## Abstract

Three-dimensional model of fracture propagation is proposed. The model simultaneously accounts rock deformation in the vicinity of a fracture and a cavity, fluid flow inside the fracture and its propagation in the direction that is selected by a growth criterion. The results of the sensitivity analysis of model solution to the variation of model parameters are presented.

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## Acknowledgments

Authors gratefully acknowledge the financial support of this research by the Russian Scientific Fund under grant number 14-11-00234.

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## Appendices

### Appendix

### Crack propagation algorithm

### 1.1 Quasi-static crack growth

Let us consider the initial fracture with the front defined by vertices \({{\mathrm{\mathbf x}}}_i^0, i=1,\ldots ,N_{fr}\). The case of an unloaded fracture in a stretched media and the case of a loaded fracture in a compressed media are combined in the algorithm by a generalized loading pressure *p*. A step-by-step fracture propagation is indicated with the superscript *n*. The general scheme of fracture propagation algorithm is displayed in Fig. 43. The iterative process

is introduced to achieve the fulfillment of the condition

The iterations

provide the fulfillment of conditions

at each vertex of the crack front at propagation step \(n+1\). The iterative schemes (73) and (75) are based on the methods of solving equations (74) and (76) respectively. The criteria (19) and (17) are implemented iteratively with the desired accuracy, at each vertex of the crack front, at every step of propagation algorithm shown in Fig. 43. If the scaling law of crack incrimination is taken as a propagation criterion, and the direction of propagation is defined by the formula (16) itself, then the algorithm becomes essentially simpler (Fig. 44). The crack trajectories calculated using the algorithms in Fig. 43 and Fig. 44 are compared in the section 4.2.2 “Fatigue crack growth under cyclic loading – scaling law for crack front increment”.

### 1.2 Viscous fluid crack growth

Let the fracture be loaded by the pressure of viscous flow. The fluid front (labeled with its vertices \({{\mathrm{\mathbf x}}}_{f \ i}^n\)), the fracture front with vertices \({{\mathrm{\mathbf x}}}_{r \ i}^n\), and the lag between fluid and fracture front \(L_{r \ i}\) are included into the algorithm. In the algorithm there is a fluid volume \(V^n\) calculated from the fracture width

The hydrodynamics-elasticity problem in the algorithm in Fig. 45 provides the relation between the fracture width \(W^{n+1 \ s}\) and the pressure \(p^{n+1 \ s}\) which is produced by the fluid flow in the fracture in the crack front position \({{\mathrm{\mathbf x}}}_{r \ i}^{n+1 \ s}\) and the fluid front \({{\mathrm{\mathbf x}}}_{f \ i}^n\). The scheme of the solution algorithm for the hydrodynamics-elasticity problem is shown in Fig. 46. The iterative process \(\Delta t^{k+1} = \mathbb {T}(\Delta t^k)\) is implemented to fulfill the condition

which provides the equivalence of the fluid velocity and the fracture front velocity \(v_f = L_f^0 / \Delta t\), where \(\Delta t\) is calculated using the fracture volume dynamics.

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Cherny, S., Lapin, V., Esipov, D. *et al.* Simulating fully 3D non-planar evolution of hydraulic fractures.
*Int J Fract* **201**, 181–211 (2016). https://doi.org/10.1007/s10704-016-0122-x

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DOI: https://doi.org/10.1007/s10704-016-0122-x