A New Method to Calculate the Relative Permeability of Oil and Water in Tight Oil Reservoirs by Considering the Nonlinear Flow

Su, Haibo; Wang, Duocai; Zhang, Ping; An, Yongsheng; Fu, Yaping; Lu, Jun; Huang, Famu; Zhang, Hong; Ren, Zhongxing; Li, Zhuo

doi:https://doi.org/10.1155/2022/9450967

Geofluids

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Abstract Introduction Analysis Conclusions Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 9450967 | https://doi.org/10.1155/2022/9450967

A New Method to Calculate the Relative Permeability of Oil and Water in Tight Oil Reservoirs by Considering the Nonlinear Flow

Haibo Su,¹Duocai Wang,¹Ping Zhang,¹Yongsheng An,²Yaping Fu,¹Jun Lu,¹Famu Huang,¹Hong Zhang,¹Zhongxing Ren,¹and Zhuo Li²

Academic Editor: Bailu Teng

Received12 Apr 2022

Revised25 May 2022

Accepted10 Jun 2022

Published29 Jul 2022

Abstract

The theoretical research on the percolation mechanism and oil-water relative permeability of low-permeability oil reservoirs is the focus and hotspot of international researchers. Oil-water relative permeability is an important parameter that describes the characteristics of oil-water two-phase flow and is widely used in the dynamic analysis of development and numerical simulation technology in the reservoir. The traditional calculation method of oil-water relative permeability (i.e., the JBN method) is based on the Darcy flow law. When the velocity and the displacement pressure gradient do not follow the linear flow law, there are some errors in the results of oil-water relative permeability calculated by the JBN method. In this paper, the traditional JBN method is improved, and we establish a new processing method of experimental data for oil-water relative permeability considering the effects of nonlinear flow, capillary pressure, and gravity in tight oil reservoirs. The experiments on the flow characteristics of single-phase oil and oil-water relative permeability under nonsteady-state conditions were carried out. The flow velocity and displacement pressure gradient show nonlinear characteristics when single-phase oil is passed through a tight core. At the same time, when the air permeability decreases from to , the nonlinear characteristics are becoming more and more obvious. Compared with the traditional JBN method, when the nonlinear flow characteristics are considered, the oil phase relative permeability increases, and the water phase relative permeability is slightly lower. If nonlinear flow characteristics are considered, when the water saturation increases from 0.605 to 0.699, the difference of oil phase relative permeability calculated by the JBN method and this method is gradually decreasing and, with the increase of water saturation, decreases from 0.0029 to 0.0001. In addition, the effects of capillary pressure, gravity, and the nonlinear flow coefficient on the oil-water relative permeability in tight oil reservoirs are studied. The capillary pressure also has a great influence on the relative permeability of the oil phase, and the relative permeability of the oil phase increases with the increase in capillary pressure. In the development process of the low-permeability reservoir, the seepage characteristics of each flow area are different, and the oil-water relative permeability is also different. Therefore, the research results play an important role in guiding the understanding of seepage characteristics of low permeability of tight oil reservoirs.

1. Introduction

The oil-water relative permeability is widely used in oilfield development dynamic analysis, development scheme optimization, and numerical simulation technology in the reservoir, which comprehensively reflects the flow law and the basic characteristics of oil-water relative permeability in the oil reservoir [1–3]. Many methods obtain the oil-water relative permeability. Some classic methods have been summarized and shown in Table 1. Methods to obtain the oil-water relative permeability include laboratory measurement [4–6], theoretical model [7–10], and numerical simulation [11–13]. Test in the laboratory is the most flexible and convenient way to obtain the curve of oil-water relative permeability. Two experimental methods are widely used to establish the curve of oil-water relative permeability: the unsteady-state method and the steady-state method. Following in-depth studies of the development of low-permeability oil reservoirs and further understanding of the performance of reservoirs, the unsteady state method has been more widely used because of its better ability to simulate actual reservoir dynamics. The main theoretical basis of the unsteady-state method is Darcy’s flow law and Buckley–Leverett’s one-dimensional water flooding equation [14]. Experimental data processing methods that are commonly used include the JBN method [15] (method for calculating oil-water relative permeability, it was first proposed by Johnson, Bossler, and Naumann in 1959), and the improved JBN method that aimed at improving calculation accuracy [16–19].

