Introduction

The electrical properties of porous media are essential characteristics in materials, rocks, geology, water resources, and other fields (Waxman and Thomas 1974; Weiss et al. 2012; Song al et. 2018). Due to an electric current can pass through a formation because it contains water with enough dissolved ions to be conductive, the conductivity of sand-based porous media depends on pore water conductivity, amount of water present, and pore structure and geometry (Schön 2011; Azizoglu et al. 2021; Chinh 2000).

Scholars have studied that when porosity decreases, pore structure can affect the electrical properties of sand-based porous media heavily (Grattoni and Dawe 1994; Mao and Gao 2000; Yue 2019,). Meanwhile, the Non-Archie phenomenon has been widely found in experimental data analysis of porous material (Verwer et al. 2011; Dunlap et al. 1991; Cai et al. 2017) . It is mainly manifested as the non-linear relationship between \(\log F\) and \(\log \phi\), and non-linear relationship between \(\log I\) and \(\log {S}_{w}\) (Al-Gathe 2009; Clavier et al. 1984, Yue and Tao 2013).

To quantitatively characterize the effects of pore structure, some conductivity models have been proposed and have increased the technology of applications dramatically (Zhang et al. 2017; Hunt 2004; Xiao et al. 2013). Winsauer has reported a method for determining the tortuosity of pore channels in porous rock (Winsauer et al. 1952). Shang has established an equivalent rock element model (EREM) by a two-orthogonal components pore model (Shang et al., 2004, 2008). Hu et al. proposed a conductivity model upon trapezoidal pore (Shengfu et al. 2017). However, the pore structure is a comprehensive parameter of pore connectivity, microstructure, tortuosity, etc. (Suman and Knight 1997; Attia 2005; Man and Jing 2000). Studies to quantitatively and systematically characterize the effects of pore structure on electrical properties are rare.

This paper intends to clarify the effects of pore structure on the measured electrical response of sand-based porous media. We improve the pore equivalent model of sand-based porous media. And based on it, a new conductive model quantitatively characterizing the integrative effects of pore heterogeneity, pore geometry, and tortuosity on electrical properties is proposed. To verify the new model, experimental data of 39 rock samples, partial exhibits of non-Archie phenomena, are analyzed. Moreover, comparative analysis between the accuracy of important parameters, i.e., formation factor and resistivity index, is calculated by the new and Archie models, EREM models. Further, according to the new model, significant factors causing four kinds of non-Archie phenomenon referring to experimental data are analyzed.

The conductive model of sand-based porous material

The conductive model of sand-based porous material

In rock-based porous media, pores include micropores, small pores, and macropores. Pores are commonly equivalent to circular tubes in existing conductivity models (Herrick and Kennedy 2009; Abousrafa et al. 2009).

Figure 1 shows a SEM (scanning electron microscope) image of a complex pore structure sandstone, and pores are filled with blue glue. It can be observed that effective connecting space is composed mainly of macropores and large throats; their pore geometry can be equivalent to a tubular-like shape. However, small and micropores and micro throats are mainly thin curved sheet shapes, small film shapes, and micro-nano crack network shapes. Their pore geometry should be equivalent to a micro membrane-like shape (Xie et al. 2020).

Fig. 1
figure 1

Pore geometry characteristics based on SEM image

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Figure 2 shows the diagram of the improved pore equivalent model, and pore geometry is equivalent to tubular-like shape and micro membrane-like shape.

Fig. 2
figure 2

Conductive diagram

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In the rock per volume, the electric conductivity and total porosity are expressed as

$$\left\{\begin{array}{c}{\sigma }_{0}=\frac{{\sigma }_{c}}{{\int }_{0}^{1}\frac{1}{{\Delta s}_{0}\left({S}_{t}\cos {\theta }_{1}+{S}_{m}\cos {\theta }_{2}\right)}dl}\\ \phi ={\int }_{0}^{{\tau }_{0}}{\Delta s}_{0}dl\end{array}\right.$$
(1)

where \({\sigma }_{0}\) is the rock conductivity, \(\phi\) is the total porosity,\({\sigma }_{c}\) is the conductivity of pore water, and \({\Delta s}_{0}\) is the cross-sectional area of pore. \(dl\) is the rock section length. \({S}_{t}\) is the area ratio of tubular-like pores. \({S}_{m}\) is the area ratio of micro membrane-like pores. \({\theta }_{1}\) is the angle between the vertical direction of tubular-like pores section and electric field direction. \({\theta }_{2}\) is the angle between the vertical direction of micro membrane-like pores section and electric field direction. Generally, \({\theta }_{1}>{\theta }_{2}\).

