Study on Stress Sensitivity of Ultra-Low Permeability Sandstone Reservoir Considering Starting Pressure Gradient

Abstract

When studying the influence of nonlinear seepage on the water flooding development of ultra-low permeability reservoirs, it is difficult to accurately characterize the nonlinear seepage state of ultra-low permeability reservoirs using conventional reservoir numerical simulation methods. A large number of field development and tests in the oil fields show that there are indeed starting pressure gradients and stress sensitivity effects in ultra-low permeability sandstone reservoirs. In this study, taking the tight sandstone of the Chang 6 Member in the Yanchang Formation as an example, the rock displacement vector was utilized to equivalently characterize the stress-sensitive effect of the reservoir based on a novel numerical simulation software tNavigator. Furthermore, the starting pressure gradient and the feasibility of a new stress-sensitive equivalent characterization method were verified combining the poroelastic media physics equations. In addition, we systematically studied the impact of stress-sensitive effects on oil well productivity under the influence of starting pressure gradient and considering petrophysical properties. The results show that the existence of the starting pressure gradient can improve the stress sensitivity of the matrix reservoir. Considering the starting pressure gradient, when the rock shear modulus is 14.29 GPa, the cumulative oil production decreases by 4.1%; when the rock shear modulus is 11.36 GPa, the cumulative oil production decreases by 11.2%. Finally, a numerical simulation was conducted with Block B in the Zhouwan-Wugucheng area, and the model can accurately predict the reservoir stress sensitivity based on the starting pressure gradient. When both the starting pressure gradient and stress sensitivity are considered, the crude oil recovery degree of the target layer decreases by 11.7%; when only the starting pressure gradient is considered, the crude oil recovery degree decreases by 8.8%; and when only the stress sensitivity is considered, the crude oil recovery degree decreases by 0.5%.

1 Introduction

In recent years, the proportion of ultra-low permeability tight sandstone reserves in China’s newly-added proven oil-in-place reserves has reached 60% (Meng and Li., 2013; Chen et al., 2019; Qian et al., 2021). Tight sandstone oil and gas has become the main energy field in the future utilization and exploitation of oil and gas resources (Kuuskraa., 1996; Lu and Wang., 1999; Ruan., 2002). Therefore, the efficient development of ultra-low permeability sandstone reservoirs is of great significance to the sustainable development of oil industry (Hansbo., 1960; Geng et al., 2017; Gao., 2021; Wu et al., 2021). Non-Darcy seepage problems are often involved in actual reservoir development (Francesconi., 1995; Shi et al., 1999; Wang et al., 1999; Dong et al., 2015; Zhao et al., 2021). If the fluids seep very slowly, they do not satisfy Darcy’s law.

Previous studies have systematically studied the effect of stress sensitivity on rock porosity and permeability (Hu et al., 2006; Jiao et al., 2011; Xu et al., 2021; Xue et al., 2021). These studies showed that with the increase of effective stress, rock porosity and permeability decreased to varying degrees (Mitchell, 1967; Von E and Tunn, 1995; Lei et al., 2007; Tian, 2014). Moreover, there are several studies showing that the initial displacement pressure required for oil flow increases as the permeability becomes effective (Liu et al., 2001; Cai et al., 2017; Li et al., 2020; Mahmud et al., 2020; Li and Li., 2021).

