A Three-dimensional Poroelastic Model for Water Injection into a Geothermal Reservoir

A three-dimensional poroelastic model is developed to investigate the poroelastic effect of fluid injection into a geothermal reservoir. In the model, the fluid flow in the fracture is assumed to be lubrication flow and modeled by the finