Homogenization and Porous Media

Roughly speaking, homogenization is a mathematical method that allows us to “upscale” differential equations. This method not only offers formulas for up-scaling but also provides tools for producing rigorous mathematical convergence proofs.

Recent progress in understanding the effective transport properties of percolation models for porous and conducting random media is reviewed. Both lattice and continuum models are studied. First, we consider the random flow network in ℤ ^N , where the pipes of the network are open with probability p and closed with probability 1 — p. Near the percolation threshold p _c ,the effective permeability κ*(p) ~ (p — p _c )^e, p → p⁺_c, where e is the permeability critical exponent. In the limit of low Reynolds number flow, this model is equivalent to a corresponding random resistor network. Here we discuss recent results for the resistor network problem which yield the inequalities 1 ≤ e ≤ 2, N = 2, 3 and 2 ≤ e ≤ 3, N ≥ 4, assuming a hierarchical nodelink-blob (NLB) structure for the backbone near p _c . The upper bound t = 2 in N = 3 virtually coincides with a number of recent numerical estimates. Secondly, we consider problems of transport in porous and conducting media with broad distribution in the local properties, which are often encountered. Here we discuss a continuum percolation model for such media, which is exactly solvable for the effective transport properties in the high disorder limit. The model represents such systems as fluid flowing through consolidated granular media and fractured rocks, as well as electrical conduction in matrix-particle composites near critical volume fractions. Moreover, the results for the model rigorously establish the widely used Ambegaokar, Halperin, and Langer critical path analysis [AHL71].

This section is devoted to the derivation of Darcy’s law for an incompressible viscous fluid flowing in a porous medium. Starting from the steady Stokes equations in a periodic porous medium, with a no-slip (Dirichlet) boundary condition on the solid pores, Darcy’s law is rigorously obtained by periodic homogenization using the two-scale convergence method. The assumption of the periodicity of the porous medium is by no means realistic, but it allows casting this problem in a very simple framework and proving theorems without too much effort. We denote by e the ratio of the period to the overall size of the porous medium. It is the small parameter of our asymptotic analysis because the pore size is usually much smaller than the characteristic length of the reservoir. The porous medium is contained in a domain Q, and its fluid part is denoted by TE. From a mathematical point of view, 513E is a periodically perforated domain, i.e., it has many small holes of size ε which represent solid obstacles that the fluid cannot penetrate.

In the preceding chapter, the flow of Newtonian fluids through porous media was discussed in detail. However, non-Newtonian fluids are extensively involved in a number of applied problems involving the production of oil and gas from underground reservoirs. There are at least two typical situations, the flow of heavy oils and enhanced oil recovery (EOR). In EOR applications, non-Newtonian fluids, such as low concentration polymer solutions are injected to increase the viscosity of agents that displace the oil. Similarly, many heavy oils behave as Bingham viscoplastic fluids.

In practical applications, the problem of modeling two-phase flow behavior is of greatest importance. In petroleum engineering the so-called enhanced oil recovery (EOR) process is based on displacing a fluid (oil) by another one (gas or water, for example). In soil science, water and contaminant or water and air are involved in many underground flows in the unsaturated zone. Domestic gas, before delivery, is kept in natural reservoirs and moved by water and a cushion gas.

Fluid flow in porous media was discussed in the preceding chapters. How to derive macro Darcy-type laws from micro laws, such as Stokes equations or its generalizations was demonstrated. It is natural to ask how one can use those results and apply them to miscible displacement problems. In soil physics or soil chemistry, e.g., it is of great importance to combine the homogenization techniques, described so far, with problems of diffusion, dispersion, and convection of chemical species that are transported in the fluid flowing in the pore space of a porous medium. In principle, it turns out that all the mathematical methods developed can easily be applied to such problems. In particular, there is no difficulty in taking chemical reactions into consideration, because, usually, these are described by undifferentiated terms in the differential equations. Therefore, they are basically unmodified during the homogenization process.

Porous materials, such as sand and crushed rock underground, are saturated with water which, under a local pressure gradient, migrates and transports energy through the material. The transport of heat in the presence of exterior forces, if the fluid is in motion, is called convection. Further examples of convection through porous media may be formed in man-made systems, such as fiber and granular insulations, the cores of nuclear reactors, or chemical industries.

In this chapter, we will investigate the acoustic macroscopic behavior of saturated porous media. The macroscopic quasi-static behavior will be deduced classically by neglecting inertial terms. The material of the porous matrix is elastic and the saturating fluid is viscous, Newtonian, and incompressible. The local structure is assumed periodic with a period Y of characteristic length 1 The characteristic length L at the macroscopic scale can be associated with the wave length λ. For simplicity, we will use L = λ2/π.

Macroscopic microstructure models of the dual-porosity type were introduced in several places earlier, such as Chapter 1, section 1.5, Chapter 3, section 3.4, and Chapter 9, section 9.3. Among other things, they model, the flow of fluids in highly fractured porous media; that is, media comprised of porous matrix rock divided into relatively small blocks by thin fractures [BZK60], [WR63], [Arb89a], [ADH90], [DA90], [ADH91], and [Arb93a]. Mathematically, dual-porosity models are a relatively complex system of partial differential equations in seven variables, (t, x, y). At first glance, it may not be apparent that there is any advantage to the macroscopic description versus the mesoscopic (i.e., the Darcy-scale description that explicitly models flow within the fractures and matrix). However, the dual-porosity model explicitly captures the length scales of the physical problem and is, thus, much easier to approximate computationally.