## Abstract

Determining rock resistivity for saturation estimation in reservoirs is challenging due to the complex nature of pores in the rock. This paper aims to establish a computational relationship between formation factors (*F*) and permeability (*K*) by combining theoretical and experimental data. Firstly, the relationship between the permeability of the curved capillary model and formation factors, as well as the relationship between the permeability of the complex curved capillary model and formation factors, are deduced. Theoretical analysis proved that the formation factors (*F*) have a power relationship with permeability (*K*) and porosity ($\phi $), and confirms the existence of additional resistivity (*R _{x}*). To validate the theoretical study, we conducted a model analysis using open experimental data from 35 sandstone cores with different porosities and permeabilities from the tight gas sandstone in the Western US basins, which measured resistivity data in saline at 20, 40 and 80 Kppm, respectively. We confirmed the existences of additional resistivity (

*R*) by fitting the relationship between the rock resistivity of saturated formation water (

_{x}*R*

_{0}) and the formation water resistivity (

*R*

_{w}). We then fitted the formation resistivity change factor (

*F*

_{d}) with permeability (

*K*), the formation resistivity change factor (

*F*

_{d}) with porosity ($\phi $), the additional resistivity (

*R*) with permeability (

_{x}*K*), and the additional resistivity (

*R*) with porosity ($\phi $). Both the changeable formation resistivity change factor (

_{x}*F*

_{d}) and additional resistivity (

*R*) showed a strong linear relationship with permeability (

_{x}*K*) in logarithmic coordinates.

We also verified the existence of a suitable equation using available experimental data by changing the formation parameters and permeability. The study shows that the fitting equations may be utilized to determine the changeable formation resistivity change factor (*F*_{d}), additional resistivity (*R _{x}*) and the rock resistivity of saturated formation water (

*R*

_{0}) with varying permeability. The predicted rock resistivity of saturated formation water (

*R*

_{0}) strongly correlates with that measured in the laboratory, providing better precision for future reservoir evaluations using saturation estimations.

In the field of petroleum exploration, accurately determining the porosity, permeability and water saturation of a reservoir is critical for conducting effective evaluation. Archie's formula (1942), developed by G.E. Archie, is a widely used method for evaluating saturation through resistivity analysis of experimental data. These parameters are influenced primarily by reservoir geology, including depositional conditions and genesis. An evaluation of water content saturation is one of the most critical elements in reservoir evaluation. Archie proposed a model for quantitatively evaluating saturation by resistivity through comprehensive analysis of experimental data (Archie 1942). Many other equations have been derived using Archie's formulation as the basic equation. Archie (1947) used an equation to measure the fluid saturation (*S*_{w}) of the rock using porosity ($\Phi $), the cementation coefficient (*m*), the saturation index (*n*), resistivity (*R*_{w}), the true formation resistivity of the rock (*R*_{t}) and the tortuosity factor (*a*). After Archie's formula was introduced, more academics such as Waxmanand Smits (1968), Waxman and Thomas (1974), Roberts and Schwartz (1985), Kumar *et al.* (2011) and Nazemi *et al.* (2019) conducted in-depth investigations using experimental and theoretical analyses to understand the rock conduction mechanism. They discovered that several factors, such as pore geometry, shale volume and roughness of the pore wall, influence the formation factor (*F*). The relationships of *F* v. $\Phi $ and resistivity index (RI) v. *S*_{w} are not always clearly linear in a double logarithmic coordinate system. The parameters in Archie's formula are related to the formation lithology and pore structure, and they vary with the salinity of the formation, and the surrounding pressure and temperature. However, in low-porosity and low-permeability reservoirs in particular, the traditional Archie formula seems ineffective. Waxman and Smits (1968) discovered the additional conductivity of clay minerals and proposed their Waxman–Smits (W–S) model based on cation-exchange capacity calculation. Clavier *et al.* (1977) divided the conductive layer on the surface of clay minerals into the surface layer close to the clay minerals that is affected by the composition of the clay minerals and the free electron layer away from the clay minerals that is unaffected by clay, and developed an improved W–S model. As the understanding of reservoir genesis has deepened, the study of the saturation model based on the conductive mechanism of complex pore structures has gradually become a hotspot in reservoir evaluation. Shang *et al.* (2003) proposed an equivalent rock component saturation model that divided the pores and throat-forming pore space into parallel and series pores, defined the pore structure coefficient and saturation scale factor, and provided a clear parameter calculation method that effectively improved the saturation calculation accuracy for reservoirs with a complex pore structure. Xiao *et al.* (2013) classified the physical characteristics of different capillary pressure curve morphological types according to their porosity range, and established a statistical relationship between the ‘J function’ and saturation (Xiao *et al.* 2013; Nazemi *et al.* 2021; Tavakoli *et al.* 2022). Numerous scholars, both domestic and international, have conducted significant research on resistivity models and established various models suitable for sandstone, including the effective medium resistivity model, which is entirely composed of conductive minerals, the pure conduction model of rock, and the skeleton composed of part of the conductive particles of the conductive pure rock conductive model (Hunt 2004; Montaron 2009; Liu *et al.* 2013; Revil *et al.* 2013; Tang *et al.* 2015). Different algorithms have been proposed for different saturation models (Glover *et al.* 2010; Wang and Liu 2017). However, all the existing resistivity models face a challenge when describing the conductivity pattern of muddy sandstone with partial conductive minerals in the skeleton.

