Image-based effective medium approximation for fast permeability evaluation of porous media core samples

Abstract

An image-based effective medium approximation (EMA) is developed so as to permit very fast transport properties evaluations of 3D porous media. From an image-based porous network (IBPN) built upon digital image processing of 3D binary images, we focus on throat’s local geometrical properties at the pore scale, for being the most sensible structural units which build up the local pressure. This approach is a 3D image–based extension of the critical point approach proposed in 2D fractures. We show, from analyzing various core rock samples available in the literature, that the asymptotic assumptions associated with the preeminence of critical points in throats are indeed geometrically relevant. We then describe how the image-based EMA evaluated from the conductances computed from the discrete IBPN can be reliably evaluated. The proposed method is evaluated upon the estimation of core sample permeability from binarized image obtained using X-ray tomography. Since it combines digital image treatments with statistical data post-processing without the need of computational fluid dynamics (CFD) computation, it is extremely cost efficient. The results are compared with a micro-scale Stokes flow computation in various rock samples. The sensitivity to the pore discretization also is discussed and illustrated.

Appendices

Appendix

A conductance of an axisymmetrical tubular throat

In this section, we show how the leading order approximation for the conductance inside a tubular throat is given by Eq. (6). We consider Stokes equation associated with the velocity field u, the pressure p, and the dynamic viscosity μ

$$ \mu{\Delta}\textbf{u}-\nabla p=0, $$

(9)

inside an axisymmetrical tubular throat having a variable radius R(z) along the longitudinal direction z, considering cylindrical coordinates (r, θ, z) (as the one represented in Fig. 2a). We chose the origin of the coordinate system at the throat minimum radius r₀, so that R(z = 0) = r₀. We consider a finite length throat where a fixed pressure drop is applied at edges z = ±L/2. Using the usual lubrication non-dimensionalization u = U(𝜖u_r, 𝜖u_θ, u_z), where U is the main longitudinal velocity along the z direction. From axisymmetry u_θ = 0. In the vicinity of the constriction, the following Taylor expansion of the local radius is

$$ R(z)=r_{0}(1+\frac{1}{2}\frac{R_{zz}}{r_{0}}z^{2}+...). $$

Considering tilde dimensionless variables, $r=\tilde r r_{0}$, $R=\tilde R r_{0}$ and $z=\tilde z \frac {r_{0}}{\sqrt {r_{0}R_{zz}}}$, one finds

$$ \tilde{R}(\tilde{z})=(1+\frac{1}{2}\tilde{z}^{2}+...). $$

Using $\epsilon =\sqrt {r_{0}R_{zz}} \ll 1$ as a small parameter, one can seek for an asymptotic expansion the velocity and pressure

$$ \tilde{u}_{\tilde{r}}=\tilde{u}_{\tilde{r}}^{0}+\epsilon\tilde{u}_{\tilde{r}}^{1}+.. $$

$$ \tilde{p}=\tilde{p}^{0}+\epsilon\tilde{p}^{1}+.. $$

when inserted into the expansion in power of 𝜖 of the operators of the Stokes problem (9) leads to the following leading order problem

$$ \frac{{1}}{r_{0}\tilde{r}}\frac{\partial}{\partial\tilde{r}}\left( \tilde{r}\frac{\partial\tilde{u}^{0}}{\partial\tilde{r}}\right)-\frac{\epsilon}{r_{0}}\frac{\partial\tilde{p}^{0}}{\partial\tilde{z}}=0. $$

The solution of this problem can be found easily

$$ \tilde{u}^{0}=\frac{\epsilon}{4}\frac{\partial\tilde{p}^{0}}{\partial\tilde{z}}(\tilde{r}^{2}-\tilde{R}^{2}). $$

Computing the resulting leading order flux $\tilde q^{0}$ from integrating the velocity in cylindrical coordinates between 0 and $\tilde R(\tilde z)$ leads to

$$ \tilde{q}^{0}=-\frac{{\pi}\epsilon}{8}\frac{\partial\tilde{p}^{0}}{\partial\tilde{z}}\tilde{R}^{4}. $$

Looking for the head pressure loss gradient $\partial _{\tilde {z}}\tilde {p}^{0}$,

$$ \frac{\partial\tilde{p}^{0}}{\partial\tilde{z}}=-\frac{8\tilde{q}^{0}}{\pi\epsilon}\tilde{R}^{-4}, $$

integrating between 0 and $\tilde {z}$, considering $\tilde {p}(0)=0$ as pressure reference,

$$ \begin{array}{@{}rcl@{}} \tilde{p}^{0}(\tilde{z})&=&\left[\frac{5}{16}\sqrt{2}\arctan\left( \frac{\tilde{z}}{\sqrt{2}}\right)\right.\\&&\left.+\frac{15\tilde{z^{5}}+80\tilde{z}^{3}+132\tilde{z}}{24(\tilde{z}^{2}+2)^{3}}\right]\left( -\frac{8\tilde{q}^{0}}{\pi\epsilon}\right), \end{array} $$

and considering limits for an infinitely long throat,

$$ \underset{\tilde{z}\rightarrow+\infty}{\lim}\tilde{p}^{0}=\frac{5\epsilon}{\sqrt{2}}\tilde{q}^{0}, $$

finally provides the following conductance for the whole throat:

$$ G_{n}=\frac{2\sqrt{2}}{5}\frac{r^{4}}{\epsilon}. $$