At present, existing commercial softwares cannot effectively simulate stress-sensitive effects in low-permeability reservoirs by setting the minimum pressure for fluid percolation (Todd., 2016; Maksim., 2019; Wu et al., 2020). The limitation of fluid flow simulation technology is that it cannot accurately characterize the non-Darcy flow state of low permeability reservoirs (Lv et al., 2002; Hong et al., 2020; Zheng et al., 2020). Fluid-structure interaction is currently recognized worldwide as an equivalent characterization method for stress-sensitive effects (Xu et al., 2014; Wang et al., 2016; He et al., 2020; Guo et al., 2021; Hower and Groppo., 2021; Mohammed et al., 2021). Most of the current commercial numerical simulation software only considers the pressure sensitivity of permeability and porosity, and does not consider the rock physical parameters (elastic modulus, Poisson’s ratio, etc.) under high temperature and pressure conditions (Shang., 2004; Song., 2012; Wang et al., 2012; Ren et al., 2020; Chen et al., 2021). In this study, taking the tight sandstone of the Chang 6 Member in the Yanchang Formation as an example, the rock displacement vector was utilized to equivalently characterize the stress-sensitive effect of the reservoir based on a novel numerical simulation software tNavigator. It can introduce rock mechanics parameters into the simulation of ultra-low permeability sandstone reservoir. In addition, a new generation of numerical simulator, the calculation kernel is optimized, the parallel computing efficiency and acceleration ratio are improved. The GPU was used to accelerate the calculation, which significantly reduces the time cost of numerical simulation. By applying this technology to actual oil reservoirs, the non-Darcy flow conditions of low permeability reservoirs can be simulated more accurately.

2 Materials and Methods

2.1 Geological Background and Reservoir

The Triassic strata in the Ordos Basin have experienced complex evolutionary stages of lake basin expansion and eventual shrinkage and extinction. The study area is located in the Zhouwan-Wugucheng area (Figure 1), where tight sandstone reservoirs are developed in the Lower Triassic Yanchang Formation. Moreover, the main reservoir is the Chang 6 Member. The Chang 6 Member belongs to the delta front subfacies deposition, and the underwater distributary channel, channel flanks and inter-distributary bay microfacies are developed.

FIGURE 1

The experimental procedure of casting thin section: low viscosity epoxy resin impregnated with pigment, casting agent and dyeing agent are injected into the pores of rock under certain temperature and pressure, and the linear epoxy resin is crosslinked into huge molecules with network structure by curing reaction to form hard solid epoxy resin. Then the cast thin sections are made by sticking and grinding procedures. According to the identification results of thin sections and SEM (scanning electron microscope) of 107 sandstone samples taken from 27 wells, the pore types of Chang 6 reservoirs in the study area mainly include intergranular pores, feldspar dissolved pores, and detrital dissolved pores. Intergranular pores are the main type of pores, followed by feldspar dissolved pores, and detrital dissolved and intercrystalline pores are locally visible. The shapes of these pores are mostly triangular and quadrilateral, and the pores are mostly connected by narrow and short throats (Figure 2). Affected by the stress-sensitivity effect, the radius of some narrow and short throats will gradually decrease during the waterflooding development process.

FIGURE 2

In the study, the profile of the Chang 6 Member shows that the subaqueous distributary channel has well continuity and long extension distance in the SW-NE direction, and multi-staged superposed channels are developed. In the NW-SE direction, the underwater distributary channel is cut by interdistributary bay and has poor sand body continuity (Figure 3).

FIGURE 3

2.2 Equivalent Simulation of Starting Pressure Gradient

2.2.1 Equivalent Simulation of Starting Pressure Gradient

The simulation of the starting pressure gradient in the ECLIPSE Software is based on the equilibrium partitions to achieve the minimum pressure setting for fluid percolation between adjacent partitions (Wu et al., 2017). If we set a separate partition for each grid, it will result in a huge number of partitions, the software cannot obtain results through normal operations, and the computational workload is also huge (Xiong et al., 2009; Wang et al., 2020; Yang et al., 2021). However, the tNavigator Software can assign the grids in the work area through the keywords PTHRESHI, PTHRESHJ, and PTHRESHK, and then calibrate the starting pressure gradients in the three directions of I, J, and K. This method can be used to study the variation law of the production index in the work area under different starting pressure gradients (Yang et al., 2010).

The starting pressure gradient of the target layer of the work area measured by the experiment is 0.04 MPa/m, the grid size of the work area is 100 × 80 × 30, and the total number of grids is 240,000. In addition, the starting pressure gradient in the I, J, and K directions between grids in the work area is 0.04 MPa/m, and the keywords are PTHRESHI 240000*0.04, PTHRESHJ 240000*0.04, and PTHRESHK 240000*0.04, respectively.