Zhang *et al.* (2020) derived equations for the resistivity change factor (*F*_{d}) of rocks and additional resistivity (*R _{x}*) without making any assumptions. The validity of these equations was tested using thousands of experimental data points and previously published data (Zhang

*et al.*2020).

Archie proposed a functional expression between the formation factor and porosity but noted the need to clarify the relationship between permeability and the formation factor compared to porosity. However, he did not conduct further analysis on this issue. Permeability is one of the most important parameters required in reservoir characterization. When there is no physical or chemical change in the pore fluid and porosity of the rock, the value for permeability and the formation factor is determined by the pore structure.

## Geological setting

This study utilizes open experimental data from tight gas sandstone in the western USA to examine the relationship between core and log petrophysical properties and lithofacies sedimentary characteristics in Mesaverde Group tight gas sandstones. The study examines 40 cores in the Washakie, Uinta, Piceance, Greater Green River, Wind River and Powder River basins (see Fig. 1). The shaly intervals of the Mesaverde Group are dominated by mudstones and burrowed silty shales, as well as lenticular and wavy-bedded very shaly sandstones, and wavy-bedded to ripple cross-laminated shaly sandstones. The sandstone intervals are dominated by ripple cross-laminated, cross-bedded, planar-laminated and massive, very-fine- to fine-grained sandstones (Byrnes *et al.* 2008).

This paper utilized a collection of open experimental data from tight gas sandstone in the western USA for study. To gain insight into the internal pore structure connectivity of the cores in the two basins, we first analysed the core images (Figs 2–5).

Analysing core images from the same area can also aid in comprehending the impact of pore structure on interconnectivity. It is evident from the core images that the connectivity between the two basins differs significantly. The Green River Basin exhibits well-developed intergranular dissolution pores with good pore connectivity and permeability, while the Piceance Basin is mainly characterized by isolated intergranular pores with poor pore connectivity and low permeability.

## Methodology

Archie's extensive experimental data revealed that the formation factor, *F*, and porosity, $\Phi $, could be approximated effectively and modelled as a straight line in logarithmic coordinates. The exponent *m* value, according to Archie, typically ranges from 1.3 to 2. The coefficients in Archie's formula are derived from a specific physical model and adjusted through real rock sample experiments. However, if a region's rock samples have complex lithological characteristics, the experimentally derived parameters may not satisfy engineering needs. Hence, researchers need to modify the values of *m* and *n* when employing Archie's formula in various regions with distinct lithologies. Therefore, the coefficients in Archie's formula may differ depending on the region.

Shaly sandstones can present difficulties in logging analysis due to the presence of clays with a high cation-exchange capacity (CEC) and formation brines with a low salinity (Kukal *et al.* 1983). To address these challenges, several algorithms have been proposed to calculate the water saturation in shaly sands. These include the empirical Simandoux model (1963), and the more theoretical dual water and Waxman–Smits models (Waxman and Smits 1968; Waxman and Thomas 1974). These models attempt to account for the influence of clay minerals and low-salinity brines on the electrical resistivity of the formation, and the subsequent measurements obtained from logging tools.

In earlier studies, various conductive models were used by Waxman and Smits (1968), Clavier *et al.* (1984) and Silva and Bassiouni (1988) to investigated the additional conductance of clay-exchange cations and to assess saturation with fixed and variable parameters of Archie's equation.

*et al.*(2020) discussed the additional resistivity and conductivity of natural reservoirs. They demonstrated that any core (

*R*

_{0}) resistivity could be expressed as a function of the equivalent additional resistivity (

*R*),

_{x}*F*

_{d}and

*R*

_{w}, as shown in equation (3):

*R*is the additional resistivity ($\Omega $·m) and

_{x}*F*

_{d}is formation resistivity change factor (dimensionless) (see Table 1).

### Relationship between the permeability of the curved capillary model and the formation factor

*r*. The tortuosity of the curved capillary is defined as $\tau w$, and $Lw$ is the actual length of the capillary channel, while $L0$ is the straight length of the capillary channel.

