Fluid Dynamics, Vol. 35, No. 5, 2000

DYNAMICS OF FLOW THROUGH POROUS MEDIA WITH UNSTEADY PHASE PERMEABILITIES D. N. Mikhailov and V. N. Nikolaevskii

UDC 532.546

The influence of nonequilibrium effects developing due to the formation of an emulsion of each phase (gas bubbles in the water and water dust in the gas) on the flow dynamics is investigated with reference to the displacement of water by a gas. The nonequilibrium effects manifest themselves in a change in the shape of the phase permeability curves (they become "convex") and the threshold phase saturations in the course of flow through the porous medium. A kinetic equation in which the relaxation time is proportional to the seepage rate is used to describe such effects. The case in which the liquid displaced by the gas is itself gassed and the volume concentration of the gas bubbles is constant is considered.

In the "classical" description of two-phase flow through a porous medium the movement of immiscible fluids is assumed to occur locally in the steady-state regime. This should be tested in the course of measuring the phase permeabilities on small samples of porous media. However, in many cases of the displacement of oil by water an emulsion is formed [1, 2]. For example, quite frequently when a two-phase mixture is pumped through porous media we can observe pressure drop oscillations corresponding to extraneous droplets being "pushed" through constrictions (throats) in the pore channels. Upon entrainment of the droplets the state and motion change suddenly over the entire system of pore channels. The dynamics of this motion is determined by the pore space geometry, the pore size, the pressure gradient, and the capillary forces. In the presence of gas bubbles the latter grow to the size of the individual pores and the gas flow takes place by jumps in the form of little jets between these bubbles. In this case the corresponding pulsatory flow regime has been noted (see review [3] and [4]). Disintegration of the droplets is also possible under the action of an electric field when the surfactant distribution becomes nonuniform and the droplets break up into smaller droplets [5]. Thus, under reservoir conditions the emulsion nature of the fluid mixture transport can be reinforced due to various natural or technical effects. Ultrasound can also lead to strong emulsification of the water-hydrocarbon system. If a small amount of gas is present, this will further facilitate emulsification [6, 7]. In real reservoirs, ultrasound is generated by seismic noises, including the noise initiated by the flow itself [8, 9]. The reservoir ultrasound is also intensified by vibroseismic action on the reservoir [1] owing to the strong nonlinearity of the rocks. The high values of the nonlinearity coefficient are related to the fracturing and porosity of real geophysical media [10, 11]. Ultrasound generation has also been illustrated theoretically [1] with reference to nonlinear elastic excitation in the presence of natural (pendular) oscillations of the grains or viscoelastic excitation of seismic noise associated with creep of the block rocks. The amplification of a weak ultrasound pulse in a strong low-frequency pumping wave field was also noted in the experiments [12]. We will consider the emulsified particle mobility conditions in the flow of a carrier phase under the conditions of a porous medium. In study [13] the oil droplet mobility was estimated using the notion of a capillary number Ca =

wµ k = grad p σ σ

Here, k is the permeability of the reservoir, p is the pressure, σ is the surface tension, µ is the viscosity of the fluid, and w is the seepage rate.

Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 103–113, September–October, 2000. Original article submitted July 7, 2000. 0015-4628/00/3505-0715$25.00 © 2000 Kluwer Academic/Plenum Publishers

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Fig. 1. Phase permeability curves for the process of displacement of water by gas (f1(s) and f2(s) are the phase permeabilities of the gas and the water, respectively). The broken and continuous curves (convex and concave, respectively) correspond to the phase permeabilities in the case of the formation of an emulsion in the flow and the case of "pure" phases.

On the other hand, as a criterion of the disintegration of a droplet (of diameter d), when the ratio of the dynamic pressure ρu2 (u is the flow velocity of the other fluid past the droplet) to the surface tension σ is fairly large, the Weber number We =

ρu 2d σ

(0.1)

was used. In order to estimate the relative role of the vibrational forces the mobility criterion [8] was formulated as the ratio of the retaining capillary forces to the sum of the gravity force, the hydrodynamic buoyancy, and the acoustic dynamic pressure M=

ρσ cos θ ≤ 0.3 (γ sin α + ∂p/∂x + A ρ ω 2)r l ∆ρ

(0.2)

