Determining Triaxial Stress Sensitivity of Oil Reservoir Rocks without Fluid Flooding

The sensitivity of oil reservoir rocks to stress is the basis for oilfield development, which determines the production method employed in the field. Therefore, it is critical to understand the stress sensitivity behavior of oil reservoir rocks in an oilfield. In this paper, a novel method for determining the stress sensitivity of oil reservoir rocks by triaxial stress testing without fluid flooding was proposed. It measures the triaxial stress and strain of the core rock samples, and based on which, the core porosity and permeability under stress can be evaluated by theoretical model. In the model, the pores of the core were assumed to be a bundle of capillaries and the necessary relationship was derived to calculate the changes of porosity and permeability of the core samples caused by the strain. Through comparison with and analysis of experimental results obtained for various rock core samples under different stress and strain conditions, it is observed that the theoretical model match well with that of the experiments. This method provides a new approach for the stress sensitivity analysis of oil reservoirs without fluid flooding.


Introduction
The variation of oil reservoir core porosity and permeability under stress is called the reservoir rock stress sensitivity, which is the basis for reservoir production mode selection. If an oil reservoir rock is relatively stress sensitive, it is necessary to take immediate actions to avoid porosity and permeability reduction. The production potential of gas hydrates mainly depends on permeability characteristics of the bearing sediments as high permeability could promote the production rate [1]. And the enhancement of gas effective permeability with increased gas saturation weakens with higher complexity and lower discreteness of a pore network. A less complex and more discrete pore network better benefits the gas injectivity index [2]. Therefore, in the late 1980s, scientists began to carry out research on the stress sensitivity of oil reservoirs, mainly by core flooding experiments [3,4]. In the experiments, a certain confining pressure is applied to a core sample in three directions [5,6]; then, crude oil or nitrogen gas is driven to flow through the core sample, and the pressure and flow rate at both ends of the core are [7][8][9][10][11]. The permeability of the core sample is calculated based on the Darcy equation, through which the permeability of the core samples under different stress and strain can be obtained by varying the confining pressure. Then, the characteristics of stress sensitivity of the rock core are obtained by analyzing the empirical law of permeability under different confining pressures [12]. It is the conventional method of obtaining reservoir rock stress sensitivity which has been widely used by oilfield scientists [13][14][15].
However, such method faces a few problems which still need to be tackled with urgently: First, it is difficult to apply stress simultaneously in three different directions (x, y, and z) during the actual production stage. In realistic producing oilfield, the stress on rocks is always applied from the top, which is different from the laboratory situation where the stress is from three directions [16][17][18]. Second, the method involves measuring the pressure and flow rate of the flooding liquid or gas at both ends of the rock core, while the permeability is calculated by the Darcy formula. If the sealing between the core sample and the core holder becomes defective during the experiment, the fluid would leak rather than fully pass through the core sample. The fluid may flow through the gap between the core sample and the core holder instead, and the measured flow rate will be different from that of the real situation. Thus, the evaluated permeability may be deviated from the actual value of the reservoir core sample [19,20].
The changes of porosity and permeability of artificial tight core with stress were studied by Cao and Lei [21] who performed lithologic flow experiment. And the stress sensitivity of the process of pressurization and unloading of tight reservoir is analyzed through the laboratory experiment of stress sensitivity hysteresis of core. The research shows that in the early stage of compression, under the joint action of rock structure deformation and bulk deformation, core permeability decreases rapidly with the increase of effective stress. When the effective stress exceeds a certain value, rock structure deformation tends to be stable, only bulk deformation occurs, and core permeability changes tend to be flat. In the unloading stage, the deformation of the rock body is gradually restored with the reduction of effective stress, while the structural deformation cannot be restored, core permeability cannot be completely restored, and stress sensitivity hysteresis phenomenon is obvious.
Chen et al. [22] mainly study the stress-strain characteristics and permeability variation of rock under confining pressure unloading condition and obtain the understanding of the by-correlation between permeability and volume strain during loading and unloading process.
Yaser et al. [23] studied the pore structure, porosity, and permeability of carbonate rocks under pressure with the combination of experiments and microscopic images. The results showed that the porosity and permeability of calcalite samples decreased with the increase of pressure load.
Liu et al. [24] conduct an experimental study of salt rock deformation and strength under different confining pressures using the salt rocks from Pingdingshan, China. The results show that the pure salt rock shows great brittleness and failure in uniaxial test. However, the presence of impurities improved the strength of the sample, and the strength of rock salt showed strong alkaline strain hardening with the increase of impurity content. Experiment shows that the higher the confining pressure, the more obviously lag expands occur.
Sui et al. [25] research the deformation of the core by using the digital core method, and the simulated pores in the digital core are simplified into three abstract types: long ellipsoid, flat ellipsoid, and ellipsoid. Based on the theory of microscopic mechanics, the three-dimensional spherical pore model of digital core is established. The constitutive relation of pore microstructure deformation of different types is studied by using Eshelby equivalent medium theory, and the influence of pore structure on pore volume compressibility under the condition of single pore and porous elastic deformation is studied. The results show that the numerical core of pore volume compression coefficient is closely related to the porosity, pore aspect ratio, and volume ratio of different types of pores. The compressibility of the long ellipsoid hole coefficient is positively correlated with the hole width ratio, while the compressibility of the flat ellipsoid hole coefficient is negatively correlated with the hole width ratio. When the mean value of hole width ratio satisfying Gaussian distribution is the same, the larger the hole width ratio is, the larger the pore aspect ratio range is, the higher the compression coefficient of long ellipsoid and flat ellipsoid hole is, and the greater the deformation is under the same stress condition. Table 1 summarizes the existing experiments.
In order to avoid the problems discussed above, in this paper, a new testing method without fluids involved during the process was proposed. The method is to test the triaxial stress and strain of rock core samples, which eliminates the challenge of sealing between the annulus core and its holder and prevents fluids from leaking out therefrom. And to make the testing happen, a novel mathematical model is established to calculate the porosity and permeability of the oil reservoir rocks based on applied stress and strain data. Actual applications demonstrate that the method is an effective approach for stress sensitivity analysis of rocks and is simpler and more advantageous than the conventional methods.

