# Predicting Resistivity and Permeability of Porous Media Using Minkowski Functionals

## Abstract

Permeability and formation factor are important properties of a porous medium that only depend on pore space geometry, and it has been proposed that these transport properties may be predicted in terms of a set of geometric measures known as Minkowski functionals. The well-known Kozeny–Carman and Archie equations depend on porosity and surface area, which are closely related to two of these measures. The possibility of generalizations including the remaining Minkowski functionals is investigated in this paper. To this end, two-dimensional computer-generated pore spaces covering a wide range of Minkowski functional value combinations are generated. In general, due to Hadwiger’s theorem, any correlation based on any additive measurements cannot be expected to have more predictive power than those based on the Minkowski functionals. We conclude that the permeability and formation factor are not uniquely determined by the Minkowski functionals. Good correlations in terms of appropriately evaluated Minkowski functionals, where microporosity and surface roughness are ignored, can, however, be found. For a large class of random systems, these correlations predict permeability and formation factor with an accuracy of 40% and 20%, respectively.

## Keywords

Permeability Formation factor Minkowski functionals Archie equation Kozeny–Carman equation## 1 Introduction

*f*is a factor that represents the shape of grains, and \(\tau \) is the tortuosity which is the ratio of a typical flow path length to the sample length. The reasoning leading to (1) is based on bundle-of-tubes analogues, and the two parameters

*f*and \(\tau \) do not have any strict definition.

*F*, only depends on pore space geometry.

*F*is often expressed in terms of the Archie equation (Archie 1942, 1950):

*m*is called the “cementation index” which may vary between rock types. The formation factor can also be expressed in terms of porosity and the electric tortuosity \(\tau _e\)

*F*based on models for \(\tau _e\). By comparing (2) and (3), we see that the Archie equation is consistent with describing the electric tortuosity as

Since permeability and formation factor only depend on pore space geometry, it is natural to propose that there should exist a small set of properly defined geometric measures that can, at least to very good approximation, be used for estimating these properties. The identification of these geometric measures remains an unsolved problem. In practice, *f* and \(\tau \) in (1), and *m* in (2) often play the role of mere fudge–factors/fudge–functions for fitting experimental data, and more rigorous approaches are needed. Recently, it has been proposed (Lehmann et al. 2008; Mecke 2000; Mecke and Arns 2005; Vogel et al. 2010; Scholz et al. 2012, 2015; Liu et al. 2017; Armstrong et al. 2018) that permeability and formation factor may be estimated in terms of a set of geometric measures known as Minkowski functionals or intrinsic volumes. Two of these measures, \(\phi \) and \(\sigma \), are already represented in the Kozeny–Carman equation (1). In the present paper, we will investigate the predictive power of the full set of Minkowski functionals on the permeability and formation factor of two-dimensional computer-generated pore spaces.

## 2 Minkowski Functionals

The Minkowski functionals are integrals over geometrical sets, and they are also well defined for polyhedra with singular edges (Mecke 2000). In *d*-dimensions, there are \(d+1\) functionals \(W_0 \ldots W_d\), where \(W_0\) is the hyper-volume and the rest are integrals over the \(d-1\)-dimensional bounding hyper-surface(s). A recent comprehensive overview of the application of Minkowski functionals for the characterization of porous media and porous media transport can be found in Armstrong et al. (2018). In two dimensions, the surface integrals are replaced by integrals over the pore space perimeter.

Our aim is to apply the Minkowski functionals for characterizing permeability and formation factor, which, for a representative elementary volume (REV) of a homogeneous medium, is independent of system size. The relevant quantities are then the *specific* functionals. Note also that what is relevant for the transport properties is the *connected pore space*. This implies that any isolated pores should be excluded from the calculation of the pore space functionals.

*porosity*

*A*is the total area,

*specific pore perimeter length*

*specific Euler characteristic*

*R*is the radius of curvature of the connected pore space perimeter. Note that (in 2D) \(\chi \) for the connected pore space is equal to the number of grains (unconnected pores are counted as being part of the enclosing grains) since we have \(\int \frac{1}{R} {\hbox {d}}L = -2\pi \) when we integrate along the perimeter of each single grain. The Euler characteristic is also a measure of topology, and for a connected periodic pore space, we have

*g*is the number of redundant connections, or, in an equivalent pore network, the number of links that can be broken keeping inlet and outlet connected.

### 2.1 Permeability and Hadwiger’s Characterization Theorem

The Minkowski functionals are especially useful for characterizing physical properties due to Hadwiger’s characterization theorem (Klain 1995), which states that certain valuations over geometric sets can be parameterized as a linear combination of the Minkowski functionals (Mecke 2000). Examples of such valuations are free energies in thermodynamics. Accordingly, claims have been made that it should be possible to estimate permeability from these functionals (Scholz et al. 2012, 2015). Minkowski functionals also correlate well with correct transport properties when reconstructing pore spaces (Schlüter and Vogel 2011; Mosser et al. 2017, 2018).^{1}

Permeability is not an additive quantity (as required by Hadwiger’s theorem), so the theorem is not directly applicable, but if the permeability is a function of a set of additive measures, then permeability is also a function of the Minkowski functionals since any additive property depends linearly on the functionals. Since permeability is not measurable by downhole logging tools, it needs to be inferred from correlations based on a combination of measurements. These measurable quantities are typically additive, and thus, if the permeability is a function of the Minkowski functionals, it should be possible to find good generally applicable correlations, while such correlations will be unobtainable if permeability cannot be determined based on these functionals.

