Edinburgh Research Explorer Linking elastic and electrical properties of rocks using cross-property DEM

Combining electrical and elastic measurements can be instrumental in lowering the uncertainty of subsurface characterisation. Many commonly used rock physics relations for joint electrical-elastic properties are at least partly empirical, and often rely on the estimation of porosity as an intermediate step. We combine differential effective medium schemes which relate respectively elastic and electrical properties to porosity and pore shape. The resulting expressions are independent of porosity, depending only on pore aspect ratio. Analysis of published joint electrical-elastic data shows that a single aspect ratio model performs well for clean sandstones, allowing us to model V p /V s ratios as a function of resistivity. Clay-bearing sandstones are more complex, but our modelling can still identify the correct trends. We speculate about the potential to extend our approach to produce additional cross-property relations.

Electrical-elastic multiphysics modelling, for example through the integration of controlled-source electromagnetic (CSEM) data into marine seismic reservoir characterisation workflows, can lead to reduced uncertainty in reservoir characterisation (Alcocer et al. 2013). However, a simple, physicsbased, and accurate multiphysics model which links a porous rock's electrical and elastic properties remains elusive. Many existing electrical-elastic relationships are at least partly empirical (Carcione et al. 2007). In addition to this, the majority of workflows which relate a rock's elastic and electrical properties require the estimation of porosity as an intermediate step (e.g., Sevostianov & Kachanov (2002); Carcione et al. (2007); Engelmark (2010); Werthmüller et al. (2013)), which can be uncertain away from well control. Carcione et al. (2007) presented a set of electrical-elastic cross-property models with no explicit porosity terms by substituting pre-existing resistivity-porosity models into pre-existing velocityporosity models; a method originally used by Sevostianov & Kachanov (2002), and more recently by Han (2018), who modelled anisotropic electrical-elastic measurements made on artificial porous sandstones with aligned fractures. Chen & Dickens (2009) and Werthmüller et al. (2013) assessed the intrinsic (theoretical) and extrinsic (parametrisation) uncertainties related to the models of Carcione et al. (2007), which were generally found to be significant. In fact, Kwon & Snieder (2011) showed the uncertainty in these multiphysics models had a larger contribution to overall uncertainty than that associated with the data. Electrical-elastic cross-property bounds exists which have no explicit porosity terms and are derived from first principles, such as the Hashin-Shtrikman bounds (Carcione et al. 2007), but these can be too widely spaced to predict the physical properties of any rock in particular.
Estimating the Earth's shear modulus is an important aspect of geophysics at many scales, from understanding the composition of the Earth's mantle (Kennett et al. 1998), to estimating reservoir fluids using V p /V s ratios (Hamada 2004), to geotechnical soil studies (Hussien & Karray 2015). Estimating a rock's shear modulus from its electrical resistivity, however, is scarcely addressed in the geophysical literature, and has been done predominantly using empirical methods (Jones et al. 2013;Yasir et al. 2018). The problem of modelling a rock's shear modulus and hence V p /V s ratio from electrical resistivity measurements alone by a simple model derived from first principles remains unsolved.
Joint electrical-elastic modelling with a single model parameter has been attempted with mixed success (Han et al. 2011a;Wang & Gelius 2010;Jensen et al. 2013). Notwithstanding, a scrutiny by Han et al. (2016) concluded a new multiphysics model was required to accurately relate the electrical and elastic properties of a porous rock using a single set of model parameters. Werthmüller et al. (2013) note that there is no known direct link between a rock's velocity and resistivity.
Recently, Han et al. (2020)  stone's pore shape and grain shape parameters, which are used in elastic and electrical differential effective medium (DEM) models respectively when modelling independently the elastic and electrical properties of rocks. In doing this, Han et al. (2020) answered the outstanding question of whether there was some unknown pore-to-grain shape correlation which would enable existing DEM models to become suitable for joint electrical-elastic modelling. Thus it was shown that standard DEM models can be used for cross-property modelling, however the method relies upon an empirical, laboratory-derived relation. It currently seems the standard DEM models do not allow for cross-property modelling without incorporating empirical relations in some way, and so their reformulation may be necessary to derive a cross-property DEM model from first principles.
