The relationship between water production rate and resistivity in sand-based porous materials under two-phase fluid flow and its application

Abstract

Accurately predicting the water production rate of multiple-phase fluid flows through porous rock is important for many engineering and geological applications. Taking into account irreducible water and capillary tortuosity, the equivalent rock element model from previous studies has been improved. Based on the improved capillary equivalent rock element model, this study proposes a relationship model between the water production rate in two-phase fluid systems and the resistivity index. The relationship model is verified using rock samples, and the result shows that the water production rate calculated by the new model closely matches experimental values, improving parameter calculation accuracy. In addition, applications of the new model to well logging data show that the calculated water production rate is consistent with the actual production situation. This approach is more efficient and accurate in reservoir evaluation. The new model provides a new idea for studying the seepage characteristics of rocks with multiple-phase fluids.

1 Introduction

The seepage characteristics of porous media with multiple-phase fluids are a research hot spot (Chen and Yao, 2017; Behrang et al., 2016). Understanding the seepage characteristics of rocks in oil–water two-phase flow and oil–gas–water three-phase flow is fundamental for reservoir development planning, production forecasting, and reservoir evaluation (Tiab and Donaldson, 2024; McKenna et al., 2020; Yushu Wu, 2015).

The water production rate, defined as the water volume ratio during reservoir exploitation, is an important seepage characteristic parameter and serves as a dynamic indicator for evaluating the reservoir’s oil-bearing properties. The theoretical results show that there is a clear functional relationship between water production rate and relative permeability (Schön, 2011). In order to accurately calculate the water production rate, several researchers have developed methods for relative permeability. Honarpour et al. derived a relationship model between water production rate and water saturation based on the empirical relative permeability model proposed by Honarpour (2018), Kasha et al. (2023), and Corey (1954). This model is one of the most widely used models, but it only considers the general relationship between water production rate and water saturation, and the accuracy is low in low porosity and low permeability reservoirs (Bear, 2013; Rutqvist et al., 2002). Chima et al. (2010) analyzed the relationship between water production rate and relative permeability during natural gas exploitation in a fractured reservoir. Clarkson described the variation in the water production rate during coal-bed gas exploitation using relative permeability analysis (Clarkson et al., 2011). A theoretical model of gas–water relative permeability in gas hydrate reservoirs is established by Lei et al. (2020). These models can accurately predict the water production rate in fractured formations, coal-bed gas formations, and gas hydrate reservoirs, but their application range is limited. Schlachter et al. examined the water production rate using Modular Dynamics Tester (MDT) testing and logging (Schlachter, 2007; Whittle et al., 2003; Wu and Zhang, 2018). Wang et al. used machine learning and deep learning to predict relative permeability and the oil content ratio during reservoir exploitation, achieving good results (Wang et al., 2022; Zhong et al., 2021).

Previous studies have shown that relative permeability is a key parameter in determining the water production rate, and relative permeability itself is a function of water saturation (Iscan, 2021; Xu et al., 2014). Therefore, the relationship between water production rate and water saturation plays a crucial role in determining the water production rate (Pius and Olamigoke, 2020). Water saturation is a static parameter describing the oil-bearing properties of a reservoir. A common method for determining water saturation is using rock electrical properties, with widely used existing models such as the Archie model and the three-water model (Archie, 1942; Rangelov and Nassiri, 2018; Xie et al., 2023; Jin et al., 2020). The equivalent rock element model (EREM), which incorporates the impact of the pore structure on rock resistivity, is also extensively utilized in the investigation of rock seepage (Shang et al., 2003; Shang et al. 2004; Shang et al. 2008). Consequently, using rock electrical properties to determine the water production rate proves to be a valuable approach. However, research on this topic is limited.