### 2.2 Dimensional Analysis

*K*is an unknown dimensionless function. Note the similarity with the Kozeny–Carman equation (1), which can also be expressed in terms of \(\sigma \) and a dimensionless permeability \(K_{\text {KC}}\):

## 3 Generating Pore Spaces with Prescribed Minkowski Functionals

Most previous work on two-dimensional pore spaces has investigated synthetic models built from grains or sub-grains of equal shape (typically circles of a given size) (Gebart 1992; Koponen et al. 1996; Yazdchi et al. 2012; Ebrahimi Khabbazi et al. 2013; Matsumura and Jackson 2014; Scholz et al. 2015; Kundu et al. 2016; Liu et al. 2017; Zarandi et al. 2019). Our investigations indicate that the resulting pore spaces do not have sufficient coverage of the \((\phi ,\lambda )\) parameter space; therefore, we need to build models using differently shaped and sized sub-grains. We will also need to generate several pore spaces for each set of Minkowski functional values, which requires an algorithm that can generate pore spaces with prescribed Minkowski functionals.

*E*is assigned to each specific configuration of grains (realization). This algorithm ideally creates realizations with probability

*T*. \(w_n\) are weights that may be tuned in order to improve convergence.

Periodic boundary conditions are applied in order to reduce edge effects both on the pore space itself and on the flow calculations.

The calculation of change in Minkowski functionals as a result of adding or removing a grain is local, and on a pixel representation (image), it amounts to recalculating the frequency of certain \(2\times 2\)-pixel configurations (Vogel et al. 2010). We have implemented a slightly modified version, taking into account the periodic boundary conditions, of the code found in the QuantIm library (Vogel 2017).

*i*and

*a*and

*b*are tunable constants. The first term is important for porosity, while the second term governs the amount of sub-grain overlap.

*connected*pore space only, and in the present work, we have selected to only generate pore spaces without any isolated pores. Isolated pores may be created when a sub-grain is added or removed (see Fig. 3), so a sub-grain should not be added or removed if this would create an isolated pore. The calculation of all energy terms used in the MCMC algorithm, including the Minkowski functionals, is local on the image so that the resulting algorithm should be very fast both in 2 and 3 dimensions. The connectivity check, however, requires non-local calculations. In our implementation, we first calculate the energy change, and the time-consuming connectivity check is performed only if the change should be accepted based on this. The non-local calculations substantially slow down the code and would especially do so in 3 dimensions. This is the main reason why our analysis presently is limited to 2D models. This is in contrast to pore space reconstruction algorithms, which can be based on the total pore space including isolated pores (Schlüter and Vogel 2011).

Based on the types of sub-grains used, we have generated four classes of models, with circular, square, triangular, and a mixture of sub-grain types, respectively.

### 3.1 Representative Elementary Volume

The concept of permeability belongs in a continuum theory for flow in porous media. The porous medium, which is highly heterogeneous on the pore scale, is replaced by a *homogeneous* medium at a larger scale. The scale at which the porous medium is homogeneous is related to the concept of a representative elementary volume (REV). Only REV-sized models should be used in the analysis, and hence, we are seeking a REV criterion based on the Minkowski functionals.

*d*as the characteristic small-scale length. As seen in Fig. 5, the resulting criterion for the REV size in terms of \(\delta = L/d\) is strongly dependent on porosity. However, porosity is scale invariant, and a good criterion for determining the REV size should not depend strongly on it. While

*d*might be a good measure for small-scale structure for low porosities, at high porosities the characteristic scale is related to the distance between particles, and thus, the REV criterion should be related to the number of particles in a REV-size box. Forming a length scale using the parameters \(\sigma [\hbox {m}^{-1}]\), and \(\chi _s[\hbox {m}^{-1}]\), a general REV criterion is

*x*- and

*y*-direction and nonzero cross-components. Based on the calculated permeability tensor, we estimate the isotropic infinite-size permeability as

## 4 Results

*a*and

*b*in (16) without enforcing the Minkowski functionals (weights \(w_n=0\) in (15)). We have then subsequently generated pore spaces using

*different*shapes, sizes, and parameters while enforcing the Minkowski functional values. This produces sets of pore spaces that can be visually quite different but have the same characterization in terms of Minkowski functionals. The simulated parameter values are shown in Fig. 6. In total, 219 pore spaces were used in the analysis.

