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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.

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 [

In hydraulic fracture of concrete and concrete gravity dams, there are some studies on it. Brühwiler and Saouma [

Hydraulic fracture is a complex problem, involving four coupled processes: (

There are many methods for the simulation of hydraulic fracture, such as phase-field method [

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: (

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 [

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. [

In Figure

Stress versus crack width curve and fracture energy

Fracture energy per unit crack width

From (

The relationship between the fracture energy

It can be determined from (

According to the principle of equivalent strain proposed by Lemaitre [

The following is thus obtained:

From the perspective of damage mechanics, the nonlinearity of the stress-strain relationships of rock and concrete is due to the formation and propagation of microcracks in the materials through load-induced continuous damage. The consequent brittleness is more obvious under tension. It is thus appropriate to use an elastic damage constitutive model to describe the mechanical properties. It has been confirmed that the results of an elastoplastic damage model do not significantly differ from those of an elastic damage model [

Based on (

The concept of the effective stress as applied to rock and concrete saturated by a single-phase fluid can be regarded as the development of Terzaghi’s effective stress principle for soil. Based on the effective stress principle, Biot first introduced a scalar parameter (known as Biot coefficient) to reflect the effect of the pore pressure on the effective stress [

Tensile stresses are assigned a positive sign. As shown in Figure

Decomposition of stresses in saturated porous media.

For the undamaged element,

Effects of cracking on pore-pressure-influence coefficients and permeability.

Concrete is composed of solid skeleton and microscopic pores. This structural characteristic may cause changes in the microscopic geometry and pore structure when the material is loaded or disturbed, resulting in variation of the porosity and permeability [

When the element is in the elastic state (i.e.,

When the first principal stress reaches the tensile strength, the element is damaged (i.e.,

The permeability matrix

Assuming that the water is incompressible, according to Darcy’s law, the calculation of the steady seepage field with free surface (without internal sources) can be reduced to solving the quasiharmonic equations that satisfy the boundary conditions

In this study, the cracking problem is formulated in effective stresses, namely, total stresses minus the pore pressures. And the effective stresses are used to perform the computation. In the finite element analysis, the equilibrium between the external load and the internal load is as follows:

The seepage, stress, and damage fields mutually affect and are coupled to each other, and solving the problem using the fully coupled equations would require a significant amount of calculation. The weak coupled method is used to solve each equation independently. In the solution of the seepage field of the

Flowchart of coupling program.

During the calculation, the loads are applied by the incremental method and the load increment is set to be quite small. The total quantity method is used for the iteration in each load step. The damage degree could increase or decrease during an iteration process but could not reduce below that corresponding to the last load step. It is therefore necessary to preserve the damage degree of the last load step. The load of a current iteration step is the sum of the load increment of the current load step and the equivalent load increment of the unbalanced stress caused by the damage in the last iteration step. Further, the equivalent load increment of the unbalanced stress is the load increment due to the difference between the computational stress and the bearing stress. It should be noted that the load of the first iteration step is the sum of the external load increment of the current load step and the residual unbalanced force of the last load step. The equivalent load increment of the unbalanced stress and the residual unbalanced force can be calculated using (

The permeability stiffness matrix

Let us assume that I is the seepage module, II is the damage iteration module, and III is the coupling module. In this study, the computations are performed in a weakly coupled form, in a sequential manner by applying I-II-III without iteration. However, there are iterations in the damage iteration module. To guarantee the convergence of the results, a small load step should be applied. The smaller the load step is, the more accurate the results are. When the load step is small enough, the results of the weakly coupled algorithm can be regarded as those of the strongly coupled algorithm (applying I-II-III with iteration).

The coupling program can also be used to investigate the crack propagation without consideration of the stress-seepage-damage coupling effect. By inputting the control parameters, the seepage and coupling modules can be skipped, and the damage iteration module directly calculated.

Carpinteri et al. [

Material parameters of gravity dam models.

