A Feasibility Study of the Pore Topology Method (PTM), A Medial Surface-Based Approach to Multi-phase Flow Simulation in Porous Media

Abstract

Computationally efficient microscale models designed to simulate multi-phase flow and characterize low-porosity media are challenged in thin, highly porous materials, primarily due to large, irregular pore spaces and inability to satisfy representative elementary volume requirements. In this article, we describe the pore topology method (PTM) and explore its capabilities to characterize a set of isotropic fibrous materials and to simulate multi-phase flow. PTM is a fast, algorithmically simple method that reduces the complexity of the 3-D void space geometry to its topologically consistent medial surface and uses it as a solution domain for single- and multi-phase flow simulations. Our results in permeability calculations, pore size distribution, and quasi-static drainage and imbibition simulations are in very good agreement with other numerical methods and analytical solutions. We expect that incorporating detailed spatial information about the porous media structure into the medial surface will enable a more accurate representation of the void space structure and of physical phenomena involved in multi-phase flow, thus expanding the applicability of PTM to a broader range of porous media, including non-fibrous materials.

Appendices

Appendix 1: Medial Surface Extraction Algorithm

In this appendix, we demonstrate the medial surface extraction algorithm on a simple, constant cross-sectional prism shown in Fig. 10. The algorithm outlined here is based on the formulation provided in Vera et al. (2012a). As mentioned earlier, we have modified the last step of the algorithm, medial surface thinning, to ensure full connectivity in the final medial surface.

1.1 Step 1: Obtain a Binary Image of Porous Media

Since the medial surface extraction algorithm described here is a voxel-based approach, obtaining a voxelized binary image of porous media is essential. Binary images for porous materials can be obtained either by using 3-D imaging of actual materials, or by using commercial packages like GeoDict. In the binary image shown in Fig. 10, we labeled voxels of the solid matrix as one, and of void space as zero.

Fig. 10

3-D model of the prism used in this example and its 2-D binary representation

1.2 Step 2: Generate Euclidean Distance Map (D)

From the binary image of the porous medium, we generate the distance map inside the void space. Distance map D, also known as distance field or distance transform, is a representation of a digital image based on the distance between each voxel and the closest boundary (solid phase). Assuming solid voxels (1-value voxels) as boundary voxel, the distance map of the void space can be generated by computing the distance between each void voxel and the closest solid voxel. Figure 11 shows the distance map of the sample prism.

Fig. 11

2-D slice of the distance map generated in the prism of Fig. 10

1.3 Step 3: Compute Gradient of Distance Map

The gradient of distance map can be computed by convolving the distance map with gradients of Gaussian kernel:

$$\begin{aligned} {\nabla }D= \left( \partial _{x}D_{\sigma } ,{ \partial }_{y}D_{\sigma }, {\partial }_{z}D_{\sigma } \right) =\left( \partial _{x}g_{\sigma }*D ,\partial _{y}g_{\sigma }*D ,\partial _{z}g_{\sigma }*D \right) \end{aligned}$$

(8)

where $\partial _{x}g_{\sigma }$, $\partial _{y}g_{\sigma }$, and $\partial _{z}g_{\sigma }$ are partial derivatives of the Gaussian kernel g with standard deviation of $\sigma$ and $*$ is the convolution operator. To avoid over-smoothing, we used $3\times 3\times 3$ voxels window kernels with $\sigma =0.5$.

1.4 Step 4: Compute Structure Tensor (ST)

The structure tensor (ST) summarizes the dominant directions of the gradients in a neighborhood of each voxel and is derived from the gradient field of a function. Smoothed ST at each voxel can be computed as