Introducing \(p=\frac{{S}_{m}}{{S}_{t}}\frac{\cos {\theta }_{2}}{\cos {\theta }_{1}}\), the electric conductivity is obtained.

$${\sigma }_{0}=\frac{{\sigma }_{c}}{{\int }_{0}^{1}\frac{1}{{\Delta s}_{0}{S}_{t}\cos {\theta }_{1}\left(1+p\right)}dl}$$
(2)
$$\text{As }{\int }_{0}^{1}\frac{1}{\cos {\theta }_{1}}dl={\tau }_{0}$$
(3)

Equation (2) can be written as

$${\sigma }_{0}=\frac{{\sigma }_{c}{\phi }_{e}\left(1+p\right)}{{{\tau }_{0}}^{2}}$$
(4)

where \({\phi }_{e}\) is the tubular-like porosity (\({\phi }_{e}=\phi {S}_{t}\)). Considering pore area heterogeneity, defining \(g=\frac{{\Delta s}_{0}}{{\Delta s}_{1}}\), \({\Delta s}_{1}\) is the minimum pore area.

The electric conductivity and total porosity in Eq. (1) can be rewritten as

$$\left\{\begin{array}{c}\frac{{\sigma }_{0}={\sigma }_{c}{\phi }_{e}\left(1+p\right)}{{{\tau }_{0}}^{2}{\int }_{0}^{1}\frac{1}{g}dl}\\ \phi ={\Delta s}_{1}{\tau }_{0}{\int }_{0}^{1}\frac{1}{g}dl\end{array}\right.$$
(5)

The weighted average \(G\) and harmonic average \(\bar{G }\) can be obtained by integrating \(g\) and \(\frac{1}{g}\) from 0 to 1.

$$\left\{\frac{{\int }_{0}^{1}gdl=\underset{n\to \infty }{\lim }\sum_{n\to \infty }^{\infty }g\left(i\right)\Delta l=\frac{1}{n}\underset{n\to \infty }{\lim }\sum_{n\to \infty }^{\infty }g\left(i\right)G}{{\int }_{0}^{1}\frac{1}{g}dl=\underset{n\to \infty }{\lim }\sum_{i=0}^{\infty }\frac{1}{g(i)}\Delta l=\frac{1}{n}\underset{n\to \infty }{\lim }\sum_{i=0}^{\infty }\frac{1}{g(i)}{~}^{1}\!\left/ \!\!{~}_{\bar{G }}\right.}\right.$$
(6)

Defining \({\chi }_{0}\) as pore area heterogeneity coefficient. \({\chi }_{0}=\bar{G }/G\) (generally, \({\chi }_{0}\leq 1\)). When pore area heterogeneity becomes stronger, \({\chi }_{0}\) decreases.

Defining \({f}_{0}\) as pore geometry heterogeneity coefficient. \({f}_{0}=\left(1+p\right)\) (generally, \({f}_{0}\geq 1\)). When pore geometry gets more complicated, \({f}_{0}\) increases.

Introducing \({\chi }_{0}\) and \({f}_{0}\) into Eq. (4), the rock conductivity is rewritten as

$${\sigma }_{0}=\frac{{\sigma }_{c}{\phi }_{e}}{{{\tau }_{0}}^{2}}{\chi }_{0}{f}_{0}$$
(7)

According to Eq. (7), a conductive model quantitatively characterizing the effects of pore structure on electrical conductivity is proposed.

Formation factor F

The definition of formation factor F is the ratio of rock conductivity with saturated-water and pore water conductivity. Defining \(H={\chi }_{0}{f}_{0}\). As tortuosity \({\tau }_{0}\) can be expressed as a power of porosity (Rangelov and Nassiri 2018; Meng and Liu 2019). According to Eq. (7), the formation factor F can be obtained

$$F=\frac{{\upsigma }_{\mathrm{c}}}{{\upsigma }_{0}}={{\phi }_{e}}^{-\upalpha }{H}^{-1}={{\phi }_{e}}^{-\upalpha }{\left({\chi }_{0}{f}_{0}\right)}^{-1}$$
(8)

where \(\alpha\) is the power coefficient.

Resistivity index I

In water-wet rock, non-wetting fluids (oil and gas) generally have lower conductivity than pore water. When non-wetting fluid entering into pores, the distribution of pore water and conduction path changes. The pressure of tubular-like pores is generally smaller than that of membrane-like pores. When pressure increases, non-wetting fluid first enters into tubular-like pores and then enters into micro membrane-like pores.