Compared with the medium or high-permeability reservoir, the low-permeability reservoir has a smaller pore throat radius. There is a strong molecular pressure between the formation of crude oil and the pore throat wall. Scholars at home and abroad have carried out extensive research on the seepage mechanism of porous media [20, 21]. The results show that the flow of fluid in the low-permeability reservoir has obvious starting pressure gradient [22, 23] and nonlinear flow characteristics [24, 25]. Practice in mining areas shows that the displacement pressure gradient of the low-permeability reservoir is mainly in the nonlinear flow and linear flow occurs only in the area near the well. Therefore, when the fluid flow does not follow Darcy’s seepage law, the JBN method is bound to produce a large error when calculating the oil-water relative permeability in low-permeability reservoirs.

Given the shortcomings of the JBN method, researchers have put forward an improved JBN model to meet the needs of oil-water seepage analysis in low-permeability reservoirs. Song and Liu [26] established an oil-water relative permeability model by introducing the pseudo-starting pressure gradient into Darcy’s flow model. Givan [16] and Li et al. [17] established a means to calculate the oil-water relative permeability while considering capillary pressure. Their results showed that the capillary pressure greatly influences the oil-water relative permeability. Based on the Li model [17], Wu et al. [18] employed the starting pressure gradient and established an oil-water relative permeability model that considers the influence of capillary pressure, gravity, and the starting pressure gradient. They found that the effect of gravity on the oil-water relative permeability cannot be ignored. Furthermore, Ren et al. [19] noticed the nonlinear flow characteristics of the fluid in the low-permeability reservoir and proposed an experimental data processing method of oil-water relative permeability considering the nonlinear seepage characteristics.

However, the implicit method was used to process the pressure gradient at both ends of the core. Thus, although their work promoted understanding of the oil-water flow law for low-permeability reservoirs, the method used to calculate the oil-water relative permeability did not fully consider the influence of nonlinear flow characteristics such as oil-water gravity and capillary pressure. Scholars have also directly introduced the pseudo-starting pressure gradient into the displacement pressure term of Darcy’s percolation equation to study the effect of the starting pressure gradient on oil-water relative permeability. However, the results of theoretical, experimental, and field tracking studies show that the non-Darcy percolation model of the pseudo-starting pressure gradient is not in line with the actual situation of the reservoir.

In this paper, firstly, a new nonlinear flow model is presented by analyzing the seepage characteristics in the tight oil reservoir. The model has the characteristics of continuity, which is compared with the typical non-Darcy seepage model. Secondly, the nonlinear flow model of oil-water in a low-permeability reservoir is established, referring to the method of extending the Darcy flow model from a single-phase to a two-phase flow model. Thirdly, by considering the influence of nonlinear flow characteristics, capillary pressure, and gravity, the typical JBN method is improved, and a new method for calculating the relative permeability of oil and water in a low-permeability oil reservoir is established. Finally, the flow characteristics of single-phase oil in a low-permeability reservoir under the irreducible water condition(Swi) are experimentally studied, based on the unsteady oil-water relative permeability. The effects of capillary pressure, gravity, and nonlinear flow on oil-water relative permeability are analyzed.