*L*

_{w}is expressed in equation (2) as:

*Q*is the flow rate (cm

^{3}s

^{−1}), $\Delta $

*P*is the pressure difference between the two ends of the capillary tube (10

^{5}Pa) and $\mu $ indicates the viscosity of water (10

^{−3}Pa s).

*Q*, can be expressed as:

*R*

_{0}is the rock resistivity of fully saturated brine only ($\Omega $·m),

*R*

_{w}is fluid resistivity ($\Omega $·m) and

*R*

_{t}is rock resistivity of fully saturated brine and gas ($\Omega $·m).

*F*, can then be expressed as:

*c*represents the dimensionless coefficient related to the pore structure. Equation (11) shows the exponential relationship between the formation factor, permeability and porosity in a single capillary model.

### Relationship between permeability and the formation factor in a complex curved capillary model

A conductive pathway can be formed in mud with a conductive skeleton due to the presence of conductive minerals in the rock, dispersed clay and residual movable water in micropores. Thus, two conductive pathways in pores would exist in parallel. This model is illustrated in Figure 7.

*L*and the tortuous degree as $\tau x$, and subsequently $Lx=\tau xL0$, so the volume in this pathway is:

_{x}*Q*is the flow rate (cm

^{3}s

^{−1}), $\Delta $

*P*is the pressure difference between the two ends of the capillary tube (10

^{5}Pa) and $\mu $ indicates the viscosity of water (10

^{−3}Pa s).

*Q*, can be expressed as:

*R*

_{w}is the rock fluid resistivity ($\Omega $·m) and

*R*is additional resistivity ($\Omega $·m).

_{x}*F*

_{d}is the formation resistivity change factor (dimensionless) and

*D*denotes the coefficient related to the pore structure (dimensionless).

Equation (24) shows that the rock resistivity (*R*_{0}) can be expressed as a primary function of the formation resistivity change factor *F*_{d} and the additional resistivity *R _{x}* when fully saturated with brine in the complex curved capillary model . The formation resistivity change factor

*F*

_{d}is influenced by the effective porosity and tortuosity of the rock.

*F*

_{d}can be obtained as follows:

*a*and

*b*represent the dimensionless coefficient related to the pore structure. A power relationship exists between permeability and the formation factor and between porosity and the formation factor in complex curved capillary models.

## Results

We selected 35 core samples from two basins with significantly different pore connectivity (Tables 2 and 3).

The core samples chosen for this study have undergone electrical resistivity measurement experiments and are representative regarding geographical location, lithology, porosity and permeability. To evaluate the possible W–S cation-exchange effects, data analyses were conducted at different NaCl concentrations of 20 (*n* = 138), 40 (*n* = 310) and 80 Kppm (*n* = 198). Obtaining a known salinity in low-permeability rocks can be challenging due to their low brine permeability, which makes it difficult to achieve flow-through displacement. To address this challenge, the samples were dried at 70°C for several hours and then immersed in a methyl alcohol bath for 24–48 h. The samples were then extracted with methyl alcohol for 3 days, dried and vacuum/pressure saturated with methyl alcohol before being immersed in the methyl alcohol bath for a minimum of 3 days. The samples were then dried at 70°C in a convection oven for not less than 24 h and weighed to confirm that the sample weights had returned to their original pre-saturation weights for clean, dry samples. The samples were then vacuum/pressure saturated with brine of different salinity level and left immersed for 2–8 weeks. This process was repeated for each change in salinity (Byrne *et al.* 2008).

Through our analysis, we have verified the existence of additional resistivity (*R _{x}*) by fitting the relationship between the rock resistivity ($\Omega $·m) of fully saturated brine (

*R*

_{0}) and formation water resistivity (

*R*

_{w}) (Figs 8 and 9).

Figures 8 and 9 provide clear evidence of the variation in formation water resistivity (*R _{w}*) and saturation across different permeability cores in the Green River Basin and the Piceance Basin, spanning several orders of magnitude. The strong linear relationship observed in the data for formation water resistivity (

*R*

_{0}) confirms the accuracy of equation (3) and the existence of additional resistivity (

*R*). Analysis of the core images suggests that the dominant factors influencing additional resistivity (

_{x}*R*) are pore structure – that is, porosity and pore connectivity. It is important to note that additional resistivity (

_{x}*R*) is always present and does not reach 0.

_{x}## Discussion

To better understand the relationship between the resistivity change factor (*F*_{d}) and the additional resistivity (*R _{x}*) with rock pore structure, we conducted a fitting analysis in this study. Specifically, we analysed the relationship between the formation resistivity change factor (

*F*

_{d}) with permeability (

*K*), as well as the relationship between resistivity change factor (

*F*

_{d}) and porosity ($\Phi $), and analysed the relationship between

*R*and

_{x}*K*and the relationship between

*R*and porosity ($\Phi $) (Figs 8–11).

