Research Article  Open Access
Peng Gong, Xiaoyan Ni, Zhanqing Chen, Yu Wu, Jiangyu Wu, "Experimental Investigation on Sandstone Permeability under Plastic Flow: Permeability Evolution Law with Stress Increment", Geofluids, vol. 2019, Article ID 2374107, 19 pages, 2019. https://doi.org/10.1155/2019/2374107
Experimental Investigation on Sandstone Permeability under Plastic Flow: Permeability Evolution Law with Stress Increment
Abstract
Permeability is an important physical property of rock. In rock and rocklike materials, permeability after yielding is closely related to plastic flow behavior. Theoretical analysis and experimental investigations are effective and reliable ways of studying the evolution of permeability. In this paper, fluid flow tests of sandstone samples under 11 stress states were conducted using the MTS 816.02 rock mechanics testing system and a selfdesigned permeation system. A new plastic flow rule was proposed based on the MohrCoulomb yield criterion and a nonassociated flow rule. The applicability of the flow rule was verified by comparing the estimated and experimental values of the equivalent shear strain of the Mohrâ€“Coulomb criterion. Two new coefficients were defined to reflect the influence of the volume deformation and shear deformation on the permeability. A permeability model for plastic flow was established, which can be used to calculate both singlestep and multistep permeability estimates of the sandstone samples during plastic flow. The errors of both estimation methods were analyzed. The experimental results showed that the plastic multiplier for the sandstone samples was positive during unloading conditions and negative during loading conditions. The experimental value error of the singlestep permeability estimation was less than 16.5%. The experimental value errors of the multistep permeability estimations varied between 15.6% and 16.7%, indicating that the iterative format of the multistep permeability estimation method was generally stable and highly precise. A comparison of the permeability influence coefficients indicated that the influence of the equivalent shear strain on the permeability was smaller than that of the volumetric strain.
1. Introduction
Sandstone is the primary type of sedimentary rock in which rock engineering problems arise, such as water disasters in coal mining engineering. Water inrush through roof or floor sandstone aquifers has become a serious problem that occurs in some deep coal mines [1].
Water inrush can happen in coal mines when roof or floor sandstone aquifers have wellconnected major flow paths. Fluid flow research of sandstone roof or floor under mininginduced stress conditions is the fundamental study on water inrush. The permeability evolution of sandstone is the key to studying the fluid flow behavior under mining. In the process of coal mining, deformation on the sandstone roof and floor is dominated by plastic displacements. Therefore, investigating the permeability evolution of sandstone under plastic deformation is essential.
Permeability, the ability to transmit fluid is one of the most important physical properties of sandstone [2â€“4]. The evolution of permeability is associated with certain rock characteristics, and numerous researchers have attempted to correlate permeability with these characteristics. Some studies noted that the permeability of rocks is closely related to their porosity characteristics [5â€“9].
Kenyon [10] proposed a powerlaw relationship between permeability and porosity: where is porosity, is permeability, is a power exponent, and is the limit of permeability when porosity tends to zero, i.e., .
Pape et al. [11] identified a relationship between the porosity and permeability of Rotliegend sandstone from a large dataset of experimental test results:
Currently, the permeabilityporosity relationship defined by rational fractions is the most widely used [12]: where is the initial porosity, and the initial permeability, i.e., .
Rock deformation can result in a permeability change. Some experimental studies were conducted by researchers to investigate the deformation and permeability characteristics of rocks and to try to establish a relationship between permeability and strain. Main et al. [13] presented a simple geometrical model of the permeability changes due to increasing inelastic axial strain based on laboratory experiments. The model quantitatively explained the entire permeability cycle observed in the sandstone samples. Wang and Park [14] investigated the permeability of sedimentary rocks during the complete strainstress process in a triaxial compression test. From the experimental results, they characterized the permeability into five stages according to the axial strain.
Many researchers have investigated the relationship between rock permeability and stress components [15â€“22].
The empirical expression between permeability and the effective normal stress was obtained by Louis [23] from waterdrilling test data at a dam site: where is the effective normal stress and is the coefficient to be determined.
Snow [24] conducted a set of seepage tests of parallel fractures and proposed the following relationship between permeability and stress: where is the initial stress.
When rock is deforming plastically, the stressloading path has a clear effect on the variation in permeability [25â€“28]. The primary factor influencing the variation in rock permeability under a complex loading path is not only porosity but also fracture size, density, and direction.