Here, γ is the specific weight, r and l are the characteristic dimensions of a droplet, α is the inclination of the reservoir, ∆ρ is the density difference in the emulsion, which depends on the mass of dissolved gas, θ is the wetting angle, and A is the amplitude of the oscillations of the frequency ω, that can create the necessary velocity (in this case, acoustic) head. In order to find the dimensionless number M experiments on the excitation of the motion of droplets wetting small glass spheres with blown air were used [8]. In accordance with (0.1), the dimensions of the droplet r and l can vary under conditions of intense ultrasound action. The critical value of M equal to 0.3 can be reached under reservoir conditions in the presence of smallamplitude oscillations at an ultrasonic frequency of the order of 103–104 Hz. In the laboratory experiment [7] frequencies of the order of 1 MHz were used, the effect (increase in residual oil production from sandstone) increasing with decrease in the frequency. In reservoirs the alternatively necessary high amplitudes A can be realized briefly — in explosions or earthquakes. However, cases of change in the oil-water ratio in production wells during repeat earthquakes are known [14]. 1. The experimental study of the motion of a pre-formed oil-water-bubble emulsion through sand packings [15] showed that the water permeability decreases with pumping. This is quite similar to the phase permeability measured under steady-state conditions without the pre-formation of a stable emulsion.

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In this connection, we will consider the possibility [16] of introducing these effects into a phenomenological, in other words, averaged description of two-phase flow through a porous medium, when the velocity of the ith phase wi is determined by Darcy’s law wi = −

∂p k fi (s) i µi ∂x

(1.1)

where fi(s) are the curves of the relative phase permeabilities in the steady-state case; µi are the viscosities; s is the saturation of the pore space by the first (displacing) phase; and pi are the phase pressures. In this case fi(s) ≠ 0 if s > s*, i.e., when the saturation s is greater than its threshold value s*. The characteristic features of the emulsion flow can be reduced to two considerations. Firstly, emulsion droplet capture in the fluid and pore system is not absolute: both an increase in the velocity of the coexisting phase and the influence of other factors taken into account in (0.2) can restore their mobility. The threshold saturation is a constant parameter of the system only under steady-state conditions. This quantity can generally be considered to be a dynamic parameter which only asymptotically reaches the steady-state value. The phase permeability curve measured under static conditions changes somewhat due to changes in the value of the threshold saturation. This reflects the mobility of part of the residual oil(or bound water). Secondly, experiments with micellar solutions [1] show that in the presence of an emulsion the effective phase permeabilities qualitatively change shape (they turn out to be convex, not concave, which means higher mobility of the droplets of the emulsified phase for average saturation values, see Fig. 1). The transition of the mixture to the emulsified state when water with surfactant additives is injected into the reservoir was discussed in [18]. If this effect is relatively weak, the existence of the emulsion will be of short duration, i.e., in the course of ordinary two-phase flow the phase permeabilities measured under steady-state conditions are most likely gradually reached. Earlier, it was proposed to introduce a kinetic factor (nonequilibrium) into the concepts of capillary pressure (phase pressure difference) and phase permeabilities [19–21]. This reflects the time spent on for the capillary displacement of the phases through the pore channels. In principle, the approach developed in the present paper also reflects this effect due to introduction of the time spent in capturing or releasing the emulsion droplets. In study [22] (on the basis of experiments [15]) an equation of the kinetics of accumulation of stationary droplets was introduced but the volume concentration zi of the emulsion droplets of the ith phase captured in the pore space was not identified with the threshold permeability in Buckley-Leverett theory, as proposed below. It was only noted that the effective permeability of the coexisting phase decreases linearly with increase in zi. In study [23] the effect of the absorbed matter on the flow through a porous medium was taken into account outside the framework of Buckley-Leverett theory. The effect of added surfactants on the phase permeabilities themselves in water flooding (needed for the hydrodynamic calculations [24]) was discussed in [17]. In this case the notion of a capillary number increasing with decrease in the interphase surface tension was used. In the limiting case of the absence of surface tension, the phase permeabilities degenerate into intersecting straight lines, as assumed earlier in [19]. Thus, if we identify zi with the residual saturation si* , the latter will enter into the relative phase permeability * N (1.2) f (s) = [(s − si ) /A ] i

