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Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*17 (04): 1231–1245.

Paper Number: SPE-154024-PA

Published: 05 December 2012

Abstract

Summary Creation of low-mobility foam for enhanced oil recovery (EOR) is triggered by an increase in superficial velocity; thereafter, injection rate can be reduced to lower values, and strong foam remains at velocities at which weak foam was previously observed. Here, we consider whether strong foam created near an injection well can propagate to large distances from the well where superficial velocity is much smaller. We study strong-foam propagation with finite-difference simulations and Riemann solutions, applying a population-balance foam model that represents the multiple steady states of foam. Our simulations show that strong foam cannot displace directly the initial high-water-saturation bank initially in the reservoir at low superficial velocities; it pushes a weak-foam state with lower velocity that in turn displaces the bank ahead. Our traveling-wave solutions show that strong foam propagates more slowly as superficial velocity decreases and stops propagating at yet lower superficial velocities, in agreement with the experiment. Failure of propagation occurs at superficial velocities greater than that at which the strong-foam state disappears; it raises concerns for long-distance propagation of strong foam created near the injection well. In the context of the model, it is not extraordinary destruction of foam at the front that slows the propagation of strong foam, but failure of foam (re-)generation at the front. Our model also represents for the first time a process where strong foam is created near the exit of a core and then propagates upstream, as seen in some experiments.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Improved Oil Recovery Symposium, April 14–18, 2012

Paper Number: SPE-154024-MS

Abstract

Foam reduces gas mobility and thereby improves sweep in enhanced-oil-recovery processes. The "strength" of foam describes the degree of gas-phase mobility reduction. Experiments show that foam can exist in three different states (weak foam, intermediate foam, and strong foam) at the same injection rate. Strong foam with fine bubbles achieves better sweep efficiency than weak foam with coarse bubbles. In experiments, creation of strong foam is triggered by an increase in superficial velocity; thereafter injection rate can be reduced to lower values and strong foam remains at velocities at which weak foam was previously observed. It is important to determine whether strong foam created near an injection well can propagate to large distances from the well, where superficial velocity is much smaller. Here, we study strong-foam propagation with finite-difference simulations and Riemann solutions, applying a population-balance foam model that represents the multiple steady states of foam. Our simulations show that strong foam cannot directly displace the initial high-water-saturation bank initially in the reservoir at low superficial velocities; it pushes a weak-foam state with lower velocity that in turn displaces the bank ahead. Our traveling-wave solutions show that strong foam propagates more slowly as superficial velocity decreases and stops propagating at yet lower superficial velocities, in agreement with experiment. Failure of propagation occurs at superficial velocities greater than that at which the strong-foam state disappears; it raises concerns for long-distance propagation of strong foam created near injection well. In the context of the model, it is not extraordinary destruction of foam at the front that slows the propagation of strong foam, but failure of foam (re-)generation at the front. Our model also represents for the first time a process where strong foam is created near the exit of a core and then propagates upstream, as seen in some experiments.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE/DGS Saudi Arabia Section Technical Symposium and Exhibition, May 15–18, 2011

Paper Number: SPE-149068-MS

Abstract

The use of foam for mobility control is a promising mean to improve sweep efficiency in EOR. Experimental studies discovered that foam exhibits three different states (weak foam, intermediate foam, and strong foam). The intermediate foam state is found to be unstable in the lab whereas the weak- and strong-foam states are stable. The model of Kam (Colloids Surfaces A, 2008) is the only mechanistic foam model that can fit a variety of steady-state experimental data including multiple steady states. This model is modified from a previous mechanistic foam model to resolve the intrinsic instability of the strong-foam state. Simple finite-difference simulations have found that an arbitrary perturbation grows for the unstable intermediate foam but diminishes for the strong- and weak-foam states. The issue of the stability of foam states, especially the strong-foam state, is a serious concern in application of foam in EOR. Instabilities may rule out one or more states and consequently have considerable effect on reservoir sweep efficiency and injection pressure. Here, for the first time the stability of the various foam states is investigated by a semi-analytical stability-analysis method. We demonstrate the instability of most intermediate states, consistent with the laboratory observations. However, fine analysis reveals a slight instability of the strong-foam state. We show that the diffusion, whether introduced artificially by the finite-difference scheme or representing physical dispersion, damps this slight instability. We obtain good agreement with finite-element simulations with and without diffusion.

