A Class of Physically Stable Non-linear Models of Flow Through Anisotropic Porous Media

Abstract

A linear stability analysis of the single-phase conservation equation in multidimensional porous media is performed, for both weakly compressible and compressible fluids. Non-Newtonian and non-Darcy effects are accounted for using a non-linear Darcy-like form for the superficial velocity, where the mobility tensor is velocity-dependent and proportional to the permeability. It is found that under this hypothesis, flows at an angle with respect to the principal axes of the permeability tensor can be unstable, unless the mobility is a function of the velocity magnitude in terms of the inverse permeability norm. As shown by previous authors, for steady-state incompressible flows this is also the condition ensuring that the governing equation derives from the minimization of a dissipation potential.

Appendices

Appendix

Criteria for Acceptable Mobilities

1.1 Weakly Compressible Case

Within the weakly compressible approximation with gravity, using our definition of fluid potential given by Eq. (23), the closure relationship for the superficial velocity can be cast in the non-Newtonian case as (see Eq. (24)):

$$\begin{aligned} \mathbf{u}= -\frac{\rho }{\rho _0}\lambda \left(\left|\underline{\underline{\kappa }}^{\alpha -1} \cdot \mathbf{u}\right|\right)\underline{\underline{\kappa }}\cdot \mathbf{\nabla }\psi . \end{aligned}$$

(79)

Due to our “small” compressibility assumption, we have $\rho /\rho _0=1+o(1)$ and to leading order

$$\begin{aligned} \mathbf{u}= -\lambda \left(\left|\underline{\underline{\kappa }}^{\alpha -1} \cdot \mathbf{u}\right|\right)\underline{\underline{\kappa }}\cdot \mathbf{\nabla }\psi , \end{aligned}$$

(80)

where $\lambda >0$ is the velocity-dependent mobility factor. To leading order this equation formally also applies to the non-Darcy case. For Eq. (80) to yield a physically solvable flow problem, there must be a bijective mapping between potential gradient and velocity. In other words, for a given velocity there should be a unique potential gradient driving the flow, and conversely. In particular, according to the inverse function theorem, local invertibility is guaranteed provided the Jacobian of

$$\begin{aligned} \mathbf{f}(\mathbf{u}) = -\frac{1}{\lambda \left(\left|\underline{\underline{\kappa }}^{\alpha -1} \cdot \mathbf{u}\right|\right)}\underline{\underline{\kappa }}^{-1}\cdot \mathbf{u} \end{aligned}$$

(81)

is non-singular. Differentiation of $\mathbf{f}$ leads to the following Jacobian expression:

$$\begin{aligned} \underline{\underline{\text{ df}}}(\mathbf{u}) = -\frac{1}{\lambda \left(\left|\underline{\underline{\kappa }}^{\alpha -1}\cdot \mathbf{u}\right|\right)}\left(\underline{\underline{I}}- \frac{\lambda ^{\prime }\left(\left|\underline{\underline{\kappa }}^{\alpha -1}\cdot \mathbf{u}\right|\right)}{\lambda \left(\left|\underline{\underline{\kappa }}^{\alpha -1}\cdot \mathbf{u}\right|\right)} \frac{(\underline{\underline{\kappa }}^{-1}\cdot \mathbf{u})(\underline{\underline{\kappa }}^{2\alpha -1}\cdot \mathbf{u})^t}{\left|\underline{\underline{\kappa }}^{\alpha -1}\cdot \mathbf{u}\right|}\right)\underline{\underline{\kappa }}^{-1}, \end{aligned}$$

(82)

where $\underline{\underline{I}}$ is the identity tensor. For invertibility we require that $\det (\underline{\underline{\text{ df}}}(\mathbf{u}))\ne 0$, which yields the condition

$$\begin{aligned} 1-\frac{\lambda ^{\prime }\left(\left|\underline{\underline{\kappa }}^{\alpha -1}\cdot \mathbf{u}\right|\right)}{\lambda \left(\left|\underline{\underline{\kappa }}^{\alpha -1}\cdot \mathbf{u}\right|\right)} \left|\underline{\underline{\kappa }}^{\alpha -1}\cdot \mathbf{u}\right|\ne 0. \end{aligned}$$

(83)

Clearly, there must exist a continuous path driving the flow velocity from 0 to $\mathbf{u}$, hence the above expression should remain strictly positive, and since $\lambda >0$ we can write that for $v\ge 0$

$$\begin{aligned} \lambda (v)-v\lambda ^{\prime }(v) > 0, \end{aligned}$$

(84)

i.e.,

$$\begin{aligned} \eta (v) =\frac{v\lambda ^{\prime }(v)}{\lambda (v)}< 1. \end{aligned}$$

(85)

Multiplying Eq. (80) on the left by $\underline{\underline{\kappa }}^{\alpha -1}$ and taking the norm, we obtain

$$\begin{aligned} \displaystyle \left|\underline{\underline{\kappa }}^{\alpha -1}\cdot \mathbf{u}\right| = \lambda \left(\left|\underline{\underline{\kappa }}^{\alpha -1}\cdot \mathbf{u}\right|\right)\left|\underline{\underline{\kappa }}^{\alpha }\cdot \nabla \psi \right|, \end{aligned}$$

