**Accurately predicting the hydraulic fracturing process remains a challenging task. The aim of this project is therefore to develop a numerical model for the simulation of the hydraulic fracturing process. With this model, we hope to better understand and optimize the fracturing process.**

We developed a two-dimensional finite element model for hydraulic fracturing. The fracture is incorporated in the finite element mesh by exploiting the partition of unity property of finite element shape functions. This is a proven technique in solid mechanics and is commonly referred to as the Extended Finite Element Method (X-FEM). The novelty of our model is that pressure in the fracture is calculated separately from the pressure in the rock formation. This ensures that all fluid flow goes exclusively in the fracture. We calculate the fluid leakage with an analytical solution. With this model we simulated fracture propagation (Fig. 1) and nucleation (Fig. 2) from a circular hole.

###### Figure 1: Fracture propagation from a circular hole

Currently we are extending our model to 3 dimensions. With such a model we can simulate more realistic fracture patterns such a fracture bending. The final goal of this project is to deliver a working finite element model to optimize the hydraulic fracturing process.

Synergy with other 2F2S project exists in delivering input parameters to our model such as stiffness and permeability values of shale rock, orientation of the natural fracture network, and the possibility to validate our model with experimental data.

###### Figure 2: Fracture nucleation from a circular hole. The situation before nucleation is shown in the left image. The nucleated fractures are shown in the right image.

**Figure 3: Fracture interaction and merging due to hydraulic fracturing.**

### Publications

**Journal publications:**

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2015). An investigation of the step-wise propagation of a mode-II fracture in a poroelastic medium. Mechanics Research Communications, in press.

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2015). The enhanced local pressure model for the accurate analysis of fluid pressure driven fracture in porous materials. Computer Methods in Applied Mechanics and Engineering, 286, 293-312

Remij, E.W., Remmers, J.J.C., Pizzocolo, F., Smeulders, D.M.J. & Huyghe, J.M.R.J. (2014). A partition of unity-based model for crack nucleation and propagation in porous media, including orthotropic materials. Transport in Porous Media, 1-18.

**Conference papers with presentation **

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2014). Numerical modelling of hydraulic fracturing. In F. Oka, A. Murakami, R. Uzuoka & S. Kimoto (Eds.), Conference Paper : Proceedings of the 14th international conference of international association for computer methods and recent advances in geomechanics, Kyoto, Japan, 22-25 September 2014, CRC Press.

Remij, E.W., Pizzocolo, F., Remmers, J.J.C., Smeulders, D.M.J. & Huyghe, J.M.R.J. (2013). Nucleation and mixed mode crack propagation in a porous material. In C. Hellmich, B. Pichler & P. Adam (Eds.), Conference Paper : Poromechanics V: Proceedings of the Fifth Biot Conference on Poromechanics, July 10-12, 2013, Vienna, Austria, (pp. 2260-2269). Reston: American Society of Civil Engineers

**Conference presentations**

Remmers, J.J.C., Remij, E.W., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2015). The Three-Dimensional Enhanced Local Pressure Model for the Simulation of Hydraulic Fracturing (2015). 9th European Solid Mechanics Congress, 6-10 July 2015, Madrid, Spain.

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2015). A 3-Dimensional Model for the Simulation of Hydraulic Fracturing. SIAM Conference on Mathematical and Computational Issues in the Geosciences, 29 June- 2 July 2015, Stanford, U.S.

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2015). An enhanced local pressure model for three-dimensional simulation of hydraulic fracturing. Engineering Mechanics Institute Conference, 16-19 June 2015, Stanford, U.S.

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2015). Numerical modeling of hydraulic fracturing under the influence of natural faults. 4th CFRAC congress on **Computational Modeling of Fracture and Failure of Materials and Structures, 3-5 June 2015, Cachan, France**: IACM / ECCOMAS.

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2014). A partition of unity based cohesive zone model for hydraulic fracturing. Conference Paper: 11th World Congress on Computational Mechanics (WCCM XI), 20-25 July 2014, Barcelona, Spain, Barcelona: IACM / ECCOMAS.

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2014). A partition of unity based model for hydraulic fracturing. Conference Paper : 76th EAGE Conference & Exhibition 2014, Amsterdam RAI, The Netherlands, 16-19 June 2014, AEGE.

**Journal publications:**

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2015). An investigation of the step-wise propagation of a mode-II fracture in a poroelastic medium. Mechanics Research Communications, in press.

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2015). The enhanced local pressure model for the accurate analysis of fluid pressure driven fracture in porous materials. Computer Methods in Applied Mechanics and Engineering, 286, 293-312

Remij, E.W., Remmers, J.J.C., Pizzocolo, F., Smeulders, D.M.J. & Huyghe, J.M.R.J. (2014). A partition of unity-based model for crack nucleation and propagation in porous media, including orthotropic materials. Transport in Porous Media, 1-18.

**Conference papers with presentation **

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2014). Numerical modelling of hydraulic fracturing. In F. Oka, A. Murakami, R. Uzuoka & S. Kimoto (Eds.), Conference Paper : Proceedings of the 14th international conference of international association for computer methods and recent advances in geomechanics, Kyoto, Japan, 22-25 September 2014, CRC Press.

Remij, E.W., Pizzocolo, F., Remmers, J.J.C., Smeulders, D.M.J. & Huyghe, J.M.R.J. (2013). Nucleation and mixed mode crack propagation in a porous material. In C. Hellmich, B. Pichler & P. Adam (Eds.), Conference Paper : Poromechanics V: Proceedings of the Fifth Biot Conference on Poromechanics, July 10-12, 2013, Vienna, Austria, (pp. 2260-2269). Reston: American Society of Civil Engineers

**Conference presentations**

Remmers, J.J.C., Remij, E.W., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2015). The Three-Dimensional Enhanced Local Pressure Model for the Simulation of Hydraulic Fracturing (2015). 9th European Solid Mechanics Congress, 6-10 July 2015, Madrid, Spain.

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2015). A 3-Dimensional Model for the Simulation of Hydraulic Fracturing. SIAM Conference on Mathematical and Computational Issues in the Geosciences, 29 June- 2 July 2015, Stanford, U.S.

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2015). An enhanced local pressure model for three-dimensional simulation of hydraulic fracturing. Engineering Mechanics Institute Conference, 16-19 June 2015, Stanford, U.S.

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2015). Numerical modeling of hydraulic fracturing under the influence of natural faults. 4th CFRAC congress on **Computational Modeling of Fracture and Failure of Materials and Structures, 3-5 June 2015, Cachan, France**: IACM / ECCOMAS.

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2014). A partition of unity based cohesive zone model for hydraulic fracturing. Conference Paper: 11th World Congress on Computational Mechanics (WCCM XI), 20-25 July 2014, Barcelona, Spain, Barcelona: IACM / ECCOMAS.

Remij, E.W., Remmers, J.J.C., Huyghe, J.M.R.J. & Smeulders, D.M.J. (2014). A partition of unity based model for hydraulic fracturing. Conference Paper : 76th EAGE Conference & Exhibition 2014, Amsterdam RAI, The Netherlands, 16-19 June 2014, AEGE.