In this paper, the theoretical analysis and experimental results show that the oil-water relative permeability of low-permeability oil reservoir is comprehensively affected by nonlinear flow, the gravity of oil and water, and capillary pressure. When establishing the analysis method of oil-water relative permeability in a low-permeability reservoir, experiments of capillary pressure and nonlinear flow are necessary. The parameter values of capillary pressure and nonlinear flow are obtained, and these data are different. This leads to the uncertainty of the experimental results of oil-water relative permeability. However, the limitation of the research results of this paper is that the causes and description of nonlinear seepage in low-permeability oil reservoirs are very complex. The values of nonlinear seepage parameters under different water saturation (nonlinear seepage parameters in the model established in this paper) are different. This paper only considers the values of nonlinear seepage parameters under irreducible water conditions. The research results play an important role in guiding the understanding of seepage characteristics of low-permeability oil reservoirs and the development of reservoir numerical simulation technology and also provide theoretical support for the effective production and development of such reservoirs.

2. Nonlinear Flow Model of Tight Oil Reservoir

The fact that the flow of single-phase fluid in a tight oil reservoir is non-Darcy has been confirmed by many scholars. Studies have shown that due to the complexity of the pore throat structure, the particularity of fluid properties, the influence of the interaction between the fluid and the porous media wall, and the flow characteristics of the fluid in low-permeability porous media did not adhere to the traditional Darcy flow law [20, 27–29]. There are obvious nonlinear flow characteristics and starting pressure gradient. Based on a theoretical analysis of the non-Darcy flow of single-phase fluid in the tight oil reservoir, a theoretical model is proposed to describe the non-Darcy flow.

Figure 1 shows the typical non-Darcy flow curve of fluid in a tight oil reservoir. The flow velocity and pressure gradient of the fluid no longer follow the traditional Darcy law. When the pressure gradient is greater than the maximum , the relationship between velocity and pressure gradient is linear. The intersection of the extrapolation and the pressure gradient axis is , which is defined as the quasi-starting pressure gradient. When the pressure gradient is less than , the relationship between flow velocity and pressure gradient is a nonlinear curve, and the intersection point between flow velocity and pressure gradient axis is , which is defined as the minimum starting pressure gradient.

The nonlinear flow model of fluid through porous media can be described as (1)The precondition of fluid flow in Equation (1) is . Its critical value is . According to the flow theory in porous media, when the pressure gradient is equal to the starting pressure gradient, the critical value of flow is

Therefore, (2)When , the nonlinear flow curve coincides with the flow curve considering the pseudo-starting pressure gradient

Therefore, the formula (2) is introduced into Equation (1) and,

Therefore,

Namely,

Therefore, the nonlinear flow model of the low-permeability oil reservoir is where is flow velocity, cm/s; and are flow parameters, 10^-1 MPa/cm; is minimum pressure gradient that needs to be overcome when fluid passes through porous media, 10^-1 MPa/cm; is maximum starting pressure gradient that fluid needs to overcome when passing through a porous medium, 10^-1 MPa/cm; is pressure gradient to start, 10^-1 MPa/cm; is absolute permeability of core samples, μm²; and is pressure gradient of displacement, 10^-1 MPa/cm.

In the above formula, when = 0 and = 0, Equation (5) is a typical Darcy flow model. Therefore, the typical Darcy flow model is a special form of the nonlinear flow model established in this paper. The flow of Newtonian fluid in porous media is characterized by nonlinear flow and a certain starting pressure gradient. Hence, the parameters can be calculated according to Equation (4), and the nonlinear flow curve of the core can be obtained.

3. A New Model for Calculating the Oil-Water Relative Permeability in the Tight Oil Reservoir

The following assumptions are made for tight reservoirs: (1)The reservoir is homogeneous and isothermal(2)The flow belongs to one-dimensional horizontal flow, considering the influence of gravity and capillary pressure(3)There are only oil and water phase fluids in the reservoir, which are immiscible(4)The fluid is Newtonian, and the flow equation of single-phase fluid satisfies the equation(5)Capillary pressure and gravity of the oil phase and water phase are considered

Considering that Darcy’s percolation is extended from the single-phase flow method to the two-phase flow method, the relative permeabilities of the water phase and the oil phase are introduced into the water phase and oil phase motion equations, respectively. Therefore, when the single-phase seepage Equation (5) is extended to oil-water two-phase, the equations of oil-water flow in the low-permeability oil reservoir considering the characteristics of gravity, capillary pressure, and nonlinear percolation are obtained:

And ; where is the nonlinear flow coefficient of the water phase and is the nonlinear flow coefficient of the oil phase and is the viscosity of the fluid, mPa·s; is the viscosity of oil, mPa·s; is the viscosity of water, mPa·s; is the density of oil phase, 0.001 kg/m³; is the density of water phase, 0.001 kg/m³; is the acceleration of gravity, 9.8 m/s²; is formation dip angle, radians; is the relative permeability of oil phase; is the relative permeability of water phase; is oil phase pressure, 10^-1 MPa; is water phase pressure, 10^-1 MPa; is minimum start pressure gradient of oil, 10^-1 MPa/cm; is minimum start pressure gradient of water, 10^-1 MPa/cm; is flow velocity of oil, cm/s; is flow velocity of water, cm/s; is the gradient of water phase pressure, MPa/cm; and is the gradient of oil phase pressure, MPa/cm.

According to the nonlinear flow model of oil phase (6) and water phase (7), Equation (A.23) is the relative permeability of the water phase at the outlet of the core and Equation (A.30) is the relative permeability of the oil phase at the outlet of the core

According to Equation (A.39), the expression of the capillary pressure gradient at the core outlet is obtained as follows:

Equation (8) and Equation (9) calculate the water-oil relative permeability in tight oil reservoirs by considering the effects of fluid gravity, capillary pressure, and nonlinear flow characteristics. It is found in these equations that when the effects of gravity, capillary pressure, starting pressure gradient, and nonlinear flow are not considered, the calculation method of oil-water relative permeability established in this paper is the classic JBN model. It can be seen from Equation (8) that the relative permeability of the water phase is affected by the starting pressure gradient and the gravity of the water phase and capillary pressure. From Equation (9), it can be seen that the relative permeability of the oil phase is not only affected by the percolation capacity of the water phase and the oil-water viscosity ratio, but is also affected by the starting pressure gradient of the oil phase, the gravity of the oil phase, the capillary pressure, the displacement pressure gradient, and the influence of nonlinear flow.

4. Experiments and Analysis

To determine the influence of nonlinear flow, capillary pressure, and gravity on the oil-water relative permeability in tight and low-permeability oil reservoirs, a physical experiment of unsteady flow was carried out, and the experimental data were analyzed.

4.1. Parameters of Core and Fluid Sample

Core samples are taken from the lower Sha2 formation of the Shanghe reservoir in an oil field in eastern China, and the lithology is fine sandstone. The core is saturated with simulated formation water, the injected water is KCl solution, the salinity is 30000 mg/L, the density is 1.0183 g/cm³, and the viscosity is 0.5763 mPa·s, and the test temperature is 50°C. White oil No.3 is used in the experiment; it has a density of 0.792 g/cm³, a viscosity of 2.15 mPa·s, and a test temperature of 50°C. The size, porosity, air permeability, and other parameters of the three core samples are shown in Table 2.

4.2. Test Conditions

The oil-water relative permeability curve is measured by the unsteady state method. The test temperature is the formation temperature (50°C), the effective overburden pressure is 17.2 MPa, and the displacement velocity is 0.6 ml/min under the original formation condition.