Bear [29] and Witherspoon et al. [30] successively derived the basic equations describing flow through a fracture. The simplest model of fracture flow is the wellknown cubic law [31]. Javadi et al. [32] proposed the cubic law relationship between fracture permeability and fracture characteristics. where is the fracture width, is the aperture of the idealized parallel smooth fracture, and is the crosssectional area.
Relative displacement between fracture surfaces can induce changes in the volumetric strain. Several studies were conducted to determine the relationship between permeability and volumetric strain [28, 33, 34]. The relationship between the permeability and volumetric strain of fractured rock was given as follows [35]: where the subscript 0 indicates initial values, is the pore volume modulus, is the fluid pressure in the fracture network, is Youngâ€™s modulus, is Poissonâ€™s ratio, and is the volumetric strain.
However, a rock sample undergoing plastic deformation is subjected to not only volumetric deformation but also shear deformation. Shear strain has an important and significant effect on permeability. In this paper, we report the results of conventional triaxial compression tests and fluid flow tests on sandstone samples using an MTS 816.02 rockmechanics test system with a selfdesigned permeation system. These tests were conducted to investigate the permeability evolution law under complex loading path. Our aim was to explore the influence of plastic flow on the variation in permeability, based on the MohrCoulomb yield criterion and a nonassociated flow rule. This study proposes a new permeability prediction model that correlates the volumetric strain and the shear strain. The model was verified by comparing the predicted and experimental values of permeability.
2. Theoretical Principles
The relationship between permeability increments and stress increments can be established based on the strain incrementstress increment relationship and the permeability incrementstrain increment relationship. In the following text, we briefly introduce the theory of plastic flow in rocks and the permeability prediction model.
2.1. ElastoPlastic Constitutive Model of Rocks under Triaxial Compression
Axial strain , radial strain , and circumferential strain occur in a rock sample under axial stress , intermediate stress , and confining stress . Due to axis symmetry, it holds that , and . In the elastic deformation, the stress and strain components of a rock sample follow the generalized Hookeâ€™s law [36], i.e., where is the volumetric strain, , and and are the Lame coefficients , and , respectively.
Shear yielding in rock obeys the MohrCoulomb criterion. If the compressive stress is positive, the yield condition can be expressed as equation (9) if : where is the cohesion of rock, is the angle of shear failure, , and is the internal friction angle.
In the event that plastic flow occurs in the rock sample after yielding, the flow paths can be represented by a directed line segment on the deviatoric plane, as shown in Figure 1. In principal strain space, the flow paths can be categorized into nine cases, as shown in Figure 1(a): (1) flow from one point on the yield surface to another point on the yield surface (), (2) flow from one point on the yield surface to one point outside the yield surface (), (3) flow from one point on the yield surface to one point inside the yield surface (), (4) flow from one point outside the yield surface to one point on the yield surface (), (5) flow from one point outside the yield surface to one point outside the yield surface (), (6) flow from one point outside the yield surface to one point inside the yield surface (), (7) flow from one point inside the yield surface to one point on the yield surface (), (8) flow from one point inside the yield surface to one point outside the yield surface (), and (9) flow from one point inside the yield surface to another point inside the yield surface ().
(a)
(b)
In principal stress space, the stress state must be located on the yield surface. Therefore, there are only four cases possible, with respect to plastic flow, on the deviatoric plane in principal stress space, as shown in Figure 1(b): (1) flow from one point on the yield surface to another point on the yield surface (), (2) flow from one point on the yield surface to one point inside the yield surface (), (3) flow from one point inside the yield surface to one point on the yield surface (), and (4) flow from one point inside the yield surface to another point inside the yield surface (). The dashed line in Figure 1(b) indicates impossible flow paths.
Plastic potential is related to not only the stress states but also the loading history. For rock, the plastic flow rule is nonassociated, and the plastic potential function can be described as a MohrCoulomb relationship, where and are no longer constant but a function of the loading history. If the plastic internal variable is used to represent the loading history, the plastic potential function can be expressed as follows [37â€“40]: Equation (10) is the plastic potential function with an isotropic hardening (softening) model, which is a group of potential functions that change in terms of and . is used to describe the plastic deformation history. The plastic volumetric strain of rock can be selected as the internal variable:
When , we have and , which correspond to equation (9). The values of and can be obtained using a triaxial compression test.
Then, the flow rule is as follows: where and are the plastic strain increments and is the plastic multiplier.
When the stress state induces flow on the yield surface, the plastic multiplier can be expressed as follows: where
Equation (13) shows the plastic flow rules for geotechnical materials and is commonly used in geotechnical engineering [41].
