J. Cent. South Univ. Technol. (2009) 16: 0149-0153

DOI: 10.1007/s11771-009-0025-3

Seepage-stress coupling constitutive model of anisotropic soft rock

ZHANG Xiang-xia(张向霞)^{1, 2}, YANG Lin-de(杨林德)², YAN Xiao-bo(闫小波)^{2, 3}

(1. Research Center for Urban Safety and Security, Kobe University, 1-1 Rokkodai-cho, Nada-ku,Kobe 657-8501, Japan;

2. Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education,Tongji University, Shanghai 200092, China;

3. College of Civil Engineering, Fuzhou University, Fuzhou 350002, China)

Abstract: To provide a seepage-stress coupling constitutive model that can directly describe the seepage-stress coupling relationship, a series of one-dimensional seepage-stress coupling tests on two kinds of soft rock (argillaceous siltstone and brown mudstone) were performed by using an MTS-815.02 tri-axial rock mechanics test system, with which the stress—strain curves according to the seepage variation were obtained. Based on the experimental results and by employing Hooke’s law, the formulation of the coefficient of strain-dependent permeability was presented and introduced to establish a coupling model. In addition, the mathematical expression and the incremental formulation for coupling model were advanced, in which five parameters that can be respectively determined by using the experimental results were included. The calculated results show that the proposed coupling model is capable of simulating the stress—strain relationship with considering the seepage-stress coupling in the nonlinear elastic stage of two kinds of soft rock.

Key words: soft rock; stress—strain relation; seepage-stress coupling; constitutive model; coefficient of permeability

1 Introduction

The coupling between the process of the fluid flow and the stress/deformation in geomaterial has become an increasingly important subject in soil mechanics and engineering design in recent years, mainly due to the modeling requirements for the design and performance assessment of underground facilities such as the storage for liquid, waste deposits and traffic rules, in which fluids play an important roles [1-2]. Especially in China, the plans of “underground space engineering” and “southwest development” carried in recent yeas have resulted in an increase of projects of large-scale structure constructed in different geotechnical conditions with complex configurations such as saturated soft rock with water effects. The numerical analysis of the interaction of these processes requires a suitable mathematical model.

The soil/water coupling theory was firstly proposed by TERZAGHI [3] in 1925 as a one-dimensional consolidation theory of soil, and then generalized by BIOT [4-5] to three-dimensional condition theory. Since then, there has been an increasing amount of literatures on the theoretical and experimental studies on the seepage-stress coupling in geomaterial [6-10]. The seepage-stress coupling models have been developed according to the poroelasticity theory, and the coupling sets of conservation equation need to be solved. However, in light of recent ever increasing complexity in the coupling models, the issue has grown in importance that the solution of the complicated coupling sets of conservation equation is not computationally efficient and generally causes some calculation problems such as the accumulation of the error [11].

In the face of this problem, a new method to consider the seepage-stress coupling in the constitutive model was proposed. A coupling constitutive model to directly describe the seepage-stress coupling relationship was established, and the correspondingly mathematical expression was obtained.

2 Experimental

A series of one-dimensional seepage-stress-strain tests on two kinds of anisotropic soft rock (argillaceous siltstone and brown mudstone) were performed by using an MTS-815.02 tri-axial rock mechanics test system at Rock Mechanics Laboratory in China University of Mining and Technology in Xuzhou. By using this test system, the stress—strain curves according to the variation of coefficient of permeability can be obtained.

The research was based on the project of the Sixiao Highway, and rock samples taken from in-suit included two kinds of soft rock: argillaceous siltstone was taken from the Madi River No.1 Tunnel and brown mudstone was from Daganba Tunnel. The test samples from the two kinds of rock were cored in two orthogonal directions in blocks. One coring direction was perpendicular to the bedding plane, whereas the other direction was parallel to the bedding plane. The parameters of the samples are shown in Table 1.