When non-wetting fluid enters into tubular-like pores, the water saturation of tubular-like pores is expressed as

$${S}_{w1}=\left({S}_{w}-{S}_{m}\right)/\left(1-{S}_{m}\right),({S}_{w}\geq {S}_{m})$$
(9)

With the accumulation of non-wetting fluid, equivalent conductivity tortuosity increases. The tortuosity coefficient can be expressed as

$$f=\left({{S}_{w1}}^{{\beta }_{1}}+p\right),({S}_{w}\geq {S}_{m})$$
(10)

where \({\beta }_{1}\) is the saturation index of tubular-like pores.

When pressure increases, non-wetting fluid enters into micro membrane-like pores, rock wettability changed, and equivalent conductivity tortuosity significantly increased. The water saturation of micro membrane-like pores and tortuosity coefficient can be expressed as

$${S}_{w2}={S}_{w}/{S}_{m},({S}_{w}<{S}_{m})$$
(11)
$$f={{S}_{w2}}^{{\upbeta }_{2}}p,({S}_{w}<{S}_{m})$$
(12)

where \({\beta }_{2}\) is the saturation index of micro membrane-like pores.

The resistivity index I is defined as the ratio of rock conductivity with partially water-bearing and rock conductivity with saturated water. Then bringing Eq. (8) and Eq. (10) into Eq. (7) obtains

$$\left\{\begin{array}{c}I=\frac{{\upsigma }_{0}}{{\upsigma }_{0}({S}_{w})}=\frac{1+p}{{{S}_{w1}}^{{\beta }_{1}}+p}, ({S}_{w}\geq {S}_{m})\\ I=\frac{{\upsigma }_{0}}{{\upsigma }_{0}({S}_{w})}=\frac{1+p}{{{S}_{w2}}^{{\beta }_{2}}p}, ({S}_{w}<{S}_{m})\end{array}\right.$$
(13)

where \({\upsigma }_{0}({S}_{w})\) is the rock conductivity under saturation \({S}_{w}\).

Model validation

In this section, the Archie model (Eq. (15)) and EREM model (Eq. (16)) are selected for comparative analysis. Table 1 shows comparative models of electrical conductivity.

Table 1 Comparative models of electrical conductivity
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In Eq. (14), \({S}_{m}\) is approximately equal to irreducible water saturation \({S}_{\text{irr}}\) (pore water in micro membrane-like shape is mainly irreducible water). Parameter \(p\) can be calculated from experimental I when only bound water existed in rock (\({S}_{w1}\) = 0); then, \(\upalpha \) and \({\beta }_{1}\) can be determined by the least square method with experimental data F and \({\phi }_{e}\), and I and \({S}_{w1}\), respectively. Further, values of \({\chi }_{0}\) can be calculated from experimental F.

In Eq. (15), a and b are constant coefficients, usually is 1. m is the cementation index of Archie’s formula, which is usually related to the degree of sandstone cementation. n is the saturation index. m and n are usually calculated by the least square method with experimental data F and \(\phi\), and I, and \({S}_{w}\) respectively.

In Eq. (16), k is pore structure efficiency; \({k}_{w}\) is pore structure efficiency for conductive water phase. \(k\) and \({k}_{w}\) can be determined by the least square method with experimental data F and \(\phi\), and I and \({S}_{we}\), respectively.

Relationship between \(\text{logF}\) and \(\log\upphi \)

A total of 27 fine sandstone and 12 glutenite samples from Dongying formation in Huanghua Depression, China, are carried out for resistivity experiment. All of these samples are water-wetted quartz sandstone (Guo et al. 2021), and low clay content (0.01–0.06 v/v) can avoid clay’s influence on resistivity. The resistivity of these 39 rock samples is measured at different water saturation states.

Figure 3 shows the logF-log \(\upphi\) cross plot of 27 fine sandstone samples (black points) and 12 glutenite samples (yellow points). It can be observed that when porosity is bigger than 0.15 v/v, logF-log \(\upphi\) relationship conforms to the Archie model, while when porosity decreases, the non-Archie phenomenon appears. At the right side of Fig. 3, sample a belongs to fine siltstone; its SEM image shows a high content of micro membrane-like pores, indicating strong pore geometry heterogeneity, while its CT (computerized tomography) section scanning face rate shows low pore area heterogeneity. Its experimental F (red star point) is below the Archie calculated F (black line). Sample b is a glutenite sample. Its SEM image shows a low content of micro membrane-like pores, while its CT section scanning face rate indicates strong pore area heterogeneity. Its experimental F (blue star point) is above the Archie calculated F (black line).