In this paper, we present a new functional form of a DEM model which relates the electrical and elastic properties of an isotropic, porous rock. The model is derived using the technique of Sevostianov & Kachanov (2002) and Carcione et al. (2007), where an electrical model is substituted into an elastic model through the common independent variable, porosity. The theoretical advantage of the proposed model on existing models is that it is derived from first principles, has no porosity terms, is correct in the high and low porosity asymptotes, has a single, intuitive parameter, and can be expressed in terms of the geometrical functions Q ( * 2) , P ( * 2) , and R ( * 2) , proposed by Berryman (1980Berryman ( , 1995. We test the forward and inverse model on the wet, mixed sandstone core measurements of Han et al. (2011b) and show both are accurate in the case of clean sandstones and less accurate in the presence of clay for this data set. We also show the single model parameter, equivalent pore aspect ratio, is weakly sensitive to clay volume fraction in this data set. In addition to this, we demonstrate that one can model a V p /V s trend for this collection of mixed, wet sandstones using only electrical conductivity measurements and an empirical velocity-density relation once the model is parameterised, with accuracy comparable to the V p /V s model of Han et al. (1986).
To begin, we reformulate the electrical differential effective medium (DEM) model of Mendelson & Cohen (1982) for the case when ellipsoidal fluid-filled pores are embedded in a background of mineral matrix, contrasting with the typical expression which embeds grains into a background of water. Following this, we combine this electrical DEM expression with the analogous elastic DEM expressions of Berryman (1992) using the technique of Sevostianov & Kachanov (2002) Chapman content on the model's parametrisation and uncertainty. Following this, we predict a V p /V s trend for mixed, wet sandstones in the absence of porosity and density measurements using the modelled elastic moduli and the Gardner et al. (1974) velocity-density relation for sandstones. Finally, we solve the inverse problem numerically to estimate electrical resistivity from measured elastic moduli and assess the effect of clay content on the inverse solution.

Electrical modelling
Electrical inclusion modelling was first popularised by Maxwell Garnett (1904), who proposed an effective medium approximation to calculate the electrical properties of a material containing spherical inclusions. This was followed by Bruggeman (1935), who proposed an electrical DEM model for the same material. By devising a DEM model which embedded fractal-layered insulating spherical grains into a background initially made of water, Sen et al. (1981) derived Archie's (Archie 1942) first law with cementation exponent m = 1.5 and tortuosity a = 1.
Recognising it is unrealistic to approximate pores, grains, and inclusions in general with spheres, ellipsoidal inclusions became favoured when analytically modelling the electrical properties of composite materials due to their ability to approximate reality more accurately while still producing analytically tractable models. Polder & Van Santeen (1946) estimated the average electrical properties of an isotropic medium containing randomly oriented ellipsoidal inclusions which are assumed to be non-interacting, before Frank (1963) calculated the effective electric field in a medium containing a single homogeneous ellipsoid with principal axes arbitrarily aligned with respect to the incident electric field. Extending the work of Sen et al. (1981), Mendelson & Cohen (1982) developed a DEM model to calculate the effective electrical properties of a medium containing many arbitrarily oriented ellipsoidal inclusions in a uniform, static electric field.
Arguably the most renowned aspect of the DEM model of Mendelson & Cohen (1982), abbreviated to "M&C" henceforth, is a subsidiary result: by setting the grain conductivity to zero, fluid conductivity to that of water, and letting ellipsoidal grains be randomly oriented, M&C's electrical DEM model takes the form of Archie's (Archie 1942) first law: Parameter σ * is the rock's overall conductivity, σ w is the saturating water's conductivity, and m is the cementation factor. We note tortuosity factor a, a common coefficient of Archie's first law (Glover 2016), can be said to be unity in equation 1, as is also derived in the electrical DEM model of Sen Electrical-elastic modelling 5 et al. (1981) for spherical grains. Importantly, M&C showed m is a function of grain aspect ratio α, agreeing with observations that m depends on grain shape (e.g., Salem & Chilingarian (1999)).
