Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 8523213, 15 pages

https://doi.org/10.1155/2017/8523213

## Modeling of Hydraulic Fracture of Concrete Gravity Dams by Stress-Seepage-Damage Coupling Model

^{1}Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China^{2}State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100038, China

Correspondence should be addressed to Guoxin Zhang

Received 28 November 2016; Revised 14 February 2017; Accepted 22 February 2017; Published 23 April 2017

Academic Editor: Giovanni Garcea

Copyright © 2017 Sha Sha and Guoxin Zhang. 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

High-pressure hydraulic fracture (HF) is an important part of the safety assessment of high concrete dams. A stress-seepage-damage coupling model based on the finite element method is presented and first applied in HF in concrete dams. The coupling model has the following characteristics: (1) the strain softening behavior of fracture process zone in concrete is considered; (2) the mesh-dependent hardening technique is adopted so that the fracture energy dissipation is not affected by the finite element mesh size; (3) four coupling processes during hydraulic fracture are considered. By the damage model, the crack propagation processes of a 1 : 40 scaled model dam and Koyna dam are simulated. The results are in agreement with experimental and other numerical results, indicating that the damage model can effectively predict the carrying capacity and the crack trajectory of concrete gravity dams. Subsequently, the crack propagation processes of Koyna dam using three notches of different initial lengths are simulated by the damage model and the coupling model. And the influence of HF on the crack propagation path and carrying capacity is studied. The results reveal that HF has a significant influence on the global response of the dam.

#### 1. Introduction

Hydraulic fracture is a phenomenon of crack propagation after high-pressure water or other kinds of fluid entering into an existing crack. Hydraulic fracture is an important issue in the hydraulic engineering, the petroleum engineering, mining, and geotechnical industries. In the hydraulic engineering, mass concrete dams are likely to experience cracking on their upstream, downstream, and base surfaces due to the low tensile strength of concrete and the action of internal and external temperature changes, shrinkage of the concrete, differential settlement of the foundation, and other factors. Concrete gravity dam is a type of concrete structure that interacts with high-pressure water. With time, cracks are filled with water and penetrate deep into the dam under the action of high water pressure, resulting in reduced carrying capacity and safety of the dam. Therefore, for the safety of high or ultrahigh concrete dams, it is necessary to consider the influence of hydraulic fracturing [1].

In hydraulic fracture of concrete and concrete gravity dams, there are some studies on it. Brühwiler and Saouma [2, 3] conducted hydraulic fracturing tests on concrete specimens of different gradations and examined the water pressure distribution in their cracks. They found that the hydrostatic pressure in the cracks was a function of the crack opening displacement, decreasing from the maximum pressure to zero along the fracture process zone. Slowik and Saouma [4] investigated the pressure distribution inside a crack with respect to time and the crack opening rate. The effect of sudden crack closure on the pressure distribution was also investigated. Using the fluid mass and momentum conservation theories, Li et al. [5] established a differential equation of the water pressure distribution in rock and concrete fractures caused by hydraulic fracturing and derived a formula for calculating the pressure at an arbitrary time during the crack propagation process. Bary et al. [6] presented an approach using mechanics of saturated porous media to model strongly coupled hydromechanical effects in concrete. Fang and Jin [7] studied the fracture process of concrete under the action of water pressure in fissure based on the extended finite element method (XFEM). Barpi and Valente [8] simulated hydraulic fracturing of the dam-foundation joint and analyzed the effect of fracture process zone on the path of crack formation. Dong and Ren [9] simulated the propagation of the crack at the gravity dam heel under uniform pressure by the XFEM. Wang et al. [10] studied hydraulic fracturing in concrete gravity dam considering fluid-structure interaction by the XFEM and finite volume method.

Hydraulic fracture is a complex problem, involving four coupled processes: () the deformation of the surrounding medium induced by the water pressure on the fracture surface; () fluid flow within the fracture; () propagation of the fracture; () the leak-off of the fracturing fluid from the fracture into the surrounding medium [11]. However, among numerical studies of hydraulic fracturing in concrete gravity dams mentioned above, the fourth coupling process is not taken into account. Further, concrete is a quasibrittle material. Fracture in concrete is characterized by the existence of a nonlinear fracture process zone at the front of the real crack tip. Depending on how the fracture process zone of the concrete is assessed, crack analysis can be conducted using linear elastic fracture mechanics (LEFM) or nonlinear fracture mechanics (NLFM). LEFM may produce inaccurate results because of the neglect of the fracture process zone in the concrete. Theoretically, NLFM is more reasonable, being based on the fictitious crack model with the application of the strain softening law of the fracture process zone [12]. However, among the numerical studies of hydraulic fracturing mentioned above, LEFM is used mostly.

