Finite element modeling of hydraulic fracturing in 3D

Abstract

A procedure based on the finite element method is suggested for modelling of 3D hydraulic fracturing in the subsurface. The proposed formulation partitions the stress field into the initial stress state and an additional stress state caused by pressure build-up. The additional stress is obtained as a solution of the Biot-equations for coupled fluid flow and deformations in the rock. The fluid flow in the fracture is represented on a regular finite element grid by means of "fracture" porosity, which is the volume fraction of the fracture. The use of the fracture porosity allows for a uniform finite element formulation for the fracture and the rock, both with respect to fluid pressure and displacement. It is demonstrated how the fracture aperture is obtained from the displacement field. The model has a fracture criterion by means of a strain limit in each element. It is shown how this criterion scales with the element size. Fracturing becomes an intermittent process and each event is followed by a pressure drop. A procedure is suggested for the computation of the pressure drop. Two examples of hydraulic fracturing are given, when the pressure build-up is from fluid injection by a well. One case is of a homogeneous rock and the other case is an inhomogeneous rock. The fracture geometry, the well pressure, the new fracture area and the elastic energy released in each event are computed. The fracture geometry is three orthogonal fracture planes in the homogeneous case and it is a branched fracture in the inhomogeneous case.

Simple models of the hydrofracture process

“…We want to treat the mechanics of fracture in greater detail than in a percolation model such as Quinn, Turcotte, and Rundle [15]. On the other hand, given the desire to treat multiple interacting cracks over scales of hundreds of meters, we are forced to include less detail than in customary hydraulic fracture simulations [10]. The design goals for the numerical method are as follows:…”

“…This approach to the problem was pioneered by Clifton [6][7][8][9]. In this and more recent numerical work [10,11] it is customary to assume that fluid flow in the crack is laminar, although assumptions such as uniform height H and elliptical cross section are dropped.…”

“…[10]) it is customary to adopt the lubrication approximation for fluid flowing into the crack, as shown in Fig. 4.…”

“…(10). Dissipation in the fluid is typically 10 times the fracture energy of the rock, which is hundreds of Joules per square meter.…”

Universal hydrofracturing algorithm for shear-thinning fluids: Particle velocity based simulation

“…These models were used for decades in practical applications to design the hydrofracturing treatments. However, a continuous increase in both the number and size of fracking installations necessitated further development of the mathematical models describing the process, which resulted in the introduction of the so-called pseudo 3D models [45], planar 3D models [4,9,65] and recent attempts to develop fully 3D models [2,11,66,70]. A comprehensive review of the topic can be found in [2].…”