Table 1 Physical parameters of rock samples

Figs.1 and 2 respectively show the relationship among the deviatoric stress, axial strain and the coefficient of permeability of two kinds of saturated soft rock samples based on the experimental results. In Figs.1 and 2, ε₁ expresses the axial strain; (σ₁-σ₃) expresses the deviatoric stress for the rock samples, where σ₁ is the maximum principal stress, σ₃ is the minor principal  stress; K denotes the coefficient of permeability for the rock samples, which can be obtained based on the experimental results measured by using the pulse method with transient. In Figs.1 and 2, the σ—ε curves describe the relationship between the deviatoric stress (σ₁-σ₃) and axial strain (ε₁), and the K—ε curves describe the relationship between the coefficient of permeability (K) and axial strain (ε₁).

Fig.1 Relationship among stress, strain and coefficient of permeability for argillaceous siltstone: (a) Axis of sample is parallel to bedding plane; (b) Axis of sample is perpendicular to bedding plane

Fig.2 Relationship among stress, strain and coefficient of permeability for brown mudstone: (a) Axis of sample is parallel to bedding plane; (b) Axis of sample is perpendicular to bedding plane

The test principle and physical properties of the studied rock formations were presented in detail in Ref.[12].

As seen in Figs.1 and 2, for the two kinds of soft rock, the axial strain increases with increasing deviatoric stress. As for the relationship between the coefficient of permeability and the strain, Fig.1 shows that, for the argillaceous siltstone, the coefficient of permeability increases with increasing axial strain, while for the brown mudstone shown in Fig.2, with the increase of the axial strain, the coefficient of permeability correspondingly decreases. Different but regular variations of the relationship between the coefficient of permeability and the axial strain for different soft rocks [12] show that the deviatoric stress, the axial strain and the coefficient of permeability are inter-dependent. The seepage-stress coupling model to directly describe the seepage-stress coupling relationship is obtained by introducing the coefficient of the strain-dependent permeability to the stress—strain equation and the coefficient of the strain-dependent permeability can be expressed by using the axial strain.

3 Seepage-stress coupling model and its experimental verification

By theoretically analyzing the seepage-stress coupling mechanism and based on the above-mentioned one-dimensional seepage-stress coupling experimental results, the mathematical formulation of one- dimensional seepage-stress coupling constitutive model was proposed.

3.1 Strain-dependent permeability

The influence of the porous medium deformation on the flow in pores can be analyzed from two aspects: (1) the variation of pore volume leads to the quantity change of the flow filled in pores; and (2) the pore deformation induces the change of the force against the flow. The macro-manifestation of this influence is that the permeability and the corresponding coefficient are variable, which can be mathematically expressed as [13-18]:

K_ij=f(σ_ij) or K_ij=f(ε_ij)                           (1)

In this work, based on the one-dimensional seepage-stress coupling test and by the regression analysis, the relationship between the coefficient of permeability and axial strain is proposed as

K(ε₁, k₀)=aexp(-ε₁/b)+k₀                       (2)

where  ε₁ is the axial strain, k₀ is the initial coefficient of permeability, a and b are the test parameters that can be obtained by using the test results.

3.2 Expression of seepage-stress coupling model and its experimental verification

By incorporating the strain-dependent permeability to the stress—strain equation, the expression to directly describe the seepage-stress coupling in the constitutive model can be obtained.

3.2.1 Assumption

To establish the seepage-stress coupling constitutive model in accordance with the mechanical and hydraulic characteristics of the saturated geomaterial, some assumptions on the seepage-stress coupling model are given as follows.

(1) For the saturated geomaterial, the pore air pressure can be negligible.

(2) Solid material is homogeneous, continuous, and incompressible.

(3) Pore water is steady, non-viscous, and incompressible flow, which obeys the Darcy’s law.

(4) Physical quantities of representative elementary volume (REV) is representative.

(5) There are no sources and sinks in seepage field.

(6) There exists certain relationship among strain, stress and the coefficient of permeability.

3.2.2 Mathematical expression

Based on the above assumption, the seepage-stress coupling relationship can be expressed as

σ=f(ε, K(ε))                                  (3)

Considering the nonlinear behaviour of the uniaxial sample under the compression of stress that increases uniformly, in this work, the general Hooke’s law was employed to establish the nonlinear seepage-stress coupling constitutive equation of the soft rock. Such a model can be written as

σ=G(ε)K(ε, k₀)                                (4)

where  ε is the strain of the solid material, K(ε, k₀) is the function of the coefficient of strain-dependent permeability, and G(ε) is a function to express the relationship between the stress and strain-dependent permeability.