Mathematical Model for Simulating Porosity and Permeability Changes under Stress
Here, The mathematical model is established for calculating the oil reservoir rock porosity and permeability from the experimentally measured triaxial stress and strain data. And how to process and incorporate the experimental data into the model is a critical factor.

Model for the Relationship between Volumetric Strain
and Stress. During the production stage of an oil reservoir, the formation fluid pressure decreases. This disturbs the static equilibrium of the formation and causes rock deformation under stress, which creates volumetric strain to the core rock, as illustrated in Figure 1.
Assuming that the stresses in the x, y, and z triaxial directions are σ x , σ y , σ z , then they can be analyzed in two situations: one is the elastic deformation, and the other is the elastoplastic deformation.

Analysis for Elastic Deformation.
According to the laws of elastic mechanics, the relationship between volumetric strain and stress can be obtained by the general Hooke's law. For the x, y, and z directions, the volumetric strain can be expressed as where E is Young's Modulus and μ is Poisson's ratio, and the volumetric strain ε v is expressed as Combining equations (1) and (2) gives where b and q represent the nonlinear elastic deformation index, if b and q are equal to 1, which represent elastic deformation.
According to the theory of elastic mechanics, From equation (4), Substituting equations (6) and (7) into equation (5) gives Integrating the both sides of equation (8) gives Similarly, for y and z directions, it gives From equations (2), (9), (10), and (11), the relationship between volumetric strain and stress is obtained as When q is equal to 0 in equation (12), it becomes a simplified expression of equation (3), which is an expression of an elastic medium. Therefore, equation (12) is universally applicable regardless of elastic or elastoplastic deformation.

Model for the Relationship between Porosity and
Volumetric Strain. The oil reservoir rock is composed of framework and pore space. It is assumed that when the oil reservoir rock is subjected to effective stress, the core framework is not deformed, and only the pore space will be compressed, resulting in volumetric strain, as shown in Figure 2.
According to the definition of porosity, where V b is the total volume of the rock sample and V r is the pore volume of the rock sample. When external stress acts on the porous core, the porosity changes from the initial state of ϕ 0 to the current state of ϕ, with a volumetric strain of ε v generated. The following equation indicates the relationship.
At constant temperature, equation (14) can be substituted into equation (13), which gives an expression for the porosity after the core volume change is obtained.

Geofluids
And substituting equations (13) and (14) into equation (15) gives Then, the change in porosity at different volumetric strains can be calculated by equation (16).