2.2.2 Optimization of Starting Pressure Gradient

FIGURE 4

Notes: PERMX, PERMY and PERMZ are the permeability in x, y and z directions, respectively. The ARITHMETIC PTHRESHI, the ARITHMETIC PTHRESHJ, and the ARITHMETIC PTHRESHK are the starting pressure gradient values in the x, y, and z directions, respectively.

Furthermore the pressure differential flow method was adopted in the experiment. The basic principle is: after different displacement pressure differences are stabilized, the flow rate of fluid through the core is measured, the relationship between flow rate and pressure gradient is drawn, and the starting pressure gradient is calculated by regression curve (Table 1).

The starting pressure gradient values for each grid in three dimensions can be obtained, based on the different permeability assignments on each grid throughout the model.

2.3 Equivalent Simulation of Stress-Sensitive Effects

Subsurface porous media rocks are composed of rock particles and pore fluids. During the development stage of the reservoir, the compression of the rock will generate compound stress, which will lead to the deformation and displacement of the rock skeleton (Figure 5). When the pore fluid migrates with the rock skeleton, relative seepage will also occur (Wang et al., 2019).

FIGURE 5

Dai., 2006). Notes: σ_x, σ_y, σ_z are normal stresses in x, y, and z directions; τ_xy and τ_yx are shear stresses on x and y planes; τ_xz, τ_zx are shear stresses on x and z planes; τ_yz and τ_zy are shear stresses on y and z planes.

As shown in Figure 5, the reservoir rock can be equivalently regarded as a single hexahedral unit. (Dai., 2006; Ewing et al., 1980; Feng et al., 2018). The opposite sides of the regular hexahedral elements are parallel to each other. And, along the x, y, and z axes, the lengths of each side of the hexahedral unit correspond to dx, dy, and dz, respectively. The nine components on the reservoir rock are shown in Eq. 1. Moreover, the size of the nine components is not only related to the direction of the coordinate axis, but also to the force of the point. The stress state at a certain point inside the rock can be represented in the form of a matrix:

Chen and Ewing., 1999 used the elastic physical equation of porous media is used to describe the relationship between the effective stress and strain of rock, and its general expression is:Where {σ_eff} is the stress vector of the rock skeleton, m; {ε} is the strain vector of the rock skeleton, m.

For isotropic elastic media, the generalized Hooke’s law satisfies:Where E_h and E_v are the elastic moduli of the rock skeleton in the horizontal and vertical directions, respectively, GPa; G_h and G_v are the horizontal and vertical shear moduli of the rock skeleton, GPa, respectively; μ_hh and μ_hv are the Poisson’s ratios in the horizontal and vertical directions, respectively; μ_hv is the Poisson’s ratio of the horizontal deformation caused by the stress in the vertical direction; τ_eff is the effective shear stress, MPa.

Equation 3 regards the rock skeleton as a plane isotropic medium (homogeneous rock), with E_h = E_v = E, μ_hv = μ_hh = μ_vh = μ, G_h = G_v = G.

Then the physical equation is:

This physical equation can be written in matrix form:Where [D_e] is the elastic matrix:Of which:Where E, μ, G, E_s, and λ are the elastic modulus, Poisson’s ratio, shear modulus, compressive modulus and Lame constant of the homogeneous rock skeleton, respectively.

It can be seen from Eq. 4 that in the development stage of low-permeability reservoirs, the effective stress on the formation rock in the x, y, and z directions remains unchanged when the same pressure drop is maintained. With the increase of rock elastic modulus E and Poisson’s ratio μ, the displacement vectors of rock unit body in x, y and z directions decrease.

In the stress-sensitive simulation based on tNavigator Software, it is necessary to consider the influence of the stress-sensitive effect on the oil well productivity and the swept range of the water injection well. By introducing the elastic modulus E and Poisson’s ratio μ of the rock, the one-way coupling method is used to couple the stress sensitivity and geomechanics to equivalently characterize the rock displacement vectors. This method can be used to analyze the production variation law of different lithologic reservoirs, and visually present the variation of rock displacement vectors on each grid.