4.3. Test Procedure

(1)After the core is vacuumized, the formation water is saturated and loaded into the core holder(2)The irreducible water saturation is achieved through oil displacement until no water is produced. The oil saturation and irreducible water saturation of the core are measured, the water quantity at the outlet is recorded, and the oil saturation, irreducible water saturation, and effective permeability of the oil phase are calculated(3)The steady-state method is used to measure the relationship between pressure difference and oil flow. Under a constant injection flow, we record the pressure at the inlet and outlet at different times until it is stable and then calculate the stable differential pressure; when we set different flows, we get a series of stable differential pressure and flow data and draw the flow velocity curve of single-phase oil with different displacement pressure gradients(4)The experiment on the relative permeability of oil and water is carried out by the unsteady state method. The displacement pressure, oil production, and water production are recorded until the water cut is close to 100% or the volume of water injection reaches 30 times the core pore volume

4.4. Experimental Results and Analysis

4.4.1. Flow Curve of Single-Phase Oil under Irreducible Water Condition(Swi)

Figure 2 shows the relationship between oil flow and the pressure gradient in core samples with different permeability. Using the parameters of two core samples and the parameters of simulated oil given in Table 1, the percolation curve of single-phase oil is fitted with formula (1).

Analysis of the core displacement experiment results shows that there is no fluid outflow at the outlet end of the core under a small pressure gradient. When the pressure gradient increases to greater than the minimum starting pressure gradient, the velocity increases with the increase in pressure gradient, and the pressure gradient continues to increase. Both the velocity and pressure gradient show the linear flow characteristics; only at the small pressure gradient will the flow velocity and pressure gradient in the gradient range show nonlinear flow characteristics.

Table 3 shows the flow parameter data of two core samples. In the experiment, the irreducible water saturation of two cores and the effective permeability of the oil phase through low-permeability cores are obtained. The results show that the lower the permeability, the lower the effective permeability of the oil phase. When the air permeability decreases from to , the effective permeability of the oil phase decreases from to .

The minimum starting pressure gradient of the oil phase passing through the low-permeability core is obtained by extrapolation of the flow curve; the quasi-starting pressure gradient is obtained by the intersection of the straight-line section and the abscissa axis in the reverse extended flow curve; and the maximum starting pressure gradient is obtained by the separation of the straight-line section and the nonlinear section. With the decrease in absolute permeability, the minimum starting pressure gradient, quasi-starting pressure gradient, and maximum starting pressure gradient of the oil phase increase. When the air permeability decreases from to , the minimum starting pressure gradient of the oil phase increases from to . The minimum starting pressure gradient of the water phase is very small () and can be ignored. The nonlinear flow coefficient of the oil phase decreases from 0.1486 to 0.2683 MPa/cm. At the same time, it is found that the flow parameter increases with the decrease in air permeability, and the nonlinear degree becomes more serious.

4.4.2. Oil-Water Relative Permeability Curve of the Tight Oil Reservoir, Considering Nonlinear Flow Characteristics

Table 4 shows the production data such as the accumulated oil volume, accumulated liquid volume, and water cut obtained at the core outlet during the water drive. By applying the percolation parameters given in Table 2, the core displacement data obtained in Table 3, and the second capillary pressure curve presented in Figure 3, the core outlet is inclined upward by 45°. Both the traditional JBN method and the improved JBN method (Equations (8) and (9)) are used to calculate the oil-water relative permeability curve of the low-permeability core. The calculation results are shown in Table 5, and a comparison of the oil-water relative permeability curves is shown in Figure 4.

It can be seen from Table 5 and Figure 5 that, compared with the oil-water phase relative permeability calculated by the traditional JBN method, the relative permeability of the oil phase increases when considering the nonlinear flow characteristics; moreover, the relative permeability of the water phase is slightly lower. Under the condition of considering nonlinear flow, with the increase of water saturation, the relative permeability of the oil phase is calculated by the two methods decreasing gradually from 0.0029 to 0.0001. The results of relative permeability of the oil phase are different in the low water cut stage. For the traditional JBN method, the flow of fluid in the tight oil reservoir is regarded as Darcy flow, which overestimates the effect of displacement pressure without considering the nonlinear characteristics of flow and the additional pressure loss caused by the minimum starting pressure gradient. Compared with the displacement pressure gradient, the minimum starting pressure gradients of the oil phase and water phase cannot be ignored.

