Acoustic and Electrical Properties of Tight Rocks: A Comparative Study Between Experiment and Theory

Abstract

The acoustic-electrical (AE) response of subsurface hydrocarbon reservoirs is highly affected by rock heterogeneity. In particular, the characterization of the microstructure of tight (low-permeability) rocks can be aided by a joint interpretation of AE data. To this purpose, we evaluate cores from a tight-oil reservoir to obtain the rock mineralogy and pore structure by X-ray diffraction and casting thin sections. Then, ultrasonic and resistivity experiments are performed under different confining pressures to analyze the effects of pores, microcracks and mineralogy on the AE properties. We have developed acoustic and electrical models based on effective-medium theories, and the Cole–Cole and triple-porosity equations, to simulate the response to total and soft (crack) porosities and clay content. The results show that these properties play a significant role. Then, a 3D rock-physical template is built and calibrated by using the core samples and well-log data. The template is applied to tight-oil reservoirs to estimate the rock properties, which are validated with log data. The good match between the predictions and these data indicates that the model can effectively explain the effects of the heterogeneous microstructure on the AE data.

Article Highlights

• Tight rock microstructure is analyzed with X-ray diffraction, thin sections and ultrasonic and electrical resistivity tests

• Rock acoustic-electrical properties are obtained by the effective-medium and triple-porosity theories

• Practical application is given based on an acoustic-electrical rock physics template

1 Introduction

With the depletion of conventional oil resources, tight-oil reservoirs become the next hydrocarbon prospects (Kathel and Mohanty 2013; Ghanizadeh et al. 2015; Behmanesh et al. 2018; Lin et al. 2022). Tight-oil rocks have a complex pore structure and mineral composition, with a high clay content (Lu et al. 2019; Pang et al. 2021a; Gao et al. 2022). Pores and cracks, as well as clay interstitial minerals, result in a strong microstructure heterogeneity (Siitari-Kauppi et al. 1997; Solano et al. 2017; Ba et al. 2017, 2019; Kumar et al. 2019; Ghasemi et al. 2020; Ma et al. 2021), thus greatly affecting the reservoir response to AE fields (Kozlovskaya and Hjelt 2000; Kazatchenko 2004; Bakar, et al. 2019; Cilli and Chapman 2018; Dutilleul et al. 2020; Leger and Luquot 2021).

Microstructure heterogeneity results in a diversity of petrophysical behaviors (Anguy et al. 1996; Nover 2005; Heise and Ellis 2016; Pang et al. 2019; Mahanta et al 2020; Iqbal et al. 2021; Ozotta et al. 2021; Iwamori et al. 2021; Zhang et al. 2021, 2022; Shen et al. 2022; Wu et al. 2022), including infiltration, acoustic velocity and attenuation, and electrical properties. Sun et al. (2019) built 3D digital cores of tight sandstones with different crack diameters, lengths and dip angles using computed tomography (CT) and studied the effects of the pore geometry on the infiltration characteristics. Fliedner and French (2021) used a differential-equivalent-medium (DEM) model and rock images to quantify the contribution of mineralogy, mineral fabric, porosity and pore shape to the P- and S-wave velocities. Pang et al. (2021a) studied the pore structure and mineralogy of tight-oil rocks by scanning electron microscopy (SEM) and ultrasonic experiments and concluded that the microstructure greatly affects the inelastic properties. More recently, Wang et al. (2022) obtained multi-component digital rocks based on CT scans, maps and SEM multi-scale experiments, estimating the resistivity by a finite-element method and studying the influence of the pore geometry and mineral composition on the electrical properties.

Joint AE properties are generally applied to estimate the characteristics of subsurface rocks (Heinson 1999; Hacikoylu et al. 2006; Carcione et al. 2007; Han et al. 2011, 2020; Jensen et al. 2013; Gomes et al. 2017; Cilli and Chapman 2018; Attias et al. 2020; Amalokwu and Falcon-Suarez 2021). Many experimental and theoretical works estimate the effect of pore structure and mineralogy on the AE properties of rocks (Gabas et al. 2016; Falcon‐Suarez et al. 2019; Cilli and Chapman 2020). Gomez et al. (2010) experimentally established relations among resistivity, elastic velocity, porosity and permeability of sandstones and used the self-consistent approximation to simulate the AE properties of rocks. Pride et al. (2017) studied changes in geophysical properties caused by fluid injection, specifically seismic velocity and conductivity of sandstone and granite samples. Wu et al. (2020) proposed a hybrid stochastic (HS) algorithm, combined the quartet structure generation set and discrete finite-element method to establish multi-scale and multi-composition digital models and simulated the rock resistivity and elastic moduli. Pang et al. (2021b) established a 3D rock physics model (RPM) to simulate the effect of mineral composition, pore structure and fluid saturation on AE properties and calibrated the model with well-log data.

In order to characterize the rock microstructure, we extract cores from a tight-oil reservoir of Yanchang Formation in the Q Area, Ordos Basin, China. Then, X-ray diffraction (XRD), casting thin sections (CTS), ultrasonic wave and electrical resistivity experiments under different confining pressures are performed. The pore structure and mineralogy of the rocks is analyzed, and the relation between the AE properties and porosity, pressure (crack) and mineral composition is studied. Based on the mineral composition and thin section analysis, we use a triple-porosity model to describe the acoustic properties of the rock (Sun et al. 2016; Ma et al. 2019).

In the acoustic RPM, the Hashin–Shtrikman equation (Hashin and Shtrikman 1963) is used to estimate the elastic modulus of the mineral mixture, and the acoustic DEM (A-DEM) equation (Berryman 1992) to add clay minerals, pores and cracks into the minerals. The final (saturation) model is obtained with the triple-porosity equation. We implement the electrical HS (E-HS) (Hashin and Shtrikman 1962), DEM (E-DEM) (Cilli and Chapman 2021) equations and Cole–Cole model (Cole and Cole 1941; Pelton et al. 1978) to estimate the electrical properties of the mineral mixture and rock frame containing the clay minerals, pores and cracks to obtain the electrical RPM. Thus, the AE models with the same microstructure are used to analyze the response to porosity, crack porosity and clay content. A 3D AE rock physics template (RPT) is built with the elastic attributes and resistivity and calibrated with core and well-log data and applied to the tight-oil reservoirs.

2 Geological Characteristics and Rock Microstructure

The prospect is located in the Q area of Ordos Basin, which has a complex tectonic history and warm and humid climate, resulting in the deposition of hydrocarbon source rocks rich in organic matter (Liu et al. 2021). The burial depth of Member 7 of the Yanchang Formation is located between 1200 and 2350 m. The sediments of the tight-oil reservoirs in this formation are mainly lacustrine facies with high-quality source rocks widely distributed. The reservoir lithology is mainly tight sandstone intercalated with mudstone and mud shale intercalated with thin sandstone. The sandstone layer is the main section of the reservoir in Member 7, which is characterized by developed microcracks and micro-/nanopore throats. The pore radius is mainly 2–8 μm. The pore structure favors the reservoir capacity and contributes to the hydrocarbon accumulation (Fan et al. 2021; Ji et al. 2022).

