Abstract. 1 Introduction

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1 On the Use of Pore-Scale Computational Models for Two-Phase Porous-media Flows M.A. Celia, P.C Reeves, H.K. Dahle Environmental Engineering and Water Resources Program, Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ USA karst.princeton. edu Abstract Pore-scale network models provide insights into two-fluid porous-media flow systems. Their ability to simulate laboratory experiments allows continuumscale constitutive relationships to be derived from these models. Because all fluid-fluid interfaces are modeled explicitly, quantities that are difficult or impossible to measure, such as interfacial areas, can be calculated within the model. This allows new constitutive relationships to be investigated, and theoretical conjectures to be tested. In one such test, a functional dependence between interfacial area, capillary pressure, and saturation is observed, but the extended interfacial area function still exhibits hysteretic behavior. Additional calculations involving contact-line length, mass-transfer phemonena, and dynamics of interface motion allow additional relationships to be developed and tested. 1 Introduction Traditional models for two-phase flow in porous media require several important constitutive relationships as input. These include the relationship between capillary pressure (Pc) and saturation (S), and between relative permeability (kr) and saturation. These relationships are generally nonlinear and hysteretic. However, they can be measured, and

2 398 Computer Methods in Water Resources XII when they are provided to the traditional set of flow equations based on balance equations for each phase, the system can be solved numerically. While the traditional descriptions of flow in porous media have shown many applications and benefits, open questions remain regarding the form of the governing equations, and the kinds of constitutive relationships that are necessary to describe properly the multi-phase flow system. For example, recent work by Hassanizadeh and Gray (see, for example, [7]) has highlighted the potential importance of interfacial areas in the mathematical description of multi-phase flow systems. One obvious difficulty with such a theory is that interfacial areas, especially fluid-fluid interfacial areas, are very difficult to measure. Therefore some alternate approaches to determine interfacial areas must be found. In addition to the role of interfacial areas, other questions related to constitutive relationships may also be raised. For example, debate continues regarding the role of dynamics in the capillary pressure - saturation relationship. Some independent means to assess the influence of dynamics in capillary displacements experiments is needed. These questions may be addressed through development and application of pore-scale models for two-fluid porous media systems. Such models represent the pore-space explicitly, and then track each fluidfluid interface within a network of tens- to hundreds-of-thousands of pores. Continuum-scale variables are then computed by volume averaging, and relationships between those continuum variables may be determined. In the following sections, we outline the essential elements of several porescale models we have developed, and demonstrate their utility by focusing on a specific question regarding interfacial area. We also outline other areas of application, and discuss future applications. 2 Pore-Scale Models Most pore-scale models are based on idealizations of pore-space geometry, so that analytical expressions for the movement of fluid-fluid interfaces within the pore space may be used. One of the earliest examples of such a simplified geometry is the bundle-of-tubes model, which involves a series of parallel, non-intersecting tubes of different radii. Bundles-of-tubes models fail to predict several important features of two-phase flow. Beginning with the pioneering work of Fatt [5], pore-scale models have been developed with an underlying interconnected pore structure, often corresponding to simple lattice structures. Each component of the lattice has a sufficiently simple geometric shape to allow analytical expressions

3 Computer Methods in Water Resources XII 399 to be used for meniscus geometry. This allows physically-based displacement rules to be formulated. Application of such expressions allows each fluid-fluid interface to be tracked through the network. A typical network configuration is a cubic lattice, with spherical pore bodies located at each junction, and cylindrical or bi-conical pore throats connecting adjacent pore bodies. The radii of the pore bodies and pore throats are assigned from separate probability distributions, often with some cross-correlation between the throats and bodies (see, for example, [8], [9], [10]). Boundary conditions are often imposed to simulate physical experiments. Celia et al. [2] provide a review of recent advances in pore-scale modeling. 2.1 Quasi-Static Displacements By analogy to quasi-static physical experiments, such as typical pressurecell measurements of the Pc-S relationship, pore-scale models are often based only on the determination of stable interface configurations for an imposed capillary pressure. This is done through explicit tracking of all individual menisci in the system. Each meniscus is examined for mechanical stability using the Young-Laplace equation [4]. If an interface is unstable, it is displaced to the next pore element. This procedure is continued until all menisci are found to be stable at the current capillary pressure. Given the final, equilibrium positions of the menisci, the percentage of the pore-space occupied by wetting fluid may be calculated, thereby providing a measure of fluid saturation. In addition, because the geometry is known explicitly, fluid-fluid and fluid-solid interfacial areas may be computed, as well as fluid-fluid-solid contact-line lengths. Furthermore, given the pore-by-pore fluid occupancy, resistance to flow across the network may be calculated for each fluid phase, thereby providing direct estimates of relative permeabilities (kr). The ability of quasi-static pore-scale network models to predict Pc-S and kr-s relationships for unconsolidated media has been tested recently by [1], [6], [9], and [10], among others. 2.2 Dynamic Models While quasi-static models are consistent with many of the experimental methods used to determine Pc-S relationships, they do not capture any dynamic effects of the displacement process. A number of models have appeared in the literature that allow dynamics to be incorporated into the