$$\begin{aligned} {ST}_{\rho ,\sigma }\left( D \right) =\left( {\begin{array}{ccc} g_{\rho }*{\partial _{x}D_{\sigma }}^{2} &{}\quad g_{\rho }*\partial _{x}D_{\sigma }\partial _{y}D_{\sigma } &{}\quad g_{\rho }*\partial _{x}D_{\sigma }\partial _{z}D_{\sigma }\\ g_{\rho }*\partial _{y}D_{\sigma }\partial _{x}D_{\sigma } &{}\quad g_{\rho }*{\partial _{y}D_{\sigma }}^{2} &{}\quad g_{\rho }*\partial _{y}D_{\sigma }\partial _{z}D_{\sigma }\\ g_{\rho }*\partial _{z}D_{\sigma }\partial _{x}D_{\sigma } &{}\quad g_{\rho }*\partial _{z}D_{\sigma }\partial _{y}D_{\sigma } &{}\quad g_{\rho }*{\partial _{z}D_{\sigma }}^{2}\\ \end{array} } \right) \end{aligned}$$

(9)

where $g_{\rho }$ is a $3\times 3\times 3$ Gaussian kernel with standard deviation of $\rho$. Here we chose $\rho =1$.

1.5 Step 5: Find Principal Eigenvectors

ST is a positive definite tensor; therefore, it has three nonnegative eigenvalues for a 3-D problem. For this case, since we have built the structure tensor from the gradients of the distance map, the eigenvectors at each voxel are the directions of principal gradients of distance map at the neighborhood of that voxel. To find the direction of the maximum gradient at each voxel, one needs to find the principal eigenvector of ST (i.e., the eigenvector corresponding to the eigenvalue of largest magnitude) at that voxel. For the sample prism of Fig. 10, the 2-D view of the principal eigenvector field $(\mathbf{V})$ is shown in Fig. 12.

Fig. 12

2-D view of the principal eigenvector field $({\mathbf{V}})$ of the prism

1.6 Step 6: Reorient Principal Eigenvectors in the Direction of Gradient

To prepare the vector field for step 7, we need all the principal eigenvectors to point toward the downward gradient. Reorienting the principal eigenvectors toward the downward gradient can be done by multiplying each vector by the sign of the scalar product of principal eigenvector and the gradient at each voxel:

$$\begin{aligned} {\varvec{V}}=\hbox {sign}\left( {\varvec{V}}\cdot {\nabla } D\right) \cdot {\varvec{V}} \end{aligned}$$

(10)

As shown in Fig. 13, besides reorienting the principal eigenvector field, this assigns zero to zero-gradient voxels.

Fig. 13

2-D view of the reoriented principal eigenvector field $({\mathbf{V}})$ of the prism

1.7 Step 7: Compute Normalized Ridge Map (NRM)

Since the magnitude of all vectors in ${\varvec{V}}$ is less than or equal to one, taking the divergence of ${\varvec{V}}$ will assign a value between N and ${+}{N}$ to each voxel, where N is the dimension of the problem. For a 3-D geometry, this value will be bounded between −3 and $+3$. We will refer to this value as the normalized ridge map (NRM). Positive NRMs correspond to ridge voxels and negative to valley voxels; therefore, the magnitude of NRM value represents the degree of ridgeness or valleyness. NRM can be calculated as

$$\begin{aligned} \hbox {NRM}=\hbox {div}\left( {\varvec{V}} \right) \end{aligned}$$

(11)

A 2-D view of NRM for the sample prism is shown in Fig. 14. For the purpose of medial surface extraction, we are only interested in ridges (positive NRMs); therefore, non-positive values can be ignored at the end of this step.

Fig. 14

2-D slice of the normalized ridge map generated for the sample prism

1.8 Step 8: Medial Surface Thinning: Hessian-Based Non-maxima Suppression

The last step is to apply a thinning algorithm on the positive NRM values to extract a voxel-wide medial surface. One of the most common thinning algorithms is non-maxima suppression (NMS). In NMS, one first needs to find a suppression direction, which is the direction perpendicular to the medial surface. The common practice is to use the direction of principal gradient of the NRM at each voxel as the suppression direction. It should be noted that the direction of principal gradient of NRM is the principal eigenvector of the structure tensor of NRM. However, it has been shown that implementing NMS on NRM map will introduce holes (disconnectivities) in the medial surface (Vera et al. 2012a). Although this drawback can be ignored in many image processing cases, for our application it will change the connectivity of the void space of the porous media, which is undesirable. To overcome this drawback, we propose a modified NMS algorithm based on using the direction of maximum curvature of NRM as the suppression direction. The direction of maximum curvature of NRM is the principal eigenvector of Hessian tensor of NRM. Hessian matrix is a symmetric matrix that summarizes the dominant directions of curvatures at a neighborhood of a voxel. Instead of using the structure tensor of NRM, modified NMS uses the Hessian tensor of NRM map to find the suppression direction. Throughout this paper, we will refer to this non-maxima suppression method as the Hessian-based non-maxima suppression (HNMS).

Fig. 15

a Suppression directions for Hessian-based NMS $({\varvec{V}}_{\mathrm{HNMS}})$ and b enlarged view of the area marked by the red square in Fig. 15a

Fig. 16

Extracted medial surface for the prism in Fig. 10

In HNMS, the principal eigenvector of Hessian tensor of NRM at each point is chosen as the suppression direction. We computed the Hessian matrix at each voxel as below:

$$\begin{aligned} H_{\rho ,\sigma }\left( \hbox {NRM} \right) =\left( {\begin{array}{ccc} \partial _{xx}g_{\rho }*\hbox {NRM} &{}\quad \partial _{xy}g_{\rho }*\hbox {NRM} &{}\quad \partial _{xz}g_{\rho }*\hbox {NRM}\\ \partial _{yx}g_{\rho }*\hbox {NRM} &{}\quad \partial _{yy}g_{\rho }*\hbox {NRM} &{}\quad \partial _{yz}g_{\rho }*\hbox {NRM}\\ \partial _{zx}g_{\rho }*\hbox {NRM} &{}\quad \partial _{zy}g_{\rho }*\hbox {NRM} &{}\quad \partial _{zz}g_{\rho }*\hbox {NRM}\\ \end{array} } \right) \end{aligned}$$

(12)

where $\partial _{x_{i}x_{j}}g_{\rho }$ is the second derivative of the Gaussian kernel with respect to coordinates $x_{i}$ and $x_{j}$, and $\rho =1$. Let ${\varvec{V}}_{\mathrm{HNMS}}$ be the principal eigenvector of the Hessian matrix at each voxel. The non-maxima suppression can be computed as

$$\begin{aligned} \hbox {NMS}\left( x,y,z \right) =\left\{ {\begin{array}{ll} 1 &{} \quad \hbox {if NRM}\left( x,y,z \right) >\hbox {max}\left\{ \hbox {NRM} \left( {\varvec{V}}_{\mathrm{HNMS}}^{+}\right) , \hbox {NRM}\left( {\varvec{V}}_{\mathrm{HNMS}}^{-} \right) \right\} \\ 0&{}\quad \hbox {otherwise} \\ \end{array}} \right. \qquad \end{aligned}$$

(13)

where $\hbox {NRM}\left( {\varvec{V}}_{\mathrm{HNMS}}^{+} \right)$ and $\hbox {NRM}\left( {\varvec{V}}_{\mathrm{HNMS}}^{-} \right)$ are NRM values in the immediate neighborhood of a voxel in the direction of ${\varvec{V}}_{\mathrm{HNMS}}$.

Figure 15 shows the suppression directions in the proposed Hessian-based NMS. It should be noted that the ideal suppression direction should be perpendicular to the medial surface. Finally, Fig. 16 shows 2-D and 3-D views of the extracted medial surface for the sample prism.