All pore spaces are generated as \(2024\times 2024\)-pixel pictures, and the permeabilities were calculated using the LIR solver for the stationary Stokes equation (Linden et al. 2015) implemented in the commercial GeoDict software (Math2Market 2018). The formation factors were calculated using a simple finite difference Laplace solver we have implemented in Python NumPy. Both solvers use periodic boundary conditions.

### 4.1 Formation Factor

*F*.

\(\mu = {2.6}\)

\(c_1={-1.7}\), \(c_2={0.79}\), \(c_3={-0.071}\), \(c_4={0.31}\), \(c_5={0.0015}.\)

The residual plot (Fig. 10) also shows a systematic trend for the triangle-based models, while the other models tend to spread within a \(\pm \, 20\%\) range in a random pattern. Based on these observation, we conclude that the formation factor is not a function of the Minkowski functionals alone, but for pore spaces built from sufficiently smooth sub-grains, the functionals may be used to determine *F* with an accuracy of 20%.

*m*that is a function of \(\lambda \) and \(\phi \):

### 4.2 Permeability

\(\alpha = {3.1},\)

\(c_1={-3.2}\), \(c_2={-0.019}\), \(c_3={1.8}\), \(c_4={0.059}\), \(c_5={-0.0026}.\)

Examples of permeability as a function of porosity for a fixed \(\lambda \) are shown in Fig. 12. We see that the models built from triangular and, to a lesser degree square, sub-grains fall outside of the anisotropy-based error estimates (21). Thus, the difference is significant and cannot be attributed to the models being smaller than a REV. The residual plot (Fig. 13) also shows a systematic trend for these models. The models with circular sub-grains tend to spread within a \(\pm 20\%\) range in a random pattern, while the models with square and mixed sub-grain types are predicted by the trend within \(\pm 40\%\). Based on these observation, we conclude that the permeability is not a function of the Minkowski functionals alone, but for pore spaces built from sufficiently smooth sub-grains the functionals may be used to determine *k* with an accuracy of 40%.

### 4.3 Describing Permeability in Terms of Electric Tortuosity

The electric tortuosity is defined by (3) as \(\tau _e = \phi F\) and is a more readily measurable quantity than the Euler characteristic. Since the permeability cannot be fully determined using the Minkowski functionals, we have investigated whether a better predictor might be found using \(\tau _e\) as an alternative to the dimensionless number \(\lambda \).

\(\alpha = {2.7}\), \(\beta = {0.34},\)

\(c_1={-3.3}\), \(c_2={2.1}\), \(c_3={-0.040}.\)

*k*with an accuracy of 45%.

### 4.4 Pore Spaces with Equally Sized and Shaped Grains

*K*and

*F*should, according to (10) and (12), be uniquely determined by porosity. We know that this is not the case, and the permeability is, for instance, dependent on lattice type for circular grains placed on a regular lattice (Gebart 1992). Thus, the validity of a Minkowski functional-based description is limited to classes of random models, especially when the porosity is close to the critical porosity where these regular models get jammed and, in 2D, permeability reach zero for nonzero \(\phi \). An approximate expression for the permeability of a system consisting of equally sized circular grains on a regular lattice is (Gebart 1992)In Fig. 15, we have plotted a number of numerically calculated permeabilities for circular grains on a regular lattice, and we see that (24) fails to give a good prediction for these systems. The prediction is much better for randomly distributed non-overlapping grains (green circles).

## 5 Conclusions

In order to investigate whether it is possible to derive good correlations for permeability an formation factor based on the Minkowski functionals, we have generated and analyzed two-dimensional computer-generated pore spaces covering a wide range of Minkowski functional value combinations. In general, due to Hadwiger’s theorem, any correlation based on any additive measurements cannot be expected to have more predictive power than those based on the Minkowski functionals.

Different surface roughness and microporosity will influence the functionals without significantly affecting the transport properties. However, also in the generated pore spaces, which do not contain micro porosity and rough surfaces, the Minkowski functionals do not determine permeability and formation factor uniquely.

Pore spaces with equally sized and shaped grains placed on a regular grid do not fall on the same trend as the random models, and models built using triangular sub-grains show a different trend than the others, possibly due to a different apparent surface roughness. Correlations that predict permeability and formation factor with an accuracy of 40% and 20%, respectively, for the other random models we have considered may, however, be found. Accuracy on the 40% level is considered very good in the context of predicting permeability from downhole measurements.

Permeability correlations where the Euler characteristic is replaced with the more readily measurable, and nonadditive, electric tortuosity may be found for a larger class of systems than the Minkowski functional-based correlations. The accuracy of these correlations is, however, found to be on the same level; we have derived a correlation accurate to 45%.

## Footnotes

- 1.
Note that in general permeability is a tensor, but in the present work we will only consider isotropic pore spaces so that permeability may be treated as a scalar. Hadwiger’s characterization theorem is not applicable to anisotropic pore spaces, which should be characterized in terms of Minkowski tensors instead of the Minkowski functionals (Schröder-Turk et al. 2013).

## Notes

### Acknowledgements

This study was funded by Research Council of Norway (Grant No. Centers of Excellence funding scheme, project number 262644, PoreLab.)

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