Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 3012794, 13 pages

https://doi.org/10.1155/2017/3012794

## Back Analysis of Geomechanical Parameters of Rock Masses Based on Seepage-Stress Coupled Analysis

School of Civil and Architecture Engineering, Xi’an Technological University, Xi’an 710032, China

Correspondence should be addressed to Dongyang Yuan

Received 8 October 2016; Revised 13 December 2016; Accepted 23 January 2017; Published 19 February 2017

Academic Editor: Roberto Fedele

Copyright © 2017 Xianghui Deng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Given that rock masses are complex, the geomechanical parameters of rock masses are hard to determine in underground engineering. In this paper, the inverse model and method are established to identify the parameters based on the coupled stress and fluid flow model combined with the finite element method and adaptive genetic algorithm. Moreover, the model and method are applied in the Lianghekou highway tunnel, and the initial permeability coefficients of the stratum and the lateral pressure coefficients of the initial ground stress are identified by the back analysis with relative errors of the measured and fitted values at measuring points below 5%. Results show that the inverse model and method are effective and sound.

#### 1. Introduction

Rock masses exist in certain geological environments. Groundwater and ground stress are the most important factors that influence the geological environment. Water flow in rock mass changes the initial ground stress status of rock masses, and the change of their stress status influences the characteristics of the fluid flow in the rock masses. The fluid flow and stress affect each other and cause coupling, regardless of whether the fluid flow changes first or vice versa [1].

The phenomenon of coupled stress and fluid flow has been widely studied in academic and engineering circles, and scholars have conducted extensive research on the seepage-stress coupled model. In 1969, Snow first proposed the mathematical equations that show the effect of normal stress on the permeability coefficient by conducting a single-fracture permeability test [2]. In 1974, after conducting a pumping test and theory analyses, Louis proposed that the seepage discharge of fractured rock masses decreases with the increase of the normal stress, and he proposed a corresponding empirical formula [3]. In 1975, Nelson presented the empirical formula of permeability coefficients based on the Navajo sandstone sample seepage-stress test [4]. Nelson’s empirical function first displayed the influence of effective stress on the rock mass permeability. Through laboratory tests, Y. Z. Zhang and J. C. Zhang [5] concluded that if the initial fracture width was small, then the seepage discharge and the stress of the fractured rock mass would not have a negative exponential function relationship with each other but a biquadrate relationship. They also suggested that the seepage discharge of the fractured rock mass declined with the increase of the compressive stress and that the seepage discharge increased with the unidirectional compressive stress parallel to the fracture surface. By the investigation and theoretical analysis, Su and coauthors proposed a negative correlation between fracture permeability and fracture normal effective stress [6]. Through the single-fracture 3D stress test, Chang et al. discovered that the fracture fluid flow was influenced by the normal stress and was strongly affected by the fracture lateral deformation caused by the fracture lateral stress [7]. Furthermore, the negative exponential formula was established between the fluid flow and the normal stress. Many researchers have also realized similar relationships in subsequent research [8, 9]. In general, seepage-stress coupling is reflected in the following aspects: on the one hand, effective stress (normal and lateral) controls the fracture width and other geometric shapes of rock masses or the porosity of porous mediums. The effective stress determines the characteristics of fluid flow, indicating the effects of the stress field on the seepage field [10, 11]. On the other hand, the groundwater influences or changes the rock mass structure by imposing hydrostatic and hydrodynamic pressure, thereby changing the stress status of the rock mass that shows the effect of seepage field on the stress field. The fluid flow and stress interaction enable the rock mass to maintain a dynamic balance [12].