3 Results

3.1 Parameter Settings

FIGURE 6

3.2 Equivalent Method Validation

The seepage velocity expression of the quasi-start pressure gradient model is:Where v is the seepage velocity, m/s; K is the reservoir matrix permeability, mD; μ is the crude oil viscosity, mPa s; c is the pseudo starting pressure gradient, MPa/m; Δp is the pressure difference between the two ends of the grid at distance L, MPa.

According to the theory of limit well spacing, Douglas et al. (1983) used the pressure gradient out of the maximum flowable distance should be greater than the starting pressure gradient. The pressure difference Δp satisfies:

For a reservoir with a matrix permeability of 1 mD, when the production pressure difference is 4.5 MPa, the maximum production range when the starting pressure gradient is considered is 100 m. Furthermore, the limit production range predicted by the model set by this method is also 100 m. Thus, the accuracy of this method is verified. When the starting pressure gradient is not considered, the production range predicted by the model is more than 300 m, which has a large error (Figure 7).

FIGURE 7

Rock mechanics parameter experiment: a solid cylindrical specimen with the same height and diameter is subjected to a relative compressive linear load in the diameter direction to fracture along the plane controlled by the load. Then, the tensile strength of the rock is obtained according to the elastic theory. In this study, a total of 8 standard cores were used to verify the effect of rock brittleness on reservoir stress sensitivity. Through rock mechanics experiments, the Poisson’s ratio and mineral composition of eight groups of samples were measured (Table 2).

It can be seen from Table 2 that the elastic moduli of the rock samples are distributed in the range of 20.75–30.80 GPa, and the Poisson’s ratio is distributed in the range of 0.285–0.318. The internal components of the tight sandstone in the target layer are mainly quartz, potassium feldspar, plagioclase and eruptive rocks.

The strain characteristics of matrix reservoirs can be analyzed according to the stress-deformation method of the fluid-solid dual medium model (Wu et al., 2014). Then, the changes of permeability and porosity caused by rock mass deformation are:Among them, φ_mi and k_mi are the original porosity and permeability, respectively.

The mean effective stress is:

The existence of stress-sensitive effect makes oil well production decrease (Liu., 2014). The formula for calculating the oil recovery index considering the stress-sensitive effect is:

Among them, J_o is the oil production index, m³ day⁻¹ MPa⁻¹; h is the effective reservoir thickness, m; B_o is the coefficient of crude oil formation volume; r_e is the oil supply radius, m; r_w is the wellbore radius, m; k_mi is the original permeability, mD.

The relationship between the production pressure difference and the change in permeability is:Where α is the stress sensitivity coefficient.

Affected by stress sensitivity, the reservoir matrix undergoes elastic deformation as the production pressure difference increases. Compared with the conventional stress-sensitive equivalent characterization method, it can be seen that the oil recovery index obtained by the method used in this study is closer to the actual data, and the model accuracy is significantly improved (Figure 8). It is also known from Eqs 10 and 13 that when the permeability change value is constant, the production pressure difference is negatively correlated with the average effective stress.

FIGURE 8

4 Discussion

In this study, we established two numerical simulation models: a homogeneous and a heterogeneous permeability model, and they both adopted a diamond-shaped inverse nine-spot well pattern. The influence of stress-sensitive effects on the productivity of ultra-low permeability reservoirs has been systematically studied, taking into account the starting pressure gradient. The rock is sensitive to stress, and the larger the elastic modulus value E, the smaller the deformation of the rock under stress. Poisson’s ratio expresses the ratio of the strain ε₁ in the vertical direction to that in the loading direction. Within the elastic deformation of the rock, μ is a coefficient that measures the lateral strain capacity of the rock. Therefore, under the same elastic modulus of rock, the larger the Poisson’s ratio, the weaker the stress-sensitive effect of the rock.

4.1 Effects of Stress Sensitivity on Reservoir Development

In this study, a shear modulus G = E/[2(1+μ)] was introduced. The value ranges of elastic modulus and Poisson’s ratio of different types of sandstone are shown in Table 3. The shear modulus is used to characterize the stiffness of the rock in the numerical model. The larger the shear modulus value, the smaller the deformation of the rock during compression or tension. The results of rock mechanics parameters are shown in Table 4.