We select 9 core samples (A–I) from the reservoirs to perform the CTS and XRD experiments. Figure 1 shows the thin sections of sample I. The reservoir space is mainly dominated by intergranular and dissolved pores and microcracks. The interstitial materials in the rocks are mainly clay and a small amount of cement such as carbonate and siliceous minerals. The existence of pores, cracks and fillings results in strong heterogeneity of the microstructure. Figure 2 shows the mineral compositions of the samples. The minerals are mainly quartz, feldspar, carbonate minerals, clay and a small amount of ore mineral. The quartz content is high, and feldspar is mainly plagioclase and potassium feldspar, where the plagioclase content is higher. The clay minerals are mainly laumontite and chlorite, and ore minerals are siderite and pyrite.

Fig. 1

Thin sections of the tight-oil rock sample I, showing the intergranular and dissolved pores, microcracks and interstitial materials. Figure 1a, b shows thin section I in plane polarized light (PPL), and 1c and 1d in cross-polarized light (XPL), respectively

Fig. 2

Mineral components of the rock samples. Please note that pyrite occurs only in Sample B with 0.35%

3 Acoustic and Electrical Properties of the Tight-Oil Rocks

3.1 Laboratory Experiments

Nine samples with different porosity and clay content are used to perform the ultrasonic and electrical experiments. The contribution of microcracks (soft pores) cannot be ignored. Experiments and theories have proved that microcracks are highly affected by effective pressure (David and Zimmerman 2012; Zhang et al. 2019). Thus, ultrasonic-wave and resistivity measurements are performed to analyze the influence of cracks on the AE properties by applying different confining pressures. Cylindrical rock specimens with a diameter of 25.077–25.147 mm and a length of 48.803–50.093 mm are extracted from the initial rock samples (see Table 1).

Table 1 Properties of the samples

Firstly, the ultrasonic experiments are carried out. The equipment (Fig. 3) is used to measure the elastic velocities by the pulse method at 25 °C and a pore pressure of 15 MPa within ultrasonic frequency range (MHz). The samples are saturated with water and then sealed with a rubber sleeve and placed in the equipment. Confining pressures of 20, 30, 40, 50 and 60 MPa are applied to the specimens, and the waveforms are recorded. The elastic velocities are calculated from the first arrivals.

Fig. 3

Ultrasonic equipment. a Transmitter; b Specimen; and c Receiver

Then, based on the two-electrode method (Fig. 4) and resistivity experimental apparatus shown in Fig. 5, the electrical resistivity is measured with an alternating current at 120 Hz and a voltage of 1 V. The samples are saturated with brine (the salinity is 56.5 g/L). The above confining pressures are applied to the specimens to measure the resistivity ρ (reciprocal of electrical conductivity) which can be computed as

$$ \uprho = \frac{{RS}}{L}$$

(1)

where R is the resistance, S is the cross-sectional area and L is the sample length.

Fig. 4

Schematic diagram of the resistivity measurement principle, A is electric current, ΔU is voltage (potential difference), and M and N are electrodes

Fig. 5

Experimental apparatus to measure rock resistivity. M and N denote electrodes

3.2 Experimental Results

The effects of the microstructures on the AE properties of tight rocks are analyzed based on the experiments. Figure 6a, b shows the ultrasonic P- and S-wave velocities of the samples as a function of the effective pressure (confining minus pore). The velocities increase with pressure. This is due to the microcracks gradually closing, increasing the elastic moduli and velocity of the skeleton. Due to the same reason, the resistivity increases and the conductivity decreases (Fig. 6c, d). The results show that the crack density greatly affects the AE properties.

Fig. 6

Ultrasonic P- and S-wave velocities a, b, resistivity c and conductivity d as a function of the effective pressure

In the reservoir, the pore and effective pressures are both about 15 MPa. Then, we select the experimental data of 30 MPa confining pressure to analyze the AE properties with porosity and clay content under in-situ conditions. Figure 7 shows the P- and S-wave velocities, Poisson’s ratio, resistivity and conductivity as functions of porosity and clay content at an effective pressure of 15 MPa, respectively. The elastic attributes (velocities and Poisson’s ratio) have no clear relations with porosity due to the influence of the clay minerals. However, there is a relatively good correlation between the electrical properties and porosity. The resistivity of the samples basically decreases with increasing porosity, and the conductivity increases as expected. In Fig. 7b, d and f, there is no evident trend between the elasticity and electrical behaviors and the clay content. It can be inferred that the AE properties are affected by the combination of pore structure and clay minerals.

Fig. 7

Ultrasonic P- and S-wave velocities (blue and red) a, b, Poisson's ratio c, d and electrical resistivity and conductivity (blue and red) e, f of the samples with an effective pressure of 15 MPa as a function of porosity and clay content. The symbols are corresponding to Fig. 6

We use 3D schematic diagrams to analyze the behavior of the AE properties with porosity and clay content, as shown in Fig. 8, where the colors represent different samples. The velocities decrease with porosity and clay content (Fig. 8a, 8b). Sample A shows the highest velocity due to its low porosity and clay content. Samples B and C have also have low porosity, but low velocity because of the high clay content. The other samples have similar clay content and the velocities are well correlated with porosity.

Fig. 8

Ultrasonic P- and S-wave velocities a, b, Poisson’s ratio c, resistivity d and conductivity e as a function of porosity and clay content

Poisson's ratio shows a poor correlation with both porosity and clay content (Fig. 7c, 7d); however, the trend is better in Fig. 8c. Poisson’s ratio decreases with porosity and increases with clay content when the effect of one of them is removed. This is why the Poisson’s ratio of the sample I with high porosity and low clay content is the lowest. In Fig. 8d, 8e, samples A and D to I with similar clay content show that the electrical properties are well correlated with porosity. As porosity increases, resistivity decreases and conductivity increases. Similarly, the behavior of samples A, B and C, with similar porosity, indicates that the resistivity decreases with clay content, which explains the divergences between the electrical attributes and porosity of sample B in Fig. 7e due to its high clay content.

4 Rock Physics Models (RPM)

Figure 9a shows a thin section of a tight-oil core, with intergranular pores, microcracks and interstitial fillings containing laumontite and chlorite, which results in strong heterogeneities. The rock microstructure can be modeled by three isotropic frames, associated with pores, cracks and clay minerals (Fig. 9b). Thus, we consider a triple-porosity structure (Fig. 9c) to build the AE RPMs of the tight-oil rock.