The analysis of the hydromechanical behavior of rock masses remains an important topic in rock mechanics. It is a critical phenomenon in ongoing challenging issues such as tunneling under high groundwater pressures and the extraction of hydrocarbons from deep and pressurized petroleum reservoirs. Despite continuing and extensive efforts, such analysis continues to be difficult. First, establishing a reasonable hydromechanical constitutive model is difficult. The major difficulty in modeling the fluid flow in fractured rocks involves handling the solid–fluid interaction. The equivalent continuum approach and the discrete fracture network approach have been developed based on the mechanical and hydraulic natures of the rock mass for such modeling [13, 14]. The continuum and discrete approaches have been combined to propose the dual permeability model [15]. In this model, flow in natural pores and cracks are governed by different equations, which may or may not be coupled [16]. Many hydromechanical models have been proposed to overcome these problems. For example, Shao et al. established a coupled constitutive model for anisotropic damage and permeability variation in brittle rocks under deviatoric compressive stresses. The formulation of the model is based on experimental evidences, and the main physical mechanisms involved in the scale of microcracks are considered [17]. From a phenomenological point of view, on the one hand, a microscopic approach is often used to analyze the permeability evolution by the fluid flow through cracks [18]. On the other hand, a macroscopic approach is appropriate for studying the mechanical characteristics of materials, such as the stress-strain relationship after damage. Therefore, Pereira and Arson established the double-porosity model based on a relationship between the microscopic and macroscopic damage tensors, which can simulate the flow through the network of cracks/porosity and evaluate the equivalent permeability [19]. Moreover, Pereira and Arson believed that the pore size distribution (PSD) of the material is coupled to the mechanical behavior of the rock and models the influence of deformation and damage on the permeability and retention properties of cracked porous media [20, 21]. De Bellis et al. referred to the above research to simplify the damaging porous material model through consistent linearization [22]. Their model consists of nested families of equispaced frictional and cohesive faults in an otherwise elastic matrix material. The linear kinematic model preserves the main microstructural features of the finite kinematic one but offers superior computational performance. Meanwhile, determining the parameters of the hydromechanical model is very difficult for actual engineering projects. Three methods are commonly used, that is, theoretical analysis, field measurement, and the back analysis method [23, 24]. First, assumptions are generally made for the derivation, which vary greatly from the actual condition, causing difficulties in applying the derived formula. Test methods consist of laboratory and in situ tests. Laboratory tests usually have an obvious “size effect,” and the accuracy of parameters cannot be guaranteed. Meanwhile, in situ tests have a limited measurement range, and the measurement result only indicates the characteristics of rock masses near the sampling point. Several uncertainties occur in identifying the parameters caused by the disturbance of sampling. In addition, in situ tests have disadvantages such as data divergence, less representation, and high costs. Comparatively, the back analysis method is based on the measured physical information (the displacement, strain, water level, and other factors) which reflects the systematic mechanical behaviors. The inversion model can be used to obtain initial parameters of the surrounding rock, and even the inverse model is analyzed occasionally. Now, the back analysis method is widely used because it is a relatively easy and cost-effective technique.

According to the mechanical behavior, the seepage-stress coupled model can reflect the real stress and fluid flow characteristics of rock masses. At present, the back analysis mostly concentrates on the inversion of the model parameters in the single field (the single field is the uncoupled problem, such as the seepage field, stress field, and temperature field), considering that the coupled interaction is obviously different from that of the single field. If the result of a single field back analysis is directly applied to the coupled stress and fluid flow model, then large errors may be generated. Therefore, identifying the parameters with the coupled model is necessary.

Until now, identifying parameters has been the problem when the seepage-stress coupling is considered. Three research ideas have been proposed, one of which is the parameter inverse of the equivalent continuous model for coupled stress and fluid flow analysis, that is, conducting a back analysis on the parameters of the model based on various types of monitoring data and coupling the forward analysis method in a continuous medium or a region considered as a continuous medium [25]. The credibility of the inversion result can be improved significantly compared with that in a single field. The second idea involves considering the rock mass as a discrete fracture medium when the back analysis is conducted. The fractured rock mass or a certain region is considered as a discrete medium, and the discrete fracture model is established in the fluid flow analysis. Meanwhile, the field monitoring data are employed for fitting, and the minimum function is taken as the objective function to identify the parameters of the coupled model [26]. Given the distribution complexity and randomness of the fractures in the rock mass, conducting numerical simulations for the discrete fracture distribution in actual engineering is difficult [27, 28]. Therefore, this idea is rarely applied in actual engineering projects. The third idea is the dual permeability coupled model [29, 30]. The actual monitoring information is applied to identify parameters, which is similar to the process of the two aforementioned ideas. According to this idea, the continuous medium model is adopted when the region has less fracture and poor permeability. The discrete fracture model is adopted when the region has a large fracture or good permeability, which largely depends on the fractures. The third idea is theoretically reasonable. However, the numerical model is complex, and numerous factors have to be considered. Therefore, applying it in practical engineering is complicated.

In this paper, the inverse model is established based on the equivalent continuum coupled stress and fluid flow model and is combined with the finite element method (FEM) and adaptive genetic algorithm (AGA). Meanwhile, the optimal parameter combination of the coupled model is identified.

