Viscous fingering in periodically heterogeneous porous media. II. Numerical simulations

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1 Viscous fingering in periodically heterogeneous porous media. II. Numerical simulations A. De Wit Service de Chimie Physique and Centre for Nonlinear Phenomena and Complex Systems, CP 231 Université Libre de Bruxelles, Campus Plaine, 1050 Brussels, Belgium G. M. Homsy Department of Chemical Engineering, Stanford University, Stanford, California Received 24 June 1997; accepted 4 September 1997 We study nonlinear viscous fingering in heterogeneous media through direct numerical simulation. A pseudospectral method is developed and applied to our spatially periodic model introduced in Paper I J. Chem. Phys. 107, The problem involves several parameters, including the Peclet number, Pe, the magnitude and wave numbers of the heterogeneity,, n x, n y, respectively, and the log of the viscosity ratio R. Progress is made by fixing R at 3.0 and then working first with layered systems n x 0 and finally with checkerboard systems in which both wave numbers are nonzero. Strongly nonlinear finger dynamics are compared and contrasted with those occurring in the homogeneous case. For layered systems, it is found that very low levels of heterogeneity leads to an enhancement of the growth rate of the fingered zones, and that both harmonic and subharmonic resonances between the intrinsic scale of nonlinear fingering and those of the heterogeneity occur. We also find that the fingering regime of layered systems can be completely disrupted by modest levels of heterogeneity, leading to a channeling regime and dispersive behavior which is identified as a Taylor dispersion mechanism. The effective axial dispersion coefficient in this regime is found to be strongly dependent on the viscosity ratio. The situation becomes more complex for the checkerboard case. The channeling regime can in turn be disrupted by the axial dependence of the heterogeneity, which stimulates tip splitting and a return to complex nonlinear finger dynamics in regions of parameter space, including very large, that would otherwise be strongly dispersive. The effectiveness of the axial variation in stimulating tip splitting is studied by a short parametric study in n x, and is found to be maximized for certain axial frequencies in a manner similar to that found in Paper I. All our results are found to be in general qualitative agreement with available but limited experimental visualizations American Institute of Physics. S I. INTRODUCTION We continue our study of viscous fingering in heterogeneous porous media begun in the companion paper, hereafter referred to as Paper I. We are generally interested in pursuing analytical/numerical studies of heterogeneous fingering. In Paper I, we reviewed the relevant literature, put forth our model of spatially periodic permeability heterogeneity, established the basic equations and scalings, and performed an analysis of the linear stability of a base state flow established through the combination of a pressure gradient and the permeability field the so-called cellular flow. It was found that certain subharmonic and sideband resonances were possible whereby the heterogeneities modified the exponential growth rates of disturbances. Analytical results were obtained for the subharmonic resonance case for very short times. These results indicated a general increase in the degree of instability as a result of heterogeneities, with long waves in the axial direction being most effective in coupling with the viscous instability, and a secondary increase in growth rates in a band of axial wave numbers and viscosity ratios. These results are limited in scope and applicability, as they pertain to short times and low levels of heterogeneity. However, they are very instructive in identifying mechanisms of interaction and the parameter regime over which they occur. The present paper extends our study into the strongly nonlinear regime of heterogeneous fingering, which we study by direct numerical simulation. We refer the reader to Paper I for a general literature review. Our goals remain those of exploiting the spatially periodic model to the fullest in discovering and understanding the mechanisms of interaction between heterogeneities and viscous fingering. We will draw on our previous experience in the numerical solution of fingering problems and will summarize the relevant dynamics and mechanisms of nonlinear fingering in Sec. IV. The paper ends with a short discussion and qualitative comparison of our findings with previous experimental visualizations. II. FORMULATION A. Basic equations We briefly recapitulate the basic equations and consider the additional geometric parameters that arise in the process of direct numerical simulation of the nonlinear problem. We consider a finite sized two-dimensional porous medium of J. Chem. Phys. 107 (22), 8 December /97/107(22)/9619/10/$ American Institute of Physics 9619