Fig. 9

Tight-oil rock: a thin section, b schematic diagram of the microstructure and c triple-porosity model (v₁, v₂ and v₃ are the volume content of the crack inclusion, host and clay inclusion, respectively)

4.1 Acoustic RPM

Figure 10 shows the modeling sequence of the acoustic RPM. Firstly, on the basis of XRD experiments, the rock minerals are mainly composed of quartz, feldspar, clay, calcite, dolomite and a small amount of ore mineral. The A-HS (Hashin and Shtrikman 1963) equation is used to calculate the bulk and shear moduli of the mineral mixture (excluding the clay minerals). Berryman (1995) proposed a more general form of the bounds for more than two phases (Mavko et al. 2009),

$$\begin{array}{*{20}l} {K^{{{\text{HS}} + }} = \Lambda \left( {\mu_{\max } } \right)} \hfill & {K^{{\text{HS - }}} = \Lambda \left( {\mu_{\min } } \right),} \hfill \\ {\mu^{{{\text{HS}} + }} = \Gamma \left( {\zeta \left( {K_{\max } ,\mu_{\max } } \right)} \right),} \hfill & {\mu^{{{\text{HS}} - }} = \Gamma \left( {\zeta \left( {K_{\min } ,\mu_{\min } } \right)} \right),} \hfill \\ \end{array}$$

(2a)

$$\Lambda \left( z \right) = \left( {\sum\limits_{i = 1}^{{\text{N}}} {\frac{{f_{i} }}{{K_{i} + \frac{4}{3}z}}} } \right)^{ - 1} - \frac{4}{3}z,$$

(2b)

$$\Gamma (z) = \left( {\sum\limits_{i = 1}^{{\text{N}}} {\frac{{f_{i} }}{{\mu_{i} + z}}} } \right)^{ - 1} - z,$$

(2c)

$$\zeta \left( {K,\mu } \right) = \frac{\mu }{6}\left( {\frac{9K + 8\mu }{{K + 2\mu }}} \right),$$

(2d)

where K_i and µ_i are the bulk and shear modulus of the single phases and f_i is the corresponding volume fraction.

Fig. 10

Modeling flowchart of the acoustic RPM

Stiff pores and cracks are added into the host mineral and inclusion, with aspect ratios of 1 and 0.001, respectively, by using the A-DEM equation (Berryman 1992). Then, clay inclusions (with few pores), whose aspect ratio is 1, are added into the host frame by using the A-DEM equation to obtain the dry-rock moduli, K^* and μ^*,

$$(1 - y)\frac{{\text{d}}}{{{\text{d}}y}}\left[ {K^{*} (y)} \right] = \left( {K_{2} - K^{*} } \right)P^{(*2)} (y),$$

(3a)

$$\left( {1 - y} \right)\frac{{\text{d}}}{{{\text{d}}y}}\left[ {\mu^{*} (y)} \right] = \left( {\mu_{2} - \mu^{*} } \right)Q^{(*2)} \left( y \right),$$

(3b)

With K^∗ (0) = K₁, G^∗ (0) = G₁, where K₁ and G₁ are the bulk and shear moduli of the host phase, and K₂ and G₂ those of phase 2, y is the corresponding content, and P^*i and Q^*i represent the geometrical factors of the ith component (see Appendix 1).

The bulk modulus and density of the fluids at reservoir conditions are estimated as in Batzle and Wang (1992). Finally, the triple-porosity equation (see Zhang et al. 2017) is used to obtain the wet-rock properties (all the pores are filled with fluids),

$$\begin{aligned} N\nabla^{2} {\mathbf{u}} + \left( {A + N} \right)\nabla e + Q_{1} \nabla \left( {\xi_{1} + \phi_{2} \zeta_{12} } \right) + Q_{2} \nabla \left( {\xi_{2} - \phi_{1} \zeta_{12} + \phi_{3} \zeta_{23} } \right) + Q_{3} \nabla \left( {\xi_{3} - \phi_{2} \zeta_{23} } \right) \\ = \rho_{00} {\mathbf{\ddot{u}}} + \rho_{01} {\mathbf{\ddot{U}}}^{\left( 1 \right)} + \rho_{02} {\mathbf{\ddot{U}}}^{\left( 2 \right)} + \rho_{03} {\mathbf{\ddot{U}}}^{\left( 3 \right)} + b_{1} \left( {{\dot{\mathbf{u}}} - {\dot{\mathbf{U}}}^{\left( 1 \right)} } \right) + b_{2} \left( {{\dot{\mathbf{u}}} - {\dot{\mathbf{U}}}^{\left( 2 \right)} } \right) + b_{3} \left( {{\dot{\mathbf{u}}} - {\dot{\mathbf{U}}}^{\left( 3 \right)} } \right), \\ \end{aligned}$$

(4a)

$$Q_{1} \nabla e + R_{1} \nabla \left( {\xi_{1} + \phi_{2} \zeta_{12} } \right) = \rho_{01} {\mathbf{\ddot{u}}} + \rho_{11} {\mathbf{\ddot{U}}}^{\left( 1 \right)} - b_{1} \left( {{\dot{\mathbf{u}}} - {\dot{\mathbf{U}}}^{\left( 1 \right)} } \right),$$

(4b)

$$Q_{2} \nabla e + R_{2} \nabla \left( {\xi_{2} - \phi_{1} \zeta_{12} + \phi_{3} \zeta_{23} } \right) = \rho_{02} {\mathbf{\ddot{u}}} + \rho_{22} {\mathbf{\ddot{U}}}^{\left( 2 \right)} - b_{2} \left( \mathbf{{\dot{u}} - \dot{U}^{\left( 2 \right)} } \right),$$

(4c)

$$Q_{3} \nabla e + R_{3} \nabla \left( {\xi_{3} - \phi_{2} \zeta_{23} } \right) = \rho_{03} {\mathbf{\ddot{u}}} + \rho_{33} {\mathbf{\ddot{U}}}^{\left( 3 \right)} - b_{3} \left( {{\dot{\mathbf{u}}} - {\dot{\mathbf{U}}}^{\left( 3 \right)} } \right),$$

(4d)

$$\begin{aligned}& \frac{1}{3}\rho_{f} R_{12}^{2} \ddot{\zeta }_{12} \phi_{2}^{2} \phi_{1} \left( {\frac{1}{5} + \frac{{\phi_{10} }}{{\phi_{20} }}} \right) + \frac{1}{3}\left( {\frac{\eta }{{5\kappa_{1} }} + \frac{\eta }{{\kappa_{2} }}} \right)R_{12}^{2} \dot{\zeta }_{12} \phi_{2}^{2} \phi_{1} \phi_{10} \\ &\quad = \phi_{2} \left( {Q_{1} e + R_{1} \left( {\xi_{1} + \phi_{2} \zeta_{12} } \right)} \right) - \phi_{1} \left( {Q_{2} e + R_{2} \left( {\xi_{2} - \phi_{1} \zeta_{12} + \phi_{3} \zeta_{23} } \right)} \right), \\ \end{aligned}$$

(4e)

$$\begin{aligned}& \frac{1}{3}\rho_{f} R_{23}^{2} \ddot{\zeta }_{23} \phi_{2}^{2} \phi_{3} \left( {\frac{1}{5} + \frac{{\phi_{30} }}{{\phi_{20} }}} \right) + \frac{1}{3}\left( {\frac{\eta }{{5\kappa_{3} }} + \frac{\eta }{{\kappa_{2} }}} \right)R_{23}^{2} \dot{\zeta }_{23} \phi_{2}^{2} \phi_{3} \phi_{30} \\&\quad = \phi_{3} \left( {Q_{2} e + R_{2} \left( {\xi_{2} - \phi_{1} \zeta_{12} + \phi_{3} \zeta_{23} } \right)} \right) - \phi_{2} \left( {Q_{3} e + R_{3} \left( {\xi_{3} - \phi_{2} \zeta_{23} } \right)} \right), \\ \end{aligned}$$

