The stress intensity factor at a fracture tip in a porous medium subjected to a fluid injection is studied. This factor evolves during the transient flow phase and tends to a limit value for the steady state. For simple fracture geometries, the finite element simulations show that for constant injection pressures this factor reaches its maximum value in the steady state regime. This result allows simplifying significantly the study and modeling of hydraulic fracture propagation because the determination of the steady flow solution is much easier and faster than the transient flow. In addition, some couplings between hydraulic and mechanical problems disappear under steady state flow and make it possible to establish some closed-form approximate expressions. These can be useful especially in the context of CO2 sequestration projects where the fluid injection is pressure-controlled.
Hydraulic fracture propagation attracts a lot of attention for its applications to important problems including oil production, shale gas production, geothermal energy and CO2 sequestration. Predictive modeling of this process meets many difficulties in experimental as well as numerical approaches and even on some theoretical issues. This article focuses on numerical aspects. Indeed, even when the reservoir rock has been well characterized for its permeability and mechanical parameters, numerical modeling of fracture propagation presents real difficulties, mainly because numerous phenomena are implied in this process: fluid diffusion in the fracture and in the matrix, fracture-matrix mass exchanges, plasticity and damage in the fracture tip because of high stress concentrations in this zone and finally the fracture propagation itself. This latter phenomenon changes the continuous material to discontinuous and is not easy to handle in numerical codes. For this reason, hydraulic fracture models have always included necessarily different simplification assumptions. A basic simplification model for the fracture-matrix mass exchange was proposed by Carter (1957) considering it as a one-dimensional, pressure-independent diffusion. On the basis of this model, and within the framework of linear elastic fracture mechanics and lubrication theory for fluid diffusion in the fracture Adachi & Detournay (2008) analyzed the general solution of a plane strain hydraulic fracture in terms of dimensionless parameters for hydraulic or mechanical, conservative or dissipative phenomena. These authors determined different regimes in which one or several of these phenomena could be neglected and so the whole model could be simplified.