Estimating geometric tortuosity of saturated rocks from micro-CT images using percolation theory

Abstract

Tortuosity ($\tau$) is one of the key parameters controlling flow and transport in porous media. Although the concept of tortuosity is straightforward, its estimation in porous media has yet been challenging. Most models proposed in the literature are either empirical or semiempirical including some parameters whose values and their estimations are in prior unknown. In this study, we modified a previously presented geometric tortuosity (${\tau }_{g}$) model based on percolation theory and validated it against a methodology based on the pathfinding A* algorithm. For this purpose, we selected 12 different porous materials including four sandstones, three carbonates, one salt, and four synthetic media. For all samples, five sub-volumes at different lengths with fifty iterations were randomly selected except one carbonate sample for which three sub-volumes were extracted. Pore space properties, such as pore radius, throat radius, throat length, and coordination number distributions were determined by extracting the pore network of each sub-volume. The average and maximum coordination numbers and minimum throat length were used to estimate the ${\tau }_{g}$. Comparison with the A* algorithm results showed that the modified model estimated the ${\tau }_{g}$ accurately with absolute relative errors less than 28%. We also estimated the ${\tau }_{g}$ using two other models presented in the literature as well as the original percolation-based tortuosity model. We found that our proposed model showed a significantly higher accuracy. Results also indicated more precise estimations at the larger length scales demonstrating the effect of uncertainties at the smaller scales.

Appendix A

Sample properties and their corresponding network characteristics are reported in Table 1. For each rock, the full sample (subscript _ini) and the subsamples with decreasing length scale (subscripts _n1,… _n5) were characterized in terms of porosity ($\overline{\phi }$), average coordination number (${\overline{Z} }_{\rm ave})$, maximum coordination number (${\overline{Z} }_{\rm max}$), and minimum pore throat length (${\overline{l} }_{\rm tmin}$). For each subsample length scale, the reported values represent averages of over fifty different subsamples; the standard deviation is given in parentheses (see Table 1).

Table 1 Sample properties and their corresponding network characteristics. Values in parentheses represent one standard deviation

Results of the average geometric tortuosity estimated via Eqs. (2–5) for all subsamples and the relative error with respect to the A* algorithm are given in Table 2. The reported tortuosity values represent averages over several subsamples. Sample number, sample length (L_s), the number of subsamples used to calculate the average (No. of subsamples), and the geometric tortuosity simulated via the A* algorithm (Eq. 9) are also reported.

Table 2 Results of geometric tortuosity for all the considered samples and relative error of the percolation model with respect to the A* algorithm. Hydraulic tortuosity was computed from velocity field numerically simulated for only the largest length scale for the sake of comparison

The hydraulic tortuosity along the $i$ direction ($i=x,y,z$) was computed from the simulated velocity values as (Koponen et al. 1996):

$${\tau }_{h,i}=\frac{\langle \left|v\right|\rangle }{\langle {v}_{i}\rangle }$$

(A1)

where $\left|v\right|=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}+{v}_{z}^{2}}$ is the magnitude of the local velocity, ${v}_{i}$ is the directional component of the velocity, and ⟨⟩ indicates the average over the void space.

Single-phase fluid flow was simulated by the computational fluid dynamic modeling at low Reynolds numbers (i.e., Re < 1) to obtain steady-state velocity field in the pore space. Simulations were performed using the software OpenFOAM 11 (Greenshields 2023), based on the finite volume method (FVM). The FVM is an efficient technique for the computational modeling of single- and multi-phase flow problems in porous media and for the evaluation of hydraulic tortuosity and permeability (Ferrari and Lunati 2013; Guibert et al. 2015; Soulaine et al. 2016; Rahmanian and Kantzas 2018; Germanou et al. 2018). Using SnappyHexMesh, the native OpenFOAM mesh generator utility, a grid representing the pore space of the rock was generated starting from the raw binarized image files of the rock samples (see Sect. 3.1). The Navier–Stokes equations (NSEs) govern the incompressible single-phase flow at small Reynolds numbers. The NSEs were solved under the steady-state conditions using the SIMPLE algorithm (Patankar 1980). The boundary conditions imposed were no-slip at the fluid–solid interface and constant pressure at the inlet and outlet. We verified that, under laminar flow, the estimated tortuosity was not sensitive to the imposed value of pressure gradient (Aminpour et al., 2018).

The ${\tau }_{h}$ values computed for the largest length scale using numerical simulations are reported Table 2. The sensitivity of the proposed percolation-based approach to the voxel dimension is presented in Table 3. For five different rock types (Berea, Bentheimer, Ketton, RG, and salt), the first realization of the subsample with length scale ${L}_{1}$ was considered. For each case, the image was resized by upscaling up to a factor 3 and by downscaling up to a factor 5. The percolation-based ${\tau }_{g}$ model, Eq. (5), obtained for the resampled 3-D images is reported in Fig. 9. Sensitivity to voxel size is observable in most scenarios, in the case of both figure upscaling and downscaling. However, the interquartile range maintained always below 20%.

Table 3 Sensitivity of geometric tortuosity to voxel dimension