(4f)

where $\bf {\text{u}}$ denotes the solid displacement vector, ${\dot{\mathbf{U}}}^{(1)}$, ${\dot{\mathbf{U}}}^{(2)}$ and ${\dot{\mathbf{U}}}^{(3)}$ are the displacement vectors of the fluids in cracks, pores and clay micropores, respectively, $e$, $\xi_{1}$, $\xi_{2}$ and $\xi_{3}$ are the corresponding divergences, and $\zeta_{12}$ and $\zeta_{23}$ are the variations in fluid content owing to the local fluid flow between cracks and intergranular pores and between clay and intergranular pores, respectively. The quantities $\kappa_{1}$, $\kappa_{2}$ and $\kappa_{3}$ are the permeabilities of the crack, host and clay-mineral frames, respectively, $R_{12}$ and $R_{23}$ are the radii of the crack and clay inclusions, respectively. The volume ratio and porosity of the crack are $v_{1}$ and $\phi_{10}$, while those of the host skeleton are $v_{2}$ and $\phi_{20}$, and those of the clay inclusion are $v_{3}$ and $\phi_{30}$, respectively; v₁ + v₂ + v₃ = 1, and $\phi_{1}$, $\phi_{2}$ and $\phi_{3}$ are the absolute porosities of the crack, host and clay skeletons, respectively, and $\phi_{1}$ + $\phi_{2}$ + $\phi_{3}$ is the total porosity, and $\phi_{1}$ = $v_{1} \phi_{10}$, $\phi_{2}$ = $v_{2} \phi_{20}$ and $\phi_{3}$ = $v_{3} \phi_{30}$; $\rho_{f}$ and $\eta$ denote the fluid density and viscosity, respectively. The stiffness coefficients $A$, $N$, $Q_{1}$, $Q_{2}$, $Q_{3}$, $R_{1}$, $R_{2}$ and $R_{3}$, dissipation coefficients $b_{1}$, $b_{2}$ and $b_{3}$, and density parameters $\rho_{00}$, $\rho_{01}$, $\rho_{02}$, $\rho_{03}$, $\rho_{{{11}}}$, $\rho_{{{22}}}$ and $\rho_{{{33}}}$ depend on the rock properties (Appendix 2).

The complex wave number k is estimated from a plane-wave analysis (see Appendix 2), and the wave velocity and quality factor are

$$\it {\text{V}}_{P} = \frac{1}{{{\text{Re}} \left( {{{v}}^{ - 1} } \right)}},$$

(5a)

$$\it \it Q = \frac{{{\text{Re}} \left( {{{v}}^{2} } \right)}}{{{\text{Im}} \left( {{{v}}^{2} } \right)}},$$

(5b)

where $v = \omega /k$ and $\omega$ is the angular frequency (Carcione 2014).

4.2 Results of the Acoustic RPM

The model allows us to analyze the wave response of rock pore structure and mineralogy. The model properties are given in Table 2. The minerals of the crack inclusions and host are same, so the elastic properties of the two phases K_s1 = K_s2 = K_s, μ_s1 = μ_s2 = μ_s and ρ_s1 = ρ_s1 = ρ_s. Then the skeleton moduli of the three phases (K_b1 and μ_b1, K_b2 and μ_b2, K_b3 and μ_b3) are estimated by the A-DEM equation and then the moduli of the dry rock (K_b and μ_b). The model is set as full-oil saturation. By adjusting the total and crack porosities and clay content, the wave velocity and quality factor can be obtained.

Table 2 Medium properties

Figure 11 shows the P-wave velocity and dissipation factor at four total porosities (ϕ), crack porosities (ϕ_C) and clay content (V_sh) settings, respectively, given in the figure. Figure 11a, b shows two dispersion steps and attenuation peaks, and in Fig. 11c, d, the steps and peaks shift to higher frequencies. Increasing the total and soft porosities and clay content decreases the velocity and increases the attenuation, the latter depending on the frequency band.

Fig. 11

Wave velocity and dissipation factor of the P wave as a function of frequency at different porosities a and b (ϕ_C = 0.1% and V_sh = 10%), crack porosities c and d (ϕ = 5% and V_sh = 10%), and clay content e and f (ϕ = 5% and V_sh = 0.1%)

4.3 Electrical RPM

Figure 12 shows the modeling workflow of the electrical RPM. Similarly, based on XRD results of the samples, the E-HS (Hashin and Shtrikman 1962) equation is used to calculate the electrical conductivity of the mineral mixture by mixing quartz, feldspar, carbonate and ironstone minerals. Berryman (1995) proposed a more general form of the bounds for more than two phases,

$$\begin{array}{*{20}c} {\sigma^{{{\text{HS}} + }} = F\left( {\sigma_{\max } } \right)} & {\sigma^{{{\text{HS}} - }} = F\left( {\sigma_{\min } } \right)} \\ \end{array} ,$$

(6a)

$$F(z) = \left( {\sum\limits_{i = 1}^{N} {\frac{{f_{i} }}{{\sigma_{i} + 2z}}} } \right)^{ - 1} - 2z,$$

(6b)

where σ^HS+ and σ^HS− are the upper and lower bounds, respectively, σ_i is the conductivity of the single phases and f_i is the volume fraction, and σ_max and σ_min are the maximum and minimum values.

Fig. 12

Modeling flowchart of the electrical RPM

We consider the same microstructure (aspect ratios of pores, cracks) to build the electrical model. The E-DEM equation is used to add pores and cracks containing fluid into the host and inclusions. Then, the crack and clay inclusions are added into the host frame to obtain the conductivity σ^* by using the E-DEM equation, as follows (Cilli and Chapman 2021),

$$(1 - y)\frac{{\text{d}}}{{{\text{d}}y}}\left[ {\sigma^{*} (y)} \right] = \left( {\sigma_{2} - \sigma^{*} } \right)m,$$

(7)

with σ ^∗ (0) = σ₁, where σ₁ is the conductivity of the host phase, σ₂ is the conductivity of phase 2, y is the corresponding content, and m is a function of the depolarizing factor L_P of phase 2:

$$m = \frac{1}{3}\sum\limits_{{{\text{p}} = 1}}^{3} {\left\{ {\left[ {1 + \left( {\frac{{\sigma_{2} }}{{\sigma^{*} }} - 1} \right)L_{{\text{P}}} } \right]} \right\}} ,$$

(8)

Archie (1942) assumed a rock without clay and other conductive minerals, so that the rock conductivity only depends on the formation water in the pore space. The additional conductivity of clay minerals is not considered, since the salinity in the target formation is 56.5 g/L (Leveaux and Poupon 1971; Han et al. 2005; Wang et al. 2006). The surface conductivity is also neglected in the studies of tight siltstones (Pang et al. 2021b). Thus, the formation factor and resistivity index are

$$F = \frac{{\sigma_{{\text{W}}} }}{{\sigma_{{\text{r}}} }} = \frac{1}{{\phi^{{\text{M}}} }},$$

(9a)

$$I = \frac{b}{{S_{{\text{w}}}^{n} }},$$

(9b)

Respectively, where σ_W and σ_r are the water and rock conductivities, respectively, S_W is the saturation, b is a lithology coefficient, and M and n are the parameters (b and n are assumed to be 1 and 2 here, respectively).