(86)

which suggests that $\left|\underline{\underline{\kappa }}^{\alpha -1}.\mathbf{u}\right|$ can be considered as a function of $\left|\underline{\underline{\kappa }}^{\alpha }\cdot \nabla \psi \right|$. To verify this, we introduce

$$\begin{aligned} g(v) = \frac{v}{\lambda (v)},\quad v\ge 0, \end{aligned}$$

(87)

and compute

$$\begin{aligned} g^{\prime }(v) = \frac{\lambda (v)-v\lambda ^{\prime }(v)}{(\lambda (v))^2}. \end{aligned}$$

(88)

Assuming that $\mathbf{f}$ defined by Eq. (81) is bijective, it clearly follows that $g$ is surjective, and according to Eqs. (84) and (88), $g^{\prime }(v)>0$ and $g$ is also injective. We conclude that $g:\mathbb R ^{+}\mapsto \mathbb R ^{+}$ is bijective and $\left|\underline{\underline{\kappa }}^{\alpha -1}\cdot \mathbf{u}\right|=g^{-1}\left(\left|\underline{\underline{\kappa }}^{\alpha }\cdot \nabla \psi \right|\right)$. Defining

$$\begin{aligned} \tilde{\lambda }(w) = \lambda \left(g^{-1}(w)\right),\quad w\ge 0, \end{aligned}$$

(89)

we can now rewrite Eq. (80) as

$$\begin{aligned} \mathbf{u}= -\tilde{\lambda }\left(\left|\underline{\underline{\kappa }}^{\alpha }\cdot \nabla \psi \right|\right)\underline{\underline{\kappa }}\cdot \mathbf{\nabla }\psi , \end{aligned}$$

(90)

and Eq. (86) as

$$\begin{aligned} \left|\underline{\underline{\kappa }}^{\alpha -1}\cdot \mathbf{u}\right| = \tilde{\lambda }\left(\left|\underline{\underline{\kappa }}^{\alpha }\cdot \nabla \psi \right|\right)\left|\underline{\underline{\kappa }}^{\alpha }\cdot \nabla \psi \right|. \end{aligned}$$

(91)

Defining

$$\begin{aligned} h(w) = w\tilde{\lambda }(w),\quad w\ge 0, \end{aligned}$$

(92)

we see that $h(g(v))=g(v)\tilde{\lambda }(g(v))=g(v)\lambda (v)=v$, therefore $h^{\prime }(g(v)) = 1/g^{\prime }(v) >0$ and $h$ is strictly monotone. Since $h^{\prime }(w) = \tilde{\lambda }(w)+w\tilde{\lambda }^{\prime }(w)$, in addition to inequality (85) we have

$$\begin{aligned} \tilde{\lambda }(w)+w\tilde{\lambda }^{\prime }(w)> 0, \end{aligned}$$

(93)

i.e.

$$\begin{aligned} \tilde{\eta }(w) =\frac{w\tilde{\lambda }^{\prime }(w)}{\tilde{\lambda }(w)}> -1. \end{aligned}$$

(94)

This second inequality is also the one ensuring that the Jacobian of $\mathbf{u}=\mathbf{m}(\mathbf{\nabla }\psi )$ (Eq. (90)) is non-singular. Conditions (85) and (94) are illustrated in Fig. (1).

Fig. 1

To be acceptable, the mobility factor $\lambda$ (or $\tilde{\lambda }$) should ensure bijectivity between $\mathbf{u}$ and $\mathbf{\nabla }\psi$, implying Eqs. (85) and (94). The figure shows the function $g^{-1}$ for shear-thinning, Newtonian–Darcy, and shear-thickening/non-Darcy cases. The situation where one of the bijectivity conditions is not satisfied is illustrated by (red) dash-dotted curves

In practice, for a given velocity-dependent mobility function $\lambda$ (what is typically known from experiments or physical models is the function $\lambda$, not $\tilde{\lambda }$), we can compute $\tilde{\lambda }$ by integration of the first-order ODE

$$\begin{aligned} \tilde{\lambda }^{\prime }(w) = \tilde{\lambda }(w)\frac{\lambda ^{\prime }(w\tilde{\lambda }(w))}{1-w\lambda ^{\prime }(w\tilde{\lambda }(w))}, \quad w \ge 0, \end{aligned}$$

(95)

with the initial condition $\tilde{\lambda }(0)=\lambda (0)$. We may also write this relationship as

$$\begin{aligned} \frac{w}{\tilde{\lambda }(w)}\tilde{\lambda }^{\prime }(w) = \frac{\frac{v}{\lambda (v)}\lambda ^{\prime }(v)}{1-\frac{v}{\lambda (v)}\lambda ^{\prime }(v)}, \end{aligned}$$

(96)

where $w=g(v)$, i.e.,

$$\begin{aligned} \tilde{\eta }= \frac{\eta }{1-\eta }. \end{aligned}$$

(97)

1.2 Compressible Case

Without gravity, using our definition of fluid potential given by Eq. (33), the closure relationship for the superficial velocity can be cast in the non-Darcy case as (see Eq. (34)):

$$\begin{aligned} \rho \mathbf{u}= -\frac{M}{2RT} \lambda \left(\rho \left|\underline{\underline{\kappa }}^{\alpha -1}\cdot \mathbf{u}\right|\right) \underline{\underline{\kappa }}\cdot \mathbf{\nabla }\psi . \end{aligned}$$

(98)

By identification with Eq. (80), i.e. “replacing $\mathbf{u}$ by $\rho \mathbf{u}$” and “replacing $\psi$ by $M\psi /2RT$”, we see that the requirements of Eqs. (85) and (94), as well as the property of Eq. (97), still hold.