Young’s modulus, |
Poisson’s ratio, |
Density, ^{3}) |
Tensile strength, |
Fracture energy, |
---|---|---|---|---|

35.7 | 0.1 | 2400 | 3.6 | 184 |

Dimensions and load distribution of gravity dam models (unit: mm).

Various researchers have performed numerical simulations of the above tests using different models and approaches such as a combination of FEM and the cohesive crack model of Barpi and Valente [

Figure

Load versus CMOD.

Figure

Damage distribution.

Comparison of crack propagation paths.

In this section, the crack propagation processes of Koyna dam under overflow are simulated by the damage model and the coupling model, respectively. Koyna gravity dam was severely damaged near the dam neck by a strong earthquake [

Mechanical properties of Koyna dam concrete.

Young’s modulus, |
Poisson’s ratio, |
Density, ^{3}) |
Tensile strength, |
Fracture energy, |
---|---|---|---|---|

25 | 0.2 | 2450 | 1.0 | 100 |

Geometry and finite element mesh of Koyna dam (unit: mm).

Overflow versus horizontal displacement of dam crest.

The presently determined damage distribution for an overflow of 10.2 m is shown in Figure

Damage distribution (overflow = 10.2 m).

Comparison of crack trajectories obtained by different methods.

Overflow versus horizontal displacement of dam crest.

Damage distributions for overflow of 10.2 m. (a)

It can be seen from Figure

To investigate the impact of the coupling effect during crack growth on the crack propagation path and carrying capacity of the dam, nonlinear stress-seepage-damage coupling analyses are conducted on the dam using the three initial notch lengths in Section ^{−9} m/s and the coupling coefficient_{0} is set as 0.01. It is really true that the dam which is 100 m high is constructed in reality with drains. Due to lack of information about the drains of Koyna dam, the drains are not taken into account in this study.

In this section, in order to select an appropriate load step, five different load steps are chosen, 1.0 m, 0.5 m, 0.1 m, 0.05 m, and 0.01 m, respectively. Taking Koyna dam with an initial notch

Overflow versus horizontal displacement of dam crest.

Damage distribution for overflow of 7.0 m.

Comparison of dam carrying capacities with and without consideration of coupling effect.

Initial crack length | Without consideration of coupling effect | With consideration of coupling effect | Percentage change | |||
---|---|---|---|---|---|---|

Critical load (m) | Carrying capacity (m) | Critical load (m) | Carrying capacity (m) | Critical load | Carrying capacity | |

0.1 |
6.0 | 10.2 | 4.1 | 7.0 | −31.7% | −31.4% |

02 |
5.0 | 10.2 | 2.0 | 5.3 | −60.0% | −48.0% |

0.3 |
3.0 | 10.2 | 1.0 | 4.2 | −66.7% | −58.8% |

Damage distributions for overflow of 4.2 m. (a)

Overflow versus horizontal displacement of dam crest.

It can be seen from Figure

In this study, a stress-seepage-damage coupling model based on FEM is developed and first applied in hydraulic fracture of concrete gravity dams. The coupling model has the following characteristics: (

The damage model is first used to simulate crack propagation of a 1 : 40 scaled model dam. The crack trajectory and structural response agreed well with the experimental results. Then, the damage model is used to simulate crack propagation of Koyna dam with an initial notch on the upstream face under overflow. The numerically determined crack trajectory and structural response of Koyna dam agree well with the previously reported numerical results. These results validate the present damage model for predicting the carrying capacity and crack trajectory in 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. By comparing the results obtained by the two models, it is found that hydraulic fracture produces the following influences: (

Hydraulic fracture is a complex problem and the effects of damage on pore-pressure-influence coefficients and permeability coefficients need to be further studied.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The research is supported by the National Key Research and Development Project of China (Grant no. 2016YFB0201000), the National Key Basic Research Program of China (Grant nos. 2013CB036406, 2013CB035904), the National Natural Science Foundation of China (Grant nos. 51579252, 51439005), the Special Scientific Research Project of the State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, and the Special Scientific Research Project of the China Institute of Water Resources and Hydropower Research.