Journal Articles

Journal:
SPE Journal

Publisher: Society of Petroleum Engineers (SPE)

*SPE J.*16 (03): 513–523.

Paper Number: SPE-129904-PA

Published: 23 February 2011

Abstract

Summary There is a renewed interest in using combustion to recover medium- or high-viscosity oil. Despite numerous experimental, numerical, and analytical studies, the mechanisms for incomplete fuel combustion or oxygen consumption are not fully understood. Incomplete oxygen consumption may lead to low-temperature oxidation reactions downstream. This paper shows that these features emerge in a relatively simple 1D model, where air is injected in a porous medium filled with inert gas, water, and an oil mixture consisting of precoke, medium oil, and light oil. Precoke is a component that is dissolved in the oil but has essentially the same composition as coke. At high temperatures, precoke is converted to coke, which participates in high-temperature oxidation. At high temperatures, medium-oil components are cracked, releasing gaseous oil. Light-oil components and water are vaporized. The model possesses an analytical solution, which was obtained by a concept introduced by Zeldovich et al. (1985) . This concept, which underlies most analytical approaches such as the reaction-sheet approximation and large-activation-energy asymptotics, entails that reaction can occur only in a very small temperature range because of the highly nonlinear nature of the Arrhenius factor. For a temperature below this range, the reaction rate is too slow, and for temperatures above this range, the reaction rate is so fast that either the fuel or oxygen concentrations become zero. The model results, in the absence of external heat losses, show that there are two combustion regimes in which coke or oxygen is partially consumed. In one regime, the reaction zone moves in front of the heat wave; whereas, in the other regime, the order of the waves is reversed. There are also two combustion regimes in which the coke and oxygen are completely consumed. Also, here the reaction zone can move in front of or at the back of the heat wave. Each combustion regime is described by a sequence of waves; we derive formulas for parameters in these waves. We analyze our formulas for typical in-situ-combustion data and compare the results with numerical simulation. The main conclusion is that mainly two key parameters (i.e., the injected oxygen mole fraction and the fuel concentration) determine the combustion-front structure and when either incomplete oxygen consumption or incomplete fuel consumption occurs in the high-temperature oxidation zone.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Improved Oil Recovery Symposium, April 24–28, 2010

Paper Number: SPE-129904-MS

Abstract

There is a renewed interest in using combustion to recover difficult oil. In spite of numerous experimental, numerical and analytical studies, the mechanisms for incomplete fuel combustion or oxygen consumption are not fully understood. Incomplete fuel combustion or oxygen consumption are indicators of sub-optimal process conditions and hazardous oxygen breakthrough in the production wells. This paper shows that these features emerge in a relatively simple 1-D model, where air is injected in a porous medium filled with inert gas, water and an oil mixture consisting of precoke, medium and light oil. Precoke is a component that is dissolved in the oil, but has essentially the same composition as coke. At high temperatures, precoke is converted to coke, which participates in high-temperature oxidation. At high temperatures, medium oil components are cracked releasing gaseous oil. Light oil components and water are vaporized. The model possesses an analytical solution, which was obtained by a method introduced by Zeldovich. This method entails that reaction can only occur in a very small temperature range, due to the highly non-linear nature of the Arrhenius factor. For a temperature below this range the reaction rate is too slow and for temperatures above this range the reacton rate is so fast that either the fuel or oxygen concentrations become zero. The model results show that there are two combustion regimes in which coke or oxygen is partially consumed. In one regime the reaction zone moves in front of the heat wave, whereas in the other regime the order of the waves is reversed. There are also two combustion regimes in which the coke and oxygen are completely consumed. Also here the reaction zone can move in front or at the back of the heat wave. Each combustion regime is described by a sequence of waves; we derive formulae for parameters in these waves. We analyse our formulae for typical in-situ combustion data and compare the results with numerical simulation.