Research has shown that the seepage characteristics and electrical properties of rocks are closely related to the pore structure (Yin et al., 2020; Smirnov et al., 2019; Reza-E-Rabbi et al., 2020). Xie et al. (2020) proposed a relationship model between relative permeability and rock electrical properties, considering the effect of the pore structure. Xu P et al. developed a relative permeability model based on the fractal theory (Peng et al., 2013; Wang et al., 2019). The pore structure is an essential factor in establishing the relationship model between water production rate and rock electrical properties. Taking into account the effect of the pore structure, this paper has improved the EREM, and a relationship model between water production rate and resistivity index is proposed. Seven rock samples with different pore structure types were used to validate the new model, and the average relative error between the water production rate calculated by the new model and the experimental data is 5%. The results of both verification and application show that the new model can directly calculate the water production rate from the resistivity index and accurately predict the water production rate of rocks with different pore structures. The novel model incorporates pore structure characterization parameters to mitigate the impact of the pore structure on the water production rate, thereby enhancing the precision of water production rates. In addition, the new model provides a new idea for studying the seepage characteristics of rocks and enriches the data foundation for guiding reservoir development planning, production forecasting, and reservoir evaluation.

2 Relationship between water production rate and electrical resistivity in two-phase fluid flows

2.1 Improved capillary equivalent rock element model

According to the EREM (Shang et al., 2003; Shang et al. 2004; Shang et al. 2008), we obtainwhere in Equation 1, (v/v) is the total porosity, (v/v) is the macropore porosity, (v/v) is the throat-pore porosity, and (dimensionless) is the ratio of to . Obviously, is closely related to the pore structure. Figure 1 shows the equivalent rock element model of the porous media. Figure 1A shows the pore distribution of the porous media. Figure 1B shows the equivalent rock element model of the porous media.

FIGURE 1

In Figure 1, is the total porosity, is the macropore porosity, is the throat-pore porosity, and is the matrix volume.

Generally, the flow characteristics are mainly featured by throat pores. Figure 2 shows the flow model of the pore fluids. The equivalent capillary porosity can be expressed aswhere in Equation 2, (v/v) is the equivalent capillary porosity.

FIGURE 2

In Figure 2, is the equivalent capillary porosity and is the matrix volume.

Actually, water-wetted rocks contain irreducible water. The effective flow porosity (v/v) is expressed aswhere (v/v) is the equivalent capillary irreducible water saturation (the ratio of irreducible water porosity to equivalent capillary porosity). (v/v) is the bound water volume. The equivalent capillary irreducible water saturation can be calculated as follows:where (v/v) is the total irreducible saturation (the ratio of total irreducible water to total porosity).

By substituting Equation 4 into Equation 3, the effective porosity of flow fluids is expressed as

The equivalent rock element model assumes that the effective flow pores are straight capillaries. However, studies have shown that the flow of pore fluids in porous media is a complex process (Cai et al., 2019). In order to simulate the actual flow of pore fluids, the effective flow pores are assumed to be curved capillaries. Figure 3 shows the improved capillary equivalent rock element model (ICEREM), where is the tortuosity. The ICEREM introduces the volume of bound water, which is more consistent with the microscopic pore characteristics of the rock, and its pore structure characterization accuracy is further improved than that of the EREM model.

FIGURE 3

In Figure 3, is the effective flow porosity, is the equivalent tortuosity per unit volume of the rock, is the bound water volume, and is the matrix volume.

2.2 Relationship between water production rate and electrical resistivity in two-phase fluid flows

According to Darcy’s law and Poiseuille equation (Darcy, 1856; Song R et al., 2019), the water flow through the rock section can be expressed aswhere (m3) is the water flow, (md) is the permeability, (Pa) is the pressure difference in the rock section, (s) is the water viscosity, and (dimensionless) is the equivalent tortuosity per unit volume of the rock.

From Equation 6, it is assumed that fluids flow only in the effective flow pores, as shown in Figure 3.

As the non-wetting phase fluid entered the pores, it was mainly distributed in the central part of the capillary pores. Figure 4 shows the distribution of the two-phase flow fluids in the capillary pores.