#### 2. Equivalent Continuum Model for Coupled Stress and Fluid Flow

##### 2.1. Equivalent Continuum Mathematical Model for Coupled Stress and Fluid Flow

Generally, the water quantity is constant when water flows in the rock mass. According to the water balance principle, the mathematical equation of water balance can be established as follows:where , , and are the conductivity velocity of rock mass in , , and directions; is the water density; is the compression coefficient of rock mass; is the compression coefficient of water; is the porosity ratio of rock mass; is the hydraulic head; is time. The formula of Darcy’s law is shown as follows:

Combining (1) with (2), (3) is obtained as follows:where is the unit storage volume; , , and are the hydraulic conductivity of rock mass in , , and directions; is the source sink term; is the hydraulic head boundary condition; and is the discharge boundary condition.

According to the properties and seepage characteristic of the fractured rock mass, the seepage-stress coupled model can be classified into the equivalent continuous coupled model, the discrete fracture coupled model, and the dual permeability coupled model [31]. The aforementioned analysis shows that the discrete fracture coupled model and dual permeability coupled model are difficult to apply in the actual engineering. The representative element volume (REV) is relatively small for some rock masses. Therefore, the equivalent continuous coupled model is used for the seepage analysis.

The seepage control equation can be deduced from formula (1). The seepage control in Equation [32] is as follows:where is the seepage water pressure, including hydrostatic pressure and hydrodynamic pressure; is the total seepage matrix; is the source sink term; is the unsteady seepage storage matrix; and is the array that shows the change rate of the hydraulic head with time.

In an equivalent continuous medium, the FEM discretized process of the stress field is derived in [32]. The stress control equation is shown as follows:where is the total stress array; is the known load array; is the displacement array; is the stiffness matrix; is the elastic matrix; and is the geometric matrix.

The linear interpolation functions are used when the FEM method is adapted to solve coupled problem. By combining the seepage equations of equivalent continuous medium, stress field equations, and seepage-stress coupling formula, the mathematical model for the coupled stress and fluid flow analysis [33] is as follows:where is the empirical formula of the coupled stress and fluid flow, is the effective normal stress, and is the permeability coefficient tensor. The permeability coefficient tensor of the equivalent continuum coupled model was listed in [34]. Details are as follows:where is the component of the permeability coefficient tensor; , , and are the coefficient tensors of the , , and directions. The permeability coefficients , , and are used in seepage analysis of the equivalent continuous model. The permeability coefficient is closely related to the effective normal stress and the seepage pressure according to the current research. In fact, some scholars think the permeability coefficient is positively correlated with the water pressure but negatively related to the stress [30, 34]. The initial permeability coefficients of the , , and directions (, , and ) will be identified in this paper.

##### 2.2. Numerical Method of Equivalent Continuum Coupled Model

In the equivalent continuum coupled model, the iteration method is used for the analysis. Before the analysis process, the numerical analysis programs of the seepage field and stress field are set up, respectively. The simulation result of the single field is employed as the boundary conditions of each other and the external load. When the calculation precision meets the convergence condition, the simulation result is obtained using an iterative calculation. The basic steps are as follows:(1)The initial stress field of rock masses is calculated.(2)The initial permeability tensor of different strata in the calculation area is computed according to the calculated initial stress field .(3)The seepage field is estimated according to the boundary conditions and the initial permeability tensor.(4)The hydrodynamic pressure and hydrostatic pressure are determined according to the seepage field calculation results. Then other load increments are considered, the stress increment is analyzed, and the stress field of this moment is obtained.(5)The seepage tensor is assessed in the new stress field according to the stress field and the empirical relationship of the two fields.(6)Steps to are repeated until the seepage field and stress field calculation errors in the adjacent time satisfy the convergence precision.

#### 3. Parameter Optimization of the Inversion Method

##### 3.1. Main Parameters to Be Inversed

Many parameters are required when conducting the coupled analysis, which include the parameters of the seepage field, the parameters of the stress field, and the coupled coefficients of the two fields. In this process, parameters that need to be determined are classified into four types according to their properties: physical and mechanical parameters of the stress field, namely, the rock mass gravity , elasticity modulus , Poisson ratio , cohesion , and internal friction angle ; parameters of the seepage field, including the permeability coefficient of different strata or the facture aperture, and the normal stiffness of the facture rock masses; parameters related to the in situ stress field, for example, the lateral pressure coefficient of initial ground stress; and the coefficients of the coupled empirical relationship. The details are shown in Table 1.