According to Archie’s equation, the electrical conductivity of pores and cracks as a function of water saturation is,

$$\sigma_{2} = b^{ - 1} S_{{\text{W}}}^{n} \sigma_{{\text{W}}} ,$$

(10)

where σ₂ is the conductivity of pores or cracks.

Finally, the Cole–Cole equation (Cole and Cole 1941; Pelton et al. 1978; Mansoor and Slater 2007) is applied to consider the frequency dependence caused by the polarization effects. The complex conductivity is

$$\sigma^{*} \left( \omega \right) = \sigma_{0} \left[ {1 + {\text{mm}}\left( {\frac{{(i\omega \tau )^{c} }}{{1 + (i\omega \tau )^{c} \left( {1 - {\text{mm}}} \right)}}} \right)} \right],$$

(11)

where $\omega$ is the angular frequency, σ₀ is the conductivity at zero frequency, τ is the relaxation time, mm is the chargeability, and c is a frequency exponent ($0 \le c \le 1$). We have

$$\sigma^{*} \left( \omega \right) = \frac{1}{{{\text{Rt}}^{*} \left( \omega \right)}} = \sigma_{{{\text{real}}}} \left( \omega \right) + i\sigma_{{{\text{im}}}} \left( \omega \right).$$

(12)

where ${\text{Rt}}^{*} \left( \omega \right)$ is the complex resistivity, $\sigma_{{{\text{real}}}}^{{}} \left( \omega \right)$ and $\sigma_{{{\text{im}}}}^{{}} \left( \omega \right)$ are the real and imaginary conductivities, respectively, and $i = \sqrt { - 1}$.

4.4 Results of the Electrical RPM

Figure 13 shows the effect of frequency, based on the Cole–Cole model, for different values of the parameters. The conductivity and the dispersion effect increase with increasing σ₀ and mm, with σ₀ and mm having low and high impacts (Fig. 13a–d), respectively. The peaks shift to lower frequencies with increasing τ (see Fig. 13e, f). We can see in Fig. 13g, h that the frequency coefficient affects curves. With the increase of c, the dispersion increases and the corresponding frequency band becomes narrow.

Fig. 13

Real and imaginary conductivities as a function of frequency at different σ₀ (a and b, mm = 0.05, τ = 0.1 and c = 0.87), mm (c and d, σ₀ = 0.02, τ = 0.1 and c = 0.87), τ (e and f, σ₀ = 0.02, mm  = 0.05 and c = 0.87) and c (g and h, σ₀ = 0.02, mm  = 0.05, and τ = 0.1)

We consider the same properties of the acoustic model. The electrical and Cole–Cole models are used to analyze the responses of pores, cracks and clay mineral. The conductivity obtained by the E-DEM model is set as the zero-frequency conductivity σ₀, and mm, τ and c are 0.05, 0.1 and 0.87, respectively (Revil et al. 2018). The conductivities of the clay, brine and mineral mixture are 0.5, 4.69 and 0.01 S/m (Han et al. 2016; Pang et al. 2021b), respectively, and the rock is saturated with water. The results are shown in Fig. 14, where we can see that with the increase of the total and crack porosities and clay content, the conductivity and dispersion increase. It is shown that these are less affected by the rock microstructure.

Fig. 14

Real and imaginary conductivities as a function of frequency for different porosities (a and b), crack porosities (c and d) and clay contents (e and f), mm  = 0.05, τ = 0.1 and c = 0.87

We consider a frequency of 120 Hz and water saturation from 0 to 1 to analyze the effects of the rock microstructure on conductivity and formation factor. Figure 15 shows the conductivity and formation factor of as a function of the water saturation at five values ϕ, ϕ_C and V_sh settings, indicated in the figure. The conductivity increases and the formation factor decreases with increasing water saturation. At low saturation, the conductivity decreases with increasing total and crack porosities. However, the electrical properties show an opposite trend when the saturation exceeds a certain value (see Fig. 15a–d). This is because when the water saturation is low, the conductivity of the pores and cracks containing fluids is lower than the mineral mixture, so the rock conductivity decreases with the porosity. With increasing saturation, the conductivity of pores and cracks are higher than that of the minerals, resulting in an increase in the rock conductivity. In Fig. 15e, f, the conductivity increases with clay content, and the formation factor decreases, as expected.

Fig. 15

Resistivity and conductivity as a function of water saturation at different porosities a and b (ϕ_C = 0.1% and V_sh = 10%), crack porosities c and d (ϕ = 1% and V_sh = 10%), and clay content e and f (ϕ = 10% and V_sh = 0.1%)

5 Rock Physics Template and Application

5.1 3D Rock Physics Template

3D rock physics templates (RPT) can be built by using the elastic attributes (Poisson’s ratio (v), and acoustic impedance (AI)) and electrical property (resistivity (R_t)). Figure 16 shows the template and experimental data (symbols), where the color bar corresponds to the effective pressure, and the black, red and blue lines are isolines of total porosity, crack porosity and clay content, within the range of 3–15%, 0.005–0.625% and 1–19%, respectively. As the pressure decreases, the AE properties agree with the data.

Fig. 16

3D acoustic-electrical RPT and samples, the color is effective pressure

Similarly, data with an effective pressure of 15 MPa (in-situ condition) are selected, as shown in Fig. 17, where the color bar represents porosity and clay content, respectively. The porosity agrees very well, whereas the clay content departs for a small number of samples.

Fig. 17

3D RPT and data (symbols): a porosity; b clay content

5.2 Well Data and Calibration

Well log data of the target reservoir are used to calibrate the RPT. Figures 18 and 19 show the logs at Wells A and B. The reservoir has low porosity (less than 13%) and a wide range of clay content, being a mixture of sandstone, muddy sandstone and mudstone layers. There is a large difference in petrophysical behaviors between the different formations (the dotted box is the mudstone layer). The sandstone layer is the main oil-producing section, and well log data from this layer (a clay content less than 30%) are selected to calibrate the AE RPT.

Fig. 18

Petrophysical properties at Well A. The dashed box indicates the mudstone section

Fig. 19

Same as Fig. 18 but for Well B

The study area is almost saturated with oil, so that full-oil saturation is assumed. The frequency is 10 kHz to match the sonic log data, and the radii of the crack and clay inclusions are set to 1 mm. As shown in Fig. 20, the well log data are projected onto the template, with the color bar denoting porosity (a) and clay content (b). The data are consistent with the RPTs, and the three attributes show the same trend with increasing porosity and clay content. Thus, a quantitative characterization of the reservoir properties can be obtained by superimposing the well log data on the template.

Fig. 20

3D RPT and well-log data: a porosity; b clay content

5.3 Application to Well Log Data

The three and two sandstone layers of Wells A and B are selected, respectively, to estimate the clay content, and the total and crack porosities. The impedance, Poisson’s ratio and resistivity are projected onto the RPT, the nearest grid of each data point is searched, and its reservoir properties are assigned to the data.

