises and Hill flow surfaces with associated flow law; to simulate perfectly plastic and hardened media;

42

M. A. Trimonova and I. O. Faskheev

Fig. 2 Curve for stress relation to axial strain obtained in IPE RAS

3.

study the plastic mode of deformation in combination with the model of shear failure.

In this paper the simplest model of plastic deformation with isotropic hardening is applied, in which the stress–strain curve is set using a table with two columns: Plastic Strain and the corresponding Yield Stress. As noted above the elastic strain regime does not allow to take into account all the features of the processes development accompanying the hydraulic fracturing. Experiments carried out at the Institute of Geospheres Dynamics (IDG RAS) have shown inability of classical elastic models to adequately accurately describe the pressure distribution in the fracture. To study the plasticity effect on the principal characteristics of hydraulic fracturing the Abaqus Student Edition package, which is designed to solve problems by finiteelement method, is used here. A two-dimensional model of the experimental sample in form of a disc with a central hole is developed (see Fig. 3). Part of the boundary is rigidly fixed from the displacement, the other part is set to loads of P = 106 Pa. The Young’s module of the material is equal to E = 3.4 × 109 Pa, the Poisson’s coefficient is ν = 0.29, density is ρ = 1689 kg/m3 . The plastic part is taken into account by direct setting of stress–strain relation (Fig. 2). In addition to accounting for the plastic regime of the medium the filtration in the sample is also taken into account. Calculations are made for permeability k = 2 × 10−15 m2 and porosity m = 0.4 values which were found experimentally for the taken material. For the filtration problem the Dirichle condition at the well is used as boundary condition, i.e. constant pore pressure is assumed. The no-flow condition is set at all other boundaries, i.e. zero filtration rate is implemented. Thus the size of the model, boundary conditions and properties of the material corresponded to the conditions of the experiment carried out at the IDG RAS.

Hydraulic Fracturing in Conditions of Non-linear Soil Behavior

43

Fig. 3 Calculated two-dimensional model of the experimental sample in the Abaqus Student Edition package

The numerical experiments demonstrated that taking into account the elastic– plastic properties of the material around the central well by injecting a rupture fluid a plasticity zone is formed (Fig. 3). The onset of plasticity in the vicinity of the well in high-porous materials can lead to hardening of the material and creating a compaction zone around the well (Stefanov 2002, 2004), which causes an increase in the breakdown pressure of the hydraulic fracturing.

5 Conclusion As a result of a numerical experiment using the Abaqus Student Edition software package it is shown that the model built on the basis of experimentally obtained data tends to plasticity in the vicinity of the well due to the action of injected fluid. The appearance of plastic deformations in the vicinity of the well in highly porous materials leads to hardening and to the appearance of compaction zones, which sequentially can lead to an increase in breakdown pressure relative to the values found with the elastic model.

44

M. A. Trimonova and I. O. Faskheev

Acknowledgements The reported study was funded by RFBR, project number 20-35-80018.

References Drucker DC, Prager W (1952) Soil mechanics and plastic analysis or limit design. Q Appl Math 10(2):157–165 Garagash IA, Nikolaevskiy VN (1989) Non-associated laws of plastic flow and localization of deformation. Adv Mech 12(1):131–183 Jaeger JC, Cook NGW (1979) Fundamentals of rock mechanics, 2nd edn. Chapman and Hall, New York Kirsch EG (1898) Die Theorie der Elastizitat und die Bedurfnisse der Festigkeitslehre. Zeitshrift des Vereines Deutscher Ingenieure 42:797–807 Nikolayevsky VN (1972) Mechanical properties of soils and theory of plasticity. Mechanics of solid deformable bodies, vol. 6. The results of science and technology. USSR Academy of Sciences. VINITI, Moscow, pp 5–85 Papanastasiou P (1997) The influence of plasticity in hydraulic fracturing. Int J Fract 84:61–79 Qingdong Z, Jun Y, Shao J (2018) Effect of plastic deformation on hydraulic fracturing with extended element method. Springer, Germany. https://doi.org/10.1007/s11440-018-0748-0 Sarris E, Papanastasiou P (2011) The influence of the cohesive process zone in hydraulic fracturing modelling. Int J Fract 167(1):33–45. https://doi.org/10.1007/s10704-010-9515-4 Sarris E, Papanastasiou P (2013) Numerical modeling of fluiddriven fractures in cohesive poroelastoplastic continuum. Int J Numer Anal Meth Geomech 37:1822–1846 Stefanov YP (2002) Deformation localization and fracture in geomaterials. Numerical simulation. Phys Mesomech 5(5–6):67–77 Stefanov YP (2004) Numerical investigation of deformation localization and crack formation in elastic brittle-plastic materials. Int J Fract 128(14):345–352 Trimonova M, Baryshnikov N, Zenchenko E, Zenchenko P, Turuntaev S (2017) The study of the unstable fracture propagation in the injection well. Numerical and laboratory modeling. SPE187822-MS Trimonova M, Baryshnikov N et al (2018) Estimation of the hydraulic fracture propagation rate in the laboratory experiment. In: Karev V et al (eds) Physical and mathematical modeling of earth and environment processes. Springer International Publishing, Moscow, pp 259–268 Vermeer PA, De Borst R (1984) Non-associated plasticity for soils. Concrete and rock. HERON 29(3)

The Exact Equation for the Waveform of Potential Stationary Surface Waves A. V. Kistovich

Abstract In article the new method for the description of stationary potential surface waves is proposed. The classical boundary value problem is reformed into a single equation for the waveform by means of exact transformations. All the necessary relations are obtained that allow to uniquely determining all the physical fields of the problem under study by the waveform. The validity of the results obtained is confirmed by a number of well-known examples. Keywords Surface waves · Waveform · Stream function

1 Introduction The problem of potential surface waves description is the classical problem of hydrodynamics in the course of two centuries. The classical results (Boussinesq 1871; Stokes 1847; Rayleigh 1876) and many other works forming the basement of surface waves theory were done for approximate equation for wave’s form. Moreover, in most cases the approaches to solution of the problem announced are based on a priory assumption about a character of physical fields’ dependence on vertical coordinate. The exact integral equations for free surface form (Nekrasov 1921; Ablowitz et al. 2006; Kistovich and Chashechkin 2008) either have restrictions on the class of permissible solutions [Nekrasov’s equation (Nekrasov 1921)] or include Fourierseries expansions (Ablowitz et al. 2006; Kistovich and Chashechkin 2008) which in common case can’t be considered as more suitable functions for surface waves description. The goal of the work presented is to produce an exact equation including only the form of surface wave and to get relations permitting on the base of wave’s form to define all physical fields in medium.

A. V. Kistovich (B) Ishlinsky Institute for Problems in Mechanics, RAS, Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Chaplina (ed.), Processes in GeoMedia—Volume III, Springer Geology, https://doi.org/10.1007/978-3-030-69040-3_5

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A. V. Kistovich

2 Problem Statement The 2D problem of potential waves spreading along the surface of uncompressible inviscid liquid with the depth h is considered. In undisturbed state the liquid flows in the direction of horizontal axis x with constant velocity V. Symbol ζ relates to the deviation of free surface from its stable state z = 0. The total velocity of medium is defined by relation w = (V + v)ex + wez where ex , ez are the orts of Cartesian coordinate system (axis z is directed in opposite to gravitational acceleration vector g). It is assumed that disturbances of all physical fields also as deviation of free surface are spreading along x-axis with constant velocity c so the relations p = p(s, z), v = v(s, z), w = w(s, z), ζ = ζ(s) are valid, where s = x − ct. The constant atmospheric pressure pa and pressure p in the liquid are normalized on constant liquid’s density. The governing system and boundary conditions of the problem have a form vt + (V + v)vx + wvz = − px , wt + (V + v)wx + wwz = − pz − g  vx + wz = 0, vz − wx = 0, p|z=ζ = pa , w − (V + v)ζx z=ζ = ζt , w|z=−h = 0 (1) For the purpose of further analysis the dimensionless variables x  = x/M, z  = z/ h, t  = ct/M and s = s/M are introduced where M is characteristic longitudinal scale of disturbances. Moreover the dimensionless deviation of free surface η = ζ/a (where a is amplitude of wave), pressures p  = p/c2 , pa = pa /c2 and stream function ψ = ψ/ac (where u = ψz , v = −ψx ) are introduced. In new variables the system (1) reforms into system   ps = γεψzs − ε2 ψz ψzs + δ2 ψs ψss ,   pz = γεψzz − ε2 ψz ψzz + δ2 ψs ψzs − σ

  δ2 ψss + ψzz = 0, p|z=εη = pa , ψs + εψz ηs z=εη = γηs , ψs z=−1 = 0

(2)

where γ = 1 − V/c, ε = a/ h, δ = h/M, σ = gh/c2 and superscripts are omitted for brevity. Integration of two first equations of system (2) gives the result p = γεψz −

 ε2  2  2 δ ψz + ψs 2 − σz + pa 2

(3)

the using of which forms the mathematical model of problem in terms of stream function and disturbance of free surface  ε = ση δ2 ψss + ψzz = 0, γψz − δ2 ψz 2 + ψs 2  2 z=εη   (4) ψs + εψz ηs z=εη = γηs , ψs z=−1 = 0

The Exact Equation for the Waveform of Potential Stationary …

47

Because the Laplace equation of (4) permits the factorization of the form δ2 ψss + ψzz = (∂z − iδ∂s )(∂z + iδ∂s )ψ = 0 where i 2 = −1, then its solution can be presented in the form ψ = A(s − iδ(z + 1)) + B(s + iδ(z + 1))

(5)

where A(s−iδ(z +1)) and B(s+iδ(z +1)) are arbitrary functions of own arguments. Because the kinematics boundary conditions of (4) may be presented as     ∂ ∂ ψ|z=εη = γηs , ψs z=−1 = ψ|z=−1 = 0 ψs + εψz ηs z=εη = ∂s ∂s then they can be integrated and reformed to the equations ψ|z=εη = γη, ψ|z=−1 = 0

(6)

in accordance to condition ψ ≡ 0 when η ≡ 0. Substitution of (5) into dynamical boundary condition of (4) and into kinematics conditions (6) reforms them to the system  2εδ2 A B  + iγδ(A − B  )z=εη + ση = 0 A(s − iδ(1 + εη)) + B(s + iδ(1 + εη)) = γη, A(s) + B(s) = 0

(7)

where superscript means derivation of function with respect to own argument. From two first equations of (7) it follows ⎤ ⎡ s i ⎣ R(ρ)dρ⎦ + 0 A(s − iδ(1 + εη)) = α(s − iδεη) − 2δε ⎡ B(s + iδ(1 + εη)) = −

−∞

i ⎣ α(s + iδεη) − 2δε

s

(8)

⎤ R(ρ)dρ⎦ − 0

(9)

−∞

where R(s) = (γ2 − 2σεη)(1 + δ2 ε2 η (s)2 ), 0 is some arbitrary constant. To take advantage of condition A(s) + B(s) = 0 in Eq. (8) to variable s the such function f (s) is added that s + f (s) − iδ(1 + εη(s + f (s))) = s ∼ f (s) = iδ(1 + εη(s + f (s)))

(10)

Analogously in (9) the function s is added to g(s) so s + g(s) + iδ(1 + εη(s + g(s))) = s ∼ g(s) = −iδ(1 + εη(s + g(s)))

(11)

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A. V. Kistovich

As a result Eqs. (8, 9) are reformed into ⎡ A(s) =

i ⎣ γs − 2εδ ⎡

s+  f (s)

⎤ R(ρ)dρ⎦ + 0 −

−∞

i ⎣ B(s) = − γs − 2εδ

s+g(s) 

γ 2ε

(12)

⎤ R(ρ)dρ⎦ − 0 −

−∞

γ 2ε

(13)

and from boundary condition A(s) + B(s) = 0 it follows s+g(s) 

2iγδ +

R(ρ)dρ = 0

(14)

s+ f (s)

The differentiation (14) with respect to s gives the equation (γ2 − 2σεη(s + f (s)))(1 + 2 f  (s)) = (γ2 − 2σεη(s + g(s)))(1 + 2g  (s))

(15)

in which the unknown functions f (s) and g(s) have place for omitting of which the formal solutions of (10, 11) f (s) = s+ − s, g(s) = s− − s

(16)

are used. Here s± = s ± iδ(1 + εη(s ± iδ(1 + εη(s ± iδ(1 + εη(s ± · · ·)))))). Substitution of (16) into (15) forms the resultant equation   iδγ2 ηs (s+ ) + ηs (s− ) − σ(η(s+ ) − η(s− ))   − 2iδεσ η(s+ )ηs (s+ ) + η(s− )ηs (s− ) = 0

(17)

in which presents the unique unknown function namely the form of surface wave. This exact Eq. (17) for the form of steady surface wave is the fundamental result of presented work. The solution of (17) permits to reconstruct the stream function on the base of known wave’s form. Because in accordance to (5) the stream function is defined by relation ψ = A(s − iδ(z + 1)) + B(s + iδ(z + 1)) then on the base of (12, 13) it follows i γz − ψ= ε 2εδ

B(s,z) 

R(ρ)dρ A(s,z)

(18)

The Exact Equation for the Waveform of Potential Stationary …

49

where A(s, z) = s + iδ(1 + z) + g(s + iδ(1 + z)), B(s, z) = s − iδ(1 + z) + f (s − iδ(1 + z)). Substitution into (18) z = εη(s) (i.e. the calculation of stream function value on free surface) with regard to equivalent formulation of (10, 1) f (s − iδ(1 + εη(s))) = iδ(1 + εη(s)), g(s + iδ(1 + εη(s))) = −iδ(1 + εη(s)) leads to result ψ(s, εη(s)) = A + B|z=εη = γη which is in rigorous correspondence to boundary condition (6) on free surface. The expression (18) for the stream function defines velocity field components v = ψz =

  1 γ i  −  f + g , w = −ψs =  f − g ε 2ε 2εδ

(19)

where  f = (1 + iδεη (s − iδ(1 + z)))R(s − iδ(z + εη(s − iδ(1 + z)))) g = (1 − iδεη (s+iδ(1 + z)))R(ε + iδ(z + εη(s+iδ(1 + z)))). The pressure in environment is defined as a result of substitution of (19) into (3). It is necessary to emphasize that all received results are exact with no approximations.

3 Validation of the Wave Form Equation 17 The application of (17) is considered for the set of partial examples. In the case of infinitesimal waves (ε  1) the expansion of (17) into series with respect to mentioned small parameter leads to equation for infinitesimal waves  iδγ2 η (s + iδ) + η (s − iδ) − σ[η(s + iδ) − η(s − iδ)] = 0

(20)

If the characteristic scale M is sufficiently greater than liquid’s depth h (it means that δ  1) then it follows from (20) (γ2 − σ)η (s) + O(δ2 ) = 0 ⇒ γ2 = σ ∼ c2 γ2 = gh ∼ c = V +



gh

(21)

The received relations define well-known result for the waves on shallow water. In this case, as it follows from (21), the waveform η is an arbitrary function under condition that the smallest longitudinal scale of this function is sufficiently greater

50

A. V. Kistovich

than water depth. Thus, longwave package spreads along water surface practically with no dispersion. This is well-known result of Stokes achieved here in a common form without Fourier-series method. In the case of arbitrary values of δ for smooth wave it is possible to present (20) in the form ∞

n 2n (2n)

(−1) δ η

n=0



σ γ2 − (s) (2n)! (2n + 1)!

 =0

so from characteristic equation for solutions of class η = exp(iks) follows the well-known dispersion relation   γ2 = σth(kδ) kδ ∼ (c − V)2 = gth(kh) k In the case of small smooth waves with finite amplitude the small parameter κ = δε  1 is introduced. Equation (17) is expanded in series with respect to κ also as a value σ = σ0 + σ1 κ + σ2 κ2 + σ3 κ3 + · · · . The received solution for the case of liquid of finite depth is somewhat tedious and does not presented here but its limit for infinite deep water (δ → ∞) has a form 3κ3 κ4 κ2 cos(2s) + cos(3s) + cos(4s) (22) 2 8 3     √ and c = V + g M(1 + κ2 2) ∼ c = V + g k(1 + k 2 a 2 2), where k is wavenumber (κ = ka). Relation (22) describes well-known weakly nonlinear Stokes wave (Stokes 1847) which velocity increases with the growth of its amplitude a. The process of stream function reconstruction on the base of (18) is shown for infinitely small waves. In this case the approximations η(s) = κ cos(s) +

A(s, z) ≈ s + iδz − iδεη(s + iδz), B(s, z) ≈ s − iδz + iδεη(s − iδz)   σ R(s) ≈ γ 1 − 2 εη(s) γ are valid, which substitution into (20) leads to relation ⎡ ψ(s, z) =

iσ γ⎢ ⎣η(s − iδz) + η(s + iμz) + 2 2 δγ

For infinitesimal waves η(s) = cos(s) so

B(s,z) 

⎤

⎥ η(ρ)dρ⎦

A(s,z)

The Exact Equation for the Waveform of Potential Stationary …

ψ(s, z) = γ cos(s)ch(δz) +

51

sh(δ(1 + z)) σ cos(s)sh(δz) = cos(s) δγ sh(δ)

i.e. stream function is reconstructed correctly. The next is the case of localized waves on the surface of shallow water (δ  1). Under localization means the validity of relations  η(n) (s)s=±∞ = 0, n ∈ N

(23)

Also the mass conservation law +∞ η(s)ds = 0

(24)

−∞

is used. The expansion of (17) in series when δ  1 (no restriction on ε) leads to relation (γ2 − σ)η +

δ2 δ4 δ6 (σ − 3γ2 )η + (5γ2 − σ)η(4) + (σ − 5γ2 )η(6) + · · · 3!  5! 7!