FIGURE 4

In Figure 4, is the effective flow porosity, is the equivalent tortuosity per unit volume of the rock, is the bound water volume, is the oil porosity in the effective flow porosity, is the water porosity in the effective flow porosity, is the oil porosity in the macropore porosity, is the water porosity in the macropore porosity, and is the matrix volume.

The water flow and non-wetting phase fluid flow across the rock section can be expressed aswhere (m3) is the water flow, (m3) is the non-wetting phase fluid flow, (s) is the non-wetting phase fluid viscosity, (v/v) is the water saturation of the effective flow porosity, is the equivalent tortuosity of the water flow in the effective flow capillary pores, and (dimensionless) is the equivalent tortuosity of the non-wetting phase fluid flow in the effective flow capillary pores.

According to Equation 7, the water production rate of a two-phase fluid flow can be calculated as follows:

Rock conductivity is the parallel conductance between equivalent capillary porosity and the invalid flow part (invalid flow capillary porosity and the rock matrix). The conductivity of the invalid flow part is assumed to be 0, and the conductivities of water-saturated and partially saturated rock can be obtained as follows:

In Equation 9, (S/M) is the water-saturated rock conductivity, (S/M) is the partially saturated rock conductivity, (S/M) is the water conductivity, (S/M) is the non-wetting phase fluid conductivity, (dimensionless) is the equivalent tortuosity of the irreducible water, and (S/M) is the conductivity of the rock matrix.

The three-water model considers rock conductivity as a parallel conductance among shaly irreducible water, micropore irreducible water, and free water, and each conductance component follows the Archie model (Zhang and Pan, 2010; Mo et al., 2001). According to the three-water model, the conductivity of water-saturated and partially saturated rock can be further revised as follows:

In Equation 10, both shaly irreducible water and micropore irreducible water are treated as irreducible water and are assumed to have the same conductivity. and (dimensionless) are the cementation exponent and saturation exponent of the equivalent capillary pores, respectively. and represent the cementation exponent and saturation exponent of the irreducible water pores, respectively.

By comparing Equation 9 with Equation 10, we can derive the equivalent tortuosity for the water flow in effective flow capillary pores (dimensionless) and non-wetting phase fluid flow in the same effective flow capillary pores (dimensionless):

In addition, when the non-wetting phase fluid conductivity equals the pore water conductivity, according to Equations 9, 10, can be written as follows:However, the difference between actual non-wetting phase fluid conductivity and wetting phase fluid conductivity may cause a variation in . To eliminate the difference, according to Equations 11, 12, is defined as

By substituting Equations 4, 5, 11, 13 into Equation 8, the water production rate (v/v) of the two-phase fluid flow can be simplified as

According to Equations 9, 10, the relationship between and the resistivity index is as follows:where (dimensionless) is the resistivity index and . Generally, the conductivity of the non-wetting phase fluid (usually oil, gas, or kerosene) is much lower than that of pore water. So, we can assume that . Then, Equation 15 can be simplified aswhere , , and (dimensionless) are the exponents (functions of , , , and ).

By comparing Equation 16 with Equation 14, the relationship between the resistivity index and water production rate can be obtained.

In Equation 17, , , and are introduced to account for the difference between the equivalent tortuosity of pore fluids and that of the electrical current. Generally, , , , , , and are regional coefficients, which can be calculated using the least squares method.

In Equations 16, 17, pore structure characterization parameters and are innovatively introduced, which can accurately characterize the seepage law of the two-phase fluid in rocks with different pore structure characteristics.