2 9620 A. De Witt and G. M. Homsy: Viscous fingering in porous media. II length L and width H. An incompressible fluid is injected in a uniform manner from the left boundary with dimensionless average velocity 1. The medium is heterogeneous with a permeability r. The velocity of the fluid u (u,v) obeys Darcy s law, and the concentration of a solute obeys a convective-diffusion equation. The viscosity of the fluid is a function of the local concentration through a known relation. The dispersive scales of Paper I are used in making the equations dimensionless. Accordingly, the dimensionless domain width is a Peclet number, Pe UH/D, the aspect ratio of the domain is A L/H, 2 and the dimensionless length of the domain is the Peclet number Pe, Pe UL/D. 3 These two Peclet numbers are not independent, but are related through the aspect ratio: Pe APe. 4 Pe is the axial Peclet number often used in the petroleum engineering literature: in these conventions, Pe determines the number of fingers present across the domain, and Pe dictates the maximum time of the simulations. Thus unlike Paper I, which dealt with an infinite domain, numerical simulations on finite domains will involve additional parameters associated with these lengths, which appear as Peclet numbers. The basic equations of the system are hence u 0, p c r u, t c u c 2 c, c exp Rc, 8 with a given initial condition and where R ln( 2 / 1 )isthe mobility ratio. The boundary conditions we will consider here are u 1, v 0, c 1 at x 0, u 1, v 0, c 0 at x Pe, and the periodic conditions u,c x,0,t u,c x,pe,t. 10 The axial boundary conditions 9 state that the system is long enough not to disturb the concentration or velocity at the inlet or outlet. The simulations will cease to be characteristic of an unbounded system for times large enough that the front has reached the boundary. As an initial condition, we take u 1, v 0 for all x,y, 11 c c 0 x,y. 12 We thus consider a constant linear velocity as an initial velocity. For the concentration, all our simulations start from a step function between c 1 and c 0 at Pe /4 and the inverse step function at 3Pe /4 with a white noise 5% in amplitude added at the left front. The periodic extension is used in order to avoid Gibbs phenomena associated with our choice of spectral methods. 2 Following the approach of the companion paper, we switch to a moving frame, introduce the stream function (x,y) such that u / y, v / x and obtain the final equations 2 R c x x c y y c y F x x F y y F y, 13 c t c x y c y x 2 c, 14 where as before F cos qy cos x t. 15 The objective of the present paper is to study the nonlinear behavior of the solutions to Eqs. 13 and 14 for various values of the parameters of the problem. B. Parametrization of the problem As we have seen, the Peclet number Pe and the aspect ratio A are the new geometric parameters, defined above. These augment the parameters appearing in the basic equations themselves, namely, R the log of the viscosity ratio, the amplitude of F, and the two wave numbers q and in the transverse and longitudinal direction, respectively. The viscous fingers that develop along the transverse direction are characterized by a wave number k. In a system of width Pe, the viscous instability developing with a given wave number k will feature m fingers across Pe, where m is related to k as: k 2 m, m 0,1,2,3,4, Pe The wavelength of the fingers is then simply given by Pe/m. The permeability field on the other hand is spatially periodic and characterized by the transverse wave number q and the longitudinal wave number. To facilitate the discussion of our numerical results, it is useful to introduce the numbers n x and n y related to and q as 2 Pe n x, n x 0,1,2,3,4,..., 17 q 2 Pe n y, n y 0,1,2,3,4, The integers n x and n y relate simply to the number of cosine extensions of the permeability along Pe and Pe, respectively. In our simulations, we will consider two different types of heterogeneous porous media: layered systems n x 0, n y varying and checkerboards n x and n y both varying. Finally, let us note that as our equations pertain to a moving frame, the permeability field is also a function of time. A given location in the system will feel a maximum of F at given times separated by an interval t Pe /n x. The inten-