When the stress state is inside the yield surface, the plastic multiplier can no longer be calculated by equation (13). As shown in Figure 2, if the flow path of stress state is from to (), we can divide into and due to small deformation. The point is on the yield surface. Therefore, we propose the following plastic multiplier expression according to equation (13):
Equation (15) shows the modified plastic flow rules for rock proposed in this paper.
2.2. Permeability Model for Plastic Flow
Assuming small deformation, the rock sample porosity is related to the rock volumetric strain as follows: where is the initial porosity. Porosity is commonly measured by the mercury injection test.
The powerexponential relationship between the porosity and permeability of the rock samples in the postpeak region is given by [42] where is the nondimensional coefficient.
Equation (18) can be written by taking the logarithm on both sides of equation (17):
Equation (18) can be differentiated with respect to porosity to obtain
The permeability increment is easily obtained from equation (19): where is the porosity increment.
Equation (21) can be written by combining equations (16) and (19):
Equations (16) and (21) indicate that, in theory, only the volumetric strain changes the porosity and permeability, whereas the shear strain does not. However, the experimental results presented in this paper show that the shear deformation changes the permeability. To characterize the influence of the shear deformation on permeability, an equivalent shear strain is given by the following equation, according to the MohrCoulomb criterion:
Considering the influence of shear deformation, we have modified the equation to calculate permeability: where and reflect the influence of the volumetric strain and equivalent shear strain, respectively, on the permeability. These parameters are defined as the permeability influence coefficients and can be estimated using the Monte Carlo method.
Equation (23) describes the changes in permeability due to plastic deformation and is known as the permeability model under plastic flow.
3. Experimental Samples, Principles, and Procedures
3.1. Experimental Samples
The standard cylindrical sandstone rock samples of 50â€‰mm diameter and 100â€‰mm length were cut from sandstone blocks that were collected from a coal mine in Shanxi Luâ€™an, China. To ensure the reliability of the experimental results, rock samples without obvious macroscopic cracks and joints were chosen. The two ends of the rock samples were polished to remove uneven surfaces [1]. Parts of sandstone samples are shown in Figure 3. The mineral composition of the sandstone is mainly quartz, feldspar, calcite, oxides, and mica, which is detected by the powder crystal Xray diffraction testing machine. The initial porosity of the sandstone is 5.7%, and the order of magnitude of the initial permeability is 10^{17}â€‰m^{2}, which measured by a PoreMaster 33 mercury intrusion meter.
The triaxial compression tests were first performed to determine the sandstone sample mechanical parameters by using the MTS815.02 mechanics testing machine. The experiments were conducted under the confining pressure of 6.0â€‰MPa. The values of the failure load, peak strength, peak strain, and elasticity modulus are shown in Table 1. The mean value and variable coefficient of peak strength are 129.9â€‰MPa and 4.30%, respectively. The mean value and variable coefficient of peak strain are 1.27% and 3.27%, respectively. The mean value and variable coefficient of the elasticity modulus are 9.64â€‰GPa and 5.86%, respectively. The values of variable coefficient show that the dispersion of the sandstone samples is very small.

3.2. Experimental Principles
3.2.1. Triaxial Compression Test
In triaxial compression tests, the confining pressure remains constant. The sampling period in this study is . The axial stress , the axial strain , and the circumferential strain of each rock sample are measured at the moment , where .
We assume that the axial stress reaches a peak value at the moment , i.e., . For the elastic deformation, the relationship between stress and strain is described as follows: where is the volumetric strain of the rock samples at sampling time, and .
The strain of the rock sample can be divided into two parts, elastic strain and plastic strain, in which the elastic strain is given as follows:
Therefore, the plastic strain equation can be written as follows: where the plastic internal variable is .
We conduct the triaxial compression tests at different confining pressures, where . Then, plastic internal variable axial stress () curves can be obtained. To establish the relationship between , , and , we set typical values of and obtain the corresponding axial stress , where , under and . Using equation (9), we can obtain the values of and under different by using linear regression.
3.2.2. Fluid Flow Test
In this study, we used a transient method to measure the permeability of a sandstone sample [1]. The principle of the method is shown in Figure 4.
The volume of storage tank 1 (ST1) is equal to the volume of storage tank 2 (ST2), and the pressures of the fluids in these tanks are and , respectively. The sandstone samples are standard cylindrical specimens of height and crosssectional area . At the start of the test, the pressure at the ends of a sandstone sample are and (). During the process of the permeability tests, the fluid from ST1 flows into ST2 through the rock sample. The pressure of the fluid in ST1 continuously decreases, while the pressure of the fluid in tank 2 continuously increases. The pressure gradient gradually decreases until the two storage tank pressures reach equilibrium, as shown in Figure 5.
If the mass flow from ST1 into the rock sample is and the rock sample is presaturated, then the mass flow from the rock sample into ST2 is also . The flow velocity within the rock sample is , where is the density of the fluid; due to the compressibility of the fluid, the following relationship can be obtained: where is the compressibility coefficient of the fluid.
Since and , we arrive at the following expression: where is the volume of the storage tank.
Similarly, we can write a similar expression for the other storage tank:
From equations (28) and (29), the expression for the flow velocity of the rock sample can be written as follows:
According to Darcyâ€™s law, equation (30) is rewritten as follows: where is the permeability of the rock specimen and is the dynamic viscosity of the fluid.
Equation (32) can be derived by combining equations (30) and (31):
In the experiments, the sampling frequency is , the total number of measurements is , and can be calculated at the end of the sampling period. Equation (32) is integrated to obtain an equation in terms of the respective pressure values of the two storage tanks, and :
Deformation occurs when the rock sample is under considerable stress. The permeability variation is sensitive to the deformation of the rock samples. We test stress states of the rock sample, i.e., , corresponding to permeability values, i.e., , . is the peak stress state, and are the elastic deformation states, while are the plastic flow states.
The Lame coefficients, and , for determining the elastic constitutive relations can be obtained by using the leastsquares method, as described below.
The first step is to apply the generalized Hookeâ€™s law to define the relationship between the stress and strain components of :
The second step is to construct a function based on the residual errors of equation (34):
The third step is to define a system of linear equations related to and , which is based on the functional extremum conditions and :
The fourth step is to introduce the sign convention:
The system of linear equations (36) of and are then transformed into a matrix:
The last step is to obtain the optimal estimation of and by resolving the linear equation for the two unknowns.
The rock sample will yield at the point where the principal stresses satisfy the MohrCoulomb criterion:
From point to point , the rock sample is subjected to flow steps. Step 1 includes the flow from to , written as , and step 2 through steps are written as , ,â€¦â€¦, and , respectively.
According to equation (12), the plastic strain increment in the flow step () is calculated as follows:
The elastic strain increment is calculated as follows:
Therefore, the strain components of the rock sample under the stress state () are as follows:
Because stress state is on the yield surface, the plastic multiplier in the flow step is calculated according to equation (13): where
Since the points are not on the yield surface, the plastic multipliers in the flow steps and are calculated according to equation (15):
Equations (43) and (45) constitute the computational format of the plastic multiplier.
According to equations (40), (41), and (42), the computational format of the volumetric strain of the rock sample can be defined as follows: where is the average principal stress.
The experimental values of permeability are recorded as . The values of permeability after yielding can be calculated according to equations (17) and (23), and the iterative format is as follows: where is the estimated value of permeability in the stress state , is the volumetric strain increment during plastic flow , and is the equivalent shear strain increment during plastic flow .
3.3. Experimental System
The experimental system includes an axial loading system, a confining pressure loop, a permeation loop, and a permeameter. The main functions performed by the system are as follows: (1) applying the axial loading to the rock sample, (2) applying the confining pressure to the rock sample, and (3) applying the pore pressure (seepage pressure) to the upper and lower ends of the rock sample. The rock sample was enclosed in the permeameter (Figure 6), and the MTS 816.02 rock mechanics testing system and TestStar II were used to apply an axial load to the permeameter, as shown in Figure 7. The permeameter is equipped with three nozzles: A, B, and C. Nozzle A is connected to the confining pressure loop, and nozzles B and C are connected to the permeation loop.
(a)
(b)
The permeameter is composed of a base plate (#1), lower steel plate (#2), porous disc (#3), cylindrical barrel (#4), upper steel plate (#5, #6), bleeder screw (#7, #8), piston (#1), cover plate (#12), and other components, as shown in Figure 6. There is a cavity between the rock sample and the cylindrical barrel, and the confining pressure system injects hydraulic oil into this cavity through nozzle A to apply the confining pressure (radial pressure) around the rock sample, as shown schematically in Figure 8. Nozzles B and C of the permeameter are connected to ST1 and ST2 in the permeation loop, respectively, and apply the pore pressure at both ends of the rock sample, as shown in Figure 9. The fluid in ST1 flows into the rock sample from nozzle B through the base plate, the lower steel plate, and the porous disc and flows into ST2 from nozzle C through another porous disc, the upper steel plate, and the piston. The cylindrical barrel is connected to the base plate and the cover plate by means of threaded fasteners (#13, #14, #15, and #16 in Figure 6). Otype rubber sealing rings are used to create seals between the cylindrical barrel and the cover plate, between the cylindrical barrel and the base plate, and between the cover plate and the piston (#20, #21, #22, and #23 in Figure 6). The upper steel plates (#5 and #6 in Figure 6) use a spherical contact structure to uniformly distribute the axial force. The air in the confining pressure cavity is evacuated via the bleeder screw (#7 and #8 in Figure 6).
