The nonlinear characteristics of the rock transport properties (permeability and electrical conductivity in this study) as a function of stress are closely related to the geometry of the pore space, which consists of stiff pores, microcracks, or microfractures. We consider two behaviors of the pore space, one linear and the other exponential, related to the stiff pores and microfractures, respectively, where the relation between stress and strain can be described by the Two-Part Hooke’s Model. With this model, the relations between porosity, transport properties, and effective stress (confining minus pore pressure) can be obtained and validated with the experimental data of four tight sandstones collected from the Shaximiao Formation of Sichuan Basin, southwest China. The agreement is good. At low effective stresses, the closure of cracks is the main mechanism affecting the transport properties, whose behavior is similar in terms of their parameters. Subsequently, experimental data of nine tight sandstones from the Yanchang Formation, collected from the Ordos Basin, west China, are employed to confirm the previous results, indicating that the fluid and electrical current follow the same path in the pore space.

Reservoir rocks have pores, cracks, or microfractures and are generally heterogeneous [1-4]. The deformation under loading is different in stiff pores and microfractures, which affects the elastic and transport properties, especially in low-permeability rocks. Since cracks provide a permeability path for the flow of reservoir fluids [5-9], understanding of the relationships between the transport properties and effective stress is important for detecting and monitoring reservoir fluids.

Previous studies revealed that the exponential function describes the behavior of permeability and conductivity as a function of effective stress [10-20]. However, an important point is to describe the behavior of the sharp decrease of these transport properties when the effective stress increases at low values, especially for low-permeability rocks [21, 22]. The power law has also been adopted to describe such variation [23-27]. For instance, Jones and Owens [28] and Walsh [29] reformulated the expression of power law. On the other hand, Kaselow and Shapiro [30] applied a four-parameter exponential equation to analyze the electrical conductivity as a function of the effective pressure.

The closure of cracks with increasing effective stress leads to lower porosity, and permeability or electrical conductivity shows a similar behavior. The transport properties as a function of porosity can be studied with a power law [31, 32] or by analyzing experimental data [21]. Archie [33] established an empirical relation between the formation factor (the ratio between bulk resistivity and that of water) and porosity. Subsequently, some researchers investigated the relationships between electrical conductivity and porosity [34, 35], clay content [36-38], crack radii, aspect ratios, tortuosity, and pore surface area [39-45].

Pores and cracks undergo different deformation under stress, and then the effects on porosity are different. A rock can be conceptually divided into “hard” and “soft” parts. By assuming that the cracks are homogeneously distributed in the rock, Two-Part Hooke’s Model (TPHM) proposed by Liu et al. [46], to characterize the stress–strain relations of porous and fractured rocks, indicates that cracks contribute nonlinearly. This nonlinear deformation may be the result of the combining effects of nonuniform pore size distributions and pore geometry heterogeneity [47]. Zheng et al. [22] derived the relations between porosity, permeability, and effective stress of low-permeability rocks based on this model and verified them by using experimental data. Based on these relations, Wu et al. [48] provided a stress-sensitivity method suitable for depletion mining and water-injection development processes. The satisfactory results are achieved by using the model [49-53]. In addition, Liu et al. [46] stated that the soft part also has an effect on the electrical conductivity, confirmed by Watanabe et al. [54] in the brine-saturated granite. They found that the closure of narrow apertures (i.e., cracks) leads to a sharp decrease in the conductivity. Pang et al. [55] studied the effects of pores, cracks, and clay content on elastic wave velocity and electrical conductivity by using an acoustical–electrical model. The aforementioned studies regarding the experimental measurements of electrical conductivity and rock physical models have analyzed the significant impacts of microcracks on electrical conductivity. However, it is still a challenge to establish a robust model in describing the contributions of pores and microcracks to electrical conductivity within the whole stress range. According to the work of Walsh and Brace [56], it is pointed out that the flow of fluid and electrical current in rocks are similar and follow similar migration paths, and other studies have shown that there is a correlation between permeability and electrical conductivity [57-59]. This means that the relationships between permeability and effective stresses (e.g., a power law relationship) may be extended to the case of conductivity. As an extension of the works of Liu et al. [46] and Zheng et al. [22], this study proposes to separate the effects of pores and microcracks on conductivity based on TPHM.

In this study, we analyze the relations between transport properties, porosity, and effective stress by using the TPHM, verified experimental data of tight sandstones from the Shaximiao Formation. The related parameters are further analyzed by data corresponding to tight sandstones collected from the Yanchang Formation. The theory can be used to establish cross-property relationship between permeability and resistivity [60].