3 A. De Witt and G. M. Homsy: Viscous fingering in porous media. II 9621 sity of the spatial variation is controlled by the variance. The problem we are considering is hence spanned by a fivedimensional parameter space determined by the values of R, Pe,, and q, or equivalently n y,n x. The parameter A moreover fixes the maximum time accessible in a numerical simulation. III. NUMERICAL METHOD To numerically integrate Eqs. 13 and 14 we follow the numerical scheme described in detail in Ref. 2 as adapted to heterogeneous media in Ref. 3. It consists of a Fourier spectral method using periodic boundary conditions in both longitudinal and transverse directions. As in previous simulations of viscous fingering problems using this method, we work with the full cosine expansion of the displacement front. When the mobility ratio is unfavorable, the left interface develops a viscous fingering instability while the right interface is stable. The number of Fourier modes we use varies depending on the size of the system. We always take a ratio of 4 between the length in one direction and the number of modes used to discretize the dynamics along this direction. As an example, a simulation of a system of size Pe Pe is based on the use of modes. We moreover typically use a time step equal to 0.1. To validate our code, we have compared the growth rates measured at a time t 10 with those predicted by the linear stability analysis of Tan and Homsy, 4 with very good agreement, within 2% being typical. IV. RESULTS A. Characterization of nonlinear fingering We have conducted a large number of simulations, varying several of the relevant parameters. Before discussing the results, it is helpful to review some of the features of nonlinear viscous fingering in homogeneous media, and some of the quantities used to characterize regimes and mechanisms of fingering. Of course, one of the more powerful methods of displaying the results is to plot the concentration field as either contour or density greyscale plots. We give several such plots throughout the paper, with all density plots showing the domain Pe /2 Pe or 5Pe /8 Pe depending on how far the fingers have evolved toward the periodic extension. It has also proven advantageous to develop other, lower dimensional measures of fingering behavior. One of the more useful of these is the transversely averaged concentration profile. Such profiles are further characterized by the length of the mixing zone, defined as the distance between c 0.01 and c 0.99, and its behavior as a function of time. It is well-known that if the displacement is dispersive in nature, the mixing zone grows as t 1/2. This growth law holds at long times and long distances for either stable or neutral displacements in both homogeneous and heterogeneous media. The coefficient of proportionality is a measure of the axial dispersion coefficient of the system. Accordingly, we refer to this behavior as dispersive. On the other hand, it is equally well-known that viscous fingering in homogeneous media results in a law in which the mixing zone is linear in t. We refer to this as convective behavior. Thus the growth law is a signature of the mechanism of front propagation, and the parametric dependence of the growth rate in the dispersive or convective regimes is a measure of the importance of heterogeneity or viscous fingering, respectively. Other aspects of the parametric and time dependence of the transversely averaged concentration are related to finger dynamics, tip splitting, and inverse cascades that are important in interpreting the growth law of the mixing layer. It is known that in homogeneous systems, there exists for a given R a critical Pe above which propagating fingers are continuously unstable to a tip-splitting instability associated with the straining flow near the tip, as discovered and discussed in Ref. 2 and studied in more detail by Zimmerman and Homsy. 