3 Verification

To verify the model in this paper, the selected rock samples have the same sedimentary environment and lithology, ensuring that the resistivity and seepage capacity of the samples are mainly affected by the pore structure while minimizing the influence of other factors, such as clay content. Seven rock samples selected from the Dongying Formation, Nanpu Sag, are used to verify the relationship model between the and water production rate. The porosity ranges from 0.19 to 0.255 (v/v), the permeability ranges from 6.09 to 292 (md), and the irreducible water content ranges from 0.518 to 0.861 (v/v). The rock samples are tested using nuclear magnetic resonance (NMR) experiments and combined tests of resistivity and water–oil two-phase seepage. The seven rock samples, with low clay content, are selected to minimize the influence of shale on resistivity. Table 1 presents the characteristic parameters of the seven rock samples.

Figure 5 shows the T₂ spectrum of rock samples; the variation in the distribution of the T₂ spectra and the position of T₂ peak values indicate differences in pore structures. For example, sample No. 1 shows a double-peak right-skew type with the highest peak position of 100 ms; sample No. 2 shows a double-peak balanced type with the highest peak position of 10 ms; sample No. 3 shows a double-peak left-skew type with the highest peak position of 3 ms; sample No. 4 shows a double-peak right-skew type with the highest peak position of 30 ms; sample No. 5 shows a medium peak with the peak position of 10 ms; sample No. 6 shows a single-peak left-skew type with the peak position of 2 ms; and sample No. 7 shows a single-peak left-handed type with the peak position of 3 ms.

FIGURE 5

Figures 6–8 show the relationship among (resistivity index), (water production rate), and (water saturation). Figure 6 shows the relationship between resistivity and water saturation for the seven rock samples, which exhibit non-Archie phenomenon, and the Archie saturation index n ranges from 1.14–3.1.

FIGURE 6

FIGURE 7

FIGURE 8

It can be observed that data and data for the seven rock samples in Figures 7, 8 are highly dispersed, which is primarily caused by differences in pore structures. This means that significant errors in the water production rate can occur without considering the pore structure (Honarpour, 2018; Purcell 1949; Burdine 1953; Corey, 1954).

In Equation 16, parameter is related to the pore structure and is used to calculate the water production rate. In this section, mercury injection capillary pressure (MICP) and NMR experimental data are used to determine macropores and throat pores.

The Swanson parameter (inflexion point) of the mercury injection curve is used to distinguish large and small pores (throat pores) in rock samples (Swanson, 1981; Zhang and Weller, 2021; Xiao et al., 2017). The integral curve of the NMR T₂ spectrum corresponds well with the mercury injection curve. According to the definition of the Swanson parameter, T₂ time at the inflexion point between macropore and small pores is determined, and then parameter is determined. In this paper, T₂ time at the inflexion point is 110 ms, determined by averaging data from 25 rock samples. Figure 9 shows the classification of macropores and throat pores.

FIGURE 9

In Figure 9, the NMR T₂ spectra of the core sample (red line) and the pore size distribution of the core sample derived from the MICP experimental data (blue dash-dotted line) are presented. T₂ (ms) is the transverse relaxation time. S (T₂) (%) is the T₂ pore component. ra (μm) is the pore radius. S (ra) (%) is the pore component derived from MICP experimental data. Demarcation T₂ time (ms) is the inflexion point between macropores and throat pores, which is 110 ms in this part.

Regional coefficients , , , , , and are calculated using porosity, irreducible water saturation, p, and water production rate through the least squares method: , , , , , and .

In order to compare with previous research results, the water production rate is expressed aswhere the relative permeability of oil and water phases in the Equation 18 can be calculated according to the model proposed by Wang and Cardenas (2018).where in Equation 19, is water saturation, is bound water saturation, is residual oil saturation, is water phase relative permeability, and is oil phase relative permeability. and are regional experience indexes, and and in this region. The calculation results are shown in Figure 10.

FIGURE 10

The water production rate can be calculated using Equation 17 and Wang and Cardenas model. Figure 10 shows the comparison between the calculated water production rate using Equation 17 and experimental data (the red point in Figure 10), where the goodness of fit is 0.97945, and the average relative error is 5%. Figure 10 shows the comparison between the calculated water production rate by the Wang and Cardenas model and experimental data, where the goodness of fit is 0.7995 and the average relative error is 23%. The result indicates a strong agreement between the calculated water production rate and experimental data.