Model for Relationship between Rock Permeability and
Volumetric Strain. The permeability of a rock is closely related to its strain. Here, the complex porous medium is simplified into a series of equal diameter capillary tubes [3]. The relationship between the theoretical rock permeability and rock porosity can be obtained according to the Kozeny-Carman equation.
where k z is a constant of approximately 5, ϕ the porosity, and S p /cm 2 the pore surface area of volume of the pore medium. Equation (17) indicates that change in porosity is the main factor causing changes in the permeability of the rock sample. Substituting equation (16) into (17), the mathematical model for the changes in rock permeability with volumetric strain is derived as From equation (18), the permeability of the rock core can be calculated once the volumetric strain and the initial porosity of the rock core are known.

Application on Rock Stress Sensitivity
Analysis with Triaxial Stress Experiments 3.1. Sample Preparation. The rock core sample used in the verification experiment is from the Tahe Oilfield in Xinjiang Province in China. The carbonate oil reservoir is located at a depth of about 3,500 meters. A standard core rock sample was prepared for triaxial stress sensitivity experiment. The experimental setup is shown in Figure 3.
In order to simplify the evaluation of stress, it was assumed that the confining and the axial pressures are equal. The experiment was done in the applied pressure range from 2 MPa to 65 MPa, and the process of loading stress was in three steps, as illustrated as follows: Step 1: firstly, a low confining pressure of 2 MPa was loaded onto the side of the sample; then, the same amount of axial pressure was applied. The radial and axial strains under the confining and axial pressures are measured; Step 2: secondly, the confining pressure and its corresponding axial pressure were increased to 65 MPa, and the axial and radial strains were measured under this condition; Step 3: finally, keeping the pressure on the rock until the rock sample was crushed. The confining and axial pressures were loaded from 2 MPa to 65 MPa.

Analysis and Application of Experimental
Results. The radial and axial strain data of the rock samples under different stresses were obtained by experiments, and the corresponding volumetric strain data under the conditions were calculated by equation (12). The results are shown in Table 2.
The data in Table 2 are used to evaluate the corresponding changes in porosity and permeability according to equations (16) and (18), respectively. Meanwhile, the porosity and permeability damage rate was evaluated as well, and the results are shown in Table 3.
As shown in Figure 4, the volume strain of the core has a good power relationship with the effective stress. Figure 5 shows that for the carbonate rock, the porosity, and effective stress satisfy the relationship of ϕ = 0:0415 e −0:003p . The porosity decreases with increase in the effective stress. Meanwhile, the porosity decreased more rapidly in the initial stage when the effective stress was less than 20 MPa.
From Figure 6 it is realized that porosity and effective stress of the core sample satisfy the relationship of k = 0:022e −0:0059p . The permeability decreased with increase in the effective stress. Meanwhile, the reduction rate of the permeability increased when the effective stress is less than 20 MPa. Figure 7 shows that the dimensionless porosity and effective stress of the core sample satisfy the power relationship of ϕ/ϕ 0 = 0:9773e −0:003p .

Geofluids
As shown in Figure 8, the dimensionless permeability and effective stress of the core sample satisfy the exponential relationship of k/k 0 = 0:9552e −0:0069p .
The results show that the oil reservoir core porosity and permeability under stress can be well evaluated using the mathematical model together with experimental stress and strain data with no need of introducing flooding fluid.

Discussion
The method proposed by the paper addresses the problem of sealing and leakage of core holders in the conventional stress and flowing fluid method which provides a new approach for the stress sensitivity analysis of oil reservoirs without fluid flooding. The results of the experimental prove that the    6 Geofluids weakens with higher complexity and lower discreteness of a pore network. Therefore, for the rock with higher porosity, this method is less effective and the results would have deviation errors. The future work is to continuously improve the model by adding other influence factors, to achieve higher accuracy of soft rocks or rocks with microcracks.

Conclusion
The experimental results showed that the triaxial stress method proposed by this investigation with applying rock mechanics method to test the stress sensitivity of reservoir rocks is feasible. The experimental method is practical and simple test while the mathematical model results are reliable. It is feasible to evaluate the mathematical model for changes in permeability and porosity with stress from experimental data of triaxial stress. The calculated results matched well with the experimental results. Comparing with the conventional method of applying stress and flowing fluid through rocks by a core flooding equipment, the proposed method by this investigation avoids the problems of sealing and leakage of core holders and has other advantages as well. However, the new method is less adaptable for soft rocks or rocks with microcracks, where the results obtained would be biased.

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

Conflicts of Interest
The authors declared that there is no financial and personal relationships with other people or organizations that can inappropriately influence our work, and there is no professional or other personal interest of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in or the review