Fig. 3
figure 3

a: SEM image and CT section scanning surface porosity rate of sample a. b: SEM image and CT section scanning surface porosity rate of sample b.

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Figure 4 shows the relationship cross plot between pore area heterogeneity coefficient \({\chi }_{0}\) and porosity. \({\chi }_{0}\) value of fine siltstone samples (black points) ranges from 0.87 to 1.1, and the average is 0.9511. For glutenite samples (red points), \({\chi }_{0}\) value ranges from 0.56 to 0.83, and the relationship between porosity and parameter \({\chi }_{0}\) fitted by a least square method is \({\chi }_{0}=1-0.83855\bullet {0.00181}^{\phi }\).

Fig. 4
figure 4

Parameter \({\chi }_{0}\) as a function of porosity

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Figure 5 shows a good relationship between p-value with irreducible water saturation \({S}_{\text{irr}}\). Relationships between \({S}_{\text{irr}}\) and parameter \(p\) fitted by a least square method are \(p=3.14231{\bullet {S}_{\text{irr}}}^{2}-0.88343{S}_{\text{irr}}\) in fine siltstone samples and \(p=1.76343{\bullet {S}_{\text{irr}}}^{2}-0.21855{S}_{\text{irr}}\) in glutenite samples.

Fig. 5
figure 5

Parameter \(p\) as a function of Sirr

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Using Eqs. (14), (15), and (16) to fit the relationship between experimental formation factor and porosity of 27 fine sandstone samples respectively. Fitting result is shown in Fig. 6, and the fitting models, goodness of fit, and average relative error are listed in Table 2. Results show that the new model calculated formation factor is in accordance with experimental values and the accuracy is higher than that calculated by Archie and EREM models.

Fig. 6
figure 6

Comparison between calculated F and experimental F

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Table 2 Parameters and fitting results
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Relationship between \(\text{logI}\) and \({\text{logS}}_{\mathrm{w}}\)

Figure 7 shows the \(\log I\)-\(\log {S}_{w}\) cross plot of 39 sandstone samples (black points and yellow points in Fig. 3). Experimental n-values of these samples range from 1.1 to 3.32. Rapidly changing n-values lead to error in resistivity index I calculation by using the Archie model.

Fig. 7
figure 7

Relationship between I and \({S}_{w}\) of samples

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Using Eqs. (14), (15), and (16) to fit the relationship between experimental resistivity index I and water saturation of 27 fine sandstone samples and 12 glutenite samples, respectively. Figure 8 shows the fitting results, and Table 2 shows the fitting models, goodness of fit, average relative error, and maximum absolute error.

Fig. 8
figure 8

Comparison between calculated I and experimental I

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Figure 8 shows the comparison among the new model calculated resistivity index I, Archie calculated resistivity index I, and experimental values. Results show that the new model calculated resistivity index I is in accordance with experimental values and the accuracy is higher than that calculated by Archie model and EREM model.

Application

The new model (Eq. 14) is applied in Well NPX123 in Dongying Formation of Nanpu sag. NMR logging measurement mode of Well NPX123 is D9TWE3. Specific parameters are given as follows, Group A: TWl = 12.988 s, TEs = 0.9 ms, echo number is 500; Group B: TWs = 1 s, TEs = 0.9 ms, echo number is 500; Group C: TWc = 0.02 s, Tec = 0.6 ms, echo number is 20; Group D: TWl = 12.988 s, TEl = 3.6 ms, echo number is 125; Group E: TWs = 1 s, TEl = 3.6 ms, echo number is 125. Figure 9 shows the evaluation result. Drilling mud resistivity is 1.7OHMM//18 °C. \({\sigma }_{c}=4.34\mathrm{S}/\mathrm{m}\), parameters are listed in Table 2.

Fig. 9
figure 9

Application in well NPX123 (layers 1–7)

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In Fig. 9, the new model calculation results agree better with rock test results than Archie model calculation results. Compared with rock test results, the average relative error of Archie model and the new model are 6.4% and 3.2%, respectively. It can be seen that in layers 1–3, upper part of 4, which have big porosity and simple pore structure, both Archie model and new model have high calculation accuracy. While in bottom part of layer 4, layers 5–7, porosity decreased and pore structure becomes more complex, exhibiting as short range of NMR T2 spectra; calculation accuracy of the new model is significantly higher than Archie model.