i

i

as a dynamic parameter. Therefore, fi(s) will vary during flow development in accordance with the stationary emulsion droplet accumulation kinetics. The notation (1.2) was introduced in [25], in which si*, Ni, and Ai were assumed to be constants. In the present paper si* varies with time and only asymptotically reaches the threshold saturation si′ measured in steadystate experiments. This must be taken into account in the kinetic relation constructed by analogy with sorption kinetics [26]. In this case the relaxation time should be assumed to be inversely proportional to the seepage velocity, as a result of which both the effect of the accumulation of fluid droplets or gas bubbles on the pore walls and the effect of their entrainment can be taken into account ∂si

*

∂t

=

wα (s) di

+

*

(si − si )

(1.3)

Here, di is the effective size of the gas/oil bubbles (i = 1) or water droplets (i = 2); and wa is the velocity of 717

the flow which redistributes the "bound" part of the water. This may be the seepage velocity of the coexisting jth phase (if the washing out of the residual oil is being studied) or even a certain combination of velocities. In the displacement front calculations given below it was assumed that α = i, i.e., the accumulation rate of the "bound" part of each phase is determined by its motion with respect to the pore space. In the steady-state case the threshold saturations of the phases, which correspond to the equilibrium concentration of the retained emulsion, are equal to s1+ = 0.1 and s2+ = 0.2. The dynamic threshold saturations relax to these values in the process of two-phase flow. Taking into account the fact that the saturations are related to each other: s1 + s2 = 1 we can represent the dependences of the phase permeabilities on the saturation of the displacing phase (gas) s = s1 in the form of the following empirical relations [25]: 3.5  *  s − s1  f1 (s, s ) =   ,  0.9 

0 ≤ s ≤ s ;

* 1

* 1

f1 (s, s ) = 0 ,

s1 ≤ s ≤ 1

* 1

*

  3.5 *  (1 − s2 ) − s  * * * f2 (s, s ) =   , 0 ≤ s ≤ 1 − s2 ; f2 (s, s2 ) = 0 , 1 − s2 ≤ s ≤ 1 0.8   * 2

(1.4)

(1.5)

2. We will consider unsteady two-phase flow through a porous medium with reference to the model problem of the displacement of water by gas. Suppose that in the pore space we have both the gas bubbles in the water and water dust in the gas instead of continuous gas and water phases. The water droplets and the gas bubbles located on the rigid walls of the pores and displaced by the other phase will be also included in the emulsified mass. Solubility is neglected. In the one-dimensional case for describing the process of two-phase flow through a porous medium we will use the Buckley-Leverett equation, neglecting the compressibility of the phases: ∂s ∂ * * m = U F (s, s1 , s2 ) ∂t ∂x (2.1) * f1 (s, s1 ) * * F (s, s1 , s2 ) = * * f1 (s, s1 ) + µ* f2 (s, s1 ) where F is the Buckley-Leverett function, m is the porosity (a constant) µ*=µ1/µ2 is the ratio of the phase viscosities, and U=w1 + w2=U(t) is the total seepage velocity; without loss of generality, this quantity can be considered constant: U(t)=const. In this case the seepage velocities of the phases can be determined from the formulas [25] * * * * * * * * (2.2) w (s, s1 , s2 ) = U [1 − F (s, s1 , s2 )] w (s, s1 , s2 ) = U F (s, s1 , s2 ) ; 1

2

Thus, we obtain a closed system of equations (1.3), (2.1) describing the process of unsteady two-phase flow through a porous medium with phase permeabilities that vary during the displacement time, the dynamics of the phase permeabilities being related to the accumulation or entrainment of stationary emulsion droplets. The equations and boundary conditions introduce several characteristic scales into the problem: the characteristic dimension (of the oil or gas reservoir or laboratory apparatus) L, for which the calculations are carried out, and the characteristic displacement time T, during which the process of flow covers the entire characteristic distance L. In this case T = L/U. Accordingly, we introduce the dimensionless time and distance: τ = t/T and ξ = x/L. In solving the system (1.3), (2.1), for the threshold phase saturation kinetics equation (1.3) we used a RungeKutta method of the fourth order in time. Equation (2.1) was directly approximated on a uniform grid by means of the finite differences n

n−1

si − si ∆τ

718

n

=U

n

*

n

*

n

n

*

n

*

F (si , s1 i , s2i − F (si − 1 , s1i − 1 , s2i − 1) ∆ξ

(2.3)

Fig. 2. Gas saturation distribution in the process of the displacement of water by gas with allowance for the threshold saturation dynamics for the gas a, water b, both phases c. The broken curve corresponds to the gas saturation for constant phase threshold saturations.

