Two major arrangements describing the oil-by-water displacement process, i.e., Muskat’s model without taking into account the surface tension on the free boundary and the Buckley-Leverett model based on the surface tension, are considered. These arrangements were subject to theoretical and numerical study, which made it possible to uncover their self-contradictorines

Meirmanov, A. M.

Reshetova, G. V.

Tcheverda, V. A.

English

DSpace at Belgorod State University

990ISSN 1028-334X, Doklady Earth Sciences, 2016, Vol. 470, Part 1, pp. 990–993. © Pleiades Publishing, Ltd., 2016.Original Russian Text © A.M. Meirmanov, G.V. Reshetova, V.A. Tcheverda, 2016, published in Doklady Akademii Nauk, 2016, Vol. 470, No. 3, pp. 340–343.Analysis of Models of Oil-by-Water Displacement through the MicrostructureA. M. Meirmanova, G. V. Reshetovab, and V. A. Tcheverdac*Presented by Academician M.I. Epov September 7, 2015Received September 7, 2015Abstract—Two major arrangements describing the oil-by-water displacement process, i.e., Muskat’s modelwithout taking into account the surface tension on the free boundary and the Buckley–Leverett model basedon the surface tension, are considered. These arrangements were subject to theoretical and numerical study,which made it possible to uncover their self-contradictoriness. The focus was on construction of a numericalsolution at the microlevel using the direct method with pressure correction.DOI: 10.1134/S1028334X1609018X1. The oil-by-water displacement process is com-monly described by two models: Muskat’s modelwithout taking into account the surface tension on theboundary [1] and the Buckley–Leverett model takinginto account the surface tension on the boundary [2].They are based on Darcy’s law for an absolutely hardrock skeleton. In 1980, L. Tartar in [3] using the aver-aging technique showed that Darcy’s law could bederived from the Stokes equations in a periodicallyhard skeleton, when the size of the periodicity celltended to zero. Today, it is possible to apply theapproaches focused on the construction and justifica-tion of mathematical models of moving f luids in envi-ronments with a microstructure, which are based onthe following steps:(1) exact description of moving f luids at the micro-scopic level (on a pore scale) by the classical equationsof continuum mechanics;(2) mathematically substantiated averaging of theresulting problem and corresponding implementationof the transition from the microlevel to the macrolevel.Unfortunately, this scheme can be implementedonly for single-phase f lows [4], so it is particularlyimportant to carry out numerical experiments.2. Let us consider the two-dimensional oil-by-water displacement process in an absolutely hard skel-eton. A one-dimensional model of perfect soil is takenas the microstructure: in a rectangle with the unitheight there is a system of noncontacting parallel cap-illaries with a rectangular cross section, whose dimen-sionless width ε tends to zero in such a way that thevolume porosity remains constant. At the microlevel,the displacement process is modeled in a single capil-lary:Ω = {x = (x1, x2): –1 < x1 < 1, –ε < x2 < ε}with a moving free surface Γ(t) subdividing it into twosubregions such as Ω−(t) and Ω+(t). Further consider-ations are performed in dimensionless variables:where L and T specify the characteristic length of thecapillary and the characteristic time of the process,respectively,  is the unit pressure (for instance, oneatmosphere), and  is the unit density (for instance,water density).Inside the capillary, the f luid motion is describedby the Stokes equations:, (1), (2)where  and  are the velocity vector and the ten-sion tensor, respectively, to the left and to the right of= = =ρ = ρ ρ = ρ00 0 0' , ' , ' ,' , ,LL p p pTp Lgx x V V0pρ0± ± ± ± ± ±μ∇ + ρ = = α −0, ( ) pP g P D V I±∇ ⋅ = 0V±V ±PGEOPHYSICSa Belgorod National Research University, Belgorod, Russiab Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch, Russian Academy of Sciences, pr. Akad. Lavrentieva 6, Novosibirsk, 630090 Russiac Trofimuk Institute of Petroleum Geology and Geophysics, Siberian Branch, Russian Academy of Sciences, pr. Akad. Koptyuga 3, Novosibirsk, 630090 Russia*e-mail: Cheverdava@ipgg.sbras.ruDOKLADY EARTH SCIENCES  Vol. 470  Part 1  2016ANALYSIS OF MODELS OF OIL-BY-WATER 991the border line Γ(t),  and  are thedimensionless density and viscosity of f luids, and is the strain velocity tensor. Velocities are sub-ject to adhesion conditions throughout the capillary,while pressure is specified in the inlet to which water issupplied and in the outlet from which oil is taken from:. (3)The conditions on