In Fig. 9, tracks from left to right include Tracks 1–4: natural gamma-ray logging (GR: GAPI)/spontaneous potential logging (SP: MV), depth (meters), apparent resistivity logs (RLLD/RLLS: OHMM), acoustic-wave slowness logs (AC: us/m)/bulk density (DEN: g/cm3)/neutron porosity (CNL: %). Track 5: NMR logging T2 spectra measured with parameters TE = 0.9 ms, TW = 12,988 ms (NMR.TA: ms). Track 6: NMR logging T2 spectra measured with parameters TE = 3.6 ms, TW = 12,988 ms (NMR.TB: ms). Track 7: clay-bound water porosity computed from NMR logging (MCBW:V/V)/capillary-bound water porosity computed from NMR logging (MBVITA:V/V)/total porosity computed from NMR logging (MSIGTA: V/V)/experimental porosity of rock samples (PHIT: V/V). Track 8: permeability calculated from NMR logging (MPERM: md)/experimental permeability of rock samples (PERM: md). Track 9: irreducible water saturation computed from NMR logging (SWIRR: V/V)/experimental irreducible water saturation of rock samples (SIRR: V/V). Track 10: water saturation computed by the Archie model (SWT: V/V)/experimental oil saturation of rock samples (SO: V/V). Track 11: water saturation computed by the new model (SWX1: V/V)/experimental oil saturation of rock samples (SO: V/V). Track 12: number of the layer.

Discussion and future work

Advantages and shortages

In Eq. (8), when H = 1, α = m, and S_m = 0, Eq. (14) can be simplified into the Archie model (Eq. (15)). It indicates that the new model is unified in form with the Archie model under homogeneous pure sandstone geological conditions. Parameters \(\upalpha \) and \({\beta }_{1}\) in the new model are power exponents of porosity and water saturation respectively, similar to parameters m and n in the Archie model. Compared with the conductive models simulating pores as circular tubes (Archie model) and circular-similar tubes with heterogeneous radius (EREM model), the new model takes into account the effects of pore area heterogeneity and pore geometry heterogeneity, which can more accurately reflect actual rock pore structure. As shown in Fig. 9, the new model can provide a high-precision inversion model for reservoir pore fluid volume inversion, that improves pore fluid volume calculation accuracy in sandstone reservoirs (Lai et al. 2018, 2019). After calibrating parameters \(\upalpha \), \({\beta }_{1}\), \({\chi }_{0}\), \(p\) in the new model by rock samples, the new model can be applied in non-sandstone reservoir evaluation, i.e., tight sandstone, carbonate rock and coal seam.

In the model validation section, the main calculation error causing factors of the new model are ① \({S}_{m}\) is equivalent to irreducible water saturation \({S}_{\text{irr}}\); however, \({S}_{\text{irr}}\) is usually larger than \({S}_{m}\) in actual rocks. ② Measurement error of \({S}_{\text{irr}}\) determined by centrifugal experiments. ③ Measurement error of rock resistivity experiments.

The main shortages of the new model are as follows: ① New model parameters \({\chi }_{0}\) and \(p\) cannot be directly measured in rock physics experiments; they are commonly fitted by mathematic methods. ② In this paper, non-Archie phenomena is only discussed in complex pore structure quartz sandstone conditions, lacking verification and analysis of experimental data in other types of sandstone, such as feldspar sandstone and graywacke sandstone. ③ Actual pore structure is far more complicated; the new model in this paper needs to be further improved.

Non-Archie phenomenon

In this paper, the non-Archie phenomenon is manifested as non-linear. LogF-log \(\phi\) relationship and non-linear. logI-log \(Sw\) relationship. Major factors causing the non-Archie phenomenon can be subdivided into four kinds. Analyze by the new model as follows.

  • a. When \(H\neq 1\) and \({\chi }_{0}{<{f}_{0}}^{-1}\) in Eq. (8), strong pore area heterogeneity causes a non-linear logF-log \(\upphi\) relationship and experimental formation factor is usually smaller than Archie calculated formation factor. Taking fine siltstone samples as examples, it can be seen in Fig. 5 that p-values change greatly as \({S}_{\text{irr}}\) increases, indicating high content of micro membrane-like pores, while in Fig. 4, \({\chi }_{0}\)-values change little as porosity decreases, indicating low pore area heterogeneity; the experimental F (red star point) is below the Archie calculated F (black line) in Fig. 3.