− 3εση2 + εδ2 (3σ − 2γ2 )(ηη + η ) + εη(5σ − 2γ2 )(ηη + 2η )  7 2 + ε2 η2 σ(ηη + 3η ) + ε3 δ4 [. . .] + ε5 δ6 [. . .] + · · · = 0 3 2

2

(25)

Here in square brackets dots mean such relations which integral properties in regard to (23) are analogues to integral properties of member in the first square brackets +∞ +∞ +∞ 2  2  2 (ηη + η )dξ = η(ηη + 2η )ds = η2 (ηη + 3η )ds −∞

−∞

−∞

+∞ =

[. . .]ds = 0 −∞

For infinitesimal limit (ε = 0) from (25) follow relations presented above. In the case ε = 0 the integration of (25) with respect to s in infinite limits leads, in accordance to (23, 24), to relation +∞ η2 (s)ds = 0 −∞

(26)

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A. V. Kistovich

Because η2 (s) ≥ 0 ∀ s then (26) is valid only for η(s) ≡ 0 ∀ s that means the absence of surface wave. As a result the existence of localized stationary waves is impossible under action of mass conservation law. It should be noted that when ε ∼ δ2 , σ = γ2 (1 − bδ2 ) (standard assumptions for producing of soliton solutions (Dodd et al. 1982), where b is some constant) Eq. (26) with regard to terms of second degree on μ inclusively is reduced into equation η /3 − bη + 3η2 /2 = 0 which solution is well-known soliton of Russel-Rayleigh (1876) (this soliton also violets mass conservation law). As it follows from definitions for ε, δ and s± the equation ha place aηa (s± ) + hηh (s± ) + MηM (s± ) = 0

(27)

The using of (27) in Eq. (17) for the waveform gives the value for relative speed c˜ = c − V a c˜a

∂ c˜ + h c˜h + M c˜M − = 0 ∂a 2

(28)

which is valid for all known wave solutions of approximate theories. The application of the presented results to the case of infinitely deep water (δ → ∞) and a priory assumption about exponential decreasing of all physical fields with water depth, i.e. ψ(s, z) = ψ(s) exp(δz) ⇒ ψz (s, z) = δψ(s, z)

(29)

leads, after substitution of (18) into the second relation of (29), to the equation ψz (s, z) =

 i  γ + R(A(s, z))Az (s, z) − R(B(s, z))Bz (s, z) . ε 2εδ

(30)

On the free surface the relations z = η(s) and ψ(s, η(s)) = γη(s) are valid and expressions have place A(s, η(s)) = B(s, η(s)) = s   iδη (s) iδη (s)   , B . Az (s, z)z=η(ξ) = (s, z) = − z=η(ξ) 1 + iεδη (s) z 1 − iεδη (s) The substitution of (31) into (30) forms the equation  γ(1 − εδη(s)) =

γ2 − 2σεδη(s) , 1 + ε2 δ2 η 2 (s)

from which follow the equations for waves on infinitely deep water

(31)

The Exact Equation for the Waveform of Potential Stationary …

53

kaη(s) , 1 − kaη(s)

(32)

kaη (s) = ±i where k = 1/M. The solutions of (32) have the forms

1 ζ± (s, ζ0 ) = −a± − W(−kζ0 exp(±iks)) k ψ± (s, z) = cζ0 exp(kz) exp(±iks),

(33)

where W(x) is the Lambert function, a± and ζ0 are free parameters. The results (33) are exact but complex-valued solutions of the classical problem (1) for infinitely deep water. The combinations of these solutions permits to form the approximate results for infinitesimal and small amplitude (Stokes) waves.

4 Conclusions For 2D potential waves the exact differential-functional equation is received which includes only the form of surface wave. Also produced relations allow to calculate all physical fields on the base of known waveform. The validity of results is proved on the set of well-known problems. The presented approach permits to get a unique equation also for non stationary 2D potential surface waves.

References Ablowitz MJ, Fokas AS, Musslimani ZH (2006) On a new non-local theory of water waves. J Fluid Mech 562:313 Boussinesq J (1871) Theorie de l’intumescence liquide, applelee onde solitaire ou de translation, se propageant dans un canal rectangulaire. Comptes Rendus de l’Academie des Science, vol 72, p 755 Dodd RK, Eilbeck JC, Gibbon JD, Morris HC (1982) Solitons and nonlinear wave equations. Academic Press, Inc., London, New-York Kistovich AV, Chashechkin YuD (2008) Integral model of propagation of steady potential waves in liquid. Doklady RAS, vol 241, no 3, p 355 Nekrasov AI (1921) About the waves of stationary type. Izvestija Ivanovo-Voznesenskogo politekhnicheskogo instituta, no 3, p 52 (only in Russia) Rayleigh (1876) On waves. Philisophical Magazine 5(I):257 Stokes GG (1847) On the theory of oscillatory waves. Trans Camb Philos Soc 8:441

Identification of the Initial Spot Location of Pollution in the Kerch Strait on the Basis of Adjoint Equation Kochergin Vladimir Sergeevich

and Kochergin Sergey Vladimirovich

Abstract In the article calculations using the model of passive admixture transport for Kerch Strait area was done. The procedure of identification of the initial pollution spot on the basis of the method of adjoint equations is carried out. Solutions to a series of adjoint problems are used to determine the possible location of the initial contamination spot. Keywords Transport model · Adjoint equations · Identification of sources of pollution · The Kerch Strait

1 Introduction The increasing anthropogenic load on the water area of the Azov-Black sea basin in General and the Kerch Strait in particular it is necessary to create of environmental monitoring systems that allow rapid evaluation of the environmental situation in domains subjected to man-made impacts. These include especially areas of intensive shipping and construction of communications of various types. The solution of such problems is possible on the basis of mathematical methods for modeling passive impurity transfer processes (Eremeev et al. 2008), as well as methods for solving inverse problems (Marchuk and Penenko 1978; Kochergin and Kochergin 2010; Penenko 1981), which are based on minimizing the prediction quality functionals and solving adjoint problems. Recently, variational methods of assimilation of measurement data and the method of adjoint equations have been actively developed and used in solving Oceanological problems (Agoshkov et al. 2013; Shutyaev et al. 2015; Ryabcev and Shapiro 2009). In order to quickly obtain information about the state of the object under study, the numerical implementation of such models and algorithms must be carried out on high-performance computer technology using modern approaches, including parallelization of calculations. It should be noted that the independence of the adjoint tasks allows their integration in parallel mode using K. V. Sergeevich · K. S. Vladimirovich (B) Marine Hydrophysical Institute, RAS, Sevastopol 299029, Russian Federation e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Chaplina (ed.), Processes in GeoMedia—Volume III, Springer Geology, https://doi.org/10.1007/978-3-030-69040-3_6

55

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K. V. Sergeevich and K. S. Vladimirovich

different processors, i.e. organization of the computing process is more effectively. In Kochergin and Kochergin (2015), a variational algorithm for identifying the source power based on this approach is considered. In this paper, the method of adjoint equations (Marchuk 1982; Kochergin 2011) is used to identify the location of the initial contamination spot. The problem is solved for the water area of the Kerch Strait using the model (Fomin 2002). The results of calculations for its barotropic variant under different wind effects are input information when integrating the passive impurity transport model and the adjoint problem.

2 Adjoint Equations Method Let’s consider in σ -coordinates the transfer model of a passive impurity: ∂ DU C ∂ DV C ∂WC ∂ ∂ ∂ K ∂C ∂ DC ∂ DC ∂ DC + + + = AH + AH + ∂t ∂x ∂y ∂σ ∂x ∂x ∂y ∂y ∂σ D ∂σ (1) on the lateral boundaries we have: :

∂C = 0, ∂n

(2)

on the surface and at the bottom we use following conditions: σ =0: σ = −1 :

∂C =0 ∂σ

(3)

∂C =0 ∂σ

and initial data: C(x, y, σ, 0) = C0 (x, y, σ ),

(4)

where t—time; x0 , y0 —coordinates of the point source; D—dynamic depth; C— impurity concentration; U, V, W —components of the velocity field; A H and K — turbulent diffusion coefficients; n—normal to boundary, M—the domain of model integration; Γ —the border of the area M. Mt = M × [0, T ]. Multiplying (1)–(4) by C ∗ , integrating by parts using boundary conditions and the continuity equation: ∂ DU ∂ DV ∂W ∂D + + + = 0, ∂t ∂x ∂y ∂σ

(5)

Identification of the Initial Spot Location …

57

and choosing C ∗ , as a solution to the following adjoint problem: ∂C ∗ ∂ DC ∗ ∂ DU C ∗ ∂ DV C ∗ ∂W C∗ ∂ − − − − D AH ∂t ∂x ∂y ∂σ ∂x ∂x , ∂ ∂ K ∂C ∗ ∂C ∗ − D AH − =0 ∂y ∂y ∂σ D ∂σ

(6)

∂C ∗ ∂C ∗ ∂C ∗ = 0, σ = 0 : = 0, σ = −1 : =0 ∂n ∂σ ∂σ

(7)

−

Γ :

t = T : C ∗ = h,

(8)

get: 

 hCd M = M

C0 C ∗ d M

(9)

M

Select in the form:  h=

− in the area  , 0 − outside the area  1 m()

(10)

where m is a measure of some area . At the same time, in the left part of the expression (9), we get the average concentration C T in  at the moment of time T . By selecting  as a cell in the calculation grid, we have:  CT =

C0 C ∗ d M.

(11)

M

With the help of solution of the adjoint problem (6)–(8), this formula is used to estimate the impurity concentration in this cell of the calculation grid. Numerical experiments carried out in Kochergin and Kochergin (2015), Kochergin (2011) showed good accuracy of reproducing the concentration field from the initial data. It should also be noted that C ∗ in it formula is actually a influence function of the initial concentration. Using on its structure, you can investigate the influence of the initial data on the decision of the transfer model.

3 Results of Numerical Experiments Numerical experiments were done using a circulation model from Fomin (2002) for the Azov Sea and the Kerch Strait region. To realization the procedure for identifying

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the location of the initial pollution spot, a calculation was performed to establish a model field of currents under the influence of a constant wind of 10 m per second from the North-East. During simulation of the model, the space distribution of the model coefficients and velocity fields was obtained and used as input information when integrating the passive impurity transfer model for five days of the model time period. The initial pollution spot was set on the surface of the Strait in an area of heavy shipping. In addition, there is a possibility of other anthropogenic impacts in this area (Makarov and Lanin 2017). Figure 1 shows a rectangular area where initial concentration values were set on the surface. The result of modeling the spread of pollution under North-East wind influence is also presented. The concentration field shown in Fig. 1 is normalized on the largest value. The pollution spot at the final time moment is characterized by a high concentration near the southern coast of the Kerch Peninsula. In Fig. 1, a circle marks the location of the top value in the concentration field, and the selected points on the periphery of the contamination spot are marked with rhombuses. After five days of the model time, the admixture spread along the coast up to the Feodosian Gulf. Such dynamics of pollution distribution is typical for this area with winds from the North and North-East. Figure 2 shows the field of suspended matter concentration in the sea surface layer obtained from the sputnic data (https://earthd ata.nasa.gov/labs/worldview/?p=geographic&l=MODIS). The dynamics of agitated bottom sediments depends on the particle size. The structure of bottom sediments (Matishov et al. 2011) in the Kerch Strait and in the surrounding areas of the Azov Sea is characterized by the presence of silty and clay fractions. Therefore, for short integration periods, deposition processes are not taken into account in the models. Figures 1 and 2 show the consistency of the results with satellite data. The result of the simulation depends significantly on the time interval at which the problem is solved. To implement the procedure for identifying the location of the initial contamination

Fig. 1 The scope for setting the initial concentration field and the model field at the end time

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Fig. 2 Field of suspended matter concentration in the sea surface layer

spot, six related tasks were solved (6)–(8), (10). The calculated grid cell was selected as the . In Fig. 1, a circle (calculation 1) and rhombuses (calculations 2–6) indicate the points for setting initial data for adjoint problems. The result of integrating the adjoint problem for the first calculation is shown in Fig. 3. The figure shows that the most likely source of pollution is the area in the Kerch Strait, which is consistent with the known initial concentration field. Another likely source is the coastal area of the Black sea coast of the Taman Peninsula. From this area, the impurity can be transported to the area of the Feodosian Gulf by the Main Black sea current. Sometimes it is not possible to accurately determine the coordinates of the maximum values in the concentration field, but you can clearly

Fig. 3 Solving the adjoint problem (calculation 1)

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Fig. 4 Areas for setting the initial field of concentration and intersection of solutions of adjoint problems (calculations 2–6)

determine the boundaries of the pollution area, for example, from satellite images. Figure 4 describes the possible location of the initial spotfield obtained by solving a series of conjugate problems with initial data (10) set in grid cells on the periphery of the model concentration (Fig. 1). Thus, solving some conjugate problems, we can select the area of possible location of the initial contamination spot. For a more accurate determination of its location, it will be useful to use available l information about the nature and composition of the impurity. To identify the initial concentration field, it is useful to use variational methods for identifying the input parameters of the transfer model.

4 Conclusions Numerical experiments have shown that it is possible to find the location of the initial contamination spot by solving a series of conjugate problems. It is shown that data on the impurity distribution are useful information for such identification. Numerical experiments have shown reliable operation of algorithms based on solving conjugate problems in relation to the model of passive transport of impurities in the Kerch Strait. The results obtained can be successfully used to study the influence of contamination sources on the environmental situation in the Azov-black sea basin. The work was carried out within the framework of the state task on the theme № 0827-2018-0004 Complex interdisciplinary studies of Oceanological processes determining the functioning and evolution of ecosystems in the coastal zones of the Black and Azov seas.

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References Agoshkov VI, Parmuzin EI, SHutyaev VP (2013) Observational data assimilation in the problem of Black Sea circulation and sensitivity analysis of its solution. In: Izv. RAN, Fizika atmosfery i okeana, vol 49, no 6, pp 643–654 (2013) Eremeev VN, Kochergin VP, Kochergin SV, Sklyar SN (2008) Mathematical modeling of hydrodynamics of deep-water basins. ECOSI-Gidrophisika, Sevastopol, 363 p Fomin VV (2002) Numerical model of water circulation in the Azov Sea. In: Nauchnye trudy UkrNIGMI, Iss 249, pp 246–255 (2002) Kochergin VS (2011) Determination of the field of concentration of a passive impurity from initial data based on the solution of conjugate problems. In: Ecologicheskaya bezopasnost’ pribrezhnoj i shel’fovoj zon morya, MGI, Sevastopol, Iss 25, no 2, pp 370–376 Kochergin VS, Kochergin SV (2010) The use of variational principles and the solution of the adjoint problem, identification of input parameters for models of transport of passive tracer. In: Ecologicheskaya bezopasnost’ pribrezhnoj i shel’fovoj zon morya, MGI, Sevastopol, Iss 22, pp 240–244 (2010) Kochergin VS, Kochergin SV (2015) Identification of a pollution source power in the Kazantip Bay applying the variation algorithm. Phys Oceanogr 2:79–88 Makarov SN, Lanin VI (2017) Environmental hazard of ferrous products in the lower and upper Churbash tailings dam of the ZHRK. In: SCI-ARTICE, Ekologiya, no 43 Marchuk GI (1982) Mathematical modeling in the environmental problem. Nauka, Moscow, 320 p Marchuk GI, Penenko VV (1978) Application of optimization methods to the problem of mathematical simulation of atmospheric processes and environment. In: Modelling and optimization of complex systems, IFIP-TC7 working conference. Springer, New York, pp 240–252 (1978) Matishov GG, Golubeva NI, Sorokina VV (eds) (2011) Ecological Atlas of the sea of Azov. Rostovon-don, Izd. YUNC RAN, 328 p Penenko VV (1981) Methods for numerical modeling of atmospheric processes. Gidrometeoizdat, 350 p Ryabcev YUN, Shapiro NB (2009) Determination of the initial position of surface lenses of low salinity impurity detected in the open sea. In: Ecologicheskaya bezopasnost’ pribrezhnoj i shel’fovoj zon morya, MGI, Sevastopol, Iss 18, pp 141–157 (2009) Shutyaev VP, Le Dime F, Agoshkov VI, Parmuzin EI (2015) Sensitivity of functionals in problems of variational assimilation of observational data. In: Izv. RAN, Fizika atmosfery i okeana, vol 51, no 3, pp 392–400 (2015)

North Atlantic Oscillation and Interannual Mixed-Layer Heat Balance Variability in the North Atlantic A. B. Polonsky

and P. A. Sukhonos

Abstract The paper is devoted to the analysis of the interaction of the North Atlantic Oscillation (NAO) and the upper mixed layer (UML) in the North Atlantic using the ORA-S3 oceanic re-analysis data for the period 1959–2011s. A mechanism of the NAO maintaining at the interannual time scale has been specified. It is shown that the 6–8 yrs spectral peak in the NAO index time series is due to the interaction of the UML in the North Atlantic and the atmospheric boundary layer. It has been found that anomalies of the UML parameters generated by the NAO in the central and eastern parts of the North Atlantic reach its western part in 24–28 months. Due to advection and diffusion processes in the UML, the oceanic signal from the western part of the North Atlantic moves to the region of the Iceland Low and leads to change in the heat exchange between the ocean and the atmosphere and, hence, switch off the NAO phase. It is confirmed that the NAO variability at a scale of 6–8 years is a coupled mode in the North Atlantic ocean–atmosphere system. Keywords North Atlantic Oscillation · Upper mixed layer · Heat balance · North Atlantic

1 Introduction North Atlantic Oscillation (NAO) is one of the most important processes that determine the interannual variability of large-scale atmospheric circulation in the Northern Hemisphere (Barnston and Livezey 1987; Hurrell 1995; Mächel et al. 1998; Walker and Bliss 1932; Wallace and Gutzler 1981). This climate signal is characterized by a change in the intensity of zonal circulation and a shift in air masses between the subtropical and subpolar centers of atmospheric action in the North Atlantic. At the same time, the NAO significantly stands out in hydrometeorological and oceanic fields. An extensive literature is devoted to an analysis of the NAO key features and its manifestations in the hydrophysical parameters of the atmosphere and ocean (see, for A. B. Polonsky · P. A. Sukhonos (B) Institute of natural and technical systems, 28, Lenina str., Sevastopol 299011, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Chaplina (ed.), Processes in GeoMedia—Volume III, Springer Geology, https://doi.org/10.1007/978-3-030-69040-3_7

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example (Dzhiganshin and Polonsky 2003, 2009; Hurrell and Deser 2010; Marshall et al. 2001; Nesterov 2013; Polonsky et al. 2004; Visbeck et al. 2003) and the bibliography therein). However, despite the great attention paid by the NAO researchers, a number of issues related to the mechanism of the formation of interannual variations in the North Atlantic ocean–atmosphere system remain open. The aim of the paper is to specify the interaction between the NAO and upper mixed layer (UML) in the North Atlantic by analyzing the terms of the heat balance equation calculated according to the ORA-S3 oceanic re-analysis data for the period 1959–2011.

2 Data and Methods In this work, we used the monthly data from the ORA-S3 ocean re-analysis (Balmaseda et al. 2008) on ocean temperature, three-dimensional field of currents and mixed layer depth for the North Atlantic (0–70° N, 8–80° W) from January 1959 to December 2011. The mixed layer depth corresponds to the depth where the Richardson number attains a critical value of 0.3 (Pacanowski and Philander 1981). Monthly data on the net ocean surface heat fluxes were taken from the ERA-40 atmospheric re-analysis (Uppala et al. 2005) during the period from January 1959 to June 2002 and operative analysis by the ERA-40 model during the period from July 2002 to December 2011. The heat and momentum fluxes from the atmospheric re-analysis and the ERA-40 operative analysis were used as boundary conditions for the ocean model in the ORA-S3 re-analysis. The monthly values of the NAO index for the period 1959–2011 were taken from the website of the Climate Prediction Center (https://www.cpc.ncep.noaa.gov/data/ teledoc/nao_ts.shtml). The UML heat balance equation, on the assumption that the ocean temperature in the UML is vertically homogeneous, is as follows: Tt = −U Tx − V Ty −

 (T − T−H ) · W−H Q 0 − Q −H + + HED H ρ0 C P H

(1)

Here, ρ 0 is the density of sea water in the UML, C P is the heat capacity of sea water at constant pressure (ρ 0 C P is taken as a constant). The H, T and T –H are the mixed layer depth, the UML temperature and temperature at the UML base, respectively. The components of the current velocity vector in the zonal and meridional directions averaged within the UML are designated as U and V, respectively. The W –H is the vertical velocity of currents normal to the UML base, taking into account the horizontal inhomogeneity of the UML:  = W−H + U−H · Hx + V−H · Hy . W−H

(2)

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The T x (H x ) and T y (H y ) are zonal and meridional gradients of the UML temperature (mixed layer depth). The U –H , V –H and W –H are zonal, meridional and vertical velocity of currents at the UML base. The x axis is eastward oriented, the y axis is oriented to the north, and the z axis is oriented vertically upward. The coordinate system origin is located on the undisturbed ocean surface. In the Eq. (1) T t is a partial derivative of the UML temperature, UT x , VT y and (T − T –H ) • W –H /H are zonal, meridional and vertical heat advection in the UML, respectively. The Q0 and Q–H are heat fluxes at the ocean surface and the UML base. The HED is the horizontal eddy diffusivity of heat. In more detail, the methodology for calculating the components of the UML heat balance and the estimation of errors arising from this is described in (Polonsky and Sukhonos 2018). The total UML heat balance also contains different-type errors, only some of them can be directly estimated from the data available. A posteriori error analysis showed that their integral effect is in the order of 10% of the magnitude of the main components of the UML heat balance. Correlation and regression analysis with an appropriate assessment of statistical significance are used. Before the analysis, linear trends for the period 1959–2011 are removed from the time series of all the UML heat balance components and the NAO index. Linear trend parameters are calculated by the least squares method. The correlations of the monthly values of the NAO index for each calendar month during the period from January 1959 to December 2011 with previous and subsequent monthly fields of the UML heat balance components for time shifts up to 36 months are calculated. Only cases with significant correlation coefficients are considered (values greater than +0.37 and less than −0.37 at a 99% significance level).

3 Results Anomalies of the UML hydrophysical parameters are generated by the interaction of the NAO with the upper ocean layer in the North Atlantic. The consequence of this is a change in the ratio between the components of the UML heat balance. The interannual changes in the NAO and the UML heat balance components (using monthly data) in the eastern and central parts of the North Atlantic (including the Azores High) are most often quasi-synchronous, and in the western part of the North Atlantic they mostly appear with some phase delay. The results obtained allow us to form a block diagram of the mutual changes in the NAO and the UML heat balance components in the North Atlantic at a scale of 6–8 years Fig. 1. According to this block diagram, the positive NAO phase corresponds to the inphase intensification of the north-east trade wind and the Canary upwelling. The negative temperature anomaly arises in the eastern part of the Tropical Atlantic. Under a positive NAO, there is also an increased outflow of cold water to the North Atlantic from the Arctic. The formation of the UML parameters anomalies and as a result anomalies of the UML heat balance components in the western parts of the

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Fig. 1 A block diagram of the mutual changes in the NAO and the UML heat balance components in the North Atlantic. Solid arrows were obtained for the winter season and characterize the connections statistically significant at a level of 99%. Dashed arrows indicate the connection statistically significant at a level of 90% (Dzhiganshin and Polonsky 2003). The vertical arrows pointing up (down) indicate the increase (decrease) of the UML heat balance component in absolute value. STG—the North Atlantic subtropical gyre. SPG—the North Atlantic subpolar gyre

North Atlantic subtropical gyre and the North Atlantic subpolar gyre occurs with a phase delay of about 24–28 months. In the central part of the North Atlantic this two year lag is due to the propagation of Rossby waves that are created by the NAO variability (Watelet et al. 2017). Note that the NAO generates disturbances in the UML in the subtropical and subpolar gyres. However, the propagation velocity of these disturbances in the subpolar gyre is substantially lower than in the subtropical gyre. The zonal extent for the subpolar gyre is also less than that for the subtropical gyre. Therefore, the time delay in the reaction of the western parts of the subtropical and subpolar gyres to the NAO forcing is almost the same. This also explains the presence of a broad peak in the spectrum of the NAO index at the interannual time scale. Advective and diffusion processes in the western part of the North Atlantic lead to the formation of the UML parameters anomalies in the region of the centers of action of the atmosphere after 24–28 months, which affect the heat exchange between the ocean and the atmosphere, atmospheric circulation and, hence, switch off the NAO phase. The relationship shown by the dashed arrows in Fig. 1 is not statistically significant at a level of 99%. However, in (Dzhiganshin and Polonsky 2003) it was found that this relationship is significant at a level of 90%. Thus, according to the block diagram, the complete cycle of interaction between the NAO and the UML in the North Atlantic is about 6–8 years, taking into account the time of advective transfer of anomalies. The result obtained is consistent with the presence of a significant peak at close periodicities in the spectrum of the NAO index.

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Verification of the relationships shown in the block diagram performed using regression analysis. For this, linear equations are constructed: i+time lag

I Ni AO = α1 · heat_advGul f

i+time lag

+ α2 · heat_adv Labr

+ ε,

(3)

where I NAO i —the NAO index in January, heat_advGulf and heat_advLabr —the module of horizontal heat advection in the Gulf Stream and the Labrador Current, respectively, α 1 and α 2 —multiple regression coefficients, ε – uncorrelated random component close to white noise. The value of i + time lag is the delay in months between the values of the NAO index and the module of horizontal heat advection in the western boundary currents. Coefficients α 1 and α 2 are calculated by the least squares method. The statistical significance of the regression coefficients was estimated using Student’s t-test. Given that the module of horizontal heat advection is always positive, a change in the sign of the coefficients α 1 and α 2 with increasing delay will show a change in the sign of the NAO index. The results of the regression analysis are shown in Fig. 2. Larger values of the NAO index correspond to larger (smaller) values of the module of horizontal heat advection in the Labrador Current (the Gulf Stream) after 24 (26) months and vice versa (Fig. 2 a). This confirms the correlation relationships shown in Fig. 1. The values of the coefficient of determination of linear connection with the NAO index at this lag are R21 = 0.25 and R22 = 0.23 for the module of horizontal heat advection in the Labrador Current and the Gulf Stream, respectively. Taking into account the fact that the time series of the module of horizontal heat advection in the Gulf Stream and the Labrador Current are in-phase anticorrelated between themselves from November to March for the period 1959–2011 (the values of the correlation coefficients range from –0.33 to –0.43 after removing the linear trend), the coefficient of multiple

Fig. 2 a The relationship between the values of the NAO index in January and the module of horizontal heat advection in the Labrador Current after 24 months (gray triangles) and the Gulf Stream after 26 months (black diamonds). Straight black lines—linear approximation. b The dependence of the coefficients α 1 and α 2 on the delay value of the module of horizontal heat advection in the Labrador Current (black solid curve) and the Gulf Stream (black dashed curve) with respect to the NAO index in January 1959–2011. The black squares in Figure (b) indicate the coefficients of the linear approximation shown in Figure (a). The gray dashed lines in Figure (b) show the lower limit of the 90% confidence interval. The gray solid horizontal line in Figure (b) visually marks a zero regression coefficient

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determination was determined by the formula R2 = R21 + R22 − R1 × R2 . Its value is 0.42. Thus, more than 40% of the NAO variability at a scale of 6–8 years can be explained by fluctuations in the advective heat transfer in the western boundary layer of the North Atlantic. The horizontal heat advection in the Gulf Stream is larger in absolute value than in the Labrador Current (Fig. 2a). Multiple regression coefficients with the NAO index for the module of horizontal heat advection in the Labrador Current are almost 2 times higher than in the Gulf Stream (Fig. 2b). This means that the contribution of the variability of advective heat transfers in the western boundary currents to the interannual variability of the NAO is approximately the same with a slight advantage in favor of the Gulf Stream. The dependence of the coefficients α 1 and α 2 on the time delay is shown in Fig. 2b. With a zero lag between the NAO index and the module of horizontal heat advection in the Labrador Current a high positive regression coefficient was obtained. This confirms the presence of an in-phase correlation between these characteristics and testifies in favor of the importance of drift transport in the UML heat balance here. The signs of the multiple regression coefficients with a lag of 24–28 months confirm the nature of the correlation relationships shown in Fig. 1. With an increase in the time lag up to 48–56 months, the signs of the regression coefficients are reversed. This also means a change in the sign of the NAO index at the same lag, which confirms the analysis of Fig. 1.

4 Conclusion As a result of changes in the UML heat balance components in the western boundary layer, the intensification of the NAO after 48–56 months is replaced by its weakening and vice versa. The full cycle of the interaction of the NAO with the UML in the North Atlantic is 6–8 years, which leads to the formation of the corresponding peak in the spectrum of the NAO index at the interannual time scale. The relatively low values of the correlation coefficients between interannual variations in the UML heat balance components and the NAO index are partially related to the presence of other mechanisms in the ocean-atmosphere system. In addition, with large time lags, it is difficult to obtain a high statistical significance of the signal due to the limited duration of the analyzed time series. Nevertheless, all the results presented in the paper are statistically significant. Acknowledgements The study was supported by state assignment of Institute of natural and technical systems (Project Reg. No. AAAA-A19-119040490047-7) «Fundamental research of processes in the climate system that determine the spatio-temporal variability of the natural environment of global and regional scales».

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References Balmaseda MA, Vidard A, Anderson DLT (2008) The ECMWF ocean analysis system: ORA-S3. Mon Wea Rev 136(8):3018–3034. https://doi.org/10.1175/2008MWR2433.1 Barnston AG, Livezey RE (1987) Classification, seasonality and persistence of low-frequency atmospheric circulation patterns. Mon Wea Rev 115(6):1083–1126. https://doi.org/10.1175/1520-049 3(1987)115%3c1083:CSAPOL%3e2.0.CO;2 Dzhiganshin GF, Polonsky AB (2003) North Atlantic oscillation and variability of characteristics of the upper oceanic layer. Izv Atmos Oceanic Phys 39(4):497–505 Dzhiganshin GF, Polonsky AB (2009) Low-frequency variations of the Gulf-Stream transport: description and mechanisms. Phys Oceanogr 19:151–169. https://doi.org/10.1007/s11110-0099047-5 Hurrell JW (1995) Decadal trends in the North Atlantic oscillation: regional temperature and precipitation. Science 269(5224):676–679. https://doi.org/10.1126/science.269.5224.676 Hurrell JW, Deser C (2010) North Atlantic climate variability: the role of the North Atlantic Oscillation. J Mar Syst 79(3–4):231–244. https://doi.org/10.1016/j.jmarsys.2009.11.002 Mächel H, Kapala A, Flohn H (1998) Behaviour of the centres of action above the Atlantic since 1881. Part I: Characteristics of seasonal and interannual variability. Int J Climatol 18(1):1–22. https://doi.org/10.1002/(SICI)1097-0088(199801)18:13.0.CO;2-A Marshall J, Kushnir Y, Battisti D et al (2001) North Atlantic climate variability: phenomena, impacts and mechanisms. Int J Climatol 21(15):1863–1898. https://doi.org/10.1002/joc.693 Nesterov ES (2013) North Atlantic Oscillation: atmosphere and ocean. Gidromettsentr RF, Moscow, p 144 Pacanowski RC, Philander SGH (1981) Parameterization of vertical mixing in numerical models of tropical oceans. J Phys Oceanogr 11(11):1443–1451 Polonsky AB, Basharin DV, Voskresenskaya EN et al (2004) North Atlantic Oscillation: description, mechanisms, and influence on the Eurasian climate. Phys Oceanogr 14(2):96–113. https://doi. org/10.1023/B:POCE.0000037873.85289.6e Polonsky AB, Sukhonos PA (2018) On the contribution of the eddy transport to the annual mean heat budget of the upper layer in the North Atlantic. Izv Atmos Oceanic Phys 54(5):507–514. https://doi.org/10.1134/S0001433818050092 Uppala SM, Kallberg PW, Simmons AJ et al (2005) The ERA-40 reanalysis. Q J R Meteorol Soc 131B(612):2961–3012. https://doi.org/10.1256/qj.04.176 Visbeck M, Chassignet EP, Curry RG et al (2003) The ocean’s response to North Atlantic Oscillation variability. North Atlantic Oscillation Clim Signific Environ Impact 134:113–145. https://doi.org/ 10.1029/134GM06 Walker GT, Bliss EW (1932) World weather V. Memoirs of the R Meteorol Soc 4(36):53–84 Wallace JM, Gutzler DS (1981) Teleconnections in the geopotential height field during the Northern Hemisphere winter. Mon Wea Rev 109(4):784–812. https://doi.org/10.1175/1520-0493(1981 )1092.0.CO;2 Watelet S, Beckers JM, Barth A (2017) Reconstruction of the Gulf Stream from 1940 to the present and correlation with the North Atlantic Oscillation. J Phys Oceanogr 47(11):2741–2754. https:// doi.org/10.1175/JPO-D-17-0064.1

Marangoni Convection in a Cylindrical Container in the Absence of Gravity A. V. Kistovich

Abstract The problem of the occurrence of convection in a cylindrical container heated from the end filled with two immiscible liquids located in weightlessness is considered. The cause of convective motion is the temperature dependence of the surface tension coefficient at the interface of media. A General criterion for the beginning of convection is formulated in an explicit analytical form and its particular expressions are given in limit cases. Keywords Surface tension · Marangoni convection · Marangoni number

1 Introduction Marangoni convection (or thermocapillary convection) has been the subject of intensive experimental and theoretical research for a century. One of the goals of such studies is to determine a certain set of dimensionless numbers that characterize the intensity of the resulting convective movement. In the presence of gravity there are two such numbers namely Rayleigh number Ra and Marangoni number Ma. In systems with reduced gravity or in weightlessness the role of the main characteristic of the intensity of convection is played only by the Marangoni number. Regardless of the problem statement, the Marangoni number is introduced on the base of dimension theory methods and is mainly represented in the form σ LT Ma = Tρχν · An , where ρ is the liquid density; χ, ν are coefficients of thermoconductivity and kinematic viscosity correspondingly; σT derivative of the surface tension coefficient by temperature of the medium; L is the characteristic spatial scale (along or across the boundary surface); T is the characteristic value of temperature overheating (overcooling); A is the aspect ratio of geometric sizes of the problem under investigation; n is some integer number. Sometimes the Marangoni number σ L 2 T  is defined by the value Ma = Tρχν l · An , where Tl is the characteristic value of temperature gradient in the medium (Villers and Platten 1992). A. V. Kistovich (B) Institute for Problems in Mechanics RAS, Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Chaplina (ed.), Processes in GeoMedia—Volume III, Springer Geology, https://doi.org/10.1007/978-3-030-69040-3_8

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It is usually assumed that the onset of convective motion occurs when the Marangoni number exceeds a certain critical value, which is usually determined either by the energy method (Joseph 1976), or based on the study of instability of solutions to the linearized problem. In this paper, we propose a different approach to determining the beginning of the convective movement caused by thermocapillary phenomena. Since when there is a temperature gradient of the interface, even the smallest, there is always a liquid movement and, consequently, convective heat transfer, then objectively thermocapillary convection occurs at arbitrarily small values of the Marangoni number. If we give a comparative assessment of the contribution to heat transfer of the convective and diffusive terms of the equation of evolution of the medium temperature, it is possible to introduce a subjective criterion for the beginning of convection, as a condition for exceeding the convective contribution in comparison with the diffusive one. For this purpose, the present paper considers the problem of starting the movement of two liquid media placed in a cylindrical container heated from the ends, the side walls of which are heat-insulated. External forces are absent.

2 Problem Statement The scheme of the studied convection problem and the coordinate system used for its solution are shown on the Fig. 1a, b. In the initial equilibrium state, liquids placed in a container with a thermally insulated side surface are at a temperature T0 constant throughout the volume. At

a

b

Fig. 1 The scheme of the problem (a) and the related coordinate system (b)

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the initial moment of time t = 0, the ends of the container are brought into thermal contact with two thermostats that maintain them at temperatures T10 and T20 . It is assumed that liquids are characterized by constant and temperatureindependent coefficients of thermal conductivity χm (m = 1, 2) and kinematic viscosity νm . Since there are no Archimedean forces, and the coefficients of temperature expansion are sufficiently small, the problem is considered in the approximation of constancy of the densities ρm of both liquids. The only physical parameter that is considered variable under temperature variations is the surface tension coefficient. Strictly speaking, there are three surface tension coefficients in this problem: at the boundaries “liquid 1–liquid 2”, “liquid 1– the surface of the container”, “liquid 2–the surface of the container”. The combined action of all three of these surface tension forces determines the value of the edge angle between the interface of two liquids and the side surface of the container. Temperature gradients occur in the liquids that fill the container under the influence of temperature differences at the ends of the container, which provide heat transfer between the ends. In this case, the interface between the liquids is heated unevenly, which, due to the dependence of the surface tension coefficient on the temperature, leads to movement in the liquids and, as a result, to convective heat transfer. It is assumed that all physical fields have axial symmetry and the following notation is introduced: vm = um er +wm ez , pm , Tm are the- fields of velocity, pressure and e −ζ e temperature in the m-th liquid; ζ(r, t) is the interface between the liquids; n = √z r r2 , 1+ζr



e +ζ e s = √r r z2 are the unit normal and tangent vectors to the interface, where er , ez are 1+ζr

the unit orts of the cylindrical coordinate system. The system of equations (1), boundary (2) and initial (3) conditions of the problem in a cylindrical coordinate system have the form (Landau and Livshitz 1986; Korn and Korn 1968)

v1 |z=−h 1

   ∂um 1 ∂ pm + (vm ∇)um = − + νm um − um r 2 ∂t ρm ∂r ∂wm 1 ∂ pm + (vm ∇)wm = − + νm wm ∂t ρm ∂z ∂ Tm + vm · ∇Tm = χm Tm ∇ · vm = 0, ∂t  = v2 |z=h 2 = vm |r =R = 0, wm − ζr um z=ζ = ζt , v1 − v2 |z=ζ = 0

T1 |z=−h 1 = T10 , T2 |z=h 2 = T20 , ∂ Tm /∂r |r =R = 0, T1 − T2 |z=ζ = 0    

    (1) ζrr σ ζr   (2) n + − σ − σ − s p1 − p2 +  · ∇σ) n (s  i j i i j i j  2  2  r 1 + ζr 1 + ζr =0

(1)

z=ζ

(2) vm |t=0 = 0, pm |t=0 = pm0 , Tm |t=0 = T0 , ζ(r, t)|t=0 = ζ0 (r )

(3)

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Here ζ0 (r ) is the interface in initial equilibrium state. In the system (1) (vm ∇) f = um fr + wm f z ; in the system (2) σ is the coefficient of surface tension on the boundary «liquid 1–liquid 2», σi j (m) is the tensor of viscose tensions in the m-th liquid, which nonzero elements have the forms (m) σrr

σ(m) zz

 ∂um ∂wm ∂um (m) , σr z = ρm νm + , = 2ρm νm ∂r ∂z ∂r ∂wm = 2ρm νm ∂z

(4)

∇σ = σT ∇T , and in addition n i = n · ei , si = s · ei . To use the initial conditions (3) it is necessary to determine the values pm0 and ζ0 (r ). In the equilibrium state, the velocity field in both liquids is zero, the medium in the container is isothermal, that is Tm = T0 , so that ∇σ = 0. Then (1, 2) taking into account (4) acquire the form ∇ pm = 0, p1 − p2 + 

σ



1 + ζr 2

 ζrr ζr + 1 + ζr 2 r

   

=0

(5)

z=ζ

and the following conditions must be met

|ζ||r =0

 < ∞, ζ 

r r =R

2π = ctgθ, 0

⎡ R ⎛ ζ(r ) ⎞ ⎤   dϕ⎣ ⎝ dz ⎠r dr ⎦ = πR 2 h 1 0

(6)

−h 1

the limitation of the deviation of the interface when r = 0 providing the required value of the edge angle θ on the wall of the container and the law of conservation of mass of the first liquid (for the second liquid, the law of conservation is performed automatically in this case). The solution to the problem (5, 6) has the form  R 2R(1 − sin3 θ) r 2 cos2 θ − 1− = η(r ), ζ0 (r ) = 3 3 cos θ cos θ R2 2σ cos θ p1 = 0, p2 = R

(7)

Next, we consider the problem of starting the movement. Since the actual movement of the medium is generated by surface tension forces due to inhomogeneous heating of the interface, then at the initial moments of time the movement of liquids occurs in a narrow layer surrounding this surface. To describe the processes in this narrow layer, at the initial moments of time a coordinate system (s, n) is introduced instead of coordinates (r, z), where s is the arc length along the equilibrium surface η(r ) (7), counted from point (r = 0, z = η(0)) to point (r = ρ, z = η(ρ)), where ρ is the radial coordinate of the point on the surface. Since the equilibrium surface is known, then

Marangoni Convection in a Cylindrical Container …

ρ  s(ρ) = 1 + ηr 2 dr

75

(8)

0

The unit vector es is directed along the tangent to the equilibrium curve η(r ), as shown in Fig. 1b. The same figure shows a single vector en normal to the equilibrium curve, directed from liquid “1” to liquid “2”, along which the normal coordinate n is counted. Together with the ort eϕ , which is not shown here and is not used in the future, the entered coordinate system is the right and orthogonal one. Since there is a one-to-one relationship (8) between the coordinates s and ρ, which allows us to express the radial coordinate ρ of a point on the surface in terms of the arc length s (that is ρ = ρ(s)), and the shape of the equilibrium surface η(r ) is represented explicitly, this form is considered a function of a variable s, that is η(ρ) = η(ρ(s)) = η(s). To reduce further entries, derivatives of functions ρ and η with respect to variable s are indicated by strokes. The radius-vector of a point lying on an equilibrium surface is determined by the expression rη = ρ(s)er + η(s)ez . Since  the form of the equilibrium state is a one-parameter surface, the relation es = ∂rη ∂s = ρ er + η ez is valid, and from the condition es · es = 1 it follows the result  ρ ≡ 1 − η2 > 0. The relationship between the coordinates (r, z) and (s, n), their orts, and the rules for differentiating an arbitrary function f are determined by the relations r = ρ − nη , z = η + nρ , fr = D =1−

nη ρ

ρ  η  f s − η f n , f z = f + ρ f n , D D s (9)



 er = ρ es − η en es = ρ er + η ez ∼   ez = η es + ρ en en = −η er + ρ ez η ∂en η ∂es ρ ∂es ρ ∂en =  en = −  en , = 0, = −  es =  es , =0 ∂s ρ η ∂n ∂s ρ η ∂n

(10)

The velocity field is presented in the form v = Aes + Ben , so in accordance to (10) relations have place um = ρ Am − η Bm , wm = η Am + ρ Bm

(11)

Now it is necessary to rewrite a system of equations, boundary and initial conditions (1–3) in coordinates (s, n), for which the types of dependencies ρ(s), η(s) are defined beforehand. Substituting (7) into (8) produces the result 

 kρ cos θ 1π 1 − θ ,k = , s(R) = s(ρ) = arctg  2 2 k k 2 R 1−k ρ

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  1 1 2(sin3 θ − 1) η(s) = − + cos(ks) , ρ(s) = sin(ks), D = 1 − kn k 3 cos2 θ k

(12)

The pressures in the media are represented as pm = pm0 + p˜ m , where p˜ m is the perturbation of the equilibrium pressure, resulting in (1) taking the form k ∂ Am + vm · ∇ Am − Bm Am ∂t D   2k ∂ Bm 1 ∂ p˜ m k 2 Am + νm Am − 2 2 − 2 =− Dρm ∂s D ∂s D sin (ks)   2k ∂ Am ∂ Bm k 2 1 ∂ p˜ m k 2 Bm + vm · ∇ Bm + Am = − + νm Bm − 2 2 + 2 ∂t D ρm ∂n D ∂s D sin (ks) Am ∂ Tm ∂ Tm ∂ ∂ Tm ∂  2  ρD Bm = 0, + + Bm = χm Tm (13) (ρD Am ) + ∂s ∂n ∂t D ∂s ∂n    where vm · ∇ f = ADm f s + Bm f n ,  f = D12 f ss + f nn + Dk D1 ctg(ks) f s − 2 f n . From the system (2) of boundary conditions for the problem of the beginning of convection, those that are set at the interface of two media and at the ends of the container, where fixed values of temperature are maintained, are of particular importance. Kinematic boundary conditions in coordinates (s, n), taking into account (11) and the fact that at the initial moments of time ζt ≈ 0, take the form   Bm |n=0 ≈ 0 ⇒ ∂ q Bm ∂s q n=0 ≈ 0, A1 |n=0 = A2 |n=0

(14)

where q is any integer number. The conditions for the temperature do not change their type, and the dynamic boundary conditions with account of (4, 9, 11, 14) are written in the form   ∂ B2 ∂ B1  − ρ1 ν1 =0 p˜ 1 − p˜ 2 + 2 ρ2 ν2 ∂n ∂n n=0    ∂ A1 ∂ A2  − ρ2 ν2 k A2 + σT Ts + ρ1 ν1 k A1 + =0 ∂n ∂n n=0

(15)

From the incompressibility equation of the system (13) follows the representation Am =

1 ∂ψm 1 ∂ψm , Bm = − 2 , ψm = ψm (s, n, t) ρD ∂n ρD ∂s

(16)

where ψm are pseudo stream functions in the corresponding medium. Substituting (16) into the first two equations of the system (13), excluding pressure, linearization of the equations at the initial moments of time,  account  taking  into |kn| 1, ∂ψm ∂s  ∼ |kψm |, the relations (14) and the fact that near the boundary   2    2 |kψm | ∂ψm ∂n , generate equations ∂ − νm ∂ 2 ∂ ψ2m ≈ 0 whose solutions ∂t

∂n

∂n

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are written in the form    ψm ≈ (−1)m ϕ(s) 1 − exp(−n 2 4νm t)

(17)

reflecting the fact that viscous boundary layers near the interface, which satisfy the boundary conditions (14). The second dynamic boundary condition of the system (15) plays the main role in the formation of the flow in the studied problem, which, in accordance with (16, 17), takes the form  kϕ(s) σT Ts n=0 + (ρ1 + ρ2 ) = 0 t sin(ks)

(18)

In order to use the obtained condition (18), it is necessary to determine, at least approximately, the temperature distribution in the container in the absence of media movement. Getting an accurate solution is very difficult due to the inconvenient shape of the interface. For this reason, an approximate model of heat propagation is chosen, as if its diffusion occurred only in the direction of the axis z. In this case, an initial boundary value problem for perturbations τ(m) (z, t) = T (m) (z, t) − T0 of temperature distributions is formulated for each fixed value of the variable s   ∂τ(m) ∂ 2 τ(m) − χm = 0, τ(m) t=0 = 0, τ(m) z=η(s) = θ(η(s), t) 2 ∂t ∂z     ∂τ(1)  ∂τ(2)  −h 1 , m = 1 (m)  χ1 = χ2 , θ(η(s), 0) = 0, τ z=zm = τm , z m = h2, m = 2 ∂n  ∂n  n=0

n=0

(19) where τm = Tm − T0 ; θ(η(s), t) is the perturbation of the interface temperature, i.e., a function that also needs to be defined, along with distributions τ(m) (z, t), since it is included in the boundary condition (18), since Ts n=0 = θs (η(s), t). The solution of (19) is carried out by the method described in Koshljakov et al. (1970), as a result of which ⎡

⎤ ∞ (m)  )⎦ 2 2 sin( pπζ (−1) p e−αm p t τ(m) (z, t) = τm ⎣ζ(m) + + (1 − ζ(m) )θ(η(s), t) π p=1 p  2  sin( pπζ(m) )  (−1)q  (q) 2 θ (η(s), t) − θ(q) (η(s), 0)e−αm p t 2 q π n=1 p (αm p ) q=1 ∞

+

∞

(20) where ζ(1) = η−z , ζ(2) = z−η , αm = πHχ2m , H1 (s, t) = h 1 + η, H1 (s, t) = h 2 − η and H1 H2 m θ(q) is the derivative of the order q with respect to time t. 2

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Direct substitution (20) in (19) makes it possible to make sure that the presented solutions satisfy both the equations and the boundary and initial conditions. Crosslinking (19) of heat flows at the boundary z = η(s) defines an approximate expression for the function θ(η(s), t) θ(η(s), t) ≈

τ1 χ1 H2 θ4 (0, x1 ) + τ2 χ2 H1 θ4 (0, x2 ) , Hχ = χ1 H2 + χ2 H1 , xm = e−αm t Hχ (21)

where θ4 (0, x) is the Jacobi theta function (Abramowitz and Stegan 1964). When t → ∞ the value in (21) passes into a known limit of temperature, which is set at the border of two bodies heated from opposite ends, if there is an ideal thermal contact between the bodies  (Isachenko et al. 1975). The desired value Ts n=0 is determined, according to (21), by the ratio  η χ1 χ2 (H1 + H2 ) Ts n=0 = θs (η(s), t) ≈ (τ2 θ4 (0, x2 ) − τ1 θ4 (0, x1 )) Hχ2

(22)

      which was calculated under the condition ∂k −1 ∂s  1 ∼ σT Ts Rtgθ σ 1 that the interface surface is stationary. Now substitution of expressions for the distributions of velocity (16, 17) and temperature (20) in the heat transfer equations of the system (13) determine the relations of the convective and diffusive components of the heat flow         Convm   vm · ∇τ(m)   σT θs 2 n exp(−n 2 4νm t)      = ≈ κm =  , m = 1, 2 Diffm   χm τ(m)   (ρ1 + ρ2 )χm νm θss 

(23)

In the resulting expression (23) is not a function ϕ(s) included in Eq. (17) for pseudo stream function, which for definiteness it is convenient to choose in the form ϕ(s) = ϕ0 sin2 (ks), ϕ0 = const, providing the execution of Eq. (18), the conditions of boundedness of the velocity field in both fluids with s → 0 and equality to zero of its radial component at the axis of the container. Expression (23) is generally quite cumbersome (but using (22) is always explicitly presentable) and difficult to access for analytical research. This is because the process of heating the interface between the media and the appearance of a temperature gradient along it is dynamic. In this case, the characteristic times of heat propagation from the ends of the container to the pointof the interface with the coordinate s are determined by estimated values tm∗ ≈ Hm2 4χm , the values of which can be either close or significantly different. In addition, the fulfillment of the criterion for the occurrence of convection depends not only on the values of temperature superheats τm , but also on the relationships between them. Therefore, in most specific cases, the evaluation of the criterion is available only for numerical analysis, the use of which leads to a division of the space-time of the system under study into areas in

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79

which κm > 1 (the area of significant convective motion) and κm < 1 (the area of predominance of the conductive heat transfer mechanism). Observable analytical results are obtained for individual cases. So when t1∗ t2∗ and τ1 τ2 (23) takes the form       n exp(−n 2  4ν (t − t ∗ )) sin2 (ks)  σ τ1  m   1 T κm ≈ , t ≥ t1∗ , m = 1, 2   (ρ1 + ρ2 )χm νm  k H1 cos(ks) + 2 sin2 (ks) (24) and in the case t1∗ ∼ t2∗ of an arbitrary relation between τ1 and τ2 , (23) is converted to the expression      σ (τ2 − τ1 )2  λ2 n exp(−n 2 4νm (t − t ∗ )) sin2 (ks)  + 1 T κm ≈  , t ≥ t1∗ , m = 1, 2 (ρ1 + ρ2 )χm νm  λ+ (τ2 − τ1 )k H1 cos(ks) + 2μ sin2 (ks)   1 √ λ± = 2 χ1 χ2 Hχ ± χ1 χ2 (H1 + H2 ) Hχ  λ− Hχ (τ1 H2 + τ2 H1 ) 1 (25) μ = 3 χ1 χ2 (χ2 − χ1 )(τ2 − τ1 )(H1 + H2 ) − Hχ H1 H2 where the value k was defined in (12). As can be seen from (24, 25), the value κm depends on the spatial and temporal coordinates of the observation point and the characteristics of the environment. In principle, it may turn out that in one environment for a given moment of time there is a region of space where κ > 1 and convection is observed, and in another environment there is no such region. It is obvious that other options following from (24, 25) can also be implemented. Since at close times tm∗ the value κm ∼ |τ1 − τ2 |, and the occurrence of convection is more likely, the greater the difference in the temperatures of the container ends overheating. At close values of the superheat temperature of the ends, the heterogeneity of the temperature distribution along the interface decreases, which leads to a decrease in tangent viscous stresses near the boundary and suppression of the expected velocity field. Thus, to achieve the greatest convective effect, it is better to heat only one end or to cool the second end when heating one end. The ratio (23–25) includes a value k = cos θ R,  and the edge angle θ is determined by the expression cos θ = (σ1 (T0 ) − σ2 (T0 )) σ(T0 ), where σ1 , σ2 are the surface tension coefficients at the borders of the container and the first and second media, respectively (Sivukhin 1975). A distinctive feature of (23–25) is a monotonic decay of |θs 2 /θss | when growth equilibrium values σ(T0 ) of surface tension and the monotone increase of this ratio with increasing temperature difference |τ2 − τ1 |. The remaining thermodynamic parameters are considered fixed. This result fully corresponds to the physical meaning, since the thermocapillary effect is more significant the relative deviations of the surface tension coefficient from its equilibrium value.

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 This property is especially evident when the edge angles are close to π 2, when σ(T0 ) σ1 (T0 ) − σ2 (T0 ). In this case (24, 25) are converted to expressions, respectively    σ τ1  |σ1 (T0 ) − σ2 (T0 )|  s 2 |n| exp(−n 2 4νm (t − t1∗ )) (24 ) κm ≈ T σ(T0 ) (ρ1 + ρ2 )χm νm h 1 R    σ (τ2 − τ1 )  2|σ1 (T0 ) − σ2 (T0 )| T s 2 |n| exp(−n 2 4νm (t − t1∗ )) κm ≈ σ(T0 ) (ρ1 + ρ2 )χm νm R(h 1 + h 2 ) (25 ) in which the relative deviations ofthe surface tension coefficient of the interface   from the equilibrium value (σT τ1  σ(T0 ) and σT (τ2 − τ1 ) σ(T0 )) are explicitly present. Here and in the future, the conditions are omitted for shortening the entry t ≥ t1∗ , m = 1, 2. The results obtained allow us to calculate the estimated values of the convection criterion. Since the adhesion conditions are met on the side walls of the container and its ends, and the radial component of the velocity field turns to zero on the axis, the maximum values of κm must be reached at s ≈ R 2, n m ≈ 2νm (t − t1∗ ) < h m , which must be substituted in the formula (23). In the special case of an almost flat partition boundary from (24 , 25 ) follows     σ τ1  |σ1 (T0 ) − σ2 (T0 )| 2νm (t − t1∗ ) T κm ≈ R σ(T0 ) 2(ρ1 + ρ2 )χm νm h1     σ (τ2 − τ1 ) |σ1 (T0 ) − σ2 (T0 )| 2νm (t − t1∗ ) κm ≈ T R σ(T0 ) (ρ1 + ρ2 )χm νm h1 + h2

(24 ) (25 )

As can be seen from (24 , 25 ), the value κm is proportional to the average temperature gradient imposed on the bi-liquid system (values |τ1 |/h 1 and |τ2 − τ1 |/(h 1 + h 2 )) and the radius of the container R. It follows from (7, 12) that to the radius R is directly proportional the difference in the levels of the separation border on the sidewall and in the center of the container. As a result, the greater the imposed temperature gradient and the deviation of the surface shape from the plane, the greater the temperature differences at the points of the interface. Consequently, the greater the relative deviations of the surface tension coefficient from the equilibrium value, which leads to an increase in the convective contribution to temperature transfer.

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3 Conclusions The resulting solution of the problem of the beginning of the convection parameter κ, which is an assessment of convective contributions to heat transfer compared to conduction, it is proposed to use as a criterion of the intensity of convective motion, because this option has all the necessary features that reflect the physical nature of thermocapillary convection, and not possessed by the Marangoni number, commonly used in problems of this type.

References Abramowitz M, Stegan IA (eds) (1964) Handbook of mathematical functions. NBS, Applied math. series 55 Isachenko VP, Osipova VA, Sukomel AS (1975) Heattransfering. Energy, Moscow (in Russian) Joseph DD (1976) Stability of fluid motions I, II. Springer, New York Korn GA, Korn TM (1968) Mathematical handbook. McGraw-Hill Book Company Koshljakov NS, Gliner EB, Smirnov MM (1970) The equations of mathematical physics in partial derivatives. The Higher school, Moscow (in Russian) Landau LD, Livshitz EM (1986) Theoretical physics. V. VI. Hydrodynamics. Nauka, Moscow (in Russian) Sivukhin DV (1975) The course of general physics. V. II. Thermodynamics and molecular physics. Nauka, Moscow (in Russian) Villers D, Platten JK (1992) Coupled buoyancy and Marangoni convection in acetone: experiments and comparison with numerical simulation. JFM 234:487–510

Filtration in Gassy Coal Seams Taking into Account the Dependence of Permeability on Stresses V. I. Karev

and T. O. Chaplina

Abstract The paper discusses a model of the filtration process in a coal seam. Change of the filtration properties of gas-filled coal initially compressed to rock pressure with a decrease in compressive stresses is studying. Depending on the degree and pattern of unloading, as well as on the strength properties of the material, its structure and filtration properties change in a certain way. The paper also considers the model problem of filtering gas into a well drilled in an underworked coal seam perpendicular to the plane of its occurrence. The nonlinear filtration equation describing this process was solved by numerical method. The influence of the plasticity zone formed around the walls of the well on the possibility of emptying the coal seam from gas, and, therefore, on the possibility of preventing an outburst-hazardous situation occurrence is considered. Keywords Coal seam · Gas-filled pores and cracks · Stress–strain state · Well · Underworking of a seam · Cracks growth · Plasticity deformation zone · Filtration

1 Introduction A coal seam in the initial state contains a large number of isolated pores and cracks filled gas-methane compressed to high pressures close to rock pressure value. Under conditions of all-around compression by rock pressure, coal seam has no filtration ability (Kuznetsov and Krigman 1978). This can be explained by that due to the plasticity of coal (the ability to plastic deform under stresses over long geological time periods), there are no filtration channels in it. As a result of the unloading of the formation during mining works from gas-filled pores under the impact of gas pressure, cracks can grow. With a sufficient degree of unloading, cracks can have grown so much that a connected system of filtration channels is formed, which will ensure the appearance of the filtration ability of the formation. Free gas contained in the pores and cracks, has an opportunity to get out of the reservoir. As a result, firstly, V. I. Karev · T. O. Chaplina (B) Ishlinsky Institute for Problems in Mechanics, RAS, Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Chaplina (ed.), Processes in GeoMedia—Volume III, Springer Geology, https://doi.org/10.1007/978-3-030-69040-3_9

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a coal and gas outburst hazard during mining work is reduced significantly (Khristianovich and Salganik 1980), and, secondly, the possibility of methane production from the coal seam is provided, what is becoming increasingly important in modern conditions (Klimov et al. 2015; Yurova 2016).

2 Free and Sorbed Gas in a Coal Seam This paper did not take into account the phenomenon of gas desorption due to which the amount of free gas in the formation can increase. For the coal-methane system, the saturation pressure of sorption is approximately 30–40 at. No matter how the pressure changes in the range above this value, the amount of sorbed gas will remain at the same level. Since these studies were carried out in connection with the problem of coal and gas sudden outburst, this paper considers formations lying at depths of more than 400 m, where rock pressure exceeds 100 at. At shallower depths, coal and gas sudden outburst do not occur usually. In addition, there is a so-called “weathering zone”, its depth is usually 100–300 m: there is no gas at all in layers lying close to the earth’s surface, as indicated by the absence of gas evolution into mine workings. The coal in this zone is not compressed enough to impede of gas filtration, and for large geological times it completely leaves the reservoir. All studies in this paper relate to the pressure range from rock pressure to the saturation pressure of sorption. Below the saturation pressure, the desorption phenomenon must be taken into account, which will further complicate the task. It should be noted that in the development of salt deposits, where sudden outbursts also occur, this problem does not exist at all, since salts do not have sorption ability.

3 Cracks Growth and Formation of a Connected System of Filtration Channels in a Coal Seam To describe the model, a material that has the above properties of a coal seam is considered, and the processes that occur in such a material when it is unloaded are investigated. It is assumed that the material is deformed elastically, and that in the initial state, it is under conditions of uniform all-round compression, the gas pressure in the pores is equal to the value of the compressive load. When material is unloaded in one of the directions below a certain level, cracks in planes orthogonal to the direction of unloading begin to grow from the pores under the influence of gas pressure. An oriented system of parallel cracks is formed. The cracks growth process under these conditions was studied in Kovalenko (1980a). Here, it is considered how, as a result of the growth of parallel cracks, a connected system of filtration channels can be formed. To do this, there are two possibilities, which of them is implemented depends on the strength properties of the material and the degree of unloading.

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As calculations showed (Kovalenko 1980b), the decrease in gas pressure as a result of crack growth occurs in lesser extent than the change in compressive stress which caused it. Around a contour of each crack, the gas pressure in which is greater than the compressive load in the direction perpendicular to the plane of the crack, zones of additional tensile stresses are formed. The material in these zones is compressed to a lesser extent than outside of them, as a result the material can become permeable to gas to some extent. In a coal seam, this can be explained by the fact that microcracks which are characteristic of the coal structure are opened in these zones under gas pressure in cracks. Thus, in the vicinity of each crack, regions filled with gas are formed, which we will hereinafter call filtration zones. As long as the filtration zones of adjacent cracks do not intersect, there will be no filtration ability in the material. Under certain conditions, the filtration zones of neighboring cracks can begin to intersect, and then a system of connected filtration channels is formed, which is a system of parallel cracks linked by a system of microcracks opened within the limits of action of additional tensile stresses around each crack (Fig. 1). The sizes of these zones depend on the difference between the gas pressure inside the crack and the value of the compressive stress in the direction perpendicular to the plane of the crack, the greater this difference, the greater the zones are; and also they depend on the size of the crack itself, for larger cracks they are larger. However, the pressure in cracks decreases markedly with the growth of cracks. Evaluation calculations showed that when the radius of the crack increases by more than three to four times, the difference between compressive stress and pressure in crack decreases to a value of about one percent of the compressive stress value, and the zones of additional tensile stresses become negligible. It should be noted that under real conditions the

Fig. 1 The formation of a system of connected filtration channels as a result of the intersection of zones of additional tensile stresses around adjacent cracks

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Fig. 2 The formation of a system of connected filtration channels as a result of the intersection of grown cracks

planes of the cracks, of course, cannot be strictly parallel in the mathematical sense, and with a significant increase in the size of the cracks, they begin to intersect with each other (Fig. 2). This is the second possibility of forming a system of connected filtration channels. Thus, the first described option is implemented in stronger coals and at not very strong unloading. In less strong coals and at large degrees of unloading, when the cracks grow greater, the second option is implemented. Since in both cases the permeability of the material is ultimately ensured by cracks, these two cases are mathematically equivalent, that is, the permeability is proportional to the third degree of crack opening, which is determined by the difference between the gas pressure in the crack and the stress value. However, the permeability of the filtration zones is much less than the filtration coefficient for the cracks themselves, since it is caused by a system of microcracks, the opening of which is much less. As a result, in the case when the filtration ability of the material appears due to the intersection of the filtration zones, the effective permeability of the material will be less than in the case of the intersection of the cracks themselves. Based on the above ideas, it is easy to explain the sharp anisotropy of the filtration properties of the coal seam in the zone of influence of the opening observed in natural conditions. It is clear that the permeability along the plane of cracks will be significantly greater than the permeability in the direction perpendicular to the plane of cracks.

4 Filtration Model in a Coal Seam So, in the presence of a pressure gradient, as soon as a system of connected filtration channels is formed in the material, gas filtration begins. During the filtration process, the gas escapes from the cracks, the pressure in them drops and additional tensile stresses around the cracks decrease. This leads to a decrease in the opening of both the cracks themselves and microcracks in the filtration zones and, consequently, to a decrease in the permeability of the material.

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In the light of the physical concepts presented above about the process of gas filtration in a coal seam, to construct a mathematical model of the phenomenon, it is necessary to solve the following problems: 1.

2. 3.

To analyze the stress–strain state around a single gas-filled crack in a material containing many such cracks to the purpose of the determination of the size and shape of the filtration zones. To Assess the Permeability of the Filtration Zone, the Pattern of Its Dependence on the Stress State and Gas Pressure in the Crack. To determine the effective coefficient of filtration of the coal seam in the zone of oriented cracks, that is, non-filtering material containing a system of parallel cracks surrounded by areas of known shape and size with known filtration characteristics, given that such areas from adjacent cracks intersect.

A study of the influence of the pattern of material unloading on the growth of gas-filled cracks was made in Karev (2019). It has been shown that the growth of cracks in the material with a system of parallel cracks occurs to a greater degree, than the growth of crack in a material without cracks and in the material with a system of randomly oriented cracks to a lesser degree. These differences are smaller for the material greater initial porosity.

5 Radial Gas Filtration in a Gas-Saturated Coal Seam Working of the coal seams lying higher or lower is used widely In the practice of developing outburst-hazardous coal seams as measures to combat sudden outbursts. The effectiveness of this method is explained by the fact that the development of the protective coal seam leads to partial unloading from the rock pressure of the protected seam in the direction perpendicular to the bedding plane. As a result, an oriented system of gas-filled cracks parallel to the roof (sole) of the layer develops in the protected layer. The formation of such system of cracks is favorable in terms of preventing the development of a sudden outburst for two reasons. Firstly, and this is the main thing, when a working of protected coal seam is conducted, a system of oriented cracks in planes parallel to a seam face does not arise (the occurrence of an outburst-hazardous situation is associated with the formation of just such system of cracks), since the gas pressure causing cracks growth has already “worked out” during the development of the protective formation to form a safe system of cracks. Secondly, the formation of a system of cracks parallel to the bedding plane of the protected layer leads to connecting filtration channels in this direction, so that it becomes possible for gas filtration along the bedding plane. To assess the effectiveness of this method of degassing a coal seam, to determine the characteristic times necessary for the most complete exit of gas from the seam, to plan the frequency and location of degassing wells, it is important to be able to calculate the filtration processes in the vicinity of one well drilled across the mining coal seam.

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Fig. 3 The formation of a system of parallel cracks in the underworked coal seam: 1—a system of cracks in the protected seam, 2—a worked out section of the protective coal seam

Suppose, that due to mining the protective formation, a certain section of the protected formation was unloaded from rock pressure, so that as a result of the growth of gas-filled cracks, a system of connected filtration channels was formed in directions parallel to the bedding plane of the formation (Fig. 3). Then, a well of radius rw was drilled instantly across the protected formation at the unloaded section. Drilling a well causes a redistribution of stresses in its vicinity, which leads to the formation of a zone of plasticity deformations of radius (r p ) near the well (Fig. 4). Gas filtration into the well begins in the unloaded section of the protected formation through the filtration channels available in it. From the point of view of filtration characteristics, two zones can be distinguished in the vicinity of the well, they are the plasticity zone and the cracks zone (Karev and Kovalenko 1988). There is reason to believe the size of the filtration channels does not depend on the gas pressure in them due to the fact that, coal in the plasticity zone is severely destroyed, that is, the permeability coefficient in this zone is constant, k = const. For the permeability coefficient in the cracks zone, it can be written:

Fig. 4 The formation of a plasticity deformations zone around the well: 1—a plasticity zone, 2—a cracks zone

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89



P −σ k(P, r ) = A P0

3 (1)

where P is the gas pressure at a given point r of the seam, σ is the average value of the external stress component acting at the given point of the seam, Po is the initial gas pressure in the seam perpendicular to the bedding plane, A is a coefficient having permeability dimension. The equation of radial gas filtration in porous medium has the form   k ∂ ∂(m P) 1 ∂ P r = , r ∂r μ ∂r ∂t

(2)

where μ is dynamic viscosity coefficient of gas, m is porosity, t is time, r is the distance from the center of the well. Permeability coefficient k is a function P and r .

k(P, r ) = A



kP P+σ P0

0

3

at

rw ≤ r ≤ r p

at

r > r p, P ≥ σ

at

r > r p, P < σ

(3)

The size of the area from which gas can filter is determined by the outer boundary of the cracks zone of the radius rc  rw . Then the initial and boundary conditions have the form. The size of the area from which gas can filter is determined by the outer boundary of the cracks zone of the radius rc  rw . The initial and boundary conditions have the form P(r, 0) = P0 ; P(rw , t) = Pa ; Q(rc , t) = 0

(4)

where Pa —atmospheric pressure, Q(rc , t)—is the gas flow rate at the boundary rc of the cracks zone and the undisturbed formation. Having expressed the gas flow rate through pressure, the last boundary condition can be written as ∂ P(rc , t) =0 ∂r If dimensionless variables are introduced P  = P/P0 ; t  =

P0 κ p t; x = ln(r/rc ) μmrc2

1 at 0 ≤ x ≤ x p 3  k(P , x) = k P − σ at x > x p , P  ≥ σ 0 at x > x p , P  < σ 







(5)

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where k  = kAp , σ = Pσ0 . Equation (2) can be rewritten in dimensionless form   ∂ ∂ P ∂P kP = e2x  ∂x ∂x ∂t

(6)

The initial and boundary conditions are written in the new variables as follows P  (x, 0) = 1; P  (0, t) = Pa /P0 ;

∂ P  (xc, t) =0 ∂x

(7)

Equation (6) is nonlinear. It was solved numerically by the finite difference method.

6 Calculation Results and Analysis of Formation Degassing Efficiency by Drilling Wells in Protected Formations Using the developed computation program, calculations were performed that show how the distribution of gas pressure in the formation during filtration into the well and the gas flow rate into the well are changed over time in dependence on the stress–strain state of the formation and its strength characteristics (Karev 2020). The following values of the input parameters were taken for the calculations: P0 = 100 at., rw = 2.5 cm, rc = 100rw , μ = 10−10 at. s, m = 0.025, κ P = 10−2 mD, α = 3, coal seam thickness H = 1 m. Figures 5 and 6 show the calculation results for the case when the value of the external compressive stress characterizing the degree of formation unloading as a result of underworking is equal to half the initial gas pressure value: σ = 0.5Po , the radius of the plasticity zone is equal to ten well radius: r P = 10rw (dashed lines). The coefficient A in (5) was taken equal to 0.8k P , so that at the initial moment of time the permeability in the cracks zone would be an order of magnitude lower than the permeability in the plasticity zone, which is natural to assume due to the high degree of disturbance of the latter. Figure 5 shows the distribution curves of gas pressure in the seam corresponding to time instants of 0.3 and 3 h after the start of filtration. Figure 6 shows the change in the gas flow rate into the well over time. For comparison, solid lines at the same figures show the corresponding curves for the case when the permeability over the entire volume of the filtering region is constant and equals k P = 10−2 mD. Figure 6 shows that the gas evolution rate, which immediately after drilling the well is 6 l/min, rapidly decreases over time. In addition, it can be seen on Fig. 5 that even after large times after drilling a well, the gas pressure in the formation, with the exception of a rather small area near the well, drops a little. This fact is even more clearly manifested when the seam is less unloaded: Fig. 7 shows the pressure curves corresponding to the same points in time after drilling a well, but for σ = 0.75P0 .

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91

Fig. 5 Gas pressure distribution in the vicinity of the well. - - - - - - in the underworked coal seam r P = 10rw , σ = 0.5P0 ; ——— in the constant permeability material; 1–0.3 h after the start of filtration; 2–3 h after the start of filtration

Fig. 6 Dependence of the gas flow rate into the well on time. - - - - - - in the underworked coal seam r P = 10rw , σ = 0.5P0 ; ——— in the constant permeability material

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Fig. 7 Gas pressure distribution in the vicinity of the well in the underworked coal seam r P = 10rw , σ = 0.75P0 ; 1–0.3 h after the start of filtration; 2–3 h after the start of filtration

From the two facts noted, it follows that the method used in practice for the degassing of underworked seams by drilling degassing wells in them is, apparently, not effective enough, since it allows you to “release” gas only from a small area near the well, while at the same time the gas pressure away from the well is close to the initial one. What determines how much gas pressure in the formation can decrease as a result of drilling a well? At the time t P when the gas pressure at the boundary of the plasticity zone decreases to a value equal to σ, the permeability k at this boundary from the side of the cracks zone becomes zero. In this case, the gas pressure in the cracks is balanced by an external compressive stress, there are no zones of additional tensile stresses around the cracks, and microcracks zones connecting the cracks between themselves are closed. However, this occurs at the very boundary of the plasticity zone, with the distance from this boundary increases, the gas pressure in the cracks zone increases, and the filtration coefficient increases too. The gas flows in the direction opposite to the pressure gradient, that is, from the outer boundary of the cracks zone to the boundary of the plasticity zone. This leads to a slight increase in pressure at the boundary of the plasticity zone and allows the gas to flow from the cracks zone to the plasticity one, but since the value at the boundary is close to zero, the amount of flowing gas is very small. In fact, with a slight correction for this amount of gas, it turns out that starting from the point in time t P , the plasticity zone is emptied, having a seemingly impenetrable external boundary, and in the crack zone there is a gradual equalization of pressure without changing the amount of gas contained in it.

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Ultimately, the plasticity zone is completely emptied, and in the cracks zone the gas remaining at the time t P is evenly distributed throughout the volume, so that the pressure throughout this zone becomes the same. What determines how much gas leaves the cracks zone and what pressure is established in it? It is clear that the later the pressure at the boundary of the plasticity zone drops to a value σ, the greater the amount of gas will leave the cracks zone. And it depends on the value σ and on the size of the plasticity zone (Figs. 5 and 6). Ultimately, the plasticity zone is completely emptied, the gas remaining in the cracks zone at the time t P is evenly distributed throughout the volume, so that the pressure throughout this zone becomes the same. What determines how much gas will leave the cracks zone and what pressure will be established in it? It is clear that the later the pressure at the boundary of the plasticity zone drops to the value σ, the greater the amount of gas will leave the cracks zone. It depends on the size of the plasticity zone (Figs. 5 and 6). Figure 8 shows the pressure distribution curves the dashed lines correspond to the case when r P = 5rw , the solid curves correspond to the case when r P = 20rw , σ = 0.5P0 . It can be seen the amount of gas, which can leave the cracks zone, increases with an increase in the radius of the plasticity zone. If there was no a plasticity zone, gas could not have left from such a material with cracks. Gas flows only from cracks crossed by the borehole, and immediately, a gas-tight boundary is form around these cracks.

Fig. 8 Gas pressure distribution in the vicinity of the well in the underworked coal seam σ = 0.5P0 ; - - - - - - r P = 5rw ; ——— r P = 20rw ; 1–0.3 h after the start of filtration; 2–3 h after the start of filtration

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The size of the plasticity zone formed in the vicinity of the well depends on the depth of the formation and the strength of coal. The deeper lies a seam and the less is coal strength, the more the amount of gas can be flowed from the seam.

7 Conclusions A number of conclusions based on the calculations of gas filtration into a well in a underworked coal seam can be made. This method of seam degassing is apparently not effective enough, since it is possible to significantly reduce the gas pressure only in a small vicinity of the well. The effectiveness of degassing wells significantly depends on the degree of unloading of the reservoir as a result of its underworking. If a seam is not underworked, then the gas will flow only from the zone of plasticity deformations in the vicinity of the well. The size of the plasticity zone has a great influence on the possibility of emptying the seam; the larger this zone is, the lower the pressure of the gas in the coal seam will drop. Acknowledgements The work was supported by the Ministry of Science and Higher Education within the framework of the Russian State Assignment under contract No. AAAA--A20120011690133-1.

