Filtration law for power-law fluids in anisotropic porous media

Abstract

This work is concerned with modelling steady-state slow flow of incompressible power-law fluids in porous media. The macroscopic filtration law is derived by upscaling the pore–scale description. The upscaling technique in use is the homogenisation method of multiple scales. Then, the filtration law is investigated by means of the theory of representation of isotropic tensor function of tensor arguments. The general form of the filtration law is given for isotropic, transversely isotropic, orthotropic and fully anisotropic systems.

Introduction

Knowledge of the filtration law for the flow of power-law fluids through porous media is of first importance in numerous engineering areas, e.g., petroleum engineering, paper manufacturing, composite manufacturing, etc. However, there have been very few theoretical investigations that focused on the general form of this law. Indeed, most of them aimed at proposing one-dimensional modified versions of Darcy's law in non-algebraic forms that are obtained on the basis of phenomenological considerations. Wissler [1] used a geometric similarity condition and the homogeneity condition (see (7) below) and obtained the following one-dimensional filtration law:

$$\text{q=k}{}^{\mathrm{\prime }}\text{Δp}\text{KL}{}^{\mathrm{1/n}}\text{,}$$

where q is the Darcy flux, Δp is the pressure drop across a porous sample of length L; k^′ and K are two parameters that are dependent on the exponent n. Larson [2] obtained the same filtration law (1) by assuming that the streamlines are independent of the volumetric flow rate. This latter property is actually a direct consequence of the homogeneity condition (7), see below. Slattery [3] derived the three-dimensional isotropic law by volume averaging, considering the homogeneity condition (7) and using a dimensional analysis (Buckingham–Pi theorem). The resulting filtration law can be expressed as follows:

$$\text{∇p+K}\text{v}\text{=0,}\text{K=K}{}_1\text{|∇p|}{}^{\mathrm{(n-1)/n}}\text{,}$$

where K₁ depends on n, and on a characteristic length of the porous system. Both filtration laws , have similar structures. When n=1, they reduce to Darcy's law. More recently, there have been several works that aimed at obtaining the filtration law by upscaling the pore–scale behaviour. Getachew et al. [4] applied the method of volume averaging to derive the macroscopic isotropic filtration law. Shah and Yortsos [5] and Bourgeat and Mikelić [6] used the homogenisation method of multiple scales. In [5], the filtration law is obtained in a formal way and its structure is investigated by using physical considerations at the macro-scale. In [6] the existence of the filtration law and the convergence of the expansions are rigorously demonstrated. Other references concerning the flow of a power-law fluid in a porous medium and its applications can be found in [7].

The present work is aimed towards rigorously deriving the most general form of the filtration law in systems of different types of anisotropy. The macroscopic filtration law is obtained by upscaling the pore–scale physical description, by using the homogenisation method of multiple scales.

Let us assume the medium to be locally characterised by a representative elementary volume (REV) whose size is O(l) (Fig. 1). Let us consider the physical description at the pore scale, i.e., within this REV. It consists of the momentum balance (3), the fluid constitutive equations (4), the incompressibility condition (5) in the pores $\text{Ω}{}_{\mathrm{p}}$ and of the no-slip condition (6) over the pore surface Γ:

$$\text{∂}\text{σ}{}_{\mathrm{ij}}\text{∂}\text{X}{}_{\mathrm{j}}\text{=0,}$$

$$\text{σ}{}_{\mathrm{ij}}\text{=−pI}{}_{\mathrm{ij}}\text{+τ}{}_{\mathrm{ij}}\text{,}\text{τ}{}_{\mathrm{ij}}\text{=μ}\text{1}\text{2}\text{D}{}_{\mathrm{pq}}\text{D}{}_{\mathrm{pq}}{}^{\mathrm{(n-1)/2}}\text{D}{}_{\mathrm{ij}}\text{,}$$

$$\text{∂}\text{v}{}_{\mathrm{i}}\text{∂}\text{X}{}_{\mathrm{i}}\text{=0}\text{in}\text{Ω}{}_{\mathrm{p}}\text{,}$$

$$\text{v}{}_{\mathrm{i}}\text{=0}\text{on}\text{Γ.}$$

In these equations, p is the pressure, $\text{v}$ is the velocity, μ>0 and n>0 are two constants. D is the rate-of-deformation tensor

$$\text{D}{}_{\mathrm{ij}}\text{=}\text{1}\text{2}\text{∂}\text{v}{}_{\mathrm{i}}\text{∂}\text{X}{}_{\mathrm{j}}\text{+}\text{∂}\text{v}{}_{\mathrm{j}}\text{∂}\text{X}{}_{\mathrm{i}}\text{.}$$

As it can be seen from (4), the stress deviator $\text{τ}$ is a homogeneous function of degree n of the velocity v:

$$\text{τ}{}_{\mathrm{ij}}\text{(λ}\text{v}\text{)=λ}{}^{\mathrm{n}}\text{τ}{}_{\mathrm{ij}}\text{(}\text{v}\text{).}$$

The purpose of our work is to upscale the pore flow description , , , . We use a rigorous upscaling process to obtain the macroscopic law without any prerequisite on the form of the macroscopic behaviour. Section 2 is devoted to a short presentation of the upscaling technique, the homogenisation method of multiple scales. The upscaling is conducted in Section 3. It yields a vectorial relationship between the Darcy velocity and the macroscopic pressure gradient. We present a demonstration which is different from that carried out in [6]. This new formulation, see (26), is of practical interest since it is suitable for numerical calculations. The purpose of Section 4 is to analyse this filtration law by means of the theory of representation of isotropic tensor functions of tensor arguments and the homogeneity condition (7). Different cases of anisotropy are investigated: isotropy, transverse isotropy, orthotropy and general anisotropy.