5,6 The tip splitting gives a counterbalance to the tendency for dispersion to continuously coarsen the length scales of the fingering pattern, and provides a mechanism for an inverse cascade and a statistically stationary state of continuously merging and splitting fingers. For Pe Pe c, no tip splitting occurs while merging, fading, etc., tend to diminish the initial number of fingers. The asymptotic dynamics then always consists of one single finger, and the mixing length grows dispersively. On the other hand, for Pe Pe c, tip splittings can balance merging of fingers and several fingers may coexist in the asymptotic dynamics. The wavelength of these asymptotic fingers is fixed by the nonlinear dynamics of the system and hence is not related to the wavelength predicted by the linear stability analysis. Furthermore the growth law remains convective for very long times, as discussed in Refs. 2,5,6. For further reference, for a log viscosity ratio of 3, the relevant Pe c is approximately equal to 250, 2 and is only weakly dependent on R for R greater than 3. With this behavior for homogeneous systems established, we refer to simulations below Pe c as small systems and those above Pe c as large systems. Furthermore, we fix R at either zero or 3.0 and study first layered systems, i.e., systems for which the permeability heterogeneities are all in the flow direction (n x 0) and determine the fingering dynamics in that case as a function of. We then follow with a study of the case when both n x and n y are nonzero, which we term the checkerboard case. B. Layered systems We first consider the case of a stable displacement with R 0 and Pe 256 starting the simulation with a jump condition of concentration at time t 0. In a homogeneous porous medium, the stable front remains flat and spreads in the axial direction according to the analytical solution c erf 2 t. x 19 Thus, the dispersive behavior in the homogeneous, viscously stable case is due entirely to the intrinsic dispersion characteristics of the medium. In presence of a layered permeability field, however, the viscously stable front becomes distorted adopting the transversal periodicity of F as seen in Fig. 1. In

4 9622 A. De Witt and G. M. Homsy: Viscous fingering in porous media. II FIG. 1. Viscously stable front (R 0) deformed by a layered permeability field with 1, n y 2, and Pe 256, A 2, t 50. this figure and all other density plots in the paper, we plot the density of the concentration between c 0 white and c 1 black. Figure 2 shows the mixing length versus time for both the homogeneous and layered cases, for both the neutral case R 0 treated in this subsection and the unstable cases treated below. Accordingly, the results are indexed with (a,b) giving the values of (R, ). Figures 2 a and 2 b use t and t 1/2, respectively. In this case, time is running up to t 500, which is long enough to see asymptotic dynamics at play. But we note that especially for systems with larger Pe, mixing lengths must sometimes be studied on longer times to unequivocally establish the asymptotic dynamics. Here the results for 0, clearly show the dispersive behavior. This is to be expected from theories appropriate to the case R 0, nonzero, leading to an effective axial dispersion due to the presence of heterogeneities. 7 We next study the case of unstable displacements. In layered systems (n x 0), we deal then with a problem characterized by four parameters R, Pe,, and n y. Having fixed R at 3.0, we now study successively the influence of the spatial dependence of F on these two regimes by analyzing first small systems Pe 128 and then large systems Pe Viscously unstable small systems FIG. 2. Mixing lengths vs time a t and b t 1/2 for Pe 128, A 8 ina layered system with n y 2. The numbers in parenthesis feature the couple (R, ). We consider a small system with Pe 128. In the case of a homogeneous system, the asymptotic dynamics is indeed one single finger as can be seen in Fig. 3 a. At early times, the wavelength predicted by the linear stability analysis expresses itself, but the two fingers we see in Fig. 3 a at time t 200 quickly merge in one single finger. These dynamics are expressed by a switch in the growth of the mixing length from a diffusive to a convective behavior as can be seen in Fig. 2 for R 3, 0. This convective behavior is, of course, only intermediate, as dispersion acting over long times and long distances in the absence of tip splitting will ultimately result in the reestablishment of the dispersive behavior. Figure 3 shows the contrasting behavior exhibited in the presence of a layering of the permeability. The asymptotic dynamics maintains the periodicity of F as shown for example in Fig. 3 b for 0.1. The mixing length grows dispersively in time Fig. 2, with no hint of the convective regime found in the homogeneous case. This phenomenon relates to the so-called channeling regime well known in pe- FIG. 3. Viscously unstable fronts (R 3) in a system with Pe 128, A 8 shown at successive times t 200, 400, 600. a 0: the asymptotic dynamics is one single finger. b 0.1, n y 2: two fingers develop along the periodicity of F.

