Numerical modeling of hydraulic fracturing in a poroelastic medium using cohesive elements

From oil and gas engineering point of view, one of the challenges in low permeable or damaged wells is improving the productivity. There are different methods to increase the productivity of low permeable wells and one of the most efficient one is hydraulic fracturing. In this study, two-dimensional modeling of hydraulic fracturing using finite element method and cohesive element approach through traction-separation law has been performed. This approach avoids the singularity in the crack tip and the cohesive zone fits naturally into the conventional finite element method. Hydraulic fracture is assumed to propagate in a poroelastic and permeable medium with a constant injection rate and under quasi-static conditions and the criterion for fracture initiation is quadratic nominal stress criterion. Also as a propagation criterion, Benzeggagh Kenane (BK) approach has been considered. Two types of elements have been implemented in the model which are 4-node bilinear displacement and pore pressure reduced integration and 6-node displacement and pore pressure two- dimensional cohesive element. Cohesive elements have three degrees of freedom that two of them are in X and Y directions and one of them is pore pressure. Mesh size in the near fracture region is small enough to consider the stress and pressure distribution efficiently and avoid any problem in convergence. Meantime, to decrease the computation cost the mesh size gradually increases from fracture area to the boundaries. Also, to increase the accuracy of the model, the time steps for fracture propagation is 0.01 second. In addition, the effect of fracturing fluid has been directly included in the model which means that the fluid pressure would be applied along the fracture without any simplifying assumption. To validate the model, the results have been compared with KGD approach. The results indicate that in the initial steps the pressure at the wellbore wall is high which decreases with time significantly and eventually it gets a steady and uniform trend. In other words, in the initial steps, the fluid pressure should be high enough to overcome the hoop stress around the wellbore and after some injection periods, the fracturing fluid pressure would reach the breakdown pressure and the fracture starts to initiate and propagate. It is clearly observed that increasing the injection rate would lead to faster propagation of hydraulic fracture and in the models with higher injection rate the fracture tends to grow in the propagation direction. This indirectly means that increasing the injection rate would affect both opening and length of the hydraulic fracture which can result in increasing the productivity. The results reveal that the peak of the normal effective stress profiles corresponds to the fracture tip position, where the fracture opening is zero,and the peak value equals the cohesive strength of the material,as expected.Moreover,with increasing thedistance from the fracture tip,the stress decreases rapidly and approaches the initial stress value. The way that Young’s modulus affects the overall characteristics of hydraulic fracture implies that higher Young’s modulus would lead to longer fractures. In other words, formations with higher Young’s modulus can be fractured easily but the opening of the hydraulic fracture would reduce at the same time. This also indirectly means that Young’s modulus would play an important role in the productivity.