  • b. When \(H\neq 1\), and \({\chi }_{0}{>{f}_{0}}^{-1}\) in Eq. (8), strong pore geometry heterogeneity causes a non-linear logF-log \(\upphi\) relationship and experimental formation factor is usually bigger than Archie calculated formation factor. Taking glutenite samples as examples, it can be seen in Fig. 5 that p-values change little as \({S}_{\text{irr}}\) increases, indicating low content of micro membrane-like pores, while in Fig. 4, \({\chi }_{0}\)-values increase as porosity decreases, indicating strong pore area heterogeneity; the experimental F (blue star point) is above the Archie calculated F (black line) in Fig. 3.

  • c. As shown in Fig. 7, the dispersed n-values are manifested as the non-linear logI-logSw relationship. The ratio of membrane-like pores changes greatly in different rocks; the large range of p values will lead to dispersed Archie model’s n-values of different rock samples even under the same lithology conditions.

  • d. According to Eq. (13), when \({S}_{w}={S}_{m}\), a corner will appear in the relationship between I and \({S}_{w}\). Figure 10 shows Tayor’s experimental data (Taylor and Barker 2006; Worthington 2004); it can be seen that a concave exists in logI-logSw relationship. Using the new model and Archie model to fit the relationship respectively, the results of new model fitting are the goodness of fit \(R=0.958\) and the average relative error 0.107. Significantly, the corner of the fitting curve existed in Fig. 10 is obvious than the experimental data. The reason may be that the actual rock pore structure is more complicated; the transition from tubular-like pores to micro membrane-like pores is a dynamic process and cannot be expressed simply as \({S}_{w}={S}_{m}\).

Fig. 10
figure 10

Fitting results of Tayor’s experimental data

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Future work

Further research directions and subjects concerning electrical properties of sand-based porous material and rock conductivity model may be anticipated in:

  • a. Compared with sandstone, pore characteristics differ greatly in other types of rock, such as fracture and cave existed in carbonate rocks, high and low angle fracture existed in volcanic rocks, etc. (Ara et al. 2001; Gang et al. 2016). The new model needs further validation to be adaptive, and model parameters need to be determined by sample experiments.

  • b. The new model assumes the skeleton is not conductive; actually, some mineral skeletons in rock samples can conduct electricity, such as wet clay and pyrite (Wang and Sun 2007; Al-Sudani et al. 2020). Considering the influence of conductive minerals on rock conductivity is the next step to improving the model’s accuracy.

  • c. Extend the application of the new model in well logging research, reservoir evaluation, reservoir research, and geological research (Mabrouk and Soliman 2015; Feng Cheng et al. 2020; Xie et al. 2021).

  • d. The new model can be extended and applied to cement, coal, materials, and other types of porous media (Li et al. 2016; Yurong et al. 2020; Saxena et al. 2021). Parameters of the new model need to be re-calibrated by resistivity experiments in different types of porous media.

Conclusion

In the present paper, we try to improve our understanding of how pore structure affects sand-based porous media’s electrical properties. The conclusions from the study are summarized as follows:

  1. (1)

    According to the pore geometry characteristics, pore geometry is equivalent to tubular-like shape and micro membrane-like shape. Based on the improved pore equivalent model, a new conductivity model of sand-based porous media is established. The new model is an extension and correction of the Archie model, and it can quantitatively characterize the effects of pore structure on electrical properties.

  2. (2)

    The non-Archie phenomenon, mainly exhibiting as non-linear. LogF-log \(\phi\) relationship and non-linear. logI-log \(\text{Sw}\) relationship is well fitted by the new model. Major reasons can be subdivided into four kinds: ① strong pore area heterogeneity causes a non-linear logF-log \(\upphi\) relationship and rock formation factor is usually smaller than Archie calculated formation factor.② Strong pore geometry heterogeneity causes a non-linear logF-log \(\phi\) relationship, and rock formation factor is usually bigger than Archie calculated formation factor. ③ Dispersed n-values caused by the large range of membrane-like pore ratio cause a non-linear logI-log \(Sw\) relationship. ④ When non-wetting fluid enters into pores, from tubular-like pores to membrane-like pores, a corner appears in logI-log \(Sw\) relationship.

  3. (3)

    The accuracy of the new model calculated results is higher than those calculated by the Archie and EREM models. This research has certain guidance and reference significance for petrophysics, reservoir evaluation, porous media properties, etc.