M&C consider a rock with background material volume fraction (porosity) φ, background material conductivity σ 2 , and inclusion (i.e., grain) conductivity σ 1 . M&C's DEM model in its most general form for inclusions of a single aspect ratio is then: where angled brackets denote an average over all inclusion orientations and Θ 1 is a 3 × 3 matrix, such that: Matrix Θ 1 (Frank 1963) contains the effective depolarisation factors of an ellipsoidal inclusion of phase 1 which is arbitrarily rotated with respect to an incident electric field. Matrix L is the diagonal matrix with the ellipsoid's three depolarisation factors L p , p ∈ {1, 2, 3}, along its diagonal. Depolarisation factors L p (e.g., Osborn (1945)) are the shape-dependent scalars which map from the scalar component of an external electric field applied along the ellipsoid's p th axis, E p , to the ellipsoid's dipole moment along its p th axis, P p , through the relation: where V is the ellipsoid's volume. Matrix L has trace unity, and R is an orthogonal matrix. In the case of randomly oriented ellipsoidal grains, R = I and the rock's effective conductivity is isotropic, presuming an isotropic background conductivity. In this paper, we only consider the isotropic case and so set Θ 1 = L henceforth.
To derive equation 1, M&C make the approximation that grains are perfect resistors by setting σ 1 = 0 in equation 2 and obtain the differential equation: with cementation factor m: Taking the average of equation 6 over all grain orientations and integrating with the boundary condition σ * (φ = 1) = σ 2 yields equation 1 with σ 2 = σ w .
Our current aim is to reformulate the DEM model of equations 5 and 6 with reversed phases, embedding fluid-filled pores into a background of matrix material. To do this, we change the variable φ to (1 − φ) and interchange subscripts 1 and 2 in equation 2 to obtain: where we definem as:m Equations 7 and 8 are a reformulation of equations 5 and 6 respectively, here with pores embedded into a background of matrix material.
Changing the independent variable in equation 7, we obtain the reformulation of M&C's electrical DEM model with ellipsoidal pores embedded into a background of matrix material, which takes the form presented by Torquato & Haslach Jr (2002): By considering spheroidal inclusions, we drop the subscript p in depolarisation factors L p , calling the principal depolarisation factor L, and the two degenerate factors (1 − L)/2, as the trace of L is unity.
Parameterm from equation 8 for randomly oriented spheroidal inclusions is thus: Berryman (1995) proposed a geometrical function R ( * 2) which accounts for the effect of the aspect ratio of randomly oriented spheroidal inclusions on the effective electrical conductivity in the Clausius-Mossotti and self-consistent approximations. Function R ( * 2) is defined as: Where the superscript ( * 2) in R ( * 2) indicates the effective and inclusion properties are used for its evaluation. Thus, by equations 9 and 12, the DEM model of M&C can now be expressed in terms of the geometric function R ( * 2) : As an aside, we note the two-phase self-consistent electrical model for spheroidal pores of Berryman (1995) can be formulated in terms ofm: This may be of interest if a physically meaningful link betweenm and a rock's petrophysical properties is found, as exists with the analogous m (e.g., Salem & Chilingarian (1999)) but is not so obvious with the SCA model's typical geometrical function R ( * 2) .
We also see equation 9 reduces to the electrical DEM model of Bruggeman (1935) in the special case when L = 1/3. This result is expected as the model of Mendelson & Cohen (1982) is consistent with that of Sen et al. (1981) in the case of spherical inclusions, which in turn is consistent with that of Bruggeman (1935).