5 A. De Witt and G. M. Homsy: Viscous fingering in porous media. II 9623 FIG. 4. Viscously unstable front (R 3) in a homogeneous system 0 with Pe 1024, A 8 shown at successive times t 200, 800, 1400, troleum engineering. It is remarkable that the dispersive behavior remains even for as small as 0.01 see Fig. 2. This behavior is robust: In all the small systems we have studied, the dynamics observed in layered porous media were no longer one single asymptotic finger evolving in a convective manner but rather several fingers following the channeling paths of higher permeability. The layering dominates the flow patterns, with the result that the channeling regime is established for small. Of particular note is the increase in the value of the effective axial dispersion coefficient, related to the slope of the straight line fits to the data in Fig. 2 b, indicating that the viscously unstable situation significantly amplifies the dispersion due to heterogeneity. We interpret these results as being related to the wellknown phenomena of Taylor dispersion. It is known that the flow-averaged concentration profiles of a passive scalar in systems characterized by both velocity gradients and transverse diffusion exhibit one-dimensional dispersive behavior at long times and long distances. This results from the combined physical mechanisms of axial convection of the concentration field due to the velocity gradient, followed by the homogenization of the concentration field in the transverse direction due to weak transverse diffusion. With this perspective, it is clear that Taylor dispersion is the appropriate explanation of our results, with the permeability field providing the axial convection, and the intrinsic dispersion providing the transverse diffusive mechanism. 2. Viscously unstable large systems In this subsection, we treat Pe 1024, i.e., a Pe far larger than Pe c for tip splitting in the homogeneous case, shown in Fig. 4 for reference. At early time, about 20 fingers develop but they soon merge or fade because of nonlinear FIG. 5. Mixing lengths vs time for Pe 1024, A 8, R 3 in a layered system. The numbers in parenthesis feature the couple (,n y ). In a homogeneous medium with (,n y ) (0,0), the mixing length grows convectively in time, i.e., grows proportional to t. If 0.1, we get a channeling regime without any tip splittings and the mixing length obeys a dispersive law, i.e., grows as t 1/2.If 0.01 however, tip splittings may occur and the dynamics remains convective. couplings. 2,5 At time t 800, we see that more or less 6 fingers have survived and that some of them start splitting following a process already well understood. 2,5 At time t 2000, the number of fingers has decreased to about 3. The overall dynamics is still convective as can be seen in Fig. 5, which displays the mixing length versus time for Pe 1024, R 3, and for different values of the couple (,n y ) to be discussed below. The curve for 0 shows that indeed the mixing length grows fairly linearly in time. We now consider the effect of increasing. We first take the very low value of 0.01, together with Pe 1024 and n y 5. Figure 6 shows the finger dynamics and the mixing length is shown in Fig. 5. It is clear that the heterogeneities do not have a significant effect on the mechanisms of nonlinear fingering. Merging, fading, and tip splitting phenomena occur as in homogeneous systems. Accordingly, the mixing length evolves also convectively as shown in Fig. 5, but with a higher growth rate than in the homogeneous case. A comparison between Figs. 6 and 4 shows that the preferred wavelength when 0.01 is that imposed by the viscous instability, and that there is a favorable interaction between the layering and the tip spitting, leading to an increased growth rate, similar to that hypothesized by Tan and Homsy. 3 If the permeability field is now layered, but with a larger value of 0.1 and n y 8, we recover the channeling regime, as shown in Fig. 7. We find that the number of fingers is dictated by the periodicity of F, as it was in the viscously neutral case, and that the mixing length is dispersive, as shown in Fig. 5. It should be noted that the effect of heterogeneities is to completely eliminate tip splitting. The effect