There are many methods for the simulation of hydraulic fracture, such as phase-field method [13], peridynamics [14], cellular automata method [15], discrete element method (DEM) [16], numerical manifold method (NMM) [17], element free method [18], finite element method (FEM) [19], and boundary element method (BEM) [20]. For the excellent abilities in LEFM, the boundary element method and especially the displacement discontinuity method (DDM) have also been extensively applied to hydraulic fracture modeling. Higher order elements to DDM are introduced to improve the precision due to singularity variations near the crack tip [21]. In the DEM framework, cracks propagate along prescribed element boundaries when the stress intensity factor meets the criteria, and the crack opening is estimated by a Coulomb friction model [22]. FEM is the most mature and the most widely used. FEM is a method based on the continuum mechanics essentially. It must be improved to simulate the discontinuity. The improving methods are divided into two categories: the variable mesh method and the fixed mesh method. In the variable mesh method, the mesh needs updating as the crack tip advances. The cracked surface must be consistent with the edge of the element and a fine mesh or singular element has to be adopted at the crack tip, which could be computationally expensive. In the fixed mesh method, the mesh keeps invariant and the crack is simulated by modifying the interpolation or constitutive relation of the cracked element, such as the XFEM, the smeared crack model, and the continuum damage model. The use of the XFEM for hydraulic fracture problem can avoid remeshing. However, the XFEM also needs objective crack propagation criteria, which are usually based on quantities such as crack-tip stresses and stress intensity factors. So fine crack-tip meshes are necessary for accurate calculation of these quantities. This means that fine meshes are needed if cracks are unknown a priori, leading to high computational cost. The smeared crack model describes a cracked solid by an equivalent anisotropic continuum with degraded material properties in the direction normal to the crack orientation and no remeshing is needed [23]. In contrast, the fixed mesh method is more convenient.

Among the numerical studies of hydraulic fracture in concrete gravity dams mentioned above, the XFEM is used mostly while the continuum damage model is less used. In this study, by regarding concrete as a saturated porous medium and employing the effective stress principle of porous media, a stress-seepage-damage coupling model based on FEM is developed and first applied in hydraulic fracture in concrete gravity dams. The coupling model has the following characteristics: () the constitutive law considers the strain softening characteristic of fracture process zone, damage-dependent pore-pressure-influence coefficient, stress-dependent permeability for the prepeak stage, and deformation-dependent permeability for the postpeak stage; () based on the principle of conservation of fracture energy, the damage model is combined with fracture mechanics to prevent the fracture energy dissipation from being affected by the finite element mesh size; () four coupling processes during hydraulic fracture are considered. By this coupling model, hydraulic fracture of Koyna dam is simulated. And the influences of hydraulic fracture on the crack trajectory and the dam bearing capacity are discussed.

#### 2. Stress-Seepage-Damage Coupling Model

In this study, the dam concrete is assumed to be a saturated porous medium. In practice, dam concrete can hardly attain a saturated state owing to the small size of the pores and the consequent very low permeability in the intact condition. Except near cracks, the pore pressure of most of the zones of a concrete dam do not vary [24]. The consideration of the dam concrete as a saturated porous medium therefore constitutes a significant simplification. However, the purpose of this study is to investigate the impact of hydraulic fracture on the dam. Isotropic damage models are not appropriate for concrete because the crack trajectory follows principal compressive stresses which are perpendicular to principal tensile stresses promoting crack initiation and propagation. However, if anisotropic damage models are adopted, both theoretical derivation and numerical calculation are difficult. In contrast, isotropic damage models are easier to implement. When considering the effects of hydraulic fracturing, water flows along the crack, permeability coefficients in the directions parallel to the crack plane are different from that perpendicular to the crack plane. And the pore-pressure-influence coefficient in the direction perpendicular to the crack plane is also different from those parallel to the crack plane. Thus, in this study, an isotropic damage model and an anisotropic seepage model are used. The following nonlinear behaviors are defined in this study: () the stress-strain relationship of the damaged element, () the variation of the pore-pressure-influence coefficient caused by the damage, and () the variation of the permeability coefficients due to stress and the damage. In the calculations, it is assumed that the compressive stress-strain relationship is linear elastic because compressive stresses higher than the compressive strength generally do not occur in a gravity dam. The characteristics of the coupling model are discussed below.

##### 2.1. Damage Model

At the front of a real concrete crack tip, there is a nonlinear fracture process zone where cohesive stresses can be transferred between the crack interfaces through aggregate interlock and interface friction. The existence of the fracture process zone causes the concrete to exhibit strain softening. The mechanical properties of the fracture process zone can be well simulated using the cohesive crack model proposed by Hillerborg et al. [12]. The majority of researchers have suggested the adoption of a linear stress-strain relationship for the ascending segment under uniaxial tension. However, different modes may occur in the descending segment, including single-line descending, segmented-line descending, and curved descending [25]. Irrespective of the adopted mode, the fracture energy of the stress-strain curve should always remain the same. In this study, the negative exponential equation developed by Jiang et al. [25] is adopted for the descending segment. The stress-strain relationship under uniaxial tension can therefore be expressed aswhere is Young’s modulus of the intact material; is the tensile strength; is the uniaxial tensile strain; is the strain at cracking (); and is the softening coefficient for controlling the descending segment.

In Figure 1, the fracture energy is given by the area enclosed by the stress-crack width curve and the coordinate axes, as expressed by (2). As shown in Figure 2, the fracture energy per unit crack width is given by the area enclosed by the descending segment (represented by the line ) and the horizontal axis *ε*, as expressed by (3).where is the sum of the opening displacements of the microcracks in the fracture process zone and is the width of the distribution area of the microcracks.