Proceedings Papers

P. Bedrikovetsky, D. Marchesin, G. Hime, A.G. Siqueira, A.L. Serra, J.R.P. Rodrigues, A. Marchesin, M. Vinicius

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE International Symposium and Exhibition on Formation Damage Control, February 18–20, 2004

Paper Number: SPE-86523-MS

Abstract

Abstract Severe fall of injectivity in porous rock occurs from the practice in offshore fields of injecting sea water containing organic and mineral inclusions. In general, injection of poor quality water in a well curtails its injectivity. The injectivity loss is assumed to be due to particle retention in the porous rock. A model for porous rock damage due to retention in deep filtration during injection of water containing solid particles is formulated. The model contains two empirical functions that affect loss of injectivity - filtration coefficient and damage coefficient versus deposited particle concentration. We show how to solve the inverse problem for determining the first function based on effluent particle concentration measurements in coreflood tests. The second inverse problem is the determination of the formation damage coefficient from the pressure drop history on a core. These two methods allow determining from laboratory tests the information necessary for prediction of well impairment. Introduction Injectivity reduction due to the injection of water containing solid particles takes place in most waterflood projects to some degree. In designing a waterflood project, the level of water treatment necessary to minimize formation damage must be assessed, so it is important to know the performance of an injector as a function of injection water quality 1 . Therefore, substantial efforts have been devoted to model the injectivity decline due to injection of water with solid particles. The basic mathematical model for deep filtration with particle retention consists of a particle mass balance equation and a kinetic equation for clogging 2–4 . An analytical model for diffusion-free flow was developed in paper 2 under the assumptions that accumulation of suspended particles can be ignored and that the suspended concentration distribution is in steady state. In order to predict injectivity decline in wells, the mathematical model for radial flow requires two empirical functions - filtration coefficient and permeability versus deposited concentration. The problem of determining these functions from laboratory deep bed filtration requires solving two inverse problems. Methods for determining constant filtration and formation damage coefficients from outlet particle concentration were presented in the literature 4,5 . However, the general inverse problems for determining filtration coefficient and permeability damage versus deposited concentration from deep bed filtration laboratory tests have not been investigated. In the current paper we derive an exact analytical solution for 1-D linear problem for diffusive-free particle flow accounting for particle capture, without other limitations. The explicit solution of the direct problem is the basis for finding unique, stable solution of the inverse problems. A well-posed and stable sequence of two procedures for determining the filtration and formation damage functions from outlet particle concentration and pressure drop measurements is formulated. Closed equations are derived. The solution of the two inverse problems allows complete tuning of the model from laboratory test data enabling prediction of well injectivity behaviour. The assumptions in the model are the following. The water and particles are incompressible. The volume of the entrapped particles is negligible compared to the effective porosity (s<<f'). The kinetics of particle capture is linear, and diffusion is negligible. Analytical model for flow of water with suspended particles The model describes the laboratory test on injection of water with suspended particles in core initially saturated by particle free water (Fig. 1). The suspended concentration at the core outlet (breakthrough curve) and pressure drop on the core are measured during the test.

Journal Articles

P. Bedrikovetsky, T.K. Tran, W.M.G.T. Van den Broek, D. Marchesin, E. Rezende, A. Siqueira, A.L. Souza, F. Shecaira

Journal:
SPE Production & Operations

Publisher: Society of Petroleum Engineers (SPE)

*SPE Prod & Oper*18 (02): 119–128.

Paper Number: SPE-83673-PA

Published: 01 May 2003

Abstract

Summary Permeability decline occurs during injection of sea or produced water, resulting in well impairment. Solid and liquid particles dispersed in the injected water are trapped by the porous medium and may significantly increase hydraulic resistance to the flow. We discuss a mathematical model for deep bed filtration containing two empirical parameters - the filtration and the formation damage coefficients. These parameters should be determined from laboratory coreflood tests by forcing water with particles to flow through core samples. A routine laboratory method determines the filtration coefficient with expensive and difficult particle-concentration measurements of the core effluent, and then the formation damage coefficient is determined from inexpensive and simple pressure-drop measurements. An alternative method would be to use solely pressure difference between the core ends. However, we proved in an earlier work that given pressure-drop data in seawater coreflood laboratory experiments, solving for the filtration and formation damage coefficients is an inverse problem that determines only a combination of these two parameters rather than each of them. A new method for the simultaneous determination of both coefficients is developed here. The method's new feature is using pressure data at an intermediate core point, supplementing pressure measurements at the core inlet and outlet. The proposed method furnishes unique values for the two coefficients, and the solution is stable with respect to small perturbations of the pressure data. In this work, the proposed method is used for analysis of laboratory test data on deep bed filtration. The values of filtration and formation damage coefficients are obtained for flow of solid and liquid particle dispersions in a number of different cores. Effects of particle type and porous media wettability on permeability decline are analyzed. Introduction Injectivity decline of oilfield injection wells is a widespread phenomenon during sea- or produced-water injection. This decline may result in significant cost increases in waterflooding projects. Reliably predicting this decline is important for waterflood design as well as for choice and preventative treatment of injected water. 1,2 One of the reasons for well injectivity decline is the permeability decrease caused by rock matrix plugging from solid or liquid particles suspended in the injected water. The flow and deposition of particles in the rock matrix is called deep bed filtration. A mathematical model for deep bed filtration presented by Herzig et al . 3 and by Sharma and Yortsos 4 contains two empirical parameters - the filtration coefficient ? and the formation damage coefficient ß. Knowledge of these two parameters is essential for predicting well injectivity decline during sea/ produced water injection. These parameters are empirical; therefore, they should be determined from laboratory coreflood tests by flowing water with particles through rock. Pang and Sharma 5 and Wennberg and Sharma 6 showed that both parameters can be inferred from the combined measurements of core pressure drop and of suspended particle concentration in core outlet water. The method is based on the analytical model for linear deep bed filtration with constant filtration and formation damage coefficients. In general, the filtration coefficient lambda and the permeability are arbitrary functions of deposited concentration that can also be found from the pressure drop and the outlet concentration by solving functional and integral equations. 7 A coreflood test is usually accompanied by pressure-drop measurements. These measurements are inexpensive and simple to perform; therefore, they are widely mentioned in the literature. 1,3-6,8 Nevertheless, suspended-particle-concentration data in core outlet water during laboratory tests are almost unavailable in the literature because measuring concentration data requires special equipment and is difficult compared to pressure-drop measurements. 8,9 These difficulties are the motivation for attempting to determine the constants ? and ß from the total pressure drop along the core measured at different times during flow. Here we show that the mathematical solution to this problem has limitations. This is discussed in the main part of the text heuristically and in Appendix C rigorously. In summary, only a combination of these two parameters can be found. At best, only ranges of each of these two parameters can be obtained. This limited result is unfortunate for common engineering practice. For example, certain existing software packages for predicting well injectivity loss provide the option of adjusting the pressure- drop curve by matching both parameters ? and ß, under the implicit assumption that these two parameters can be found from the test. A method for determining the filtration and formation damage coefficients from pressure measurements at an intermediate point of the core as well as at core entrance and exit during deep bed filtration, was proposed in a previous work 10 (the so-called three point pressure method). The main part of this paper describes the laboratory procedure and the mathematical recovery method. The method is proven to furnish unique values for the two coefficients and verifies that the solution of the inverse problem is unique and stable. The precise mathematical description of the model is contained in Appendix A. Appendix B contains the expressions for impedance, and the final equation for solution of the inverse problem is derived in Appendix C. Laboratory data on deep bed filtration with pressure measurements at two intermediate points have been presented in the literature. 11,12 In the current work, the results of 34 laboratory tests 11,12 have been analyzed with the three-point pressure method. The objective of this study is to determine statistical relationships between the filtration and formation damage coefficients and the ratio of pore radius to particle size. 13 The results should help in predicting well injectivity decline with permeability and porosity without performing special coreflood tests. A system of dimensionless parameters has been selected, and the forms of dependencies on the parameters have been derived. The results are presented in Tables 1 through 4 for different groups of tests, such as water with solid, liquid, or with both types of particles.

Proceedings Papers

P. Bedrikovetsky, P. Tran, W.M.G.T. Van den Broek, D. Marchesin, E. Rezende, A. Siqueira, A.L. Serra, F. Shecaira

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the International Symposium and Exhibition on Formation Damage Control, February 20–21, 2002

Paper Number: SPE-73788-MS

Abstract

Abstract Permeability decline occurs during injection of sea or produced water, resulting in well impairment. Solid and liquid particles dispersed in the injected water are trapped by the porous medium and may increase significantly hydraulic resistance to the flow. We discuss a mathematical model for deep bed filtration containing two empirical parameters - filtration coefficient and formation damage coefficient. These parameters should be determined from laboratory coreflood tests by forcing water with particles to flow through core samples. A routine laboratory method determines the filtration coefficient from expensive and difficult particle concentration measurements of the core effluent; then the formation damage coefficient is determined from inexpensive and simple pressure drop measurements. An alternative method would be to use solely pressure difference between the core ends. However, we have proved in earlier work that given pressure drop data in seawater coreflood laboratory experiments, solving for the filtration and formation damage coefficients, is an inverse problem that determines only a combination of these two parameters, rather than each of them. A new method for the simultaneous determination of both coefficients is developed here. The new feature of the method is that it uses pressure data at an intermediate point of the core, supplementing pressure measurements at the core inlet and outlet. The proposed method furnishes unique values for the two empirical coefficients, and the solution is stable with respect to small perturbations of the pressure data. In the current work the proposed method is used for analysis of laboratory test data on deep bed filtration. The values of filtration and formation damage coefficients are obtained for flow of solid and liquid particle dispersions in a number of different cores. Effects of particle type and porous media wettability on permeability decline are analyzed. Introduction Injectivity decline of oilfield injection wells is a widespread phenomenon during sea or produced water injection. This decline may result in significant cost increase in waterflooding projects. Reliable prediction of this decline is important for waterflood design as well as for choice and preventive treatment of injected water 1,2 . One of the reasons for well injectivity decline is permeability decrease due to rock matrix plugging by solid or liquid (oleic) particles suspended in the injected water. The flow and deposition of particles in the rock matrix is called deep bed filtration. The mathematical model for deep bed filtration presented by Herzig et al. 3 and by Sharma and Yortsos 4 contains two empirical parameters - filtration coefficient ? and formation damage coefficient ß. Knowledge of these two parameters is essential for predicting well injectivity decline during sea or produced water injection. These parameters are empirical, therefore they should be determined from laboratory coreflood tests by flowing water with particles through rock. Pang and Sharma 5 and Wennberg and Sharma 6 showed that both parameters can be inferred from combined measurements of core pressure drop and of suspended particle concentration in core outlet water. The method is based on analytical solution of model 4 for linear deep bed filtration with constant filtration and formation damage coefficients. In general, the filtration coefficient ? and the permeability k are arbitrary functions of deposited concentration, and these functions can also be found from pressure drop and outlet concentration by solving functional and integral equations 7 .

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Latin American and Caribbean Petroleum Engineering Conference, March 25–28, 2001

Paper Number: SPE-69546-MS

Abstract

Abstract Severe fall of injectivity happens with the reinjection of produced water which contains oil droplets and solid particles, with the injection of sea water in offshore fields which contains organic and mineral inclusions, and in a general case of injection of a poor quality water. The mathematical model contains two empirical functions - filtration coefficient versus concentration of deposited particles and velocity and formation damage function versus concentration of deposited particles. Two inverse problems for determination of these two functions from the laboratory coreflood test are formulated. The first problem is determination of a filtration coefficient from the concentration history on a core outlet. The algorithm of solution is given, and the laboratory data treatment is presented. The second problem is determination of a formation damage function from the pressure drop history on a core. These two methods allows to determine from laboratory test the information necessary for prediction of well impairment. Introduction Injectivity reduction with the injection of water which contains solid and liquid inclusions takes place to some degree in most waterflooding projects. It becomes an important issue with respect to waterflooding of offshore fields using the sea water which usually contains different organic matters and solids. Facilities for raw sea water treatment are very limited by limitations for operations in sea platforms, so often a poor quality water is used in offshore waterflood projects. Reinjection of produced water which contains oil droplets and solid particles happens in several onshore fields and is often accompanied by injectivity decline. Amongst the advantages of produced water reinjection are that the water quality treatment for produced water is less than that for raw water from other sources; also produced water is usually compatible with the reservoir fluids and do not cause scale problems 7,8 . Nevertheless, oil droplets and solid particles may be present in the produced water, which often causes severe formation damage. Presently produced water reinjection is under consideration in offshore fields due to the possible environmental impact of the alternative of sea disposal. In designing a waterflood project, the level of water treatment necessary to minimise formation damage must be assessed, particularly whether solids or oil droplets should be removed and by how much. The different water treatment options are to be considered, so it is important to know the performance of an injector as a function of quality of injection water. Therefore, substantial efforts have been done in modelling of injectivity decline with injection of water with solid particles and oil droplets. The flow of solid-containing suspensions have been studied in many engineering branches, including the petroleum one, and was widely exposed in the literature, while there has been little published on the aspect of formation damage resulting from the flow of oily water. The basic mathematical model for deep filtration with particle retention consists on the mass balance equation and the kinetic equation for clogging 1–5 . The analytical model for diffusion-free flow was developed in the work 1 under the assumptions that accumulation of suspended particles can be ignored and that the suspended concentration distribution is steady state. The simplified assumption that filtration coefficient is constant is proposed in other works 3,4 . In this case the direct problem of prediction of concentation of particles and pressure drop on the core allows for simple analytical solution.

Proceedings Papers

Publisher: Society of Petroleum Engineers (SPE)

Paper presented at the SPE Latin America/Caribbean Petroleum Engineering Conference, April 23–26, 1996

Paper Number: SPE-36132-MS

Abstract

Abstract To describe two-phase displacement with hysteresis we use the Buckley-Leverett model with the imbibition, drainage and scanning fractional flows. Mathematical theory for the initial-boundary non-self-similar problems is developed. Structure of solutions is presented together with the physical interpretation of phenomena. Analytical solutions for the injection of the water slug with the gas drive and for the sequential injection of water and gas slugs with the water drive are obtained. The solutions show that the hysteresis decreases the gas flux in the case where the drainage relative permeability is lower than the imbibition one, which is a positive effect for the WAG injection. Introduction Hysteretic behaviour of relative permeability curves has long been recognised. Numerous laboratory studies have reported hysteresis of relative permeabilities for one-dimensional flow in cores and physical explanation of the phenomena have been presented. Non-monotonic change of phase saturations takes place in almost every process of oil recovery (waterflood in heterogeneous reservoirs, WAG injection, water coning, gravitational separation, steam and polymer huf-n-puf stimulations, etc.). Therefore, numerous researches were dedicated to the mathematical modelling of the two-phase history-dependent flow. Formulae for the drainage, imbibition and scanning relative permeability curves have been developed by C. S. Land and J. E. Killough. In these works, the hysteresis between the primary drainage and imbibition curves have been considered, however, all the imbibition curves have been assumed to be reversible. This assumption is relevant with the simulation of creation of the initial saturation and further floods in the laboratory cores. More complex formulae for hysteretic relative permeability have been derived in. The mathematical model for two-phase displacement honouring hysteresis consists on the basic mass balance equations and the modified Darcy's law. This model is the same as in the hysteresis- free model, the only difference is the hysteretic parameter which shows the direction of the saturation process. Introduction of the hysteresis into the Buckley-Leverett two-phase flow model changes the mathematical type of the governing equation as well as initial and boundary conditions. Therefore, a more detailed mathematical investigation which provides with the existence of the global solution, is required (A monograph is an example of such a study). It is important for the development of numerical methods for reservoir simulators with hysteresis. A mathematical model for the two-phase flow with hysteresis was developed by D. Marchesin, H. Medeiros and P. J. Paes-Leme. The model contains two unknowns: saturation and the number of previous drainages and imbibitions in each point of the reservoir. The complete solution of the Riemarin problem was obtained. The existence of the global solution have been established. This model considers two extreme fractional flow curves and the scanning curves are not incorporated. A mathematical model for the fractional flow with scanning curves was proposed by Kh. Furati. The system contains two continuous unknowns: saturation and the hysteretic parameter, which is the maximum value of saturation which have been reached in each point of the reservoir. The author introduced immobile saturation shocks which allowed to obtain solution of the self-similar problem of the decay of the initial discontinuity. The Riemann problem was solved and the classification of self-similar configurations was presented. Self-sharpening solutions have been found for the waterflooding (one hyperbolic equation) and for the polymer flooding (system of two hyperbolic equations). P. 557

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