We only require equation 9 to obtain our proposed, joint electrical-elastic DEM expressions. For completeness, however, we integrate equation 9 here to obtain an expression analogous to Archie's first law as derived by M&C. For simplicity, we make the approximation thatm in equation 10 is constant in σ * by assuming σ * σ 2 , as is often the case in rocks with fluid-filled pores. However, equation 9 can be integrated analytically with σ * -dependentm as defined in equation 10 if desired.

Elastic modelling
Berryman (1992) presented a DEM model which estimates the effective elastic moduli of a dilute dispersion of randomly oriented ellipsoidal inclusions in an isotropic background material by the coupled equations: where K and µ denote bulk and shear moduli respectively and φ represents inclusion volume fraction. Sub-and superscripts 1 and 2 represent background and inclusion phases respectively, while Chapman * denotes the effective properties of the composite. Functions P ( * 2) and Q ( * 2) are defined explicitly by Berryman (1980) for the arbitrary spheroidal inclusion. The form of these functions follows from the solution of Eshelby (1957) for the deformation of an elastic ellipsoidal inclusion in a matrix under stress. In the differential effective medium approximation, functions P ( * 2) and Q ( * 2) depend on the effective moduli, the inclusion moduli, and the inclusion aspect ratio α. Analogous to the boundary condition of equation 7, equations 16 and 17 are solved with boundary conditions K * (φ = 0) = K 1 and µ * (φ = 0) = µ 1 .

Electrical-elastic modelling using cross-property DEM
Dividing equations 16 and 17 by equation 9 -that is, employing the chain rule -we obtain a set of coupled differential equations which relate the electrical and elastic properties of a two-phase, isotropic, linearly elastic, electrically conductive composite: Porosity is rendered a dummy variable when the chain rule is applied, meaning equations 18 and 19 contain no porosity terms.
By equation 12, equations 18 and 19 can be expressed in terms of P ( * 2) , Q ( * 2) , and R ( * 2) : The boundary conditions of equations 18 and 19, or 20 and 21, are: The key conceptual difference between these cross-property DEM expressions and a typical electrical or elastic DEM model is that in the cross-property method, the porosity added in each rockbuilding iteration is not dφ, but the unknown pore volume fraction which increments the rock's effective conductivity by dσ * . These cross-property DEM expressions are correct in both the high and low porosity limit even though porosity is rendered a dummy variable.
Electrical-elastic modelling 9  (2016), presents the additional physical properties which can be modelled with mathematical equivalence to the electrical conductivity inclusion problem. It follows from this equivalence that modelling using cross-property DEM also theoretically extends to any combination of two properties from Table 1, as well as a rock's elastic moduli and electrical conductivity.

Elastic Moduli Modelling
To test the proposed electrical-elastic DEM model, we use the laboratory data set of Han et al. (2011b).  Han et al. (2011a) these microcracks would increase the rock's equivalent inclusion eccentricity when compared to a rock at higher pressure. Open microcracks may also allow for squirt flow over a certain frequency band (Mavko & Jizba 1991;Dvorkin & Nur 1993;Chapman et al. 2002). However, these factors are not of concern in this study as all measurements analysed are made at constant pressure and frequency.
We study only measurements from the subset of 30 cores which have quartz content greater than 60% volume fraction. We refer to this subset of measurements as the "Han" data henceforth. By considering only these measurements, we ensure mineralogy is quartz-dominated. Accordingly, we model all data assuming samples have a pure quartz matrix. The clay volume fraction is non-negligible in many samples, as is shown in Figure 1. By modelling the properties of clay-bearing rocks with the assumption of pure quartz matrix, we can test the impact of clay content on the model's parametrisation.
All electrical and elastic parameters used in modelling are shown in Table 2.
To investigate the effects of clay on the electrical-elastic model, we partitioned the Han data into clean and clay-bearing data by arbitrarily choosing a threshold of 3.5% clay volume fraction, as shown in Figure 1. This is of interest as the electrical properties of a rock are affected by the presence of clay through the double layer effect (Waxman & Smits 1968), which the proposed electrical-elastic model does not account for. In addition to this, the elastic moduli of a sandstone can also be significantly affected by only a small amount of clay (Han et al. 1986).