6 9624 A. De Witt and G. M. Homsy: Viscous fingering in porous media. II FIG. 6. Viscously unstable front (R 3) in a layered system with Pe 1024, A 8, 0.01, and n y 5 shown at successive times t 200, 800, 1400, Although the permeability field is layered, tip splittings occur and the prefered wavelength is the one imposed by the viscous instability. of strong layering and channeling is to reduce the effective Pe within a layer by an amount 1/n y, which for the cases studied, lowers the effective Pe below Pe c. These and other parametric studies not shown here suggest that in the presence of layered heterogeneities of the permeability field, a transition occurs between a convective behavior analogous to that occurring in homogeneous media FIG. 7. Viscously unstable front (R 3) in a layered system with Pe 1024, A 8, 0.1, and n y 8 shown at successive times t 200, 800, 1400, The permeability field imposes a wavelength that is not the one prefered by the viscous instability. No tip splittings occur. FIG. 8. Slope L d of the linear behavior of the mixing length vs t for different values of n y. The fixed parameters of the different runs are Pe 1024, A 2, The slope is measured in the convective regime between t 250 and t 400. and a channeling dispersive regime dominated by the heterogeneities. The transition between regimes occurs for a critical value c, dependent on both Pe and n y, that is apparently an increasing function of Pe. For Pe 1024, c lies between 0.01 and 0.1 as can be inferred from our results. This remains an issue for further study. Let us now try to understand in some more detail the convective dynamics when 0.01 as the vertical scale n y is varied. We have plotted the slope L d of the linear dependence of the mixing length versus t versus different values of n y Fig. 8. This slope is maximum when n y 6, i.e., when the periodicity of F is commensurate with the naturally preferred number of nonlinear fingering occurring in homogeneous systems. This idea behind this resonance phenomenon was already described by Tan and Homsy, 2 in which a resonant interaction between the scales of fingering and heterogeneity was hypothesized. The advantage of our periodic model is that these resonances may be convincingly exposed as indicated in Fig. 8. The increase of L d between n y 8 and 10 seen in Fig. 8 cannot be inferred to continue for higher n y. Indeed in our simulations, we see that above n y 10, the mixing length starts to behave dispersively on the times looked at because the value of the effective Pe within one layer has decreased below the critical Pe c for tip splittings. It is important to remark that even in the convective regime of layered systems, there is still a competition between the wavelength preferred by the viscous instability and that of the permeability field. In the situation of Fig. 6, i.e., with 0.01, the wavelength of the fingers was always the same as in a homogeneous system. In that case n y 5 and at early times, the viscous instability can easily accommodate 4 fingers in each layer of F so that we roughly see the 20 fingers preferred by the viscous instability. This is not al-

7 A. De Witt and G. M. Homsy: Viscous fingering in porous media. II 9625 FIG. 9. Viscously unstable front (R 3) in a layered system with Pe 1024, A 2, 0.01, and n y 9 shown at successive times t 400, 800, 1200, At early times, the wavelength follows the periodicity of F before the processes of merging and tip splittings take over again. ways the case and in some situations, transients may develop that show a number of fingers whose scale is dictated by the permeability. An example of this is shown in Fig. 9 with 0.01 but here n y 9. As in the situation depicted in Fig. 6, the viscous instability would like to develop around 20 fingers. However, there is a competition, as 9 fingers develop at early times along the nine layers of F. This leads then to a frustration, since the wavelengths of the permeability field and the intrinsic fingering are incommensurate. To accommodate the wavelength of the viscous instability, some of the fingers try to split see Fig. 9 a. However, in this range of parameters and for early times, the permeability field dominates the pattern, imposing 9 fingers. It is only at much later time that the process of merging, fading, and splittings assert FIG. 10. Viscously unstable front (R 3) in a layered system with Pe 1024, A 2, 0.1, and n y 2 shown at successive times t 200, 300, 400, 500. Fingers can develop and interact inside one layer of F. FIG. 11. Viscously unstable front (R 3) in a checkerboard system with Pe 256, A 2, 1, n x 4, and n y 3 shown at a time t 200. The three panels feature from up to down the stream function, the concentration, and the permeability field F. The viscous fingers follow the periodicity of F and the tip splittings are periodically induced by the local changes of F. themselves and that the wavelength returns to that appropriate to the viscous instability in a homogeneous system. In the convective regime, therefore, a subtle interplay between the intrinsic lengths of the viscous instability and that of the permeability field may lead to complex intermediate dynamics. Another interesting situation occurs when splitting and merging mechanisms occur inside a channeling regime as seen in Fig. 10. Here 0.1 as for the dispersive case of Fig. 7. However, n y 2 and several fingers can develop inside one channeling band of F. The usual nonlinear interactions between fingers take play inside a band as long as the effective Pe inside the layer is above Pe c. C. Checkerboards We now study the effect of permeability variation in the flow direction, characterized by n x, on the behaviors observed thus far. In order to fix ideas, we begin with the case of large 1.0, in which the effect of the heterogeneities in driving the cellular flow discussed in Paper I may be easily visualized. Figure 11 shows a viscously unstable system