In this case the derivatives with respect to time and space were replaced by a "backward difference" (this algorithm is first-order accurate in time and space). In order to determine the correctness of the choice of step (test the stability of the numerical algorithm) we used dual calculations with the chosen and doubled step [27]. If the difference between the solutions obtained did not exceed a given accuracy, the chosen step was conserved; otherwise the step was reduced by half. This algorithm was tested on a number of examples (calculation of a single triangular pulse and a "step" with a suitable choice of Buckley-Leverett function; F(s) = s2 when the analytic solutions for the velocity of the discontinuity and the length of the pulse are known [28]). The test calculation results showed that this algorithm has a small numerical diffusion. This makes it possible to carry out the calculations without introducing an additional "artificial viscosity" for smoothing the discontinuous solutions. In the course of the numerical experiment the time step was chosen so as to ensure a calculation accuracy of the order of 10−2; moreover, we tested the stability of the results obtained both with respect to a decrease in the finite-difference scheme step by more than an order of magnitude and an increase in the time base of the calculations. On the basis of this algorithm we obtained a number of numerical solutions of the problem of unsteady twophase flow through a porous medium (displacement of water by gas) with allowance for the phase permeability dynamics. The problem parameters were as follows: U = 10−5 m/s, m = 0.2, µ1/µ2 = 0.01, and di = 1 mm. The initial conditions for the system (1.3), (2.1) were specified in the form of a step in the gas saturation distribution and a uniform spatial distribution of the threshold phase saturation *

s1 (t = 0 , x) = 0.3 ;

*

s2 (t = 0, x) = 0.4

(2.4)

The boundary conditions were specified as a constant gas saturation at the left endpoint of the calculated interval s(t, x = 0) = 0.7 and s(t, x) = 0.1 on the rest of the interval. This formulation corresponds to the beginning of injection of gas into a reservoir containing a small fraction of the pore-trapping gas remaining after a previous displacement cycle. The initial condition (2.4) was chosen because at the onset of flow due to the presence of both gas bubbles in the water and water droplets in the continuous gas phase the threshold saturations of the two phases turn out to be higher than those in the steadystate case (because of pore closure and, consequently, blocking of the movement of the continuous phase by droplets of the other phase). Owing to the entrainment of trapped water droplets by the continuous gas phase (and gas bubbles by the water phase) in the process of flow the phase saturations gradually reach the threshold values characteristic of the steady-state case. The calculation results reproduced in Fig. 2 (105 iterations were performed, time step ∆τ = 0.05) indicate that taking into account the effects associated with the variation (dynamics) of the gas and water threshold saturations

719

Fig. 3. Two-phase flow through a porous medium when the phase permeability curve for the displaced phase is concave and that for the displacing phase is convex: a phase permeability curves (continuous curves) and the graph of the function F′(s) (broken curve), curves 1 and 2 correspond to the displacing and displaced phases; b saturation distribution profile. The broken curve corresponds to the saturation distribution profile in the case of classical concavity of both phase permeability curves.