2.1. Two-Part Hooke’s Model

According to Liu et al. [46], a porous and fractured rock can be divided into “soft” and “hard” parts (Figure 1), where the former is composed of microcracks or microfractures with a large degree of relative deformation and follows the natural-strain-based Hooke’s law, and the latter is the residual structure with less deformation and follows the engineering-strain-based Hooke’s law. Let subscripts “e” and “t” refer to the hard and soft parts. For the soft part, the deformation corresponds to the natural strain, with the volume change scaled to the current pore volume, and the stress–strain relation of the soft part is

dσKt=dεv,t=dVtVt
(1)

where Kt is the bulk modulus, σ is the effective stress (confining minus pore pressure), and V denotes volume. Integrating equations (1) with the initial condition V=V0,t for σ=0 , we have:

Vt=V0,t exp(σKt)
(2)

where subscripts “0” denote the unstressed state.

For the hard part, the deformation corresponds to the engineering strain, with the volume change scaled to the initial pore volume, and we have

dσKe=dεv,e=dVeV0,e
(3)

where Ke is the bulk modulus. Integrating equations (3) with the initial condition Ve=Ve,0 for σ=0 , we obtain

Ve=V0,e(1σKe)
(4)

Therefore, the stress–strain relation is given by

ε=dVV0=γedσKe+γtexp(σKt)dσKt
(5)

where V0=V0,t+V0,e , γt=V0,t /V0 , and γe=1γt .

2.2. Porosity and Permeability

The total porosity of the rock is given by [46]

ϕ=ϕe,0(1Ceσ)+γt,0 exp(σKt),
(6)

where Ce=1/Ke is the compressibility for the hard fraction of the pore volume, ϕe,0+γt,0=ϕ0, ϕe=ϕe,0(1Ceσ) is the hard-part porosity, and ϕt=γt,0 exp(σ /Kt) is the soft-part porosity.

Although the soft part represents extremely small, the apparent decrease in permeability at low-effective stresses is closely related to its deformation. Assuming that the soft-part permeability is Kt=α(ϕt)m, where σ and m are constants, Zheng et al. [22] derived the following expression for the permeability:

k=ke,0 exp[βCeϕe,0 σ]+α[γt,0 exp(σKt)]m,
(7)

where ke,0 is the hard-part permeability under an unstressed condition, and β is a constant (i.e., a stress-sensitive coefficient).

Due to experimental limitations, a measurement at zero effective stress (index “0”) cannot be performed, and it is approximated by that at the minimum effective stress (index “1”) [22]. Then, equations (6) and (7) are further reformulated as

ϕ=ϕe,1(1CeΔσ)+γt,1 exp(ΔσKt),
(8)
k=ke,1 exp[βCeϕe,1Δσ]+α[γt,1exp(ΔσKt)]m,
(9)

where Δσ=σσ1, σ1 is the minimum effective stress, ϕe,1 and ke,1 are the hard-part porosity and permeability at σ1, respectively, and γt,1 is the soft-part porosity at σ1.

2.3. Electrical Conductivity

The electrical conductivity of a rock is closely related to porosity [33, 61, 62], and cracks are the main cause of its nonlinear behavior [10, 35, 55]. Similarly, the variations of conductivity with effective stress are also attributed to the deformations of pores and cracks. By assuming that the cracks are closed at high-effective stresses, their contributions to conductivity are negligible. Then,

d(lnSe) /dσ=a(1/V0)(dVep/dσ),
(10)

where Se is the conductivity of the hard part, a is a constant, and dVep=CeVe,0dσ [46]. This part is in accordance with the theory of poroelasticity. By integrating equation (10) with an initial condition of Se=Se,0 for σ=0 , we obtain

Se=Se,0 exp[aϕe,0Ceσ],
(11)

where Se,0 is the conductivity at an unstressed condition.

From the previous section, we note that the conductivity at low-effective stresses is mainly associated with the large relative deformation of cracks. The conductivity of the soft part is then obtained by subtracting Se from the total conductivity,

St=SSe.
(12)

Similar to Zheng et al. [22], a power law function is adopted to describe the empirical relationship between conductivity and soft porosity, that is,

St=b[ϕt]n,
(13)

where b and n are constants. By combining equations (11), (12), and (13), we obtain the bulk conductivity as

S=Se,1 exp[aϕe,1CeΔσ]+b[γt,1 exp(ΔσKt)]n,
(14)

where Se,1 is the hard-part conductivity at σ1, and we have considered the minimum effective stress (index “1”). Comparison between equations (9) and (14) shows that the expressions of conductivity and permeability are similar, as illustrated in previous studies [57].

3.1. Samples

We have measured the properties of four tight-sandstone samples from the Jurassic Shaximiao Formation of Sichuan Basin, China as a function of effective stress. The porosity has been obtained with the helium expansion method, and the permeability was determined with the unsteady-state pulse transient decay technique. An impedance–capacitance–resistance meter was used to measure the electrical conductivity with brine saturation (5% concentration). Porosity and permeability are measured at the range of 2–58 MPa, and conductivity is measured at the range of 5–35 MPa. The properties are given in Table 1. Figure 2 shows thin sections of the TS4 sample, a lithic feldspar sandstone with mostly pore-contact cementation. The mineral composition is mainly quartz, with feldspar and rock fragments, and the interstitial materials include clays and calcite. Additionally, a small amount of metallic minerals, such as siderite and pyrite, can be observed.

3.2. Verification of the Porosity and Permeability Equations

Figures 3 and 4 compare the theoretical (equations (8) and (9)) and experimental porosity and permeability of the four tight sandstones as a function of effective stress, respectively. At low stresses, the properties decrease nonlinearly, followed by a linear trend at high stresses. The nonlinear deformation under low stress may be caused by the closure of the slot-like cracks corresponding to the soft part. The porosity (permeability) fitting coefficients R2 of TS1–TS4 samples are 0.97 (0.98), 0.96 (0.96), 0.92 (0.85), and 0.97 (0.88), respectively. The method to obtain the fitting parameters (see Table 2) is that of Liu et al. [46] and Zheng et al. [22]. Specifically, at high stresses, there is a linear relation between log(ke) and σ , which can fit with ke=ke,1exp[βCeϕe,1Δσ]. Then, the slope βCeϕe,1 of the straight line can be used to estimate the value of β, where the values of Ce and ϕe,1 have been obtained by fitting the porosity-stress data. The value of ke,1 is determined by extrapolating the straight line at σ1. The soft-part permeability can be obtained by using kt=kke, and the values of α and m can be estimated according to kt=α(ϕt)m.

Table 2 shows that the soft-part porosity of sample TS2 is the largest, while that of sample TS3 is the smallest. This indicates that TS2 experiences the largest decrease in porosity and permeability at low stresses. On the contrary, TS3 has the smallest decrease in these properties. Porosity and the permeability logarithm as a function of stress are linear at high stresses, indicating that the contribution of the soft part can be neglected at this range. Furthermore, the high-stress range can be set as 20–58 MPa, and 2–15 MPa can be defined as a low range, which may be used for the parameter estimation of the soft part.

3.3. Verification of the Electrical Conductivity Equation

Figure 5 shows the results for the electrical conductivity. Similar to the porosity and permeability, the conductivity decreases nonlinearly with stress. However, this behavior is not evident for sample TS4, and constant n corresponding to the soft part is significantly smaller than that of the other samples (see Figure 6). There are fewer TS3 and TS4 soft part data points. This is because microcracks or microfractures, like the soft part in TS3 and TS4, are likely to have a smaller aspect ratio, and the soft part will sharply decrease and gradually approach closure in the lower effective pressure range. The results of equation (14) agree with the experimental data, where the fitting coefficients R2 of the four samples are 0.99, 0.96, 0.99, and 0.95. The conductivity variation is significant at low stresses, while at high stresses, cracks close and do not contribute to the fluid-flow continuity, showing a linear decreasing trend.

Similar to the permeability, the decrease of the soft-part porosity of TS2 is the largest at low stresses, while that of the TS3 sample is smaller. The results indicate that the soft part contributes as a major fluid migration path and cannot be neglected at low stresses. At high stresses, there is a linear relationship between log(Se) and σ, which can be fitted with equation (11). Then, the slope aCeϕe,1 of the straight line can be used to estimate a, where the values of Ce and ϕe,1 have been obtained by fitting the porosity-stress data. The value of Se,1 is determined by extrapolating the straight line at σ1. The soft-part conductivity is obtained with equation (12), and then b and n can be estimated according to equation (13). The soft-part porosity and conductivity are shown in Figure 6, and the fitting parameters are listed in Table 2.

Table 2 shows that γt,1 accounts for a small part of the total porosity, and it decreases sharply with increasing stress. The soft-part bulk modulus Kt (9–13 MPa) is much smaller than that of a tight sandstone sample (29–31 GPa). Some parameters (Ce and Kt) have their own specific meanings and can be measured directly by laboratory experiments, but in this paper, they are calculated by fitting the experimental data. Moreover, Liu et al. [46] pointed out the rationality of the parameters such as pore compressibility and bulk modulus of soft part obtained by fitting calculation. Note that ke,1 is significantly lower than the measurement at the minimum effective stress, which means that the soft-part permeability is the main contribution in this case. The value of m, obtained by fitting the permeability-porosity data, is between 1 and 2.86, with an average of 1.85, lower than that of the silty shale in Zheng et al. [22]. This is because the stress sensitivity of the latter is significantly higher [63]. Regarding the conductivity (unlike the TS4 sample), n ranges from 1.37 to 2.85, with an average of 2.26, close to m, indicating that the flow of fluid and electrical current follow similar migration paths [56]. Furthermore, the value of Se,1 is much lower than the measurement at the minimum effective stress, suggesting that the soft-part conductivity also plays an important role. It should be noted that these fitting parameters can be affected by the selection of the stress range of the two parts.