4 Application in well logging data

The new method is applied to Well NPX49 in Nanpu Sag, Huanghua depression, China. The NMR logging measurement mode of Well NPX49 is D9TWE3. Specific parameter areas are as follows: for Group A, TW_l = 12.988 s, TEs = 0.9 ms, and the echo number is 500; for Group B, TWs = 1 s, TEs = 0.9 ms, and the echo number is 500; for Group C, TWc = 0.02s, TEc = 0.6 ms, and the echo number is 20; for Group D, TW_l = 12.988 s, TE_l = 3.6 ms, and the echo number is 125; and for Group E, TWs = 1 s, TE_l = 3.6 ms, and the echo number is 125. Drilling mud resistivity is 2.26 OHMM/18°C.

Figure 11 shows the evaluation result of Well NPX49 (layers 1–3) using the new method. The reservoir parameters and water production rate of the produced fluid from layers 1–3 are shown in Table 2.

FIGURE 11

In Figure 11, 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), and neutron porosity (CNL:%). Track 5: NMR logging T₂ spectra measured with parameters TE = 0.9 ms and TW=12988 ms (NMR.TA:ms). Track 6: NMR logging T₂ spectra measured with parameters TE = 3.6 ms and TW=12988 ms (NMR.TB: ms). Track 7: clay-bound water porosity (PWC: V/V)/ capillary-bound water porosity (PWI: V/V)/ movable water porosity (POR: V/V) computed from NMR logging. Track 8: permeability calculated from NMR logging (NMR.PERM: MD). Track 9: water saturation by the Archie model (ARCHIE.SW: V/V)/ irreducible water saturation from NMR (NMR.SIRR: V/V). Track 9: the new method calculated the water production rate of percolating fluid (FW: V/V). Track 10: the number of the layer.

In layers 1–3, the low natural gamma-ray logging values present the characteristics of sandstone with low clay content. The resistivity of layer 1 is high, with a value of 39 Ω·m. The distribution of NMR T₂ spectra for both short-TE and long-TE is broad. The calculated water saturation is 40.3%, and the bound water saturation calculated based on NMR is 38%, demonstrating excellent oil-bearing characteristics. The daily oil production of layer 1 is 14 m³ with no water (the water production rate of layer 1 is 0, as shown in Table 2; represents the volume ratio of water in the produced liquid volume). This indicates that pore water in layer 1 is in a bound state and cannot flow. The resistivity of layers 2 and 3 is low, with values of 22 and 27 Ω·m, respectively. The distribution of NMR T₂ spectra for short-TE and long-TE is also broad. The calculated water saturation is 65.8% and 61.7% for layers 2 and 3, respectively, indicating the oil-bearing characteristics of layers 2 and 3.

The distribution of long-TE T₂ spectra and short-TE T₂ spectra narrows from layer 1 to layer 3. The oil saturations of layers 1–3 are 59.7%, 34.2%, and 38.3%, respectively. The calculated water saturation reduced from layer 1 to layer 3, indicating that layer 1 is more oil-rich than layers 2 and 3. The calculated results of layers 2 and 3 show some differences. However, the daily oil production of layer 2 is 0.2 m³, with a water production rate of 95.6%. The daily oil production of layer 3 is 3.2 m³, with a water production rate of 60.4%. The main reason for the difference between layers 2 and 3 is the difference in the pore structure; for instance, the NMR T₂ spectrum in layer 2 shows obvious bimodal right-skew characteristics, while in layer 3, it shows single-modal left-skew characteristics. The application results show that oil saturation does not reflect actual oil production. The water production rate calculated by the new model is consistent with the actual production situation.

5 Discussion and future work

6 Conclusion

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