References Karev VI (2019) The influence of the pattern of unloading on the growth of gas-filled cracks. Process Geomedia 2(20):146–153 (in Russian) Karev VI (2020) Strain properties of materials with gas-filled cracks. In: Processes in geomedia, vol I. Springer geology, pp 169–179 Karev VI, Kovalenko YuF (1988) Theoretical model of gas filtration in gassy coal seams. Sov Min Sci 24(6):528–536. https://doi.org/10.1007/BF02498610 Khristianovich SA, Salganik RL (1980) Sudden outbursts of coal (rock) and gas. Stresses and strains. Preprint No. 153. IPM of the Academy of Sciences of the USSR, 88 p (in Russian) Klimov DM, Karev VI, Kovalenko YF (2015) The geomechanics of gas recovery from coal seams. Dokl Phys 60(5):214–216 Kovalenko YuF (1980a) An elementary act of a sudden outburst. An outburst into a well. Preprint No. 145. IPM USSR Academy of Sciences, 44 p (in Russian) Kovalenko YuF (1980b) Effective characteristics of bodies with isolated gas-filled cracks. The wave of destruction. Preprint of IPM USSR Academy of Sciences, No. 155, 52 p (in Russian) Kuznetsov SV, Krigman RN (1978) The natural permeability of coal seams and methods for its determination. Nauka, Moscow, 122 p (in Russian) Yurova MP (2016) Prospects and possibilities of using coal methane as an unconventional energy source. Georesources 18(4) Part 2:319–324. https://doi.org/10.18599/grs.18.4.10 (in Russian)

Development and Sampling of the Device for Collecting Liquid Hydrocarbons from the Water Surface T. O. Chaplina

and E. V. Stepanova

Abstract The description of modern methods used for oil spills liquidation in the ocean is given. The original method of liquidating hydrocarbon spills with the help of natural sorbent—natural sheep’s wool—is stated. The effect of currents onset in the resting liquid during sorption of oil products and oils by fibre materials, and in particular natural sheep’s wool, can be used to improve the efficiency of technology for the hydrocarbon spills elimination and purification of natural water reservoirs. The carried out fluorescent diagnostics of the stages of process of water purification from oil pollution with sorbent on the sheep wool based showed that wool soaks up to 89% of oil depending on its initial concentration and the ratio of sorbent and contaminant amounts. Keywords Oil spills · Liquidation of hydrocarbon spills · Sorbents · Sheep’s wool

1 Introduction Mankind still gets most of energy from natural sources such as oil, gas and coal. Calculations and statistics show that the world’s fuel reserves are composed of 70% coal and 10% oil. During peak consumption periods only about 10–15% of explored coal reserves, and up to 65–70% of oil reserves are exploited. Due to the huge volumes of hydrocarbons production and transportation, accidental emissions are inevitable and are likely to occur at all stages of production, transportation and storage (from wellhead to the processing unit). Accidents and leaks occur during production, collection and storage of oil, from reservoirs, during drainage operations, delivery of oil products to consumers, transportation by pipelines, etc. (Fig. 1). The share of losses is small, but at the same time, the absolute amount of leaks reaches large values and, according to various sources, makes up from 5 to 17% of the produced T. O. Chaplina (B) · E. V. Stepanova Ishlinsky Institute for Problems in Mechanics, RAS, Moscow, Russia e-mail: [email protected] E. V. Stepanova e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Chaplina (ed.), Processes in GeoMedia—Volume III, Springer Geology, https://doi.org/10.1007/978-3-030-69040-3_10

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Fig. 1 Oil spills in the world ocean (https://e-cis.info/news/569/85080/)

products (Vishnevskaya et al. 2016). The accidental spill of oil products leads to economic damage not only due to the loss of raw materials, but mostly due to the need to restore ecosystems destroyed by these accidents. The causes of hydrocarbons ingress into the environment, source characteristics and assessment of the pollution values at oil and gas production facilities are adequately covered in the printed sources (“Expert” Group of Companies 2010; Vladimirov 2014). The effects of oil hydrocarbons on the biological equilibrium in natural ecosystems can be revealed over decades. In order to reduce the negative consequences of accidental emissions, the methods of localization and liquidation of oil spills need to be improved, there is a need in scientifically grounded development of a set of additional measures, in the course of which the collection and utilization of hydrocarbons released into the environment will become more efficient and less costly.

2 Methods and Devices for Hydrocarbons Removal from Water Surface The challenges of eliminating the consequences of accidental oil and oil products spills are not new, but still relevant in today’s life. Currently, there are several methods to clean up the open water surface from oil contamination, the first stage of all such methods is the detection and/or localization of contaminated water (oil slick), after this stage elimination begins—collection, destruction and disposal (Fig. 2). The most widely used technology at present is the mechanical assemblement of oil products after the localization of the slick with booms (Chaplina and Stepanova 2018). This method is often used in combination with a thermal method based on burning out of oil and its components. The physicochemical method is based on the use of dispersants and sorbents. Application of dispersants leads to strengthening of natural oil dispersion that considerably facilitates clearing of water surface. The use of sorbents, on the contrary, leads to the formation of lumps of material saturated with petroleum products, which

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97

Mechanical

Thermal

Physico-chemical

Biological

Mounting of barriers, collection, soil excavation

Combustion

Sorbents and dispersants placement

Bio- and phytoremediation with special microorganisms

Thin film remains on water and other surfaces

Strict observance of fire safety measures required, incomplete combustion produces additional contamination with carcinogenic substances

Refouling, drainage, sorption

Comparatively safe and efficient, but expensive and lengthy

Fig. 2 Methods of oil spill response

are less difficult to be mechanically assembled. At the moment, the most accurate method, which allows for the fine purification of water areas is biological. This method is based on special cultures of microorganisms that feed on oil and petroleum products and, therefore, processing them. More than a hundred sorbents are now produced and used for oil and oil products spills, which are divided into 4 basic types: inorganic, synthetic, organomineral and natural organic. The main characteristics determining the efficiency of the sorbent are its capacity in relation to oil, degree of hydrophobicity, buoyancy before and after sorption, availability of the regeneration and/or useful utilization, convenience of oil desorption. Properties and characteristics of natural organic sorbents have been described in detail in numerous scientific studies. Rice husks, brown coal, wool, crushed walnut and pine nut shells, peat, straw, graphite were considered as organic sorbents (Bayat 2005; Kuzmin 2002; Rethmeier and Jonas 2003; Yu et al. 2007). One of the most effective natural sorbents by all characteristics is wool. By its sorption capacity it is comparable to the modified peat: one kilogram of wool can absorb up to 8–10 kg of oil, at the same time wool allows desorbing the most part of light oil fractions.

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3 Experimental Studies of Physical Properties of Various Sorbents The results of experiments to determine the sorption capacity of different materials when used to eliminate hydrocarbon contamination of water surface are shown below. Spills simulation is carried out using petroleum, diesel fuel, aviation oil and sunflower oils with different physical characteristics (Chaplina et al. 2016a). Sheep’s wool, peat, cellulose fibre are used as sorbents. Also there are conducted experiments on application of biological preparation “Dop-Uni” with the purpose to evaluate its effectiveness at independent application and on the background of other methods of spills liquidation. For sorbents of organic origin (sheep’s wool, peat, cellulose fibre) buoyancy measurements are also made. The method of experiment prescribed placement of various amounts of sorbent material on the surface of known water volume at rest and further observation of sorbent immersion inside liquid depth. Identical vessels with different sorbents on the water surface are placed next to each other, to register the dynamics of the process of sorbent material immersion, the experiment advancement is recorded with digital camera in automatic mode for 4 h, shooting is carried out at a rate of 1 frame per minute, the obtained sequence of frames is processed and analyzed. Positions of sorbent mass centers are calculated as a result of images photometry (Fig. 3), generated from original frames by batch processing. In experiments containers comprise 100 ml of water and 1.0 g of sorbent, from left to right wool, cellulose and peat respectively (Fig. 3). Based on the obtained data, the graphs showing the dynamics of sorbent immersion (wool and cellulose) are drawn, which are shown in Fig. 4. Both sorbents are submerging in the water column, but the immersion rates vary significantly. The mass center of cellulose (in the central vessel) moved downwards (despite the presence of sorbent residues on the free surface), the volume of wool (in the vessel on the left) only sank slightly in 60 min after the start of the experiment. The experimental studies of sorption capacity and other properties of various materials in relation to oil products are performed in Chaplina and Stepanova (2018), on prospect to be used for the elimination of water surface pollution with hydrocarbons. This studies have shown that wool effectively absorbs hydrocarbon pollution (oil, diesel fuel, etc.) due to hydrophobicity and a high sorption coefficient.

Fig. 3 Fixed sorbent positions: Initial (a) and at time t = 66 min (b) after placement on the water surface

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Fig. 4 Dynamics of sorbents immersion into water: position of mass center 1—wool, 2—cellulose, 3 and 4—interpolation of corresponding data by polynomials of 4th degree

In experiments on placing the chosen amount of sorbent (from 1.0 to 3.0 g of natural sheep’s wool) on the water surface immediately after the sorbent comes into contact with the admixture under the unevenness of the wool layer the uniform thickness of the hydrocarbon splick is disturbed. The combined action of all forces affecting the system components led to the formation of visible currents changing the general picture of the hydrocarbon slick location on the free surface (Fig. 5). Left column in Fig. 5 shows the initial conditions of the survey for hydrocarbons sorption (tinted diesel fuel, petroleum) on the water surface. Right column shows the results of sorption of hydrocarbons portion by untreated sheep’s wool fibers placed on the surface. It is easy to see that regardless of the admixture composition on the surface sorption process allows to clean from contamination relatively large part of the surface originally occupied by the admixture. The dynamics hydrocarbon sorption process with sheep’s wool is shown in Fig. 6. Characteristic times inherent to oil sorption by wool are tens of seconds. For those shown in Fig. 5 cases, the characteristic time of reduction in time of the area of hydrocarbon surface coverage in e times, calculated by approximation of the data obtained by the photometry method (Shevtsov and Stepanova 2015), is 63.4 ± 4.9 s, with the preservation of the volume of the pollutant and increasing of the sorbent mass in 1.5 times, the characteristic time is reduced by an order of magnitude and is already 6.1 ± 0.4, the graphs show the relative areas of treated water surface (Fig. 6). For the experiments illustrated in Fig. 5, the threshold values of the relative purification area are 0.55, 0.83 and 0.80 respectively (Stepanova and Chaplina 2016). Characteristic change of water surface area free of hydrocarbons in time during a long experiment in all phases of sorption process is presented in Fig. 6b. For long periods of time the rate of reduction of oil-covered water surface area is about 0.025% per second, that testifies to practical exhaustion of sorbent capacity in the given time interval (Fig. 6b). The highest rates of absorption of contaminants by sorbent, and, accordingly, the highest rates of increase in the area of water surface free from

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a

b

c

d

Fig. 5 The initial and final states of the hydrocarbon spot on the water surface in the cuvette due to the currents induced by sorption on the fibre material (diameter of area 294 mm): 50 ml of diesel fuel (a, b), sorbent mass 2.0 g, time between frames—420 s; 30 ml of petroleum (c, d), 2.0 g of sorbent, 180 s

contaminants, are observed immediately after the application of sorbent (Chaplina et al. 2016b). The sorbed hydrocarbons bind well to the sorbent material—the resulting lumps are easily extracted from the water surface. The mass of the extracted lumps allows independent assessment of the volume of sorbed material, taking into account some water capture. Water samples taken at different stages of the purification process from oil pollution are analyzed by the methods developed in the “Laser spectroscopy of water media and laser biophotonics” laboratory of the Faculty of Physics of the Lomonosov Moscow State University. The method is based on the normalization of the fluorescence intensity of petroleum products on the intensity of the Raman scattering band (CS) of light by molecules of the medium—in our case, water or

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Fig. 6 Dependence on the time of the area free of surface contamination relative to the size of the initial spill (a) V o = 50 ml, mw = 2 g; reduction of the area of contamination under the influence of sorbent (b) in the time interval t = 1200–1250 s after the sorbent placement

Table 1 Oil concentration values in water samples after two consecutive sheep wool cleaning Sample No.

Oil concentration, mg/l

Cleanup degree, %

1

368

–

2

89

76

3

4.5

89

hexane (Filippova et al. 1993). Absorption spectra are measured on Lambda 25 spectrophotometer (PerkinElmer, USA). The values given in Table 1 correspond to the water cleanup from oil contamination by two equal portions of sorbent (sheep’s wool) in series. Sample No. 1 corresponds to the water sample taken before the purification process. Samples 2 and 3—after the first and second application of 1 g sorbent respectively.

4 Devices for Assembling Liquid Hydrocarbons from Water Surface The authors of the presented research have developed a device for separation of liquid hydrocarbons from water, which belongs to the class of devices for liquidation of hydrocarbons spills (petroleum and oil products, vegetable and mineral oils) on the surface and in water volume. The device is a frame filled with natural sheep’s wool, made of hollow elements or mesh, the test results of this device are presented in Chaplina and Stepanova (2018). After further research, the pilot model sample was refined and an upgraded device for collecting liquid hydrocarbons from the water surface was developed. The new device is a frame made of hollow elements or mesh, filled with natural sheep’s wool, packing density not exceeding 15 g/dm2 . The volume of sorbent (wool) is divided

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into two parts by a waterproof septum perpendicular to the direction of the smallest size of the frame. Cellulose fibre in the amount of 10–70% of wool volume is added to one of the parts of the device separated by the septum. The use cycle of the device includes cleaning of the collected hydrocarbons and reuse (at least three times). The Patent for useful model No 169140 “Liquid Hydrocarbon Collecting Device” is obtained (Chaplina and Stepanova 2017). Model samples of “Liquid Hydrocarbon Collecting Device” were manufactured in three dimension-types: 0.2 × 0.4 m, 0.4 × 0.8 m, 1.0 × 2.0 m, which are netted cells filled with natural sheep’s wool. The volume of wool is divided by a waterproof polyethylene septum, on one side of mat the wool with the addition of cellulose fibre in amount of 20% is used (Fig. 7). To clarify the terminology further, the word “mats” will be used to describe the experiments with the samples of the device. In order to test mat in laboratory conditions, the manufactured samples are placed on the surface of resting water, to which oil and diesel fuel (winter and summer) in various quantities are preliminary added. At each stage of the experiment, liquid samples are taken to determine the degree of contamination/refinement. Tested samples are exposed on the water surface for 24 h. The samples are then extracted, weighed and squeezed using squeezing device (created independently from the washing machine squeezing nozzle). As the material passes through the rollers of

Fig. 7 Photographs of “Liquid hydrocarbon collecting devices” pilot model samples in three dimension-types: a, b, c—0.2 × 0.4, 0.4 × 0.8, 1.0 × 2.0 m

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Table 2 The results of determining the degree of purification by the extract method Type of pollution, concentration (mg/l)

Mat size, m

Cleaning degree, %

Summer diesel (25.4)

0.2 × 0.4

33

0.4 × 0.8

35

Winter diesel (12.8)

“Siberian Light” oil (2.02)

Summer diesel (41.0)

Winter diesel (20.6)

“Siberian Light” oil (3.23)

1.0 × 2.0

29

0.2 × 0.4

45

0.4 × 0.8

40

1.0 × 2.0

41

0.2 × 0.4

61

0.4 × 0.8

55

1.0 × 2.0

52

0.2 × 0.4

38

0.4 × 0.8

36

1.0 × 2.0

31

0.2 × 0.4

37

0.4 × 0.8

37

1.0 × 2.0

36

0.2 × 0.4

77

0.4 × 0.8

78

1.0 × 2.0

72

the device, the liquid flows by gravity into the 200 l barrel. Used devices are placed in vacuum packaging for further processing or disposal. It should be noted that after 24 h none of the samples drowned. Samples are analyzed by fluorescent spectroscopy (fluorimetry) calibrated by Raman scattering of solvent—water or hexane on Horiba Jobin Yvon Fluoromax-4 spectrofluorimeter. The results are presented in Table 2. The device for collection of liquid hydrocarbons from the water surface can be used repeatedly: a removable framing after filling the sorbing cells with oil/diesel fuel is unmounted and can be used with a device that has been squeezed or with another device to reposition on the water surface. Experiments have shown that the ability to sorb the hydrocarbons well is maintained for at least 3 use cycles of the device.

5 Conclusion The authors of the presented study suggest to use as sorbent comparatively cheap raw material—sheep’s wool, part of which is currently eradicated by agricultural producers along with other waste. Due to the hydrophobic properties and low density,

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sheep’s wool, together with the absorbed substance, does not sink into the liquid depth and therefore can easily be mechanically assembled. A sketch sample of a “Liquid Hydrocarbon Collecting Device” from the water surface is created. The device can be used for the elimination of surface spills of various hydrocarbons in water reservoirs, and constitutes a removable framing around hollow elements or mesh, filled with natural sheep’s wool. Cellulose fibre in the amount of 10–70% of the wool volume can be added to one part of the device separated by septum. The use cycle of the device includes cleaning (squeezing) of the assembled hydrocarbons and reuse (at least three times). Fluorescent diagnostics of water purification process from oil pollution with sorbent on the basis of sheep wool is carried out. It is shown that wool sorbates up to 89% of oil depending on initial conditions and quantity of sorbent. Tests of pilot model sample of mat from the water surface are carried out in laboratory conditions and showed that the sample effectively assembles hydrocarbon pollution (oil, diesel fuel, etc.), due to the hydrophobicity of wool and its high sorption rate and can be successfully used to eliminate surface spills of various hydrocarbons in water bodies. Acknowledgements The work was supported by the project of the Russian Federation represented by the Ministry of Education and Science of Russia No. 075-15-2020-802.

References Bayat A et al (2005) Oil spill cleanup from sea water by sorbent materials. Chem Eng Tech 28(12):1525–1528 Chaplina TO, Stepanova EV (2017) Patent for useful model No. 169140 “Device for collecting liquid hydrocarbons”, date of state registration 16.03.2017 Chaplina TO, Stepanova EV (2018) Elimination of hydrocarbons spills on water objects and fluorescent diagnostics of water pureness. In: Karev V, Klimov D, Pokazeev K (eds) Physical and mathematical modeling of earth and environment processes. PMMEEP 2017. Springer geology. Springer, Cham. https://doi.org/10.1007/978-3-319-77788-7_3 Chaplina TO, Chashechkin YuD, Stepanova EV (2016a) Two forms of the collapse of the contact surface of the immiscible liquids in a composite with a cavity vortex. Monit Sci Technol 1(26):83– 89 (in Russian) Chaplina TO, Chashechkin YD, Stepanova EV (2016b) Flows induced by sorption on fibrous material in a two-layer oil−water system. Dokl Phys 61(9):444–448 “Expert” Group of Companies (2010) Analysis of the situation of environmental pollution by oil products. https://expertyug.ru/public/index.php/analitics/33-analiz-situacii-zagrjaznenija (in Russian) Filippova EM, Chubarov VV, Fadeev VV (1993) New possibilities of laser fluorescence spectroscopy for diagnostics of petroleum hydrocarbons in natural water. Can J Appl Spectrosc 38(5):139–144 https://e-cis.info/news/569/85080/ Kuzmin OA (2002) Passion for flax fiber. Tech Text (3):15 (in Russian) Rethmeier J, Jonas A (2003) Lignite based oil binder mats: a new absorbent strategy and technology. Spill Sci Technol Bull 8(5–6):565–567

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Shevtsov NI, Stepanova EV (2015) Application of photometry method to some hydrodynamic objectives. Moscow Univ Phys Bull 70(3):208–212. https://doi.org/10.3103/S00271349150 3008X Stepanova EV, Chaplina TO (2016) Investigation of the sorbing ability of different materials for application in oil spill response. Monit Sci Technol 1(30):22–28 Vishnevskaya NS, Yavorskaya EY, Salnikov AV (2016) Energy saving technologies of transport and storage of oil and gas. USTU, Ukhta, 169 p (in Russian) Vladimirov VA (2014) Oil spills: causes, scales, consequences. Civ Defense Strategy Probl Res 1(4):217–229 (in Russian) Yu EE, Yu MR, Lebedeva AA (2007) Obtaining the sorbent from the shell of pine nuts by the method of low-temperature processing. Polzunovskii Vestn 3:35–39 (in Russian)

Physical Modeling of Deformation, Fracture, and Filtration Processes in Low Permeability Reservoirs of Achimov Deposits V. I. Karev , Yu. F. Kovalenko , and Yu. V. Sidorin

Abstract The paper presents the results of physical modeling of deformation, fracture and filtration processes in Achimov deposits reservoirs of the Urengoy gas condensate field. At the experiments, the influence of stresses arising in the vicinity of wells during their drilling and operation on the filtration properties of the rocks of the Achimov deposits was studied. The experiments were carried out on the unique Triaxial Independent Loading Test System of the IPMech RAS (TILTS) which allows in cubic rock samples to recreate any real stresses arising in a bottom hole formation zone and to study their effect on filtration properties of rocks. During the experiments, the stress states arising near an open borehole and near the tip of a perforation hole were simulated. A direct modeling of the filtration processes near an uncased well at a decrease in bottom-hole pressure was carried out. The research results confirmed the prospect of using the geomechanical approach to implement new technologies including the directed unloading formation method (DUF) for the effective development of deposits with low permeability reservoirs. The conducted studies are important both scientifically and in practice to justify technical and technological solutions in the development of deposits with hard-to-recover reserves. Keywords Rock · Well · Perforation hole · Test system · Permeability · Stress · Fracture · Anisotropy

1 Introduction Nowadays, all over the world, the problem of hydrocarbon reserves depletion is becoming increasingly acute. Large and accessible deposits are largely exhausted. Under these conditions, the development of deposits with hard-to-recover reserves is becoming increasingly important. One of the main indicators of “hard to recover” reserves is the low permeability of reservoirs.