Under different shear modulus conditions (Figure 9), the reservoir stress sensitivity is weaker when the starting pressure gradient is not considered. Affected by the starting pressure gradient, the reservoir stress sensitivity is strong, and the production range of the oil well is reduced. In addition, the formation pressure changes sharply during constant liquid production, the rock displacement vector increases, the matrix permeability decreases, and the difference in cumulative oil production increases as well.

FIGURE 9

Figure 10 different degrees of starting pressure gradient exist in the development process of ultra-low permeability oil and gas fields. Therefore, the increase of rock shear modulus G is accompanied by the change of bottom hole pressure. The rock displacement vector and variation range near the wellbore of the oil wells are reduced, which has a strong impact on the productivity of ultra-low permeability reservoirs with starting pressure gradients. It can be seen from Figure 9 that when the rock shear modulus is 14.29 GPa, the cumulative oil production decreases by 4.1%; when the rock shear modulus is 11.36 GPa, the cumulative oil production decreases by 11.2%.

FIGURE 10

4.2 Case Analysis

The main producing layer of the Chang 6 reservoir in the study area is Chang 6₁, the reservoir is buried at a depth of 1,950 m, the average reservoir thickness is 28.2 m, and the original formation pressure is 28.9 MPa. In addition, the Chang 6 reservoir has an average porosity of 11.1%, an average permeability of 0.59 mD, and a crude oil density of 0.8225 g/cm³. The model contains four layers from top to bottom: chang 6₁¹ (model layer 1), chang 6₁² (model layer 2), chang 6₂ (model layer 3–4), chang 6₃ (model layer 5–6). Figure 11 is the planar distribution of the three-dimensional porosity of the target layer.

FIGURE 11

Figures 12–14 show the comparisons of daily oil production, water cut and recovery degree of the non-Darcy seepage model equivalently characterized by the method in this study under the condition of diamond reverse nine-spot well pattern. When the two factors of stress sensitivity and starting pressure gradient are not considered, the recovery degree of the target layer is 35.0%. When only stress sensitivity is considered, the pressure of the well group remains relatively stable in the early stage of development; however, as the development period increases, the pressure drop of the well group begins to become significant, which in turn leads to a decrease in reservoir permeability. Furthermore, the daily oil production of the oil well gradually decreased, and the recovery degree was 34.8%. Usually, it is affected by differences in oil and water viscosity, when only the starting pressure gradient is considered, it is difficult for crude oil to break through its own seepage resistance. However, the impact of the starting pressure gradient on the formation water is relatively small, and the water cut rises rapidly, resulting in water channeling. At this time, the production capacity of oil wells declined, and the recovery rate of daily oil production was only 31.9%. When both factors are considered, the reservoir stress sensitivity effect is strong due to the influence of the starting pressure gradient. In addition, the rock displacement vectors in the plane increase, and the permeability decreases greatly. At this time, the daily oil production of the oil well decreased significantly, and the recovery degree was 30.9%. Therefore, comparing the results of different factors, when only the starting pressure gradient is considered, the recovery degree of the target layer decreases by 8.8%; when only the stress-sensitive effect is considered, the recovery degree of the target layer decreases by 0.5%; when both factors are considered, the recovery degree of the target layer decreases by 11.7%.

FIGURE 12

FIGURE 13

FIGURE 14

Comparing Figures 15A,B, it can be seen that when both factors are considered, due to the influence of the starting pressure gradient, the crude oil in the reservoir cannot break through its own seepage resistance, and the pressure variation range near the wellbore of some oil wells expands. Furthermore, it leads to an increase in the swept range of the variation of the reservoir rock displacement vector, and the target layer exhibits relatively strong stress sensitivity.

FIGURE 15

Based on this study, it can be seen that in the development process of ultra-low permeability sandstone reservoirs, the simulation results have higher accuracy when the stress-sensitive effects considering rock properties and starting pressure gradients are introduced. However, in low-permeability sandstone reservoirs, the influence of natural fractures is usually taken into account, which is a current research difficulty in reservoir numerical simulation. The method in this study can also be applied to the numerical simulation study of ultra-low permeability sandstone reservoirs considering natural fractures.

5 Conclusion

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