Appendix 2: Hydraulic Conductivity of Prisms

To test the validity of the planar flow assumption in computing local conductivities in Eq. (2), we computed hydraulic conductivities for a set of prisms with polygonal cross sections using the PTM permeability calculation algorithm and compared the result with exact solutions derived by Mortensen et al. (2005). We followed the procedure outlined in Sect. 2.2, but instead of Eq. (4), we used the following expression for the hydraulic conductivity K of the prism:

$$\begin{aligned} {K} = -\frac{{QL}}{\Delta {P}} \end{aligned}$$

(14)

As in Mortensen et al. (2005), we defined the dimensionless hydraulic resistivity or correction factor ${\alpha }$, and dimensionless compactness C, to represent our results. These parameters can be computed as follows:

$$\begin{aligned} \alpha= & {} \frac{A^{2}}{K\mu } \end{aligned}$$

(15)

$$\begin{aligned} C= & {} \frac{p}{A^{2}} \end{aligned}$$

(16)

where A and p are the cross-sectional area and cross-sectional perimeter of the prism, respectively; K is hydraulic conductivity; and $\mu$ is fluid viscosity. Figure 17 compares the PTM results with exact values for $\alpha$ and C computed in Mortensen et al. (2005), for rectangular and triangular prisms. As the compactness increases, the PTM results approaches the exact solution, which is consistent with our assumption of flow between parallel plates for all voxels on medial surface.

Fig. 17

Correction factor versus dimensionless compactness for triangular and rectangular prisms

Appendix 3: Discretization of Eq. (1)

As mentioned in Sect. 2.2, to find the pressure field inside the void space, the following equation is discretized and solved on the medial surface of the void space:

$$\begin{aligned} {\nabla }\cdot \left( K_{\mathrm{local}}\nabla \,{P} \right) =0 \end{aligned}$$

(17)

where $K_{\mathrm{local}}$ is the local hydraulic conductivity at each voxel of the medial surface, and its value is zero on all non-medial surface voxels in the porous media domain.

Since we are aiming to discretize and solve this equation only on the medial surface, we have used a modified version of the well-known central differencing scheme used in finite volume method (FVM).

Following the finite volume approach, we start the discretization by taking the integral of both sides of Eq. (17) over the volume of a voxel (V).

$$\begin{aligned} \int _V {{\nabla }\cdot \left( K_{\mathrm{local}}\nabla {P} \right) \hbox {d}V } =0 \end{aligned}$$

(18)

Then, using the divergence theorem, we can change the above volume integral to a surface integral:

$$\begin{aligned} \int _S {\left( K_{\mathrm{local}}\nabla {P} \right) \cdot {\varvec{n}}\hbox {d}S} =0 \end{aligned}$$

(19)

So far, we have been followed the exact procedure that one would take in FVM. Then, instead of using the six faces of our cubic voxels as our integration surfaces, we assumed an imaginary rhombicuboctahedron with 26 equal-area faces. Using this analogy, instead of three gradient directions, we will have 13 directions:

$$\begin{aligned} \mathop {\sum }\limits _{i=1}^{13} \left[ \left[ K_{\mathrm{local}}\frac{\partial P}{\partial {\varvec{n}}_{{\varvec{i}}}} \right] _{i^{+}}-\left[ K_{\mathrm{local}}\frac{\partial P}{\partial {\varvec{n}}_{{\varvec{i}}}} \right] _{i^{-}} \right] =0 \end{aligned}$$

(20)

where ${\varvec{n}}_{{\varvec{i}}}$ is the gradient direction and $i^{+}$ and $i^{-}$ shows the opposite faces along the gradient direction. This assumption is essential in capturing all curvatures and the junctions that are present in the voxel-wide medial surface.

Further simplification of Eq. (20) will result:

$$\begin{aligned}&\mathop {\sum }\limits _{i=1}^{13} \left[ \left[ \left( K_{\mathrm{local},i+1}+K_{\mathrm{local},i} \right) \left( P_{i+1}-P_{i} \right) \right] \right. \hbox {sign}(K_{\mathrm{local},i+1}) \nonumber \\&\quad \left. -\left[ \left( K_{\mathrm{local},i}+K_{\mathrm{local},i-1} \right) \left( P_{i}-P_{i-1} \right) \right] \hbox {sign}(K_{\mathrm{local},i-1}) \right] =0 \end{aligned}$$

(21)

In Eq. (21), sign($\cdot$) is used to automatically zero out the gradient between a medial surface voxel and a non-medial surface voxel.