8 9626 A. De Witt and G. M. Homsy: Viscous fingering in porous media. II FIG. 12. Viscously unstable front (R 3) in a checkerboard system with Pe 1024, A 8, 0.1, n x 10, and n y 8 shown at successive times t 200, 800, 1400, All parameters are the same as in Fig. 7 except that here there is an axial modulation of F. Tip splittings occur but here they are related to the encountering of a change in permeability along the x direction and not to the viscous dynamics. for the case of a moderately wide system and a large heterogeneity. The three panels show the stream function, the concentration field, and the underlying permeability field. It is clear that if the cellular flow is sufficiently strong and dispersion sufficiently weak, the concentration field will be convected in that flow and the permeability field will dominate what is seen. We are interested in regions of parameter space in which neither fingering nor heterogeneity completely dominates the other and new mechanisms that express themselves under these circumstances. Our study of checkerboards focuses on large systems Pe 1024, which admit a range of nonlinear finger dynamics. The case of low Pe is uninteresting, since the finger dynamics there are simple. Consider the case 0.1, for which the layered systems are strongly channeled and dispersive. In Fig. 12 we show the finger dynamics for the case n x 10 and n y 8 at successive times. Although the fingering starts highly layered, tip splitting occurs, in strong contrast to the purely layered case shown in Fig. 7. We infer that the splitting in this case is driven entirely by the encounters between the front and regions of alternatively high and low permeability. Axial variation can completely eliminate the layered, channeling regime. This is illustrated in Fig. 13, in which the layered system (n x 0) is contrasted with two checkerboard cases n x 8 and 10 at a time t 750. As we have seen, when n x 0 channeling occurs with the periodicity imposed by F. For the intermediate value of n x 8, the axial modulation of F destroy the channeling phenomenon allowing tip splittings to come back into the play. At higher n x 10, the frequency of encountering permeability changes gets too high and the FIG. 13. Viscously unstable fronts (R 3) in checkerboard systems with Pe 1024, A 2, 0.1, n y 6, and varying n x shown at a time t 750. From top to bottom: n x 0, 8 and 10. When n x 0, i.e., in a layered system, channeling occurs with the periodicity imposed by F. For an intermediate value of n x 8, the axial modulation of F destroy the channeling phenomenon allowing tip splittings to come back into the play. At higher n x 10, the frequency of encountering permeability changes gets to high and the ability of the checkerboard to induce tip splittings is reduced. ability of the checkerboard to induce tip splittings is reduced. This strongly suggests a resonance in which the frequency of encounter with axial variations is commensurate with the characteristic diffusion time, similar to the local resonances in found in Paper I. The permeability-driven tip splitting persists for very large. Shown in Fig. 14 is the case 1 and n y 6ata time t 550 for two different values of n x, n x 8 and 10. We find that when is high, the periodicity of F imposes a regular array of tip splitting phenomena. This is also expressed in the corresponding results for the mixing length, shown in Fig. 15, in which the irregular growth of the mixing layer is suggestive of an oscillation superposed on a linear trend, with a period associated with n x. In Fig and the (a,b) value refer to (,n x ) with n y 6. The frequency of encountering the axial changes of F are clearly superimposed on the linear convective growth. For a smaller 0.1 and longer times as studied in Fig. 12, the corresponding

