Pore-to-Core Upscaling of Solute Transport Under Steady-State Two-Phase Flow Conditions Using Dynamic Pore Network Modeling Approach

Abstract

We present a solute transport model, developed by employing a dynamic pore network modeling approach, to investigate dispersive solute transport behaviors in consolidated porous media. The model is capable of upscaling solute transport processes from pore to core, under two-phase fluid configurations. The governing equations of fluid flow, fluid displacement, and solute transport are solved at the pore level. A heavily parallelized computing scheme is utilized to simulate dynamic fluid displacements and transport processes in a core-scale pore network constructed from micro-computed tomography (micro-CT) images of a Berea sandstone sample. A series of solute transport simulations are conducted under the single-phase condition to validate the model by comparing the computed longitudinal dispersion coefficients against the experimental data over a wide range of Peclet numbers (Pe), i.e., $3 \times 10^{-2}{\sim}3 \times 10^5$. The model is then used to simulate solute transport under two-phase fluid configurations to examine the effects of the non-wetting phase saturation on solute transport behaviors. More specifically, solute transport is studied at different Pe and different water saturations obtained at the end of imbibition processes. We find that in a two-phase system, the longitudinal dispersion coefficient substantially increases with enhanced convective mixing under the transport regime with high Pe but decreases with restricted diffusive mixing at low Pe. In addition, the results indicate a non-monotonic dispersion–saturation relation under the convective transport condition. We illustrate that a strong correlation exists between the extent of dispersive mixing and the heterogeneity of fluid saturation profile across a core-scale network.

Appendix 1

1.1 Area Open to Flow and Phase Conductance

For a single-phase filled element, its cross-sectional area open to flow is:

$$\begin{aligned} A = {\left\{ \begin{array}{ll} 4R^2, \quad \hbox{for square elements} \\ \frac{R^2}{4G}, \quad \hbox{for triangular elements} \\ \pi R^2, \quad\hbox{for circular elements} \end{array}\right. }, \end{aligned}$$

(20)

where R and G are the inscribed radius and shape factor of the element, respectively. The phase conductance is then obtained with the following equations (Oren et al. 1998; Patzek and Silin 2001):

$$\begin{aligned} g = {\left\{ \begin{array}{ll} \frac{0.5623GA^2}{\mu }, \quad \hbox{for square elements} \\ \frac{3R^2A}{20\mu }, \quad \hbox{for triangular elements} \\ \frac{0.5GA^2}{\mu }, \quad \hbox{for circular elements} \end{array}\right. } \end{aligned},$$

(21)

where $\mu$ is the fluid viscosity.

For a two-phase filled element with oil occupying the center and water residing in the corner, the corner area open to water flow can be calculated by utilizing the geometrical relationship:

$$\begin{aligned} A^w = {\left\{ \begin{array}{ll} r^2[{\cos \theta (\cot \alpha \cos \theta - \sin \theta ) + \theta + \alpha - \frac{\pi }{2}}], \quad if \ \alpha + \theta \ne \frac{\pi }{2} \\ \frac{1}{2}{b^2}{\sin (2\alpha )}, \quad if \ \alpha + \theta = \frac{\pi }{2} \end{array}\right. } \end{aligned}$$

(22)

and the Young–Laplace equation:

$$\begin{aligned} r = \frac{\sigma ^{ow}}{P_c}, \end{aligned}$$

(23)

where $A^w$ is the cross-sectional area of the water layer in a specific corner, $\alpha$ is the corner half angle, b is the apex-meniscus distance, $\sigma ^{ow}$ is the interfacial tension (IFT) between oil and water, and $\theta$ is the angle between the oil–water interface and the solid surface toward the corner. It can be the contact angle or hinging angle depending on the local capillary pressure. In Eq. 23 and as shown in Fig. 3, r is the radius of the AM interface, whose curvature in the plane perpendicular to that of the paper is assumed to be negligible. The area open to oil flow is readily available:

$$\begin{aligned} A^o = A - \sum _{i=1}^{n} A_i^w, \end{aligned}$$

(24)

where n is the number of corners that are occupied by water.

In this case, the center oil conductance is evaluated using $A^o$ and Eq. 21, while the corner water conductance is given by (Zhou et al. 1997; Hui and Blunt 2000):

$$\begin{aligned} g^w& = \frac{(A^w)^2(1 - \sin \alpha )^2(\phi _2\cos \theta - \phi _1)\phi _3^2}{12\mu ^w \sin ^2\alpha (1-\phi _3)^2(\phi _1+\phi _2)^2}\end{aligned}$$

(25)

$$\begin{aligned} \phi _1& = (\frac{\pi }{2} - \alpha - \theta ) \end{aligned}$$

(26)

$$\begin{aligned} \phi _2& = \cot \alpha \cos \theta - \sin \theta \end{aligned}$$

(27)

$$\begin{aligned} \phi _3& = (\frac{\pi }{2} - \alpha )\tan \alpha. \end{aligned}$$

(28)