_{x}The results of the fitting equations demonstrated a strong linear relationship between *R _{x}* and

*K*when plotted on a log–log coordinate for the core data in both basins. This suggests that the fluid-flow channels within the rock pores play a significantly role in the additional resistivity (

*R*). Furthermore, the relationship

_{x}*F*

_{d}and

*K*is stronger than that between

*F*

_{d}and $\Phi $ in the Green River Basin, while in the Piceance Basin the relationship

*F*

_{d}and $\Phi $ appears to be stronger than the one between

*F*

_{d}and

*K*. These findings suggest that Piceance Basin cores have a lower porosity and permeability compared to those of the Green River Basin. It can be inferred that when

*K*is below 10

^{−3}mD, the resistivity change factor (

*F*

_{d}) is primarily influenced by porosity ($\Phi $), based on the core data and images from both basins.

Figures 10–13 demonstrate a negative correlation between the values of additional resistivity (*R _{x}*) and the formation resistivity change factor (

*F*

_{d}) with permeability (

*K*).

*R*represents the equivalent resistivity added by pore networks in series with pore expansion, and when pore connectivity is optimal the corresponding additional resistivity is low. As such, the underlying physical meaning is aligned with the observed trend. As

_{x}*F*

_{d}and

*R*exhibit a strong correlation with

_{x}*K*in the two basins, this relationship can be utilized within the fitting formula to calculate the

*F*

_{d}and

*R*values of a given core permeability. This information can then be used in equation (3) to estimate the rock resistivity (

_{x}*R*

_{0}) fully saturated by brine.

The fitting formula is as follows:

*R*is additional resistivity ($\Omega $·m),

_{x}*F*

_{d}is the formation resistivity change factor and

*K*is permeability (mD).

We conducted a new analysis using experimental data from sandstones with varying permeability in both the Green River and Piceance basins to validate the general applicability of our fitting formula (Tables 4 and 5). We used the fitting formulae (equations 26–29) and equation (3) to calculate the rock resistivity ($\Omega $·m) of fully saturated brine (*R*_{0}). We then compared the estimated values for saturated formation water (*R*_{0}) with the laboratory measurements and found them to be very similar (Figs 14 and 15).

The proposed method of calculation has provided a new approach for estimating the rock resistivity of saturated water (*R*_{0}) using permeability. In practical scenarios, the rock resistivity ($\Omega $·m) of fully saturated brine (*R*_{0}) may not be readily available for all core samples. As a result, the fitting formula and equation (3) can be used to obtain an estimation of the rock resistivity ($\Omega $·m) of fully saturated brine (*R*_{0}) within a given basin.

## Conclusions

In this study, we investigated the presence of additional resistivity (*R _{x}*) through theoretical and data analysis. Our results indicate that the pore structure of the material, particularly porosity and pore connectivity, is the primarily influence on additional resistivity (

*R*). Furthermore, our findings suggest that additional resistivity (

_{x}*R*) is consistently present and cannot be reduced to 0.

_{x}In addition, our study demonstrated the relationship between additional resistivity (*R _{x}*), the formation resistivity change factor (

*F*

_{d}) and permeability (

*K*) through an analysis of experimental data. When plotted in logarithmic coordinates, our results revealed a strong linear correlation between the formation resistivity change factor (

*F*

_{d}) and additional resistivity (

*R*) with respect to permeability (

_{x}*K*). Specifically, we observed a decline in both the formation resistivity change factor (

*F*

_{d}) and additional resistivity (

*R*) as permeability (

_{x}*K*) increased, which is consistent with our theoretical expectations.

To demonstrate the broad applicability of our fitting formula, we utilized experimental data from a sandstone reservoir in a basin in the western USA (Green River) with varying permeability levels. We applied our fitting formula to calculate the rock resistivity of saturated formation water (*R*_{0}). Our results showed that the estimated values for the rock resistivity ($\Omega $·m) of fully saturated brine (*R*_{o}) were remarkably similar to the laboratory measurements. Therefore, our study has provided a new approach for calculating resistivity (*R*_{0}) of rock in saturated formation water.

## Author contributions

**WZ**: writing – original draft (lead); **TL**: writing – review & editing (supporting); **JT**: writing – review & editing (equal); **JZ**: writing – review & editing (supporting)

## Funding

This work was funded by the National 13th 5-Year Plan of Oil and Gas Program of China (2017ZX05019-001), National Natural Science Foundation of China (41476027), PetroChina Key Technological Program (2016E-0503) and CNPC Research Projec (2017 F-16).

## Competing interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data availability

All data generated or analysed during this study are included in this published article (and, if present, its supplementary information files).