We refer to the model's parameter α as the equivalent pore aspect ratio (EPAR) when applied to real data, following Fournier et al. (2011Fournier et al. ( , 2014Fournier et al. ( , 2018 and Cilli & Chapman (2020). We differentiate between "EPAR" in applied inclusion modelling and "inclusion aspect ratio" in theoretical inclusion modelling because real pores are not ideal spheroids, while the theoretical model's inclusions are.
To test the electrical-elastic model's performance on the Han data, we first invert equations 18, 19, 22, and 23 for the EPAR associated with each core sample, using the parameters in Table 2 and the measured elastic and electrical data. Rather than solving for a single α which satisfies both bulk and shear modulus measurements in some way, we solve for both the "bulk modulus EPAR", α K , and "shear modulus EPAR", α µ , for each sample, by minimising the Root Mean Squared Error (RMSE), which is defined as the square root of the mean of each modelling residual squared. Fournier et al. (2011Fournier et al. ( , 2014Fournier et al. ( , 2018 propose α K and α µ are the EPARs which minimise the misfit between the modelled and measured bulk modulus and shear modulus data respectively. In an ideal composite material with a dilute dispersion of spheroidal inclusions, α K and α µ of the medium are expected to be equal (Eshelby 1957). In reality, deviations from the ideal spheroid, or "pore asperities", cause α K and α µ to be generally unequal, as observed by Fournier et al. (2011Fournier et al. ( , 2014Fournier et al. ( , 2018 and Cilli & Chapman (2018. We display the inverted EPARs, α K and α µ , for each sample, with their means, α K 0 and α µ 0 , and 95% confidence intervals in Figures 2 and 3. We also show inversion statistics in Table 3, including the number of samples in each data subset and the standard deviation of the inverted α K and α µ for each data subset. We allow the inversion algorithm to search over the full model space α ∈ (0, ∞) and, interestingly, find it consistently favours modelling with prolate spheroidal pores, where α > 1.
To calculate the optimal model parameters, α * K and α * µ , we used the average solutions, α K 0 and α µ 0 , as starting points in non-linear global optimisations which minimised the bulk or shear modulus misfit for all samples. These optimal EPARs are also shown in Table 3. There is a 1.8% difference in α * K and a 4.7% difference in α * µ between clean and clay-bearing model parametrisations. Thus, we   conclude α * K and α * µ are weakly influenced by clay volume fraction in this data set. The ratio of α * K and α * µ in this electrical-elastic DEM modelling exercise is also shown in Table 3, and interestingly is not unity, which is an observation previously made only in purely elastic modelling settings.
We forward modelled resistivity-bulk modulus and resistivity-shear modulus trends using α * K and α * µ respectively, as shown in Figure 4. The propagated 95% confidence intervals from the sampleby-sample EPAR inversions of Figures 2 and 3 are also shown. The joint electrical-elastic Hashin-Shtrikman bounds (Carcione et al. 2007) are displayed, with the proposed electrical-elastic modelling method obeying these bounds in all examples. Being asymptotically correct, the electrical-elastic DEM model converges to the water moduli at 100% porosity and the resistivity of water, and converges to the matrix moduli at zero porosity and the resistivity of matrix material. We note the matrix resistivity is not shown on Figure 4 and so the low-porosity, asymptotic modulus values (red diamonds) do not coincide with the forward modelled trend line in the figure.

V p /V s Modelling
The ratio of a rock's P-and S-wave velocities is of importance in reservoir characterisation as different fluids can have similar shear wave velocities but vastly different compressional velocities. The V p /V s ratio of any given rock is also highly sensitive to the rock's porosity due to the often-large differences between pore fluid and mineral velocities. Using the electrical-elastic model presented above, we now predict a rock's V p /V s ratio using only electrical measurements, an empirical velocity-density relation, and the model's known parameters, assumed a priori to be pre-calibrated.