Figures 21 and 22 show the predictions and log interpretation for Wells A and B. The main ranges of clay content, total and crack porosities are 8–30%, 2–15% and 0–0.7%, respectively. The predicted curves are basically in agreement with the porosity and clay content of the two wells. It is clear that there is a good relation between porosity and crack porosity, and it is oppositely correlated with clay content. Formations with high total and crack porosities and low clay content have better pore space and connectivity, and higher hydrocarbon storage potential.

Fig. 21

Results for Well A; a porosity b clay content c crack porosity

Fig. 22

Same as Fig. 21 but for Well B

6 Conclusions

We have analyzed the properties of a tight-oil reservoir based on a joint acoustic-electrical petrophysical model. The rocks belong to the Yanchang Formation in the Q area of the Ordos Basin, whose core samples were characterized by X-ray diffraction and cast thin sections. Then, ultrasonic and resistivity experiments were performed at different confining pressures. A 3D rock-physics template has been built, based on Poisson’s ratio, impedance and resistivity, and calibrated with the ultrasonic and well-log data. The template was applied to predict the properties of the reservoir. We conclude the following:

• (1) The rocks have low porosity and permeability, with microcracks and high clay content, which highly affect their petrophysical behavior.

• (2) Wave velocities and resistivity increase with effective pressure (crack density decreases), and the conductivity decreases. The combined effects of the stiff pores, cracks and clay minerals have to be considered to obtain reliable predictions.

• (3) The good match between the proposed joint model and data on the basis of acoustic-electrical rock-physics templates shows that these can effectively be used to perform suitable interpretations involving tight rocks.

Appendices

Appendix 1: The geometrical factors (P, Q)

The coefficients P and Q for ellipsoidal inclusions are given in Berryman (1980) and Mavko et al. (2009),

$$\begin{array}{*{20}c} {P = \frac{1}{3}T_{1} ,} & {Q = \frac{1}{5}\left( {T_{2} - \frac{1}{3}T_{1} } \right)} \\ \end{array} ,$$

(13)

with the pertinent scalars T₁ and T₂ given by,

$$\begin{array}{*{20}c} {T_{1} = \frac{{3F_{1} }}{{F_{2} }}} & {T_{2} - \frac{1}{3}T_{1} = \frac{2}{{F_{3} }} + \frac{1}{{F_{4} }} + \frac{{F_{4} F_{5} + F_{6} F_{7} - F_{8} F_{9} }}{{F_{2} F_{4} }},} \\ \end{array}$$

(14)

where

$$F_{1} = 1 + G\left[ {\frac{3}{2}(g + \theta ) - J\left( {\frac{3}{2}g + \frac{5}{2}\theta - \frac{4}{3}} \right)} \right],$$

(15)

$$\begin{aligned} F_{2} = & 1 + G\left[ {1 + \frac{3}{2}(g + \theta ) - \frac{J}{2}\left( {\frac{3}{2}g + 5\theta } \right)} \right] + H\left( {3 - 4J} \right) \\ & + \frac{G}{2}\left( {G + 3H} \right)\left( {3 - 4J} \right)\left[ {g + \theta - J\left( {g - \theta + 2\theta^{2} } \right)} \right], \\ \end{aligned}$$

(16)

$$F_{3} = 1 + G\left[ {1 - (g + \frac{3}{2}\theta ) + J\left( {g + \theta } \right)} \right],$$

(17)

$$F_{4} = 1 + \frac{G}{4}\left[ {g + 3\theta - J\left( {g - \theta } \right)} \right],$$

(18)

$$F_{5} = G\left[ { - g + J\left( {g + \theta - \frac{4}{3}} \right) + H\theta \left( {3 - 4J} \right)} \right],$$

(19)

$$F_{6} = 1 + G\left[ {1 + g - J\left( {g + \theta } \right) + H\left( {1 - \theta } \right)\left( {3 - 4J} \right)} \right],$$

(20)

$$F_{7} = 2 + \frac{G}{4}\left[ {3g + 9\theta - J\left( {3g + 5\theta } \right)} \right] + H\theta \left( {3 - 4J} \right),$$

(21)

$$F_{8} = G\left[ {1 - 2J + \frac{g}{2}\left( {J - 1} \right) + \frac{\theta }{2}\left( {5J - 3} \right)} \right] + H\left( {1 - \theta } \right)\left( {3 - 4J} \right),$$

(22)

$$F_{9} = G\left[ {\left( {J - 1} \right)g - J\theta } \right] + H\theta \left( {3 - 4J} \right),$$

(23)

with G, H and J given by

$$G = \frac{{\mu_{i} }}{{\mu_{m} }} - 1,$$

(24)

$$H = \frac{1}{3}\left( {\frac{{K_{i} }}{{K_{m} }} - \frac{{\mu_{i} }}{{\mu_{m} }}} \right),$$

(25)

$$J = \left[ {\frac{{\left( {1 - 2v_{m} } \right)}}{{2\left( {1 - v_{m} } \right)}}} \right],$$

(26)

where K_m, μ_m and v_m are the bulk and shear moduli and Poisson’s ratio of the host phase, respectively, K_i, and μ_i are the bulk and shear moduli of the phase i, and

$$\theta = \left\{ \begin{aligned} \frac{\alpha }{{\left( {\alpha^{2} - 1} \right)^{3/2} }}\left[ {\alpha \left( {\alpha^{2} - 1} \right)^{1/2} - \cosh^{ - 1} \alpha } \right] \hfill \\ \frac{\alpha }{{\left( {1 - \alpha^{2} } \right)^{3/2} }}\left[ {\cos^{ - 1} \alpha - \alpha \left( {1 - \alpha^{2} } \right)^{1/2} } \right] \hfill \\ \end{aligned} \right\},$$

(27)

for prolate (α > 1) and oblate (α < 1) spheroids, respectively, the α is aspect ratio, and

$$g = \frac{{\alpha^{2} }}{{1 - \alpha^{2} }}\left( {3\theta - 2} \right),$$

(28)

Appendix 2: The dispersion equation

The dispersion equation is given in Sun et al. (2016) and Zhang et al. (2017). The plane-wave analysis was performed by substituting a time harmonic kernel $e^{{i\left( {\omega t - k \cdot {\mathbf{x}}} \right)}}$ into Eq. (4). Then the complex wave number $k$ can be obtained as

$$\left| {\begin{array}{*{20}c} {a_{11} k^{2} + b_{11} } & {a_{12} k^{2} + b_{12} } & {a_{13} k^{2} + b_{13} } & {a_{14} k^{2} + b_{14} } \\ {a_{21} k^{2} + b_{21} } & {a_{22} k^{2} + b_{22} } & {a_{23} k^{2} + b_{23} } & {a_{24} k^{2} + b_{24} } \\ {a_{31} k^{2} + b_{31} } & {a_{32} k^{2} + b_{32} } & {a_{33} k^{2} + b_{33} } & {a_{34} k^{2} + b_{34} } \\ {a_{41} k^{2} + b_{41} } & {a_{42} k^{2} + b_{42} } & {a_{43} k^{2} + b_{43} } & {a_{44} k^{2} + b_{44} } \\ \end{array} } \right| = 0,$$