3.4. Applications to Other Tight-Sandstone Samples

To verify the validity of equations (8), (9), and (14), based on the TPHM model, we consider nine tight-sandstone samples collected from the Member 7 of Yanchang Formation in Qingyang area, Ordos Basin, west China. According to X-ray diffraction and thin sections analysis, they mainly consist of quartz, feldspar, and clay. Their porosity and permeability are in the range of 5.065%–10.165% and 0.015–0.096 mD, respectively. They are typical low-permeability rocks, whose properties are given in Table 3. The porosity and permeability are measured by the automated permeameter of the Core Measurement System at stresses of 5–45 MPa, and the conductivity is measured with the two-electrode method at 3–45 MPa. Figure 7 shows thin sections of the samples, where pores and microcracks can be seen.

Figures 8-10 show the results for porosity, permeability, and conductivity, respectively. The results show that the theoretical equation based on the TPHM theory can provide a good agreement with the experimental data. At low stresses, the cracks play an important role. According to Tables 2 and 4, the values of m and n are also similar, so it might be possible to estimate conductivity from permeability and vice versa, by using this correlation.

In general, conductivity increases with clay content [55, 64]. Comparing Figure 3(b) with Figure 8(f), samples TS2 and TS38 have similar porosity, but the clay content of the former is much higher and the conductivity lower than that of the latter (see Figures 5(b) and 10(f)), which may be related to Se,1. Comparing samples TS19 and TS40, we find that the clay and siderite content of the two samples are almost the same (see Table 3), while the former contains a small amount of pyrite, but the corresponding Se,1 is lower than that of the latter, indicating that the content of metallic minerals will not strongly affect the electrical conductivity. Samples TS41 and TS42 allow the same conclusion.

Tables 2 and 4 show that the constant m(n) of the tight sandstones of the Yanchang Formation is apparently lower than that of the Shaximiao one. According to the model, the permeability (conductivity) parameter m(n) is relevant to the properties of the soft part at low stresses. Comparing Figures 3-5 with Figures 8-10, the porosity variation of the former is smaller at low effective stresses as illustrated in the first figures, while the permeability variation has an opposite behavior, indicating that the m values of the former are higher. Similar discussions can be found in Dong et al. [63] (the corresponding m is high when the porosity change is small, while the permeability variation is high). Moreover, greater porosity may lead to more electrical paths, and we have found that Se,1 increases with increasing porosity ϕe,1. In addition, it should be noted that the pore pressure is fixed in the experiment, and the effect of the effective stress is considered by changing the confining pressure. In rocks with low permeability, the Biot coefficient may be less than 1 [65], and furthermore, it has been considered that the Biot coefficient will basically decrease with the increase of effective stress [66, 67]. In this study, the Biot coefficient is assumed to be close to 1, which allows for reasonable predictions in the low-effective stress range, while deviations might exist for modeling at higher effective stresses. Liu et al. [46] assumed the Biot coefficient equals 1 when modeling sandstones for the porosity range of 9.00%–17.52% and the stress range of 0–100 MPa, and their predictions agree well with the measured results. In this work, according to the modeling results, the proposed procedure still provides satisfactory results in comparison with the measured data. Incorporating the effect of the variations in Biot coefficient in the approach of TPHM will be considered in future work.

The TPHM establishes the relation between the transport properties (permeability and electrical conductivity) and the effective stress of porous and fractured rocks. Experimental values of these properties and the porosity of tight sandstones from the Shaximiao and Yanchang Formations are employed in the analysis, and the results show that the TPHM describes the experiments. The empirical exponential parameters of permeability and conductivity are similar, indicating that the effects of the cracks on the transport properties as a function of effective stress are similar. The analysis shows that the paths of fluid flow and electric current under are similar in the whole stress range. Cross-property relationship between conductivity and permeability will be developed in a future study, which can be used, for instance, to predict permeability from the resistivity log.

Datasets are available at https://zenodo.org/records/10526553

The authors declare that they have no conflict of interest.

This work is jointly funded by the National Natural Science Foundation of China (42104110, 12334019, and 42174161), Natural Science Foundation of Jiangsu Province (BK20210379), the Postdoctoral Science Foundation of China (2022M720989), Jiangsu Innovation and Entrepreneurship Plan.

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