We consider modelling a V p /V s ratio trend through the entire Han data set. The first step is modelling elastic moduli trends (Figures 4a,b) using parameters α * K = 16.4 and α * µ = 12.8 (Table 3). V p and V s can then be calculated using the linear elasticity equations and the empirical Gardner (Gardner et al. 1974) relation for sandstones: We use an empirical density-velocity relation here to demonstrate how a V p /V s ratio trend can be estimated solely using electrical measurements. However, the use of reliable density measurements, if possible, should lead to more accurate V p /V s predictions.  Figure 4. The elastic measurements (circles) of the Han data set and its clean and clay-bearing sample subsets are modelled by solving equations 20 and 21 (red trends) using optimal aspect ratios shown in Table 3. Bulk modulus (left column) and shear modulus (right column) are modelled using parameters α * K and α * µ respectively. The 95% confidence intervals on initial EPAR estimates, shown in Figures 2 and 3 (1986) is shown for comparison. The empirical trend provides a better fit to the data at lower velocities, however it is calibrated on many samples, with shear velocities ranging from less than 1500 m/s to over 3500 m/s. Because of this severe difference in model calibration, we cannot conclude the empirical trend of Han et al. (1986) is actually a better model than the proposed electrical-elastic model.

Electrical Conductivity Modelling
Having shown a rock's elastic moduli can be modelled from electrical conductivity measurements using cross-property DEM, we now demonstrate a solution to the inverse problem: modelling electrical conductivity from elastic moduli measurements.
We consider the case where a rock's measured bulk and shear moduli are known, as well as the elastic moduli and electrical conductivities of its matrix and pore fluid. We assume α K and α µ are known a priori. For this example, we also assume a priori that α K and α µ remain constant in both the forward and inverse mapping of equations 20 and 21. Hence we invert equations 20 and 21 for electrical conductivity using a rock's measured elastic moduli, the known physical properties of the matrix and pore fluid, and a known aspect ratio. As in the case of elastic modelling, we do not require knowledge of the rock's porosity to estimate electrical resistivity from elastic moduli measurements.
Rather than attempting to invert the strongly non-linear coupled differential equations 20 and 21 analytically, we use a numerical approach. Just as we solved for two EPARs, α K and α µ , by minimising misfit between modelled and measured bulk and shear modulus respectively, we now Electrical-elastic modelling 17 Table 4. Root mean square error (RMSE) for modelling σ K and σ µ in the Han data set and its clean and clay-bearing subsets.
The other solves for a conductivity, σ µ , by minimising the misfit in measured and modelled shear modulus using model parameter α µ . Figure 6 demonstrates the inverted resistivities calculated for each subset of the Han data using each sample's experimentally measured bulk and shear modulus, with lines connecting individual inversion results. The 95% confidence intervals on α K and α µ are also propagated to resistivities for each sample and connected by dashed lines. Table 4 shows the RMSE for each modelling scenario.
In the presence of clay, the lower scatter in measured resistivity-shear modulus data than in resistivity-bulk modulus data leads to a more accurate (lower RMSE) prediction of sample resistivity through shear modulus inversion than bulk modulus inversion. That is, σ µ is a more accurate predictor of measured σ than σ K in the Han data when at least some samples have clay. The modelling of the clay-rich data is less compelling than that of the clean sandstone data. This could be for a range of reasons, including the inappropriate modelling mineralogy and the double layer effect. In the case of clean samples ( Fig. 6c and 6d), σ K is a more accurate prediction of measured σ than σ µ , however concluding which model is preferable with confidence is hampered by the small number of samples in this data subset.

DISCUSSION
Cross-property DEM modelling has a number of advantages over other cross-property models. Namely, the model is theory-based, has no porosity terms, has a single, intuitive model parameter, is correct in the high and low porosity asymptotes, and relates inclusion modelling geometrical functions P ( * 2) , Q ( * 2) , and R ( * 2) . We have shown the model to work on a single data set in this study, however its practical advantage is yet to be seen. A practical advantage may include characterising a rock's physical properties with fewer measurements, for example estimating a rock's bulk and shear modulus in the absence of V p , V s , density, and porosity measurements.