the free boundary are specifieddepending on whether its surface tension is taken intoaccount or not.Displacement without surface tension. In this case,the consistency of velocities and the normal constitu-ent of the tension tensor is required on the border lineΓ(t):, (4)where n is the unit normal to the surface Γ(t).Displacement with surface tension. In this case, theborder conditions on the free surface are written as fol-lows:(5)where n still specifies the unit normal to the freeboundary Γ(t), R is its curvature radius, t is the unitvector of the tangential direction to the border linesurface, and  is the surface tension coefficient.3. Numerical experiments were conducted by directsolution through the pressure correction by tracing thefront [5]. The free boundary was constructed by step-wise recalculation with relevant interpolation. The fol-±± ρρ =ρ0' ±±μμα =ρ0TLg( )D V± ± ±=±ε =±= ⋅ =2 1 10,x x pV P n n + − + −Γ Γ− = ⋅ − ⋅ =( ) ( )( ) 0, ( ) 0t tV V P n P n+ − + −Γ Γσ− = ⋅ − ⋅ =( ) ( )( ) 0, ( ) ,t t RV V P n P n tσlowing averaging was used to transit from microscaleto macroscale:(1) the displacement task (1)–(3) is solved in thetotal arrangement under the condition of (4) or (5) forone rectangular capillary with hard walls;(2) the obtained solution is periodically replicatedorthogonally to the capillary axis with simultaneousreduction of its cross section  in such a way that thetotal area occupied by the capillaries remainsunchanged, thus providing the specified porosity;(3) a function with two variables is constructed; itdescribes the f low in the limit case ( ); due tosymmetry of the task, the limit function depends ononly one spatial variable, i.e., the capillary length .In case of no tension on the boundary, the resultsof the numerical experiment are given in Fig. 1. Timevariations in the form of the boundary between twounmixable f luids in the capillary are shown on the left.The boundary within the capillary is drawn as far asdesired due to no surface tension and adhesion condi-tion for the walls. In this case, below the contact(adhesion) points and above the end of the expanded“tongue,” there are areas where a volume fraction ofwater is equal to one and zero, respectively. The mix-ing zone formed between these areas is characterizedby a constant decrease in the volume fraction of waterfrom one to zero. The resulting averaging of the pro-cess by the capillary replication with the given porosityis shown on the right. It is seen clearly that the zoneformed expands, while the water concentrationdecreases.Tension on the free boundary. The calculationresults are given in Fig. 2 for the arrangement withaccount for the surface tension. As before, the timevariations in the form of the boundary are shown onεε → 01xFig. 1. Numerical modeling of oil-by-water displacement without taking into account surface tension. On the left, change in theborder line in one capillary; on the right, f low averaging by periodic replication of the capillary.x1 x1x2x1s = 00 < s < 10 < s < 10 < s < 1s = 0s = 0s = 1t = t0 t = t0 + τ t0 = t + 2τs = 1 s = 1992DOKLADY EARTH SCIENCES  Vol. 470  Part 1  2016MEIRMANOV et al.the left. It is evident that the occurrence of surface ten-sion changes the f low nature considerably. In thiscase, the conditions for the tension tensor on bound-ary (5) include the surface tension coefficient and cur-vature radius. Description of the displacement processfollows from the second condition of (5):(1) under zero pressure drop in the capillary, thecurvature radius is infinite, i.e., the boundary isdescribed as a straight line segment;(2) under increase in pressure, the curvature radiusdecreases, but while it is more than half of the capillarywidth, the boundary remains unchanged;(3) under the critical pressure value,1 when the cur-vature radius becomes less than half the thickness ofthe capillary, drops replacing f luid start forming and“shooting,” and their diameter decreases under anincrease in pressure.4. One-dimensional arrangements for Muskat’smodel and the Buckley–Leverett model allow analyt-ical solutions.Muskat’s model. In the region , two different f lu-ids occupy subregions  separated by the boundary. Their velocity  and pressure  are consistentwith the system of equations of filtration theory(Darcy’s law):(6)1 This value depends on the tension coefficient and channelwidth.