(29)

and

$$\begin{aligned} a_{11} = A + 2N + \left( {Q_{1} \phi_{2} - Q_{2} \phi_{1} } \right)M_{0}^{{\left( {12} \right)}} + \left( {Q_{2} \phi_{3} - Q_{3} \phi_{2} } \right)M_{0}^{{\left( {23} \right)}} , \hfill \\ a_{12} = Q_{1} + \left( {Q_{1} \phi_{2} - Q_{2} \phi_{1} } \right)M_{1}^{{\left( {12} \right)}} + \left( {Q_{2} \phi_{3} - Q_{3} \phi_{2} } \right)M_{1}^{{\left( {23} \right)}} , \hfill \\ a_{13} = Q_{2} + \left( {Q_{1} \phi_{2} - Q_{2} \phi_{1} } \right)M_{2}^{{\left( {12} \right)}} + \left( {Q_{2} \phi_{3} - Q_{3} \phi_{2} } \right)M_{2}^{{\left( {23} \right)}} , \hfill \\ a_{14} = Q_{3} + \left( {Q_{1} \phi_{2} - Q_{2} \phi_{1} } \right)M_{3}^{{\left( {12} \right)}} + \left( {Q_{2} \phi_{3} - Q_{3} \phi_{2} } \right)M_{3}^{{\left( {23} \right)}} , \hfill \\ a_{21} = Q_{1} + \phi_{2} R_{1} M_{0}^{{\left( {12} \right)}} ,\quad a_{22} = R_{1} + \phi_{2} R_{1} M_{1}^{{\left( {12} \right)}} , \hfill \\ a_{23} = \phi_{2} R_{1} M_{2}^{{\left( {12} \right)}} ,\quad a_{24} = \phi_{2} R_{1} M_{3}^{{\left( {12} \right)}} , \hfill \\ a_{31} = Q_{2} - R_{2} \left( {\phi_{1} M_{0}^{{\left( {12} \right)}} - \phi_{3} M_{0}^{{\left( {23} \right)}} } \right),\quad a_{32} = - R_{2} \left( {\phi_{1} M_{1}^{{\left( {12} \right)}} - \phi_{3} M_{1}^{{\left( {23} \right)}} } \right), \hfill \\ a_{33} = R_{2} \left( {1 - \phi_{1} M_{2}^{{\left( {12} \right)}} + \phi_{3} M_{2}^{{\left( {23} \right)}} } \right),\quad a_{34} = R_{2} \left( { - \phi_{1} M_{3}^{{\left( {12} \right)}} + \phi_{3} M_{3}^{{\left( {23} \right)}} } \right), \hfill \\ a_{41} = Q_{3} - \phi_{2} R_{3} M_{0}^{{\left( {23} \right)}} ,\quad a_{42} = - \phi_{2} R_{3} M_{1}^{{\left( {23} \right)}} , \hfill \\ a_{43} = - \phi_{2} R_{3} M_{2}^{{\left( {23} \right)}} ,\quad a_{44} = R_{3} \left( {1 - \phi_{2} M_{3}^{{\left( {23} \right)}} } \right), \hfill \\ b_{11} = - \rho_{00} \omega^{2} + i\omega \left( {b_{1} + b_{2} + b_{3} } \right),\quad b_{12} = - \rho_{01} \omega^{2} - i\omega b_{1} , \hfill \\ b_{13} = - \rho_{02} \omega^{2} - i\omega b_{2} ,\quad b_{14} = - \rho_{03} \omega^{2} - i\omega b_{3} , \hfill \\ b_{21} = - \rho_{01} \omega^{2} - i\omega b_{1} ,\quad b_{22} = - \rho_{11} \omega^{2} + i\omega b_{1} ,b_{23} = b_{24} = 0, \hfill \\ b_{31} = - \rho_{02} \omega^{2} - i\omega b_{2} ,\quad b_{33} = - \rho_{22} \omega^{2} + i\omega b_{2} ,b_{32} = b_{34} = 0, \hfill \\ b_{41} = - \rho_{03} \omega^{2} - i\omega b_{3} ,\quad b_{44} = - \rho_{33} \omega^{2} + i\omega b_{3} ,b_{42} = b_{43} = 0, \hfill \\ \end{aligned}$$

(30)

where the Biot dissipation coefficients (Biot 1962 and Sun et al. 2016) and the permeabilities of the three phases (Vaughan et al. 1986 and Mavko et al. 2009),

$$ \begin{array}{*{20}l} {b_{1} = \phi_{1} \phi_{10} \frac{{\eta_{f}^{{}} }}{{\kappa_{1} }},} \hfill & {b_{2} = \phi_{2} \phi_{20} \frac{{\eta_{f}^{{}} }}{{\kappa_{2} }},} \hfill & {b_{3} = \phi_{3} \phi_{30} \frac{{\eta_{f}^{{}} }}{{\kappa_{3} }},} \hfill \\ {\kappa_{1} = {{DR}}_{12}^{2} \phi_{1}^{3} ,} \hfill & {\kappa_{2} = \frac{{\kappa_{0} \phi_{2}^{3} }}{{\left( {1 - \phi_{2} } \right)^{2} }},} \hfill & {\kappa_{3} = \frac{{\kappa_{0} \phi_{3}^{3} }}{{\left( {1 - \phi_{3}^{2} } \right)}},} \hfill \\ \end{array} $$

(31)