due to the accumulation and entrainment of emulsion droplets in flow through a porous medium leads both to a change in the velocity of the saturation jump and a change in the general pattern of saturation distribution. As follows from the gas saturation profiles reproduced in the figure, the kinetics of the displacing phase (gas) play a leading part in the two-phase flow process considered, while the kinetics of the displaced phase during the calculation period had practically no effect on the resulting distribution. Generally, owing to the threshold saturation dynamics the displacement front becomes steeper and residual gas saturation is reached more slowly than in the steady-state case. This is attributable to the superposition of the asymptotics associated with relaxation on the asymptotics associated with the displacement process (the initially stationary water droplets and gas bubbles are gradually entrained in the flow through the porous medium). 3. We will consider the case of "convex" phase permeability curves (Fig. 1). This corresponds to a change in the exponent Ni in formula (1.2); thus, we now have 0 < Ni < 1. For the phases in which there is no emulsion ("pure") we will retain the typical "concavity" of the permeability curves. To simplify the presentation we will disregard the dynamics of the threshold phase saturations and set them equal to the following values measured in the steady-state experiments [25]: s1* = s1+ = 0.1 and s2* = s2+ = 0.2. From the Buckley-Leverett theory [25] it follows that the velocity of propagation of a given saturation s can be determined from the formula (3.1) V = (U /m) ∂F (s) /∂s In what follows, we will consider two cases of two-phase flow: first, the phase permeability curve of the displacing phase is concave and that of the displaced phase is convex (Fig. 3a) and, conversely, the phase permeability curve of the displaced phase is concave and that of the displacing phase is convex (Fig. 4a). The problem parameters were made the same as in the previous calculations: U = 10−6 m/s, m = 0.2, µ1/µ2 = 0.01, and di = 1 mm. For the convex phase permeability curve the exponent in formulas (1.4) and (1.5) was set equal to Ni = 0.6, while in the case of the concave curve Ni = 3.5. We performed 105 iterations with a time step ∆τ = 0.05. In the first case, the function F′(s) is nonmonotonic (Fig. 3a); therefore, starting from a certain instant of time the saturation distribution turns out to be multivalued and, as a result, the profile of the displacement front contains a saturation jump (Fig. 3b). In the second case, the function F′(s) monotonically decreases (Fig. 4a) and there is no multivalued effect (the points on the displacement front corresponding to a lower initial saturation always move more rapidly than those with a higher initial saturation). Thus, in the case considered no saturation jump is formed. This is clearly illustrated by the saturation distribution profile of the displacing phase (Fig. 4b) — the profile is flat and there is no discontinuity. 720

Fig. 4. Two-phase flow through a porous medium when the phase permeability curve for the displacing phase is convex and that for the displaced phase is concave. The notation is the same as in Fig. 3.

We will now consider the case in which the liquid displaced by the gas is itself gassed and the volume concentration of the gas bubbles is constant. As before, the solubility of the gas is neglected. We assume that the gas bubbles are small so that they can freely penetrate through the elementary channels between reservoir sand grains. Thus, in the process of two-phase flow the continuous gas phase moves with a certain velocity, while the gas bubbles in the liquid move with the velocity of the liquid itself. This scheme essentially combines the Buckley-Leverett theory (with phase permeabilities) and that proposed earlier (1932) by Leibenzon (when there is no gas bubble "fly-through" effect, both components move with the same velocity [29]). In order to determine the seepage velocity of the gas and the gassed liquid we will use Darcy’s law ki ∂pi

wi = −

µ i ∂x

(4.1)

where i = 1 and i = 2 correspond to the gas and the gassed liquid, respectively. Then for the mass flow rate Qi of each phase we obtain the formulas Qi = −

ki µi

ρi Φ

∂pi ∂x

(4.2)

where Φ is the cross-sectional area of the reservoir and ρi is the density of the ith phase. The density of the gassed liquid can be expressed in terms of the density of the pure liquid (without gas bubbles) and the density of the gas inside the bubbles (4.3) ρ 2 = ρ L (1 − n) + ρ g n where ρL is the density of the liquid (without gas); ρg is the density of the gas inside the bubbles; and n is the volume concentration of the gas bubbles. We will use the adiabatic equation of state of the gas in the form: η η (4.4) p /ρ 1 = p /ρ a ≡ β η 1

a

where η = const is the specific heat ratio, and pa and ρa are the atmospheric pressure and the gas density at atmospheric pressure. In order to calculate the pressure of the gas inside the bubbles we will use the Laplace formula [25] (4.5) pg = p2 + 2σ /R Here, R is the bubble radius. 721

Fig. 5. Phase permeabilities of the gas (curve 1) and water (curve 2) for two-phase flow (gas – gassed liquid) through a porous medium, n = 0.005. The broken curves correspond to the phase permeabilities of the gas and the water for n=0.

