Geometric state function for two-fluid flow in porous media

Models that describe two-fluid flow in porous media suffer from a widely recognized problem that the constitutive relationships used to predict capillary pressure as a function of the fluid saturation are nonunique, thus requiring a hysteretic description. As an alternative to the traditional perspective, we consider a geometric description of the capillary pressure, which relates the average mean curvature, the fluid saturation, the interfacial area between fluids, and the Euler characteristic. The state equation is formulated using notions from algebraic topology and cast in terms of measures of the macroscale state. Synchrotron-based x-ray microcomputed tomography and high-resolution pore-scale simulation is applied to examine the uniqueness of the proposed relationship for six different porous media. We show that the geometric state function is able to characterize the microscopic fluid configurations that result from a wide range of simulated flow conditions in an averaged sense. The geometric state function can serve as a closure relationship within macroscale models to effectively remove hysteretic behavior attributed to the arrangement of fluids within a porous medium. This provides a critical missing component needed to enable a new generation of higher fidelity models to describe two-fluid flow in porous media.

Figure 1

The shape of drainage and imbibition curves are determined in part by the fluid connectivity. The irreducible wetting phase and residual nonwetting phase saturations correspond to limits where fluid connectivity breaks down.

Figure 2

Sequence showing the pore-level evolution of the nonwetting fluid geometry during imbibition of the wetting fluid. As the volume fraction of the nonwetting fluid volume decreases, corresponding changes in the surface area, mean curvature, and connectivity also occur.

Figure 3

Two-dimensional analog for Eq. (9): the change in area (blue region) obtained by rolling a small ball around the boundary for a 2D object (red region) is determined by the shape of the red object boundary and can be expressed in terms of averaged measures of the boundary shape.

Figure 4

(a) The ink bottle geometry is defined based on the position of five equally sized spheres in a symmetric system with radius $R$ to create a two-pore system connected to three throats with widths $H_1$, $H_2$, and $H_3$. (b) Meniscus configurations within the ink bottle can be determined analytically.

Figure 5

(a) Apparent hysteresis in the ink bottle results from the observation of apparently different equilibrium states for drainage and imbibition; (b) the integral mean curvature is a one-to-one function of the saturation.

Figure 6

The relationship between geometric state variables is explored based on the possible fluid microstates in six porous media, each with a distinct solid microstructure (white).

Figure 7

Traditional models assume that the macroscale capillary pressure is a function only of the saturation of the wetting fluid. Two-fluid displacement simulations within a sand pack show that: (a) At fixed saturation, $s^w$, the relative mean curvature ${\acute{J}}_w^{wn}$ can attain many possible values depending on the system history; (b) ${\acute{J}}_w^{wn}(s^w,{\acute{\epsilon }}^{wn})$ is nonunique for $s^w=0.65$.

Figure 8

The geometric state function ${\acute{J}}_w^{wn}(s^w,{\acute{\epsilon }}^{wn},{\acute{\chi }}^n)$ is unique for each geometry, which is not the case for ${\acute{J}}_w^{wn}\left(s^w\right)$ and ${\acute{J}}_w^{wn}(s^w,{\acute{\epsilon }}^{wn})$. Locally smooth approximations were generated for each relationship using generalized additive models (GAMs) to compare the accuracy of macroscopic constitutive models. Error measures are given by (a) the coefficient of variation $R^2$ and (b) generalized cross validation (GCV).