V. I. Karev (B) · Yu. F. Kovalenko · Yu. V. Sidorin Ishlinsky Institute for Problems in Mechanics RAS, Moscow, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Chaplina (ed.), Processes in GeoMedia—Volume III, Springer Geology, https://doi.org/10.1007/978-3-030-69040-3_11

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The development of significant hydrocarbon reserves of low permeability deposits, including deep-seated fields, is complicated by the low productivity of production wells and their high cost. Such reserves cannot be effectively extracted using traditional development methods for geological and technological reasons. It is necessary to create new low-cost methods with the long-term effect for increasing the productivity of wells draining deposits of low-permeability reservoirs, as well as methods to reduce the risk of uncontrolled destruction of the bottom-hole formation zone. The most promising is the creation of new technologies based on the impact on the formation using a huge reserve of elastic energy of rock massif due to the action of rock pressure and reservoir pressure of the fluid. Methods and parameters of such impact are determined on the basis of the geomechanical approach. The implementation of such technologies will increase production efficiency (well flow rate, oil recoverability coefficient) and decrease natural-technological risks during development, including reducing accident rate and possible environmental damage. An analysis of the geological conditions allows concluding that the use of the geomechanical approach can be effective for the development of the Achimov deposits of the Urengoy gas condensate field. The rocks of these deposits are characterized by low permeability—the maximum permeability values reach just several dozens mD. They lie at depths of more than 3.5 km and have an abnormally high reservoir pressure (anomaly coefficient of about 1.6). The last two factors are the reason that makes it difficult to apply traditional technologies, for example hydrofracture of reservoir, for the development of the Achimov deposits, but they are a plus in terms of the application of technologies based on the geomechanical approach. Below are the results of experimental studies of core material from the Achimov deposits of the Urengoy gas condensate field by using the TILTS confirming the conclusion that the use of technologies based on the geomechanical approach is promising.

2 Directed Unloading Formation Method A new method has been developed at the Institute for Problems in Mechanics of the Russian Academy of Sciences under the guidance of Academician S. A. Khristianovich to increase the productivity of oil and gas wells—the directed unloading formation method (Khristianovich et al. 2000; Kilmov et al. 2015; Karev and Kovalenko 2013; Karev et al. 2018; Karev 2019). The technology is based on a fundamentally new approach to the extraction of oil and gas from low-permeability and unconventional reservoirs and has no analogues in the world. It consists in creating an artificial system of micro- and macrocracks in a bottom-hole zone due to directed unloading of the formation from rock pressure and using a special design of the bottom of the well. Required values of the pressure drawdown and a bottom-hole geometry are determined on the basis of testing core material at the TILTS. The installation allows to recreate in rock samples real stress

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states that occur in the reservoir when carrying out certain technological operations. In this technology, in contrast to the hydraulic fracturing method when it needs to waste a lot of energy on creating a crack of hydraulic fracture for overcoming rock pressure, the huge amount of elastic energy accumulated in the rock mass is used to create a system of cracks which become new filtration channels. In addition, such exposure is absolutely environmentally friendly. The available data on the development of wells that uncovered Achimov reservoirs of the Urengoy gas condensate field favor the fact that the DUF method can be successfully applied at this field. Facts indicate that well productivity is determined not so much by geological criteria as by technological factors, in particular, the design of the bottom hole. As an example, two adjacent wells can be cited, one of which has an open bottom, and the second one has a cased bottom with perforation. As a result, the flow rate of a well with the open hole on a 7-mm fitting was 308,000 m3 per day at the pressure drawdown of 60 at, and the flow rate of a cased well was 137,000 m3 per day on a 14-mm fitting at the pressure drawdown of 518 atm. To apply the DUF method at a particular field, as well as other technologies based on the geomechanical approach, and to select the optimal parameters for their implementation, it is necessary to determine the deformation, strength and filtration properties of the reservoir rocks and, most importantly, to study the effect of threedimensional stress states on their permeability.

3 Tested Samples and Their Preparation for Experiments Core material for testing at the test bench of the IPMech RAS was taken from wells No. 789 and No. 737 which uncovered the Achimov deposits of the Urengoy gas condensate field. Depth of core sampling from the well No. 789 is about 3630 m (Ach-7.3, Ach-7.4, Ach-6, Ach-10), from the well No. 737 is 3835 m (Ach-51).The samples were made in the form of a cube with an edge of 40 mm. They were marked as follows: axis 1 of the sample coincided with the axis of the core; the orientation of axes 2 and 3 was arbitrary. To measure the permeability of the rock, four faces of the sample were covered with an impermeable thin film, and the other two opposite faces were left free. Air flow was passed through them during the test and measured its flow rate. A total of 6 samples were manufactured. Two samples (Ach-7.3 and Ach-7.4) were made from one core piece. The propagation velocities of longitudinal waves along each of the three axes of the sample were measured before testing at the TILTS. The measurements were carried out on a specially created setup (Karev et al. 2020). In all samples, the velocities of the longitudinal waves along the core axis were a little bit lower than along the two axes in the horizontal plane, along which the velocities were approximately the same. For example, in samples Ach-7.3 and Ach-7.4, the velocity along the core axis was 2439 m/s, and along the axes 2 and 3 it was 2777 m/s. This suggests that the tested rock is weakly anisotropic.

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3.1 Sample Loading Programs A series of experiments was conducted at the TILTS on physically modeling deformation, fracture, and filtration processes in reservoirs of the Achimov deposits of the Urengoy gas condensate field to study the effect of stresses that actually arise in the vicinity of wells during their drilling and operation, on the filtration properties of the rocks. The experiments were carried out according to two loading programs: one program simulated the conditions in the vicinity of the open hole, the second one simulated the conditions near the tip of the perforation hole. During the tests, stresses corresponding to real stress states arising in the vicinity of wells at various pressures at their bottom were applied to the faces of a sample. Strains of the sample along three axes and permeability in the direction corresponding to the radius of the well were recorded during tests. As a result, the dependence of rock permeability on the real threedimensional stresses was determined. The loading programs are described in detail in (Karev et al. 2020). The program “well” modes a stress state near an open hole of a wellbore. Radial stress σr , circumferential stress σθ and stress σz along the axis of the well (σi < 0) act in the vicinity of a well. The program for testing samples in experiments simulating changes in stresses on a contour of an open hole of well at decreasing pressure at the bottom of a well for the case of an isotropic medium and equicomponent rock pressure is shown on Fig. 1a. The stresses s1 , s2 , s3 depicted on it are applied to the faces of the sample along the axes 1, 2, 3 of the loading unit of the TILTS installation. Their values correspond to the effective stresses sz , sθ , sr acting on a contour of an open hole of a well (si = σi + pc , si < 0, pc > 0, where σi are the full stresses, pc is the pressure at the bottom of the well) and which, according to the solution of the Lame problem, are defined as sr = 0, sθ = 2(q + pc ), sz = q + pc , where q is the rock pressure (q < 0), assumed to be equicomponent. In the absence

Fig. 1 Sample loading programs: a “Well”, b “Sphere”

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of explicit geological disturbances, q is equal to the weight of the overlying rocks, i.e. q = −γ h, where γ is the average specific weight of rocks, h is the depth of the reservoir. Point A corresponds to the stresses acting in the soil skeleton before drilling the well: s1 = s2 = s3 = |q| − p0 , where p0 is the initial reservoir pressure, which was taken equal to = 58.1 MPa. Point B corresponds to the state when the well has been drilled and the pressure at its bottom is equal to the reservoir pressure, pc = p0 . At points B: s1 = |q| − p0 , s2 = 2(|q| − p0 ), s3 = 0. The average stress (s1 +s2 +s3 )/3 remains constant throughout this step. The sections BC correspond to lowering the pressure at the bottom of the well. Loading at the third stage continues until the sample is destroyed. The program “sphere” models a stress state near a tip of a perforation hole. A good approximation for determining the stresses acting in the vicinity of a tip of a perforation hole is the solution of the problem for the distribution of stresses in the vicinity of a hollow sphere under the impact of internal and external pressures (Karev et al. 2020). A radial stress σr and two circumferential stresses σθ and σϕ act in the vicinity of the tip of the perforation hole having a spherical shape. Accordingly, the effective stresses are sr , sθ , sϕ . Figure 1b shows a testing program which simulates the change in stresses on the surface of the perforation hole tip while decreasing the pressure at the bottom of the well pc . The stresses s1 , s2 , s3 depicted on the graph act on the axes 1, 2, 3 of the loading unit of the TILTS. Their values correspond to the stresses sϕ , sθ , sr acting along the contour of the perforation hole tip: sr = 0, sθ = sϕ = 3/2(|q| − pc ).

3.2 Test Results of Samples The following are the test results of four samples from the well No. 789 of the Urengoy gas condensate field. The samples Ach-7.3 and Ach-6 were tested according to the “well” program, and the samples Ach-7.4 and Ach-10 were tested according to the “sphere” program. The loading program, the change in permeability during loading, and the strain curves of the sample along 3 axes are shown for each sample. Figures 2 and 3 show the test results for the samples Ach-7.3 and Ach-7.4, cut from one core piece. Figure 2a shows that the permeability of the sample Ach-7.3 on the equicomponent compression section decreased from 19 to 14 mD. This value should be taken as the initial permeability of the sample in reservoir conditions. Subsequently, when simulating a decrease in pressure in the well, the permeability of the sample slightly increased, but not significantly—up to 15.6 mD. It can be seen from Fig. 3a that the permeability of the sample Ach-7.4 on the equicomponent compression section decreased from 24 mD to about 19 mD. Subsequently, when simulating a decrease in the pressure in the well, at first, the

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Fig. 2 Results of the test of the sample Ach-7.3: a Loading program “well” and permeability, b Strain curves

Fig. 3 Results of the test of the sample Ach-7.4: a Loading program “sphere” and permeability, b Strain curves

permeability of the sample remained unchanged, but it increased sharply at intensive deformation of the sample and its destruction. Figure 4a shows that the permeability of the sample Ach-6 on the equicomponent compression section decreased from 35 to 27 mD. Figure 4 and 5 show the test results for the samples Ach-6. It can be seen from Fig. 4a that the permeability of the Ach-6 sample in the all-round compression section decreased from 35 mD to about 27 mD. The creep of the sample was measured in testing the sample Ach-6 too. For this purpose, its loading was stopped at two stress values s2 = 54.7 MPa and s2 = 61.6 MPa, and the strains of the sample were measured in three directions at the constant stress values. The first value corresponds to the pressure drawdown at the

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Fig. 4 Results of the test of the sample Ach-6: a Loading program “ well” and permeability, b Strain curves

Fig. 5 The creep curves and the permeability of the sample Ach-6

bottom of the well equal p = 2.1 MPa, and the second one corresponds to the pressure drawdown equal p = 5.5 MPa. When measuring creep, a change in the permeability of the sample was also recorded. At s2 = 54.7 MPa, the creep of the sample was not observed. The permeability of the sample did not change. When the stress value reached s2 = 61.6 MPa, the picture has changed. Figure 5 shows the creep curves of the sample along axes 2 and 3 at this stress, as well as the change in the permeability of the sample along axis 3 after stopping the loading at the value of the stress s2 = 61.6 MPa. At first, the creep of the sample was not observed for a rather long time, and the permeability slightly increased. Then of the sample has

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begun to creep at a constant speed, and the permeability of the sample continued to grow slowly. After some time, the creep of the sample sharply accelerated and the sample was collapsed. The permeability of the sample at this stage was increased sharply to 37.5 mD and became higher than initial one. Figure 6 shows the results of testing the sample Ach-10. Figure 6a presents a loading program and a change of the permeability during testing the sample. The permeability of the sample on the section of the hydrostatic compression decreased from 22.5 mD to 18.5 mD. When simulating an increase in the pressure drawdown at the bottom of the well, permeability continued to decrease gradually and dropped to 17.5 mD. At the stress of about 80 MPa, the sample began to creep, and its permeability increased. At stress s2 = 85.5 MPa, the sample began to deform more intensively and eventually collapsed. The permeability of the sample was increased sharply and reached a value of 20 mD. When testing the sample Ach-10, the creep of the sample was measured at two stress values s2 : 80.7 MPa and 85.5 MPa. The first value corresponds to the pressure drawdown at the bottom of the well p = 28.5 MPa, and second one to the pressure drawdown p = 31.7 MPa. When measuring the creep, a change in the permeability of the sample was also recorded. Figure 7 shows the creep curves of the sample along axis 3 for both stress values, as well as the change in its permeability during the stops. It can be seen from Fig. 7 that at s2 = 80.7 MPa, the creep of the sample was insignificant, while some increase in its permeability was observed. At s2 = 85.5 MPa, the creep of the sample was accelerated sharply and it led to destruction of the sample, and its permeability was increased sharply too. The results of testing samples from well No. 789 showed that creating the necessary stress field in the vicinity of a well using directed unloading of the formation

Fig. 6 Results of the test of the sample Ach-10: a Loading program “sphere” and permeability, b Strain curves

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from rock pressure and choosing the certain design of the bottom of the well can cause cracking and destruction of the rock, leading to an increase in its permeability.

3.3 Simulation of Filtration into a Perforation Hole Experiments on direct modeling of filtration into perforation holes in uncased wells were performed on the sample Ach-51 from the well No. 737 to confirm this conclusion and to achieve a greater visibility. The sample Ach-51 was prepared specially. A hole with a diameter of 6 mm was drilled along the center of the sample face in the direction of axis 3 of the sample (the radial direction for the well). Compressed air was supplied to the sample through the vents made in two tips of the opposite pressure plates located along the direction of the axis 2 of the apparatus loading unit. The opposite faces of the sample located along the axis 2 were free from impermeable coating. There was a channel in the tip of the active pressure plate of the axis 3 of the machine coincided with the hole along the axis 3 of the sample, through which the filtered through the sample air was withdrawn and its flow rate was measured. The loading program “well” shown in Fig. 1a was used in the test. Sample’s initial permeability was 1 mD. The results are shown in Fig. 8. At the stresses s2 = 81 MPa, s2 = 86 MPa and s2 = 95 MPa, the sample Ach-51 was unloaded to s2 = 55 MPa and then loaded again, which simulated a cyclic change in pressure at the bottom of the well. Limited creep of the sample accompanied by an increase in its permeability has been begun at the stress s2 = 77 MPa corresponding to the pressure drawdown of

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Fig. 8 Results of the test of the sample Ach-51. The uncased well: a Loading program “well” and permeability, b Strain curves

Fig. 9 The sample Ach-51 after the test

about 10 MPa. The gas flow into the hole at the time of sample destruction has been increased by more than 300 times, Fig. 8a. The fact is of interest that after partially unloading the sample and returning it to the state of uniform all-round compression, the flow into the hole decreased, but still remained more than 100 times higher than initial one. This increase in the permeability of the sample is associated with the formation of a multiple system of cracks. The hole has been turned from the round hole into elliptical one in the cross section after the test (Fig. 9).

4 Conclusions The results of the tests of rock samples from the Achimov deposits of the Urengoy gas condensate field allow to draw a number of practically important conclusions.

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Physical modeling of the processes of deformation, fracture and filtration in the vicinity of open boreholes and perforation holes using true threedimensional tests has shown that it is possible to cause cracking and fracture of the rock, leading to an increase in its permeability. For this purpose, it is necessary to create certain stress field in the vicinity of wells using the directed unloading the formation from the rock pressure. Experiments on the direct simulation of filtration into perforation holes in cased and uncased wells have shown the importance of choosing the right design of the bottom borehole. If when modeling a perforation hole in a cased well, creating even the deep pressure drawdowns did not lead to an increase in the sample’s permeability, then modeling a perforation hole in an uncased well led to a significant and irreversible increase in the sample’s permeability. Summarizing the results of the experiments, it can be conclude that the geomechanical approach using physical modeling of deformation, fracture and filtration processes in reservoirs can serve as a basis for the development of new efficient and environmentally friendly technologies to increase in the productivity of wells and recovery of lowpermeable reservoirs with hard-to-recover reserves.

Acknowledgements The study was supported by the project of the Russian Federation represented by the Ministry of Education and Science of Russia No. 075-15-2020-802.