9 A. De Witt and G. M. Homsy: Viscous fingering in porous media. II 9627 FIG. 14. Viscously unstable fronts (R 3) in checkerboard systems with Pe 1024, A 2, 1, and n y 6 shown at a time t 550. a n x 8. b n x 10. When is high, the periodicity of F imposes a regular array of tip splittings phenomena. mixing length shown in Fig. 16 can begin to behave dispersively as long as the channeling regime hold but after a while tip splittings come back into the play and the mixing length goes back to a linear trend with small oscillations. V. COMPARISON WITH EXPERIMENTS AND CONCLUSIONS As a complement to Paper I, we have numerically studied the influence of spatially periodic heterogeneities of the permeability on the viscous fingering instability. The main difference with regard to viscous fingering phenomena in homogeneous systems is the possibility of a channeling regime in layered systems. This channeling occurs through a FIG. 15. Mixing lengths vs time for Pe 1024, A 2, R 3 in the checkerboard systems of Fig. 16, i.e., with n y 6. The numbers in parenthesis feature the parameters (,n x ). In the checkerboard systems, the mixing length grows irregularly. The n x 8 curve clearly features a periodic modulation of the mixing length due to the axial periodicity of F. FIG. 16. Mixing lengths vs time for Pe 1024, A 8, R 3 in a checkerboard system. The numbers in parenthesis feature the parameters (,n x,n y ). In a homogeneous medium with (,n x,n y ) (0,0,0), the mixing length grows convectively in time, i.e., grows proportional to t. If 0.1 and n x 0, we get a channeling regime without any tip splittings and the mixing length obeys a dispersive law i.e. grows as t 1/2.Ifn x 10 however, tip splittings may occur and the mixing length has a more complicated evolution. Taylor dispersion mechanism and is characterized by a dispersive growth of the mixing length versus time. In checkerboard systems, this channeling is destroyed by the reappearance of tip splittings induced by the axial spatial variation of the permeability and a convective growth of the mixing length is recovered. Experiments in two-dimensional porous media with controlled and well-characterized permeability variations were conducted by Brock and Orr. 8 The experimental apparatus consisted of so-called bead packs held between solid transparent plates in a Hele Shaw arrangement. Two bead sizes were used in order to accomplish a permeability contrast of a factor of 4.0, corresponding to 0.65, i.e., a higher range than that covered in the present study. The viscosity ratios were either 40 or 80, corresponding to R , i.e., slightly higher than the value considered here. Four types of experiments were conducted and the fingering pattern was observed and analyzed by digital image techniques. Unfortunately, no average profiles, analogous to Figs. 2 and 5, were reported. The main difference between our simulations and these experiments is in the dispersion characteristics of the medium, being isotropic in the simulations and anisotropic in the experiments. The axial Peclet number was approximately 400, i.e., above the critical value for tip splitting, while the transverse Peclet number was approximately Four different permeability fields were constructed under well-characterized conditions: i a homogeneous system; ii a stratified, two-layer system; iii a three-layer, channeled system, and; iv a random block system. The results for the homogeneous system do not concern us here.

10 9628 A. De Witt and G. M. Homsy: Viscous fingering in porous media. II Their results, while limited and in a slightly different parameter regime, are in complete qualitative agreement with the simulations reported here. In the two-layer system, case ii, no flow took place in the less permeable zone and fingering was observed in the more permeable zone in a fashion consistent with behavior in homogeneous systems, analogous to the behavior shown in Fig. 10. In the three-layer channeled system, case iii, the more permeable zone was so thin as to eliminate fingering and contained all the flow, in manner completely analogous to a single wavelength of Fig. 7. Finally, in the random checkerboard system, case iv, while the flow followed the more permeable path in the mean, tip splitting was stimulated by the heterogeneity in a fashion completely analogous to that described in Fig. 12. Thus the main physical phenomena described in this study, suppression of tip splitting by small wavelength heterogeneities, quasihomogeneous fingering within more permeable zones, and the stimulation of splitting by axial permeability variations, are seen in both the simulations and experiments. ACKNOWLEDGMENTS A.D. acknowledges Alain Noullez and Hans Schepers for their help in the numerics and the F.N.R.S. Belgium and the Petroleum Research Fund for financial support through Grant No. PRF AC9. G.M.H. acknowledges support from the U.S. Department of Energy, Office of Basic Energy Sciences. 1 A. De Wit and C. M. Homsy, J. Chem. Phys. 107, , preceding paper. 2 C. T. Tan and G. M. Homsy, Phys. Fluids 31, C.-T. Tan and G. M. Homsy, Phys. Fluids A 4, C. T. Tan and G. M. Homsy, Phys. Fluids 29, W. B. Zimmerman and G. M. Homsy, Phys. Fluids A 3, W. B. Zimmerman and G. M. Homsy, Phys. Fluids A 4, L. W. Gelhar and C. L. Axness, Water Resour. Res. 19, D. C. Brock and F. M. Orr, Jr., Proceedings of the 66th SPE Meeting, Dallas, TX, 1991 unpublished, SPE Paper No