For the one-dimensional seepage-stress coupling experiment, function G(ε) can be obtained as

G(ε₁)=m(ε₁+ε₀)                               (5)

where  ε₀ is the initial strain of the sample, and m is the test parameter.

The initial strain of the rock sample should be considered due to the given axial and radial pressure to seal the test sample in the transverse plane and radial direction. On the other hand, the fracture growth is some what resisted due to the given axial and radial pressure, which can induce some errors on the test results.

By substituting Eqns.(2) and (5) into Eqn.(4), one-dimensional seepage-stress coupling constitutive model can be derived:

σ_d=(σ₁-σ₃)=m(ε₁+ε₀)[aexp(-ε₁/b)+k₀]             (6)

By differentiating Eqn.(6), the incremental formulation of the seepage-stress coupling constitutive model, that is, the relationship between the stress increment and the strain increment can be obtained as

dσ₁=dε₁                                  (7)

where   is the tangent modulus and can be expressed as

         (8)

In initial state ε₁=0, the initial value of  will be

                        (9)

Eqn.(9) is the initial tangent modulus.

3.2.3 Experimental verification

There are five parameters in the proposed model, in which ε₀ and k₀ are respectively the initial strain and the initial coefficient of permeability of the sample, and a, b and m are the tested parameters, which all can be determined by using the test results. In this work, for argillaceous stone and brown mudstone, the corresponding parameters are shown in Table 2.

Table 2 Parameters in stress—strain coupling constitutive model

It can be seen that in different directions of the soft rock, the values of parameters are different, but the orders of magnitude of which are the same. While for different soft rocks, the values of parameters are obviously different, especially for a, m, and k₀, the orders of magnitude of which for two kinds of soft rock are different.

Fig.3 shows that the fitting curves of the stress—strain relationship by using the seepage-stress coupling model (Eqn.(7)) and the test results for argillaceous siltstone. It can be seen that the fitting curves using the proposed model well simulate the experimental results. Fig.4 shows the fitting curves of the stress—strain relationship by using the seepage-stress coupling model (Eqn.(7)) and the test results for brown mudstone, indicating that the fitting curves and the test results are in good agreement. It can be concluded that the proposed model can well describe the seepage-stress coupling in constitutive model directly.

Fig.3 Comparison of fitting curves and test results for argillaceous siltstone: (a) Axis of sample is parallel to bedding plane; (b) Axis of sample is perpendicular to bedding plane

Fig.4 Comparison of fitting curves and test results for brown mudstone: (a) Axis of sample is parallel to bedding plane; (b) Axis of sample is perpendicular to bedding plane

4 Conclusions

(1) A series of one-dimensional seepage-stress coupling tests on two kinds of soft rock are performed and the stress—strain curves according to the seepage variation are obtained, from which the formulation of the coefficient of strain-dependent permeability is presented.

(2) One simple method to directly describe the seepage-stress coupling relationship is proposed, and the corresponding mathematical expression for coupling constitutive model is established, in which there are five parameters that can be determined by the test results. For different soft rocks, the values of parameters are obviously different. The corresponding incremental formulation for coupling model is developed.

(3) All test results are simulated by using the proposed model, and the results show that the proposed coupling model can describe the stress—strain relation- ship of the nonlinear elastic stage of these two kinds of soft rock.

(4) The stress—strain relationship in the plastic stage should be described by using the piecewise defined functions. On the other hand, the model is based on the tests of two kinds of soft rock, while not for all geomaterials.

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Foundation item: Projects(50378069, 50639090) supported by the National Natural Science Foundation of China; Project(50639090) supported by the Joint Fund of Yalong River Hydropower Development, China

Received date: 2008-05-09; Accepted date: 2008-06-30

Corresponding author: ZHANG Xiang-xia, PhD; Tel: +86-80-30105408; E-mail: zhangxiangxia@126.com

(Edited by CHEN Wei-ping)