Considering the influence of nonlinear flow characteristics and oil-water gravity on oil-water relative permeability, compared with the oil-water relative permeability calculated by the traditional JBN method, the oil-water relative permeability reduced, and the isoosmotic point shifted to the right (in the experiment, the outlet end of core sample inclined upward by 45°, and gravity is the oil-water flow resistance).

When considering the influence of nonlinear flow characteristics and the capillary pressure on the oil-water relative permeability, the relative permeability of oil increases, the relative permeability of water does not change, and the permeability point moves to the upper right. For the water-wet reservoir, the relative permeability of the water phase has nothing to do with the capillary pressure; the capillary pressure greatly influences the oil relative permeability, improves the flow capacity of the oil phase, and increases the relative permeability of the oil phase.

Compared with the traditional JBN method, the influence of gravity and capillary pressure on the relative permeability of oil and water is based on nonlinear flow characteristics. The results show that the relative permeability of the oil phase increases, while the relative permeability of water decreases; meanwhile, the permeability point moves up and to the right. Gravity is the resistance of oil-water flow; meanwhile, the capillary pressure drives oil phase flow and reduces the flow resistance caused by oil-water gravity. Therefore, when gravity and the capillary pressure are considered at the same time, the relative permeability of the oil phase increases, while the relative permeability of water decreases; compared with the gravity of the oil-water phase, the capillary pressure has a greater impact on the relative permeability of oil-water.

4.4.3. Sensitivity Analysis of Parameters

(1) Effects of Gravity. Figure 5 describes the effect of gravity on the oil-water relative permeability under different reservoir dip angles. The results show that when the dip angle of the reservoir is less than 90°, the relative permeability of both oil and water phases decreases with the increase in dip angle. The larger the dip angle, the more obvious the change in the relative permeability of oil and water. The main reason for this is that gravity is the resistance of oil-water flow, gravity reduces the flow capacity of the oil-water phase, and the relative permeability of the oil-water phase decreases. When the dip angle of the reservoir is greater than 90°, the oil-water relative permeability increases with the increase in dip angle. At this time, gravity drives oil-water flow and promotes the flow capacity of the oil-water phase. When the reservoir dip angle is 90 degrees, it is the inflection point that the flow capacity of the oil-water phase decreases gradually to increase gradually. Therefore, for the inclined reservoir, when there is low water injection and high oil recovery, due to the gravity effect of the injected water, the flow resistance of injected water will be reduced, the penetration of the injected water will be slowed down, and the spread of the injected water will be expanded to improve the recovery degree. However, in the case of high water injection and low oil recovery, the injected water can increase the breakthrough speed of injected water under the action of gravity, which is detrimental to the injection development of heterogeneous reservoirs. For the reservoir with bottom water, giving full play to the lifting effect of the bottom water can improve the recovery degree.

(2) Influence of the Capillary Pressure. To analyze the influence of the capillary pressure on oil-water relative permeability, three curves of the oil-water capillary pressure are selected (Figure 3). The improved JBN method is applied to calculate the curves of oil-water relative permeability under different capillary pressure, and the results are shown in Figure 6. It is clear that the capillary pressure greatly influences the relative permeability of the oil phase; furthermore, the larger the capillary pressure becomes, the more significantly the relative permeability of the oil phase increases. Therefore, during the water flooding development of the tight and low-permeability reservoir, the capillary pressure drives oil and water flow (water-wet reservoir). Additionally, its imbibition effect has been widely used in the water flooding development of tight or low-permeability oil reservoirs in China.

According to Equation (9), the shape of the oil phase relative permeability curve in the tight oil reservoir is affected by the nonlinear characteristics of flow, and the relationship between oil phase relative permeability and the nonlinear flow coefficient is inversely proportional. Figure 7 shows the influence of the nonlinear flow coefficient on the oil phase relative permeability curve. The results show that the smaller the nonlinear percolation coefficient is, the higher the oil phase relative permeability is (based on the same core, fluid properties, and production data at the outlet). In this paper, the nonlinear flow coefficient describes the nonlinear flow characteristics of the oil phase, and comprehensively reflects the influence of the displacement pressure gradient, fluid properties, and medium permeability on flow characteristics. With the increase in displacement pressure gradient, the nonlinear flow coefficient decreases, and the nonlinear degree becomes more serious. When the starting pressure gradient can be ignored relative to the displacement pressure gradient, the nonlinear flow coefficient tends to 1; i.e., the influence of nonlinear characteristics on flow can be ignored.

Practice in mining areas shows that during the water flooding development of low-permeability reservoirs, due to the influence of nonlinear characteristics of flow, the flow area can be divided into a quasi-linear flow area and a nonlinear flow area; moreover, different flow areas have different displacement pressure gradients. Additionally, the flow law of flow is different even in the same flow area at different development stages. In the development of the tight reservoir, the displacement pressure gradient is mainly in the nonlinear flow interval; only in the near well displacement pressure gradient is it in the quasi-linear flow interval. In the near well area, the displacement pressure gradient is far greater than the minimum starting pressure gradient of the oil phase. The fluid flow is characterized by quasi-linear flow, and the nonlinear flow has little effect on the relative permeability curve of oil and water. When the starting pressure gradient and nonlinear flow characteristics cannot be ignored, i.e., when the flow is in the nonlinear flow area of the low-permeability reservoir, the influence of nonlinear flow characteristics on oil relative permeability cannot be ignored, and the oil relative permeability curve is quite different under different displacement pressure gradients. Therefore, for low-permeability reservoirs, the oil-water relative permeability is different in different flow areas and different development stages; in the same development stage, the relative permeability of the oil phase in the near well area is lower than that in the far well area; in the same flow area in different development stages, the relative permeability of the oil phase in the early development stage is higher than that in the later development stage.

Figure 8 compares the oil-water relative permeability curves of different cores considering nonlinear seepage characteristics. The lower the permeability of the core, the greater the irreducible water saturation; the smaller the residual oil saturation is, the higher the relative permeability of the oil phase; and the lower the relative permeability of water phase is, the more the equivalent permeability point (the point where the relative permeability of oil and water phase is numerically equal) moves to the right.

5. Summary and Conclusions

(1)When a fluid passes through a tight core, the flow does not follow Darcy’s law, and the fluid shows a certain starting pressure gradient and nonlinear flow characteristics. The velocity and pressure gradient meet the nonlinear flow equation established in this paper. Considering the nonlinear flow characteristics, capillary pressure, and gravity, the traditional JBN method is improved, and a new method for calculating the oil-water relative permeability in a tight oil reservoir is established(2)In this work, we conduct a percolation experiment on single-phase oil under the condition of confined water. This is followed by an experiment on the oil-water relative permeability under the unsteady state. The results show that when single-phase oil (under the condition of confined water) passes through a low-permeability core, the percolation velocity and pressure gradient of the oil phase do not follow Darcy’s flow law, and nonlinear flow characteristics are observed(3)Compared with the oil-water relative permeability curve calculated by the traditional JBN method, in our method, the relative permeability of the oil phase increases when considering the nonlinear flow characteristics. The relative permeability of the water phase is slightly lower, and the smaller the nonlinear flow coefficient is, the higher the relative permeability of the oil phase becomes. When gravity is the resistance of oil-water phase flow, with the increase in reservoir dip angle, the relative permeability of oil-water decreases. The reservoir dip angle has a great influence on the oil-water relative permeability, and when gravity drives oil-water phase flow, with the increase in reservoir dip angle, the relative permeability of oil-water increases. The capillary pressure also greatly influences the relative permeability of the oil phase but does not influence the relative permeability of the water phase. The relative permeability of the oil phase increases with the increase in capillary pressure(4)During water injection in low-permeability reservoirs, at the same development stage, the relative permeability of the oil phase in the near well area is lower than that in the far well area. Meanwhile, in the same flow area, the relative permeability of the oil phase in the early development stage is higher than that in the later development stage

Appendix

A. Calculation of the Oil-Water Relative Permeability

On any cross section of the core, the total flow through this cross-section is the sum of the water and oil phases, and where is the flow velocity of fluid, cm/s, and is the cross-sectional area of the core, cm².

The flow velocity of the water phase through any cross section of the core can be described as where is water saturation at the core location and is the water content to the water saturation .

According to the Buckley–Leverett equation, the relationship between any point in the core and water saturation is

At the core outlet, where is derivative of water content to water saturation; is derivative of water content at the outlet to water saturation at the outlet; is the length from the entrance to any position of core samples, cm; is the length of the core sample, cm; is core porosity; and is the cumulative amount of water injected at time , cm³.

in Equation (A.2) is described as

According to Equation (A.4), where is the dimensionless cumulative water injection, and there are

According to Equations (7), (A.1), and (A.2),

According to Equation (A.8), the pressure gradient of the water phase at any point in the core is where

;

When there is water at the outlet of the core, the water phase is continuous, and the pressure difference at both ends of the core is expressed by the integral value of the water phase pressure gradient at both ends of the core, i.e.,

By injecting Equation (A.9) into Equation (A.10), the pressure at both ends of the core is reduced to where is the length of the micro-unit, cm, and is the injection pressure difference at both ends of the core, 10^-1 MPa.

We then multiply both ends of Equation (A.11) by : where

;

According to Equations (A.3) and (A.4),

Thus,

Equation (A.14) is brought into Equation (A.12) to yield

Define as injection capacity ratio, and

Equation (A.16) is then brought into Equation (A.15):

Derivation from both ends of the formula (A.17) to the outlet end of the core yields where

;

Set:

Equation (A.19) is brought into Equation (A.18) to produce

According to Equation (A.20),

Equation (A.21) is brought into Equation (A.19) to get where is water phase relative permeability at the core outlet and is water saturation at the core outlet at time t.

The relative permeability of the water phase at the outlet of the core is

According to formulas (6) and (7), the following deformation treatment shall be carried out:

Formula (A.25) minus formula (A.24) results in

We then consider the effect of capillary pressure: where is the gradient of capillary pressure, MPa/cm.

Equation (A.27) is brought into Equation (A.26):

The flow velocity of the water phase and that of the oil phase can be expressed as

When Equation (A.29) is introduced into Equation (A.28), the relative permeability of the oil phase at the outlet of the core is where is the oil phase’s relative permeability at the core outlet and and are the nonlinear flow coefficients of the water phase and the oil phase, respectively, and are dimensionless. . When , the water phase does not consider nonlinear flow.

Finally, the water saturation at the core outlet is where is cumulative oil production at the core outlet at time , cm³, and is irreducible water saturation.

B. Calculation of the Capillary Pressure Gradient

It can be found from Equation (A.30) that an important point in the calculation of the relative permeability of the oil phase is how to calculate the expression of the capillary pressure gradient at the outlet of the core. The gradient of capillary pressure can be expressed as follows:

Therefore, it is necessary to calculate the water saturation gradient at the end of the core before calculating the capillary pressure gradient. From the basic properties of partial derivatives,

Under the condition of one-dimensional flow, the unsteady flow equation of the water phase fluid is

Rearranging Equation (A.32) produces

Upon further transformation, the above formula is described as

Formula (A.34) and formula (A.1) and (A.2) are brought into formula (A.33), and the saturation gradient of the water drive front is

According to Equation (A.35), the saturation gradient at the core outlet is

Equation (A.36) is brought into Equation (A.32), and the expression of the capillary pressure gradient at the core outlet is obtained as follows:

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Project of PIPECHINA: Optimization technology research on key parameters design and production of W23 gas storage reservoir (SSCC202101).

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