Ω±ΩΓ( )t ±V ±p± ± ± ±±= −∇ + ρ ∇ ⋅ =μ( ), 0.k pV F VOn the unknown contact boundary , the con-ditions are as follows:, (7)where k is the permeability coefficient,  and  arethe f luid viscosity and density, and  is the given exter-nal force. The task is completed by the boundarymateriality requirement (the boundary moves alongwith the f low):,where  is the boundary movement velocity, underthe initial conditionsand under the limit conditions, (8)where  are surfaces where f luid is pumped andtaken, while  is the surface limiting the f luid f lowarea.In the case of one spatial variable, the region  is asegment of the real axis, while the boundary is specified by one point. The settask is solved explicitly, and the boundary movement is defined by the following equation:. (9)Γ( )t+ − + −Γ Γ− ⋅ = − =( ) ( )[( ) ] 0, ( ) 0t tp pV V n±μ ±ρF±= ⋅nV V nnVΓ = Γ0(0)±± ±∈ ∈= ⋅ =01 2 0( , , ) , 0x S x Sp x x t p V n±S0SΩ{ }Γ = =( ) ( )t x X t{ }Γ = =( ) ( )t x X t− + − + − +− + − +− + ρ − ρ + ρ + ρ=μ − μ + μ + μ0 0( ) ( ) ( )( ) ( )p p XdX kdt XFig. 2. Numerical modeling of oil-by-water displacement taking into account surface tension. On the left, change in the borderline in one capillary; on the right, f low averaging by periodic replication of the capillary.x1 x1x2x1s = 00 < s0 < 10 < s0 < 10 < s0  < 1s = 0s = 0s = 1t = t0 t = t0 + τ t0 = t + 2τs = 1 s = 1DOKLADY EARTH SCIENCES  Vol. 470  Part 1  2016ANALYSIS OF MODELS OF OIL-BY-WATER 993These relations are indicative of the fact that Mus-kat’s model in one dimension describes the so-called“piston” displacement, without the mixing zone.Consequently, it should not be obtained by averagingof the considered idealized model, and therefore, it isincorrect to describe the filtering process even in sucha simple environment.The Buckley–Leverett model. Let us consideragain the region , which consists of two nonintersec-tion regions  separated by the boundary . Ineach subregion, the velocity, pressure, and concentra-tion s of the f luid pumped from the left from the vol-ume with a “minus” index are related by the followingsystem of equations:and the condition relating pressures,.Functions  and  specifying permeabilityand capillary pressure, respectively, depending on theconcentration s, are considered to be predetermined.The automodel solution for this model demon-strated in Fig. 3 is given in [6]. It differs substantiallyfrom the solution obtained by one-dimensionalΩ±Ω Γ( )t−− − − −−++ + + ++∂= −∇ + ρ + ∇ ⋅ =∂μ∂= −∇ + ρ − + ∇ ⋅ =∂μ( )( ), 0,( )( ), 0k s sptk s sptV F VV F V− +− = ( )cp p P s±( )k s ( )cP snumerical averaging. Moreover, it is characterized by anonphysical artifact, i.e., f low in the direction oppo-site to the pressure difference, which is impossible inthe one-dimensional case.5. Numerical one-dimensional averaging of thetwo-phase f low in a rectangular capillary with hardwalls made it possible to determine the characteristicfeatures for two typical arrangements: with surfacetension on the free boundary and without it. In thecase of no surface tension, the mixing zone is formedand expanded and water concentration in it is con-stantly reduced from one to zero. In the case of surfacetension, two flow conditions are possible: the freeboundary is fixed, if the pressure of the supplied f luidis balanced by the surface tension; increase in pressureresults in separation of f luid drops supplied from thefree boundary and their further distribution along thecapillary. Consequently, the expanding mixing zone isformed here as well, but now with a constant waterconcentration dependent on the number of dropsdetached from the free boundary. Meanwhile, analyt-ical solutions obtained for the same idealized geome-try of the pore space using Muskat’s model and theBuckley–Leverett model yield a completely differentstructure of the f low, thus raising serious doubts as totheir physical correctness.ACKNOWLEDGMENTSThis work was supported by the Russian ScienceFoundation, project no. 14-17-00556.REFERENCES1. M. Muskat, Physics, No. 5, 250–264 (1934).2. S. E. Buckley and M. C. Leverett, Trans. AIME 146,107–116 (1942).3. L. Tartar, Lect. Notes Phys. 129, 368–377 (1980).4. A. Meirmanov, Mathematical Models for PoroelasticFlows (Atlantis Press, Paris, 2013).5. A. Prosperetti and G. Trygvasson, Computational Meth-ods for Multiphase Flow (Cambridge Univ. Press, Cam-bridge, 2008).6. A. Meirmanov and S. Shmarev, Electron. J. Differen-tial Equations 227, 1–13 (2014).Translated by E. MaslennikovaFig. 3. Automodel solution for the Buckley–Leverett one-dimensional model. The artefact that should be noted:flow against the pressure direction at the initial moment oftime.x0 x0 < s  < 1s = 1ts = 0

Analysis of models of oil-by-water displacement through the microstructure

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Analysis of models of oil-by-water displacement through the microstructure

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