where D = 50 and κ₀ = 75.54 mdarcy, and

$$\begin{aligned} S_{12} = & \frac{{ - \phi_{1} \phi_{2}^{2} R_{12}^{2} \omega \left( {\rho_{f} \omega \left( {1/5 + \phi_{10} /\phi_{20} } \right) + i\left( {\eta /\left( {5\kappa_{1} } \right) + \eta /\kappa_{2} } \right)\phi_{10} } \right)}}{3} - \phi_{2}^{2} R_{1} - \phi_{1}^{2} R_{2} , \\ S_{23} = & \frac{{ - \phi_{3} \phi_{2}^{2} R_{23}^{2} \omega \left( {\rho_{f} \omega \left( {1/5 + \phi_{30} /\phi_{20} } \right) + i\left( {\eta /\left( {5\kappa_{3} } \right) + \eta /\kappa_{2} } \right)\phi_{30} } \right)}}{3} - \phi_{3}^{2} R_{2} - \phi_{2}^{2} R_{3} , \\ M_{0}^{{\left( {12} \right)}} = & \frac{{\left( {Q_{1} \phi_{2} - Q_{2} \phi_{1} } \right)/S_{12} + \phi_{1} \phi_{3} R_{2} \left( {Q_{2} \phi_{3} - Q_{3} \phi_{2} } \right)/\left( {S_{12} S_{23} } \right)}}{{1 + \left( {\phi_{1} \phi_{3} R_{2} } \right)^{2} /\left( {S_{12} S_{23} } \right)}}, \\ M_{1}^{{\left( {12} \right)}} = & \frac{{\phi_{2} R_{1} /S_{12} }}{{1 + \left( {\phi_{1} \phi_{3} R_{2} } \right)^{2} /\left( {S_{12} S_{23} } \right)}}, \\ M_{2}^{{\left( {12} \right)}} = & \frac{{ - \phi_{1} R_{2} /S_{12} + \phi_{1} \phi_{3}^{2} R_{2}^{2} /\left( {S_{12} S_{23} } \right)}}{{1 + \left( {\phi_{1} \phi_{3} R_{2} } \right)^{2} /\left( {S_{12} S_{23} } \right)}},\quad M_{3}^{{\left( {12} \right)}} = \frac{{ - \phi_{1} \phi_{2} \phi_{3} R_{2} R_{3} /\left( {S_{12} S_{23} } \right)}}{{1 + \left( {\phi_{1} \phi_{3} R_{2} } \right)^{2} /\left( {S_{12} S_{23} } \right)}}, \\ M_{0}^{{\left( {23} \right)}} = & \left( { - M_{0}^{{\left( {12} \right)}} \phi_{1} \phi_{3} R_{2} + Q_{2} \phi_{3} - Q_{3} \phi_{2} } \right)/S_{23} ,\quad M_{1}^{{\left( {23} \right)}} = - M_{1}^{{\left( {12} \right)}} \phi_{1} \phi_{3} R_{2} /S_{23} , \\ M_{2}^{{\left( {23} \right)}} = & \left( { - M_{2}^{{\left( {12} \right)}} \phi_{1} \phi_{3} R_{2} + \phi_{3} R_{2} } \right)/S_{23} ,\quad M_{3}^{{\left( {23} \right)}} = \left( { - M_{3}^{{\left( {12} \right)}} \phi_{1} \phi_{3} R_{2} - \phi_{2} R_{3} } \right)/S_{23} . \\ \end{aligned}$$

(32)

The stiffness and density coefficients are

$$\begin{aligned} A = \left( {1 - \phi } \right)K_{s} - \frac{2}{3}N - \frac{{K_{s} }}{{K_{f} }}\left( {Q_{1} + Q_{2} + Q_{3} } \right),\quad N = \mu_{b} , \hfill \\ Q_{1} = \frac{{\phi_{1} \beta_{1} K_{s} }}{{\beta_{1} + \gamma }},\quad Q_{2} = \frac{{\phi_{2} K_{s} }}{1 + \gamma },\quad Q_{3} = \frac{{\phi_{3} K_{s} }}{{\beta_{1} \gamma + 1}}, \hfill \\ R_{1} = \frac{{\phi_{1} K_{f} }}{{{{\beta_{1} } \mathord{\left/ {\vphantom {{\beta_{1} } \gamma }} \right. \kern-\nulldelimiterspace} \gamma } + 1}},\quad R_{2} = \frac{{\phi_{2} K_{f} }}{{1 + {1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-\nulldelimiterspace} \gamma }}},\quad R_{3} = \frac{{\phi_{3} K_{f} }}{{{1 \mathord{\left/ {\vphantom {1 {\beta_{1} \gamma }}} \right. \kern-\nulldelimiterspace} {\beta_{1} \gamma }} + 1}}, \hfill \\ \gamma = \frac{{K_{s} }}{{K_{f} }}\frac{{\phi_{1} \beta_{1} + \phi_{2} + {{\phi_{3} } \mathord{\left/ {\vphantom {{\phi_{3} } {\beta_{2} }}} \right. \kern-\nulldelimiterspace} {\beta_{2} }}}}{{\left( {1 - \phi } \right) - {{K_{b} } \mathord{\left/ {\vphantom {{K_{b} } {K_{s} }}} \right. \kern-\nulldelimiterspace} {K_{s} }}}}, \hfill \\ \end{aligned}$$

(33)

$$\begin{aligned} \begin{array}{*{20}c} {\rho_{11} = \frac{1}{2}\phi_{1} \rho_{f} \left( {1 + \frac{1}{{\phi_{10} }}} \right),} & {\rho_{22} = \frac{1}{2}\phi_{2} \rho_{f} \left( {1 + \frac{1}{{\phi_{20} }}} \right),} & {\rho_{33} = \frac{1}{2}\phi_{3} \rho_{f} \left( {1 + \frac{1}{{\phi_{30} }}} \right),} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {\rho_{01} = \phi_{1} \rho_{f} - \rho_{11} ,} & {\rho_{02} = \phi_{2} \rho_{f} - \rho_{22} ,} & {\rho_{03} = \phi_{3} \rho_{f} - \rho_{33} ,} \\ \end{array} \hfill \\ \rho_{00} = v_{1} (1 - \phi_{10} )\rho_{s1} + v_{2} (1 - \phi_{20} )\rho_{s2} + v_{3} (1 - \phi_{30} )\rho_{sh} - \rho_{01} - \rho_{02} - \rho_{03} , \hfill \\ \end{aligned}$$

(34)

where K_s, K_b and K_f are the bulk moduli of the mineral mixture, skeleton and fluid, respectively, ρ_s1, ρ_s2 and ρ_sh are the mineral densities corresponding to the three phases, ρ_f is the fluid density, and

$$\begin{aligned} \beta_{1} = \frac{{Q_{1} R_{2} }}{{Q_{2} R_{1} }} = \frac{{\phi_{20} }}{{\phi_{10} }}\left[ {\frac{{{1 \mathord{\left/ {\vphantom {1 {K_{s1} }}} \right. \kern-\nulldelimiterspace} {K_{s1} }} - {{\left( {1 - \phi_{10} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \phi_{10} } \right)} {K_{b1} }}} \right. \kern-\nulldelimiterspace} {K_{b1} }}}}{{{1 \mathord{\left/ {\vphantom {1 {K_{s2} }}} \right. \kern-\nulldelimiterspace} {K_{s2} }} - {{\left( {1 - \phi_{20} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \phi_{20} } \right)} {K_{b2} }}} \right. \kern-\nulldelimiterspace} {K_{b2} }}}}} \right], \hfill \\ \beta_{2} = \frac{{Q_{2} R_{3} }}{{Q_{3} R_{2} }} = \frac{{\phi_{30} }}{{\phi_{20} }}\left[ {\frac{{{1 \mathord{\left/ {\vphantom {1 {K_{s2} }}} \right. \kern-\nulldelimiterspace} {K_{s2} }} - {{\left( {1 - \phi_{20} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \phi_{20} } \right)} {K_{b2} }}} \right. \kern-\nulldelimiterspace} {K_{b2} }}}}{{{1 \mathord{\left/ {\vphantom {1 {K_{sh} }}} \right. \kern-\nulldelimiterspace} {K_{sh} }} - {{\left( {1 - \phi_{30} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - \phi_{30} } \right)} {K_{b3} }}} \right. \kern-\nulldelimiterspace} {K_{b3} }}}}} \right]. \hfill \\ \end{aligned}$$

(35)

where K_b1, K_b2 and K_b3 are the skeleton bulk moduli of the crack inclusions, host and clay inclusions, respectively, and K_s1, K_s2 and K_sh are the mineral bulk moduli corresponding to the three phases.