4 400 Computer Methods in Water Resources XII mathematical description (see, for example, [3]). These models have focused on ganglia mobilization by viscous forces, and viscous effects on relative permeabilities, among other things. We have been interested in possible dynamic effects on Pc-S relationships, and have developed a dynamic model based on networks with cylindrical pore throats. The model uses an iterative solution scheme to solve for interface dynamics over each given time step. For each time step, a pressure is determined at each pore junction (pore body), and the flow-rates in each of the pore throats is determined from a modified Poiseuille equation accounting for interfaces. Conservation of volumetric flow at each junction leads to a coupled set of nonlinear equations for the unknown pressures. A Newtonbased iterative scheme is used to solve for the pressures and associated interface locations. Trapping mechanisms are included, leading to residual saturations for each of the fluid phases. 2.3 Investigation of New Constitutive Relations Pore-scale network models allow traditional continuum-scale variables to be computed, such as fluid saturations and relative permeabilities, and relationships between these variables to be investigated. However, porescale models also allow non-traditional continuum variables, such as interfacial areas and contact-line lengths, to be computed. Associated constitutive relationships for these variables can be derived and, in some cases, tested against theoretical conjectures. For example, [10] and [11] used a quasi-static pore-scale network model based on a cubic lattice and bi-conical throats to calculate the traditional Pc-S relationship as well as the non-traditional relationship between fluid-fluid interfacial area and saturation. Following [10], Figure 1 shows the complete hysteresis loop for Pc-S, as well as a series of drainage scanning curves. Figure 2 shows the associated curves forfluid-fluidinterfacial area (A) as a function of saturation. Both sets of curves show strong hysteresis and complex functional dependencies, especially the area versus saturation curve. Recently, Hassanizadeh and Gray [7] have used thermodynamics to argue that capillary pressure should be a function of both saturation and interfacial area. This has motivated us to plot interfacial area as a function of saturation and capillary pressure in Figure 3. That figure shows a smooth, well-behaved surface for interfacial area, as opposed to the highly complex curves given in Figure 2. While the curvature in the surface indicates that PC is not a unique function of S and A, it suggests

5 Computer Methods in Water Resources XII 401 strongly that fluid-fluid interfacial area should be functionally dependent on both saturation and capillary pressure. The surface shown in Figure 3 is derived from drainage scanning curves. A similar surface may be derived for imbibition scanning curves. The associated Pc-S and A-S curves are not shown here (see [10] for the detailed plots), but Figure 4 shows A as a function of PC and S for the imbibition curves. The imbibition surface has a similar form to the drainage surface. A conjecture of Gray and Hassanizadeh is that these surfaces should be identical, because hysteresis in the traditional Pc-S relationships results from the incomplete functional dependence, and inclusion of the interfacial area provides the appropriate functional dependencies. To examine the differences between the two surfaces, we show several cuts through the surface (at fixed values of PC) in Figure 5. This figure shows that, while the surfaces have similar shapes, they are not identical, differing by up to 20%. This observation appears to be consistent over a number of different networks that we have examined. 3 Discussion and Conclusions Pore-scale models may be used to provide insights into, and quantitative estimates of, traditional constitutive relationships such as PC vs. S and kr vs. S. However, one of the most important attributes of pore-scale network models is their ability to provide quantitative estimates for properties that are difficult or impossible to measure, such as interfacial areas. Among other things, this allows theoretical conjectures to be tested and possible new functional dependencies to be investigated. This applies not only to interfacial areas but also to contact-line length, and to dynamic effects in constitutive relationships. These new functional dependencies provide opportunities for improved descriptions of complex flow systems. References [1] Bryant, S.L. and M. Blunt, Phys. Rev. A, 46(4), , [2] Celia, M.A., P.C. Reeves, and L.A. Ferrand, Reviews of Geophysics (Supplement), , [3] Constantinides, G. and A.C. Payatakes, AIChEJ., 42(2), , [4] Dullien, F.A.L., Porous Media: Fluid Transport and Pore Structure, Second Edition, Academic Press, [5] Fatt, I., Petr. Trans. AIME, 207, , 1956.

6 402 Computer Methods in Water Resources XII [6] Fischer, U. and M.A. Celia, "Prediction of Relative and Absolute Permeabilities for Gas and Water from Soil Water Retention Curves using a Pore-Scale Network Model, under review, Water Res. Research, [7] Hassanizadeh, S.M. and W.G. Gray, Water Resources Research, 29(10), , [8] Jerauld, G.R. and S. Salter, Transport in Porous Media, 5, , [9] Rajaram, H., L.A. Ferrand, and M.A. Celia, Water Resources Research, i?(l), 43-52, [10] Reeves, P.C., PhD Dissertation, Princeton University, [11] Reeves, P.C. and M.A. Celia, Water Resources Research, 32(8), , Acknowledgements This work was supported in part by the National Science Foundation under Grant EAR We thank Lin Ferrand, Ulrich Fischer, William Gray, Rudy Held, Majid Hassanizadeh, and Carlo Monetmagno for helpful discussions and comments regarding this work. CUD C '5 c G "c3 5 cd 15

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