For the two-phase (gas-gassed liquid) the total gas mass flow rate can be found as the sum of the mass flow rate of the gas phase and that of the gas in the form of bubbles entrained by the liquid phase: QΣ = Q1 + nQ2. We will assume that the phase pressure of the gassed phase and that of the continuous gas phase are identical: p1 = p2 ≡ p. Then, with allowance for (4.2) and for η = 1 (isothermal process), we obtain the following equality from which we can determine the total phase permeability of the gas k p ∂p (4.6) QΣ = Σ µ1 β ∂x   µ1  ρL  2σ   + n 1 + kΣ = k1 + n β  k pR   2 µ2  p 

(4.7)

Assuming n << 1, we can neglect the second term in parentheses in formula (4.7): kΣ = k1 + n β

µ1 ρL µ2 p

(4.8)

k2

For two-phase flow through a porous medium the total gas saturation sΣ is equal to sΣ = s1 + ns2 = s1 + n (1

s1)

(4.9)

Taking into account the fact that in the case of the displacement of water by gas the dependences of the phase permeabilities on the displacing-phase saturation can be specified in the form of empirical relations (1.4) and (1.5), in which the total gas saturation is taken into account, we can construct a plot of the function kΣ(sΣ). In this case, the volume gas bubble concentration in the gassed liquid was specified as n = 0.005. From an analysis of the dependence (4.8) (Fig. 5) it follows that in the case of two-phase flow (continuous gas – gassed liquid) through a porous medium the phase permeability of the gas is not equal to zero, even when the gas saturation is below the threshold value. For subcritical saturation the non-vanishing of the gas permeability is attributable to the fact that in the course of two-phase flow the part of the gas contained in the water in the form of bubbles is entrained by the 722

continuous water phase and, consequently, becomes mobile. Then the effective phase permeability turns out to depend on the pressure. Summary. If under real conditions in the process of two-phase flow through a porous medium an emulsion of one phase in the other is formed, then nonequilibrium effects associated with the change in the shape of the phase permeability curves and the threshold saturation dynamics develop. In order to describe these effects we suggest using a kinetic equation in which the relaxation time is made inversely proportional to the seepage velocity. This makes it possible to take into account both the accumulation of droplets and their entrainment by the flow. Owing to the threshold saturation dynamics the displacement front becomes steeper and the residual gas saturation is reached more slowly than in the steady-state case. This is attributable to the superposition of the asymptotics associated with relaxation on the asymptotics associated with the displacement process (the initially stationary water droplets and gas bubbles are gradually entrained in the flow through the porous medium). If the phase permeability curve of the displaced phase is concave and that of the displaced phase is convex, the saturation profile is flat and has no discontinuity. In the opposite case (the phase permeability curve of the displaced phase is convex and that of the displacing phase is concave) the saturation profile retains a jump. The presence of droplets of an emulsion of one of the phases in the coexisting phase (for example, when a gassed liquid with a constant gas bubble concentration is displaced by a gas) leads to the phase permeability of this phase depending on the pressure and being non-zero even for saturations below the threshold value. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

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H. Soo and C. J. Radke, "A filtration model for the flow of dilute, stable emulsion in porous media. I. Theory," Chem. Eng. Sci., 41, 263 (1986). A. A. Barmin and D. I. Garagash, "Solution flow in a porous medium with impurity adsorption," Izv. Ros. Akad. Nauk, Mekh. Zhidk. Gaza, No. 4, 97 (1994). G. I. Barenblatt, V. M. Entov, and V. M. Ryzhik, Flow of Fluids and Gases through Natural Reservoirs [in Russian], Nedra, Moscow (1984). I. A. Charnyi, Underground Hydrogasdynamics [in Russian], Gostoptekhizdat, Moscow (1963). T. R. Camp, "Theory of water filtration," Proc. Amer. Soc. Civil. Eng., 90, No. 4, Pt. 1, 1 (1964). S. S. Chesnokov, Numerical Methods in Problems of the Theory of Vibrations and Waves [in Russian], Moscow University Press, Moscow (1980). M. B. Vinogradova, O. V. Rudenko, A. P. Sukhorukov, Wave Theory [in Russian], Nauka, Moscow (1990). L. S. Leibenzon, "Trial of the theory of flow of gassed oil through a reservoir," in: Collected Works, Vol. 2 [in Russian], USSR Academy of Science Press, Moscow (1953).