References Karev VI (2019) The influence of the pattern of unloading on the growth of gas-filled cracks. Processes in Geomedia 2(20):185–192. (in Russian) Karev VI, Kovalenko YuF (2013) Well stimulation on the basis of preliminary triaxial tests of reservoir rock. In: Rock mechanics for resources, energy and environment. Proceedings of EUROCK 2013.The ISRM international symposium. Wroclaw, Poland, 23–26 September 2013. CRC Press/Balkema, Leiden, pp 935–940 Karev VI, Klimov DM, Kovalenko YuF (2018) Modeling geomechanical processes in oil and gas reservoirs at the true triaxial loading apparatus. In: Physical and mathematical modeling of earth and environment processes. Springer, pp 336–350 Karev V, Kovalenko Y, Ustinov K (2020) Geomechanics of oil and gas wells. Advances in oil and gas exploration and production. Springer International Publishing, Cham, Switzerland, p 166 Khristianovich SA, Kovalenko YuF, Kulinich YuV, Karev VI (2000) Increasing the productivity of oil wells using the geoloosening method. Oil and Gas Eurasia 2:90–94 Klimov DM, Karev VI, Kovalenko YuF (2015) Experimental study of the influence of a triaxial stress state with unequal components on rock permeability. Mech Sol 50(6):633–640

Features of Accumulation and Spatial Distribution of Microelements in Bottom Sediments of the Crimea Coastal Regions K. I. Gurov , Yu. S. Kurinnaya , and E. A. Kotelyanets

Abstract The relevance of the study of the bottom sediments physicochemical characteristics in Crimea coastal regions, known for their resort attractiveness, is explained by the fact that these characteristics largely determine or significantly affect on the characteristics of the pollutants and trace metals accumulation in them, the oxygen regime of pore and near-bottom waters and conditions for the occurrence of oxygen deficiency and secondary sources of pollution, environmental and recreational characteristics. The paper presents the results of a comprehensive physicochemical analysis of the bottom sediments surface layer samples taken with the Ocean 0.1 dredger in the coastal zone of the Crimean Peninsula from Cape Tarkhankut to the Kerch Strait during the expedition on the R/V Professor Vodyanitsky in September 2018. A statistical assessment of the correlation between the individual fractions of bottom sediments, the content of organic matter and concentrations of trace metals was performed. The obtained results of the trace elements content were compared with the background values characteristic of the Black Sea coastal regions and Clarke elements in the earth’s crust. To summarize and level the types of bottom sediments, the average characteristic concentration (ACC) was used. It is shown that in the coastal zone, pelite-aleuritic silts predominate with the inclusion of sand material and shell detritus. An increased proportion of sand and gravel material was noted in the area of the Kerch Strait and near Cape Tarkhankut. It was noted that the maximum concentrations of organic carbon (Corg ) were observed for samples with an increased proportion of pelite-aleuritic silts, and calcium carbonate (CaCO3 ) for gravel-sand deposits of the Kerch Strait. It was noted that all of the microelements and trace elements studied are characterized by a high positive correlation with the content of organic carbon and negative with the content of calcium carbonate. It has been established that for a number of elements such as Mn, Cu, Pb, Ni and As maximum concentrations are observed in the coastal zone from the city of Sevastopol to Cape Tarkhankut. Fe and Zn accumulate in the shelf zone in the form of local maxima. Increased V content is observed on the southern coast of Crimea and in the Feodosia Gulf, and Cr—in the sea part of Feodosia Gulf. To identify the source of K. I. Gurov (B) · Yu. S. Kurinnaya · E. A. Kotelyanets Marine Hydrophysical Institute, Russian Academy of Science, Sevastopol, Russia e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 T. Chaplina (ed.), Processes in GeoMedia—Volume III, Springer Geology, https://doi.org/10.1007/978-3-030-69040-3_12

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the bottom sediments composition formation, the values of the enrichment coefficient (EC) were calculated. An indirect assessment of the redox environment using geochemical indices (Mn/Fe, Ni/Co, V/Cr, V/(V + Ni)) was carried out. Keywords Black Sea · Crimean Peninsula · Bottom sediments · Particle size distribution · Organic carbon · Carbonate · Heavy metals

1 Introduction It is known that bottom sediments are a complex multicomponent system that can absorb and accumulate various chemical substances, and under certain conditions become a source of secondary pollution of water bodies. The study of the bottom sediments physicochemical characteristics is relevant for the coastal regions of the Black Sea, and especially the areas of the Crimean Peninsula, known for their resort attractiveness. The physical (particle size distribution, humidity) and chemical (content of organic and inorganic carbon, certain microelements) characteristics of bottom sediments determine the features of the pollutants and trace metals accumulation in them. Studying the correlation between individual fractions of bottom sediments, the accumulation of organic matter and concentrations of trace metals allows us to determine and predict the ecological state of the marine environment, the conditions for the development and prevention of environmentally hazardous phenomena. This is confirmed by a number of works devoted to studying the features of the trace metals accumulation in bottom sediments of various fractional composition (Bat et al. 2017; Sabra et al. 2011; Dundar et al. 2013; Yu et al. 2012; Rusakov et al. 2017; Budko et al. 2017). The first information on the individual elements concentrations and their participation in sedimentogenesis was obtained by Egunov at the end of the nineteenth century on the example of soils and estuaries of the Black Sea (Egunov 1896). A significant contribution to the development of this direction was made by the works of Strakhov and Glagoleva, devoted to the study of the features of elements accumulation and distribution in the surface layer of the Black Sea sediments, as well as their relationship with various types of bottom sediments (Glagoleva 1961; Strakhov 1971). In (Mitropolsky et al. 1982), a review of the main natural and anthropogenic sources of trace metals entering the water area and sediments of the Black Sea basin, the features of accumulation, distribution, and the relationship with the physicochemical characteristics of bottom sediments of individual metals in various areas of the Crimean Peninsula shelf are presented. An analysis of the main sources of pollutants in the Black Sea, as well as the development of regulatory assessments of the quality and level of bottom sediments pollution, developed on the basis of data from many years of water and bottom sediments monitoring, are presented in (Klyonkin et al. 2007). Modern studies on the spatial distribution of the physicochemical characteristics of bottom sediments and the accumulation of individual elements in various regions of the Crimean Peninsula are described in the works of the Institute

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of Biology of the Southern Seas (IBSS, Sevastopol) and the Marine Hydrophysical Institute (MHI, Sevastopol). In (Gubanov et al. 2008, 2010, 2012; Kotelyanets and Konovalov 2008; Ovsyany et al. 2015), the results of the physicochemical characteristics content and accumulation features of the contaminating trace elements in the bottom sediments of the Feodosia Gulf and the Kerch Strait are presented. The data on the content of organic carbon, calcium carbonate and trace metals and their correlation to the particle size distribution in sediments of the Kalamitsky Gulf are presented in (Gurov et al. 2014). The results of long-term studies of the pollution level of bottom sediments from different coastal regions of Crimea are reflected in (Petrenko et al. 2015). Modern information on the trace metals content in bottom sediments of the coastal regions of the Crimean Peninsula, obtained during the 83th cruise of the R/V “Professor Vodyanitsky” in January–February 2016, is displayed in (Tikhonova et al. 2016). The microelement composition of the vertical section of the Black Sea bottom sediments, obtained on the basis of data from the columns selected in November–December 2016 during the 91th cruise of the R/V “Professor Vodyanitsky” described in (Nemchenko et al. 2020). The data on the vertical distribution of the microelement composition of pore water and bottom sediments obtained by the method of polarographic profiling are presented in (Orekhova and Konovalov 2019). The purpose of this work is to study the features of accumulation and spatial distribution of the trace metals in the surface layer of bottom sediments of the Crimean Peninsula coastal and shelf regions.

2 Materials and Methods Samples of the surface (0–5 cm) layer of the bottom sediments for studying the spatial distribution of physicochemical characteristics and microelements concentrations taken using the Ocean 0.1 dredger during an expedition on the R/V “Professor Vodyanitsky” in September 2018 (Fig. 1). The twenty one samples were taken on the site of the Crimean Peninsula coastal zone from Tarkhankut Cape to the Kerch Strait. The separation of the pelite-aleuritic fraction (≤0.05 mm) was carried out by wet sieving with the subsequent determination of the dry mass gravimetrically. Coarse fractions (>0.05 mm) were separated by the sieve method of dry sieving using standard sieves (GOST 12536–2014; introduced from 01.07.2015) with recommendations (Petelin 1967). The content of Corg was determined coulometrically on an express analyser—AN 7529 according to a technique adapted for marine bottom sediments (Lutzarev 1986). To determine the total content of chemical elements (Fe, Mn, Al, Si, Ti, V, Cr, Co, Ni, Cu, Zn, As, Sr, Pb), the X-ray fluorescence analysis (XRF) method was used on a Spectroscan Max-G spectrometer Spectron company (Russia). This analysis method is used to determine the total content of element concentrations from beryllium to uranium in the range from 0.0001 to 100% in substances of different origin (Spectron 2016).

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Fig. 1 Locations of bottom sediment sampling in the coastal areas of Crimea

The obtained results of the trace elements concentrations were compared with the background values characteristic of the Black Sea coastal regions according to (Mitropolsky et al. 1982), and Clarkes of chemical elements in the earth’s crust according to (Vinogradov 1962). In order to generalize the types of bottom sediments, when analyzing the level of their pollution, we used the value of the average characteristic concentration (ACC) obtained in (Klyonkin et al. 2007) for various types of bottom sediments in the northeastern part of the Black Sea. Comparison of the analysis results in absolute concentrations with ACC gives a dimensionless value—the multiplicity of ACC: Multiplicity ACC =

Ci ACC

where Ci , is the absolute concentration of the determined i-element; ACC is the average characteristic concentration of i-element for various types of sediment. The multiplicity of ACC characterizes the susceptibility of a given area to anthropogenic impact in the studied period of time. If the ACC multiplicity is 1, this region is a region of increased anthropogenic impact in a specific period of time and requires a more detailed study to establish the source of pollution (Klyonkin et al. 2007).

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3 Results and Discussion The results showed that in the coastal zone of the Crimean shelf, pelite-aleuritic sediments with inclusions of sand material and individual inclusions of shell detritus predominate. On average, the proportion of finely dispersed material in all samples was 76.5%, of which 56.5% was accounted by the pelite-aleuritic fraction and 20% by the aleurite-pelitic fraction. It was established that the increased content of silty material is observed at stations located in the seaward part (stations 1, 12.2), as well as in the area of Southern Crimea from Yalta to Feodosia. For the central part of the Feodosia Gulf the maximum content of silty material (97%) including the pelite-aleuritic fraction—86.4% was noted. The content of Corg varies from 0.5 to 0.6% in the gravel-sand deposits of the Kerch Pass to 2.5–2.7 in the silt material from the northwestern part. It was noted that increased concentrations of Corg were observed for samples with an increased (>70%) proportion of the pelite-aleuritic fraction and silty material in general. An analysis of the results obtained showed that the CaCO3 content in the surface layer of bottom sediments of the Crimea coastal regions varies from 5.27% in the coastal part of the Feodosia Gulf to 80.07% in the area of the Kerch Strait. As part of the study of the microelements spatial distribution the regions of maximum and minimum concentrations were identified and correlation dependencies between the accumulation of various elements, physical (particle size distribution, humidity) and chemical (organic carbon and calcium carbonate content) characteristics of bottom sediments were obtained. It has been established that for a number of elements such as Mn, Cu, Pb, Ni, and As, maximum concentrations are observed in the coastal zone from the city of Sevastopol to the Tarkhankut Cape. Fe and Zn accumulate in the shelf zone in the form of local maxima. The increased content of V is observed on the Southern coast of Crimea and in the Feodosia Gulf, and Cr—in the seaward part of Feodosia Gulf. It was noted that increased concentrations of the studied elements are observed at stations where fine-grained clay material predominates. Features of the spatial distribution of elements are shown in Fig. 2. The maximum positive correlations with the pelite-aleuritic fraction were noted for Ni (0.83), Zn (0.77) and Fe (0.7), the minimum—for Mn (0.39) and Pb (0.55) (Fig. 3). It was found that all studied elements are characterized by a high positive correlation with the content of organic carbon and negative with the content of calcium carbonate (Fig. 4). Minimum concentrations of all elements are observed on the Opuk Cape—Kerch Strait section, which is explained by the predominance of gravel-sand deposits in this region, the proportion of clay fraction in which is 5–10%. It was shown that a significant excess of the background concentrations obtained in (Mitropolsky et al. 1982) was noted only for Fe, Cr and Pb; for V and Zn the excess of background values was not significant; for Mn, Cu, Ni and As the excess of background values was noted only on individual stations. For such elements as Pb, As, Cr and V an excess of the content with respect to the composition of the upper part of the earth’s crust was noted (Vinogradov 1962). Exceeding the ACC

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Fig. 2 Spatial distribution of elements in the surface layer of bottom sediments

Fig. 3 Quantitative values of the relationship between the concentrations of the studied elements with the particle size distribution of bottom sediments

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Fig. 4 Quantitative values of the relationship between the concentrations of the studied elements with organic carbon and calcium carbonate of bottom sediments

value (Klyonkin et al. 2007) at all stations was noted for Cr, Zn, V and Pb; elements such as As, Ni, Cu and Fe exceed the ACC values only for gravel-sand deposits. Estimates obtained by different methods showed good correlation (from 0.7 for V to 0.95 for As). A detailed analysis of the accumulation and spatial distribution features is presented for the most priority elements, such as Fe, Zn, Cr, Cu and Pb. Iron. It was established that high concentrations of Fe are observed in bottom sediments of the coastal zone from Cape Tarkhankut to Feodosia Gulf (Fig. 2). An analysis of field data showed that the iron content in the surface layer of bottom sediments of the coastal regions of Crimea varies from 2.35% in the Kerch Strait to 6.31% in Veselovskaya Bay area (near the city of Sudak). Comparison with the background values of the Crimean shelf deposits showed that iron is characterized by an excess of the concentration level everywhere on average by 1.5 times, and for gravel and sand material by 2 times. Comparison with the ACC value (Klyonkin et al. 2007) showed that all values are significantly greater than 1, which determines the studied deposits as contaminated. For iron, the maximum negative correlation (−0.9) with the content of calcium carbonate is noted. This indicates that iron accumulates exclusively in silt sediments. Zinc. An analysis of the results showed that the Zn content in the surface layer of bottom sediments varies from 60 mg/kg in the area of the Kerch Strait to 107 mg/kg in the area of the Yalta collector outlet (station 63). It was noted that the zinc content in the sediments of the Kerch Strait is significantly lower than previously obtained (Kotelyanets and Konovalov 2012), but slightly higher than the data presented in (Tikhonova et al. 2016). For the waters of the Kalamitsky Gulf the obtained values are higher than the data presented in (Gurov et al. 2014), which is explained by the seaward location of the sampling station and the increased proportion of pelitealeuritic material. Excess of background concentrations are also observed in the area

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of Donuzlav Lake and near Veselovskaya Bay area. The average ACC value was 1.3, which indicates a slight pollution of coastal sediments. Chromium. An analysis of the data showed that the chromium concentrations in the bottom sediments of the Crimean coastal areas in 2018 vary from 63 to 93 mg/kg in the Kerch Strait to 127 mg/kg in the seaward part of the Feodosia Gulf (Fig. 2). Comparison with early data for the Kerch Strait section (Kotelyanets and Konovalov 2012) showed good agreement, but the values obtained are comparatively lower than the data presented in (Tikhonova et al. 2016). For the sediments of the Feodosia Gulf, the obtained values are in good agreement with the data of (Kotelyanets and Konovalov 2008). For Cr background concentrations were exceeded at all stations, with the exception of the Cape Opuk area. The ACC value varied from 1.41 in the seaward part of the Kalamitsky Gulf to 1.82 in the area of the Southern Coast of Crimea, the average value was 1.59. This indicates that the observed values were 59% higher than the average characteristic concentrations. Copper. Copper concentrations in modern bottom sediments of the Crimean coastal regions vary from 13–18 mg/kg in the Kerch Strait to 43–45 mg/kg in the northwestern part of the study area (Fig. 2). The minimum copper concentration was observed at station 51a (the shore of Laspi Bay)—7 mg/kg, and the maximum was recorded in the area of Donuzlav Lake—45 mg/kg. Elevated concentrations are observed in the region where ferromanganese concretion are located, which is consistent with the data of (Mitropolsky et al. 1982). Comparison with the data of (Petrenko et al. 2015) showed that for the seaward part of the Feodosia Gulf, copper concentrations reached the values noted for the port area, which indicates an intensification of the pollution level of the entire gulf. For the Kerch Strait region the obtained values also exceed the data of literary sources (Gubanov et al. 2010). However, the average ACC value for copper is low (0.83). This indicates that in recent years the proportion of finely dispersed silty material, characterized by higher average gross contents of the studied elements, including copper, has significantly increased in the Feodosia Gulf. Lead. The lead content in the surface layer of bottom sediments of the studied shelf part of the Crimean Peninsula varies from 30–39 mg /kg in the northwestern part to 8–17 mg/kg in the area of the Kerch Strait (Fig. 2). Increased values at st. 12.1, 12.2, 13.1 are apparently explained by the presence of a polymetallic deposit and lead–zinc hydrothermal veinstones in this region (Mitropolsky et al. 1982), which is also confirmed by the increased zinc concentration in this area. It is noted that, with the exception of the Kerch Strait area, concentrations exceed background levels for shelf deposits. The values obtained for Feodosia Gulf and the Kerch Strait region are several times higher than previously obtained values (Gubanov et al. 2008). An average ACC value of 1.86 indicates that, over the last time, lead input to bottom sediments has increased. To identify the contribution of lithogenic, biogenic, or other source of the bottom sediments composition formation, the enrichment coefficient (EC) values (Nemchenko et al. 2020) were calculated relative to the average composition of the earth’s crust according to the formula:

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EC =

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(El/Elind − Elind )sample (El/Elind − Elind )e.c.

where El and Elind are the content of the chemical element and indicator element in the sample of bottom sediments and the upper part of the continental crust (according to Vinogradov (Vinogradov 1962)), respectively. According to (Solovov 1990), Al, Ti, Sc and some other elements can be used as an indicator element of the lithogenic component. In this work, the enrichment coefficient values were calculated relative to Ti. Based on our calculations, for most elements 2 < KO ≤ 3, which, according to (Nemchenko et al. 2020), indicates that they have a predominantly terrigenous source close in composition to the earth’s crust. For elements such as Fe, Al, Si, V and Ni KO ≤ 2, which indicates lithogenic sources of their input. For Sr, As and Mn an elevated content (KO > 3) was noted relative to the average composition of the upper part of the earth’s crust. High values for Sr, especially in the gravel-sand sediments of the Kerch Strait and Cape Tarkhankut, are explained by its biogenic accumulation in shell material. For Mn an increased content was noted for stations located, apparently, in the occurrence sites of ferromanganese concretions (Mitropolsky et al. 1982). For an indirect assessment of the redox environment indicators in the bottom waters, various geochemical indices are used, such as Mn/Fe, V/(V + Ni), Ni/Co, V/Cr. Previously, these ratios were obtained for samples of bottom sediment columns in the coastal regions of Crimea (Nemchenko et al. 2020). By the values of the Mn/Fe ratio, we can speak of the presence of anaerobic conditions in the layer of bottom sediments (Naeher et al. 2013). It was noted in (Naeher et al. 2013) that with an increase in the values of the Mn/Fe ratio, the O2 concentrations in the bottom water layer increase and vice versa. Anaerobic conditions are typical for the Mn/Fe ratio 7—under anaerobic. The estimates obtained are in good agreement with the results of polarographic analysis obtaned in (Orekhova and Konovalov 2019). One of the most used indicators is the V/Cr ratio. According to (Bond et al. 2004), the value of this ratio >5 indicates an anaerobic conditions, and