The key advantage of the cross-property DEM model is the consistency of the physical assump- (d) Figure 6. Core measurements (circles) of the Han data set (top row) and its clean (middle row) and clay-bearing (bottom row) sample subsets are modelled by inverting equations 20 and 21 (red diamonds) using optimal aspect ratios shown in Table 3. Resistivities 1/σ K (left column) and 1/σ µ (right column) are modelled using parameters α * K and α * µ respectively. The 95% confidence intervals on initial EPAR estimates are propagated (black points). Measured sample porosity (marker colour) is also shown for context. Lines connect individual inversion results to obviate possible trends.
Electrical-elastic modelling 19 tions behind the two methods we have combined. In a more empirical approach, this is not necessarily the case.
The proposed electrical-elastic DEM modelling method's preference to approximate pores using prolate spheroids contrasts with typical elastic modelling flows, where pores are often represented by oblate spheroids. Modelling elastic properties with oblate spheroidal inclusions in a DEM framework coincides with the lower Hashin-Shtrikman bound (Norris 1985), while modelling with prolate spheroids does not. This may suggest that prolate spheroids may only be preferable for electricalelastic modelling so long as the rock's elastic moduli are sufficiently high. We note the data and modelled trends in Figures 4 and 6 all lie within the Hashin-Shtrikman bounds. Additional work on cross-property bounds has been discussed by Gibiansky & Torquato (1996)  The scatter in inverted EPARs (Figures 2 and 3) for clay bearing samples is significantly lower in the inversion for α µ than for α K . This suggests there is higher modelling uncertainty due to clay content in estimating the effective bulk modulus than shear modulus. This difference in uncertainty is a result of the difference in correlation between measured K and σ, and measured µ and σ, as shown in Figures 4 and 6. In addition to this, Table 3 shows the standard deviation in inverted α µ 0 for clean samples and the full data set are very similar. It therefore seems the uncertainty in modelled shear modulus is weakly sensitive to clay content in the Han data.
The presented model accounts for an isotropic stiffness and conductivity, where the electrical and elastic properties are fully described by scalar moduli, rather than electrical resistivity and stiffness tensors. Extending this model to the anisotropic case may be useful, as Singh et al. (2020) showed that elastic EPAR differed depending on the direction of sonic measurement in anisotropic, digital rocks. This anisotropic extension of the model could be of interest as North & Best (2014) showed that even visually isotropic sandstones can have up to 25% anisotropy due to syn-depositional and post-depositional compaction processes.
We present here the basis for a functional form which maps from any of six physical properties (elastic moduli, electrical resistivity, and those in Table 1)  cation of some of these mappings to be in other scientific fields, for example in the engineering or materials sciences.
In the unrealistic scenario of an infinite, isotropic, homogeneous material containing a dilute dispersion of isotropic, homogeneous, randomly oriented ellipsoidal inclusions acted on by a low frequency incident field with its source located at infinity, inclusion modelling theory would suggest a single aspect ratio will predict all properties from all other properties discussed in this paper. We have shown on one data set of clean, wet sandstones that the forward and inverse electrical-elastic cross-property DEM models are both accurate when using a single aspect ratio. However, the disparity between α K and α µ in real rocks is seen here and otherwise well documented. This raises the question of whether a single aspect ratio can predict all properties from all other properties in reality.

CONCLUSION
By combination of differential effective medium schemes, we derive new relationships between elastic and electrical properties of composite media. The relations are independent of porosity but correct in the high and low porosity limits, depending only on aspect ratio. Fitting this single aspect ratio allows us to accurately model joint elastic and electrical properties of clean sandstones. There is more scatter in the case of clay-bearing rocks but our model identifies the correct trend. We suggest this scatter may be related to a wider range of pore geometries, the double layer effect, or