In the confining pressure system (see Figure 8), the valve spool of the directional control valve (DCV) is an Mtype function valve. Before the fluid flow test, the spool is set to the lefthand position, and the hydraulic oil flows into the cavity via the plunger pump (PP) through both the DCV valve and the combined valve (a oneway valve and a throttle valve). At this point of the experiment, the oneway valve is not operational; the flow rate is adjusted by turning the hand wheel of the throttle valve (TV), and the confining pressure is adjusted by turning the hand wheel of the relief valve (RV). In the fluid flow test process, the confining pressure remains constant. After the test is completed, with the RV in the neutral position, the hydraulic oil flows directly to the oil tank (OT) through the PP. When the rock sample is removed, the cutoff valve (COV) is opened, and the RV will be placed in the righthand position. At this point, the oneway valve in the combined valve is operational. The hydraulic oil in the cavity then flows back to the OT through the oneway valve and the COV.
In the permeameter loop, the volume of the storage tanks is an index of the permeability measurement ranges. Before the fluid flow test, the pressure values of ST1 and ST2 are set to and . The process to set the initial storage tank pressures is as follows: turn on COV1, COV2, COV3, and COV4; start the electric test pump so that the pressure of the entire loop stabilizes to ; turn off COV1, COV3, and COV4; pressurize the pipeline with the manual pump (MP); finally, stop the MP and turn off COV2 when the pressure of ST1 is stabilized at . During the test, COV1 and COV3 are turned on at the same time, and the pressure in the two tanks is recorded. After the test, COV1, COV2, COV3, and COV4 are turned on at the same time.
3.4. Experimental Procedure
Before the fluid flow test, the rock specimens were saturated with water. The test was divided into 11 steps, corresponding to the 11 stress states and 11 strain states , (, and , ). The displacement controlling method is used in the test with constant strain rate of . Each step of the fluid flow test had a sampling time of 1000â€‰s, with a sampling rate of 1â€‰Hz. The flow chart of the procedure for the fluid flow test is shown in Figure 10.
4. Experimental Results
4.1. Triaxial Compression Test Results
We conducted a series of conventional triaxial compression tests under confining pressures of 2â€‰MPa, 4â€‰MPa, 6â€‰MPa, 8â€‰MPa, and 10â€‰MPa. The test results are presented in Table 2.

The plastic volumetric strain of rock samples RTS1, RTS4, RTS8, RTS12, and RTS15 was calculated according to the method described in Section 3.1. Then, we obtained 5 plastic internal variable versus axial stress curves, one for each of the confining pressures tested, as shown in Figure 11.
We calculated the values of and corresponding to the plastic volume strains of 0.0, 0.001, 0.002, 0.003, 0.004, and 0.005 by using equation (10), and the results are presented in Figure 12 and Table 3.

4.2. Fluid Flow Test Results and Analysis
The fluid flow tests were carried out under stress states, specifically . Each of the rock samples had a diameter of 49.6â€‰mm and a height of 99.2â€‰mm. Water was used as the flow liquid; the water had a mass density of , dynamic viscosity of at 20Â°C, and a compressibility coefficient of .
At the start of the fluid flow test, the pressures at the ends of the sandstone sample are and ; therefore, the pore pressure difference is 2â€‰MPa. In the prepeak strength state, the confining pressure is set as 6â€‰MPa. To investigate the influence of plastic flow on the permeability of the sandstone, the axial stress and confining pressure of the rock samples were both changed after the seventh step of the permeability test.
The axial stressstrain curve of the sandstone sample is shown in Figure 13. As seen in Figure 13, the axial stress reaches a peak value at point . At that point, the rock sample is considered to be deforming via shear yielding. Therefore, points describe the prepeak strength state, and points describe the plastic flow state. The rock sample went through a 6step plastic flow, represented by points in Figure 13. The values of the axial stress , confining pressure , axial strain , and circumferential strain measured during the test are presented in Table 4. The sign convention of , , and is that and in compression are positive and in dilation is positive.

For , in the elastic deformation state, the values of the Lame coefficients and of the sandstone sample were obtained according to the method described from equation (34) to equation (38), where
Then, the values of the Lame coefficients are and .
A total of 6 plastic flow steps occurred after yielding of the rock sample, (). The values of the stress increments ( and ), strain increments ( and ), plastic potential increment (), and plastic multiplier in each flow step are presented in Table 5.

As shown in Table 5, the flow steps , , , and show the unloading process, and and show the loading process. The plastic multiplier lies between and , and it is positive during the unloading steps () and negative during the loading steps.
The volumetric strain, porosity, equivalent shear strain, and permeability of each stress state after the sample yielded are presented in Table 6.

As seen in Table 7, the volumetric strain increments during the first unloading process and are positive, and the volumetric strain increments during the later unloading processes and are negative. In addition, the volumetric strain increment during the loading process is negative, and the volumetric strain increment during the loading process is zero. These changes in the sign of the volumetric strain increment indicate that there is no onetoone relationship between the volumetric strain/variation and either loading or unloading. Similarly, there is also no onetoone relationship between the equivalent shear strain variation and either loading or unloading.

Shear deformation occurred in the rock sample during the loading steps of , without volume deformation. A permeability increment magnitude on the order of was observed at this step, and a permeability increment magnitude on the order of to was observed during the other steps. Therefore, the influence of the shear strain on permeability is less than that of the volumetric strain.
The region of search for was , and the region of search for was . By using the Monte Carlo method, the optimal estimated values of the permeability influence coefficients were obtained as and . We found that , which proves that the influence of the shear strain on permeability is less than that of the volumetric strain.
From the fluid flow test results and the calculated values of and , it can be concluded that both shear strain and volumetric strain have an obvious effect on permeability. These two results both verify that the volumetric strain has a major effect on permeability.
The estimated values of permeability under every stress state after shear yielding, calculated by using equation (47), are presented in Figure 14 and Table 8.

Table 8 shows that the error between the estimated values and the experimental values in the stress state is 16.5%, which is because the axial stress decreases sharply as soon as the rock sample reaches the peak stress state. The errors in the stress states were between 0.22% and 1.20%. These small errors indicate that the iterative format for estimating permeability, represented by equation (47), is generally stable.
5. Discussion
In this study, we conducted fluid flow tests under 11 stress states. The permeability variation law for the sandstone samples after shear yielding was discussed. The strain increments after yielding included both the elastic strain increments and plastic strain increments. The former is directly given according to the generalized Hookeâ€™s law, while the latter is given by the flow rule, and the total strain increments are as follows:
According to equations (22) and (49), the volumetric strain increment and the equivalent strain increment can be written as follows:
The expression between the permeability increment and the stress increment in plastic flow can be obtained by substituting equations (50) and (51) into equation (23):
Based on equation (52), an iterative format is constructed to estimate the permeability increment in each step and the permeability at each stressstrain state, after shear yielding.
The performances of the plastic flow rule and the permeability model are discussed below.
5.1. Plastic Flow Rule
Substituting equations (40) and (41) into equation (42), we obtain the following equation: where and for .
The flow rule represented by equation (53) is a mapping of ; however, geometrically comparing the estimated value of the strain component with the experimental value is difficult. Therefore, we introduce the equivalent shear stress that corresponds to the MohrCoulomb yield criterion:
Before yielding, the equivalent shear stress is linearly related to the equivalent shear strain , i.e., . After yielding, and .
The equivalent shear stress and the equivalent shear strain , where are calculated for each stress state according to equations (54) and (22). The increment of equivalent shear strain is found to be by combining equations (22) and (54), where , and the estimated value of the equivalent shear strain is , where ; these values are also calculated for each stress state.
Figure 15 shows the estimated values and experimental values of the equivalent shear strain corresponding to the equivalent shear stress , where . Figure 15 shows that the estimated equivalent shear strain values are approximately equal to the experimental values in each state after yielding. It can be argued that the flow rule estimation of the equivalent shear strain, as proposed in this paper, is highly accurate.
5.2. Permeability Model
The permeability model represented by equation (47) estimates the permeability at the state, based on the experimental value of permeability at the state, which is a singlestep estimation. Although the singlestep estimation is highly precise, it largely depends on the test. To facilitate engineering applications, we estimate the permeability at the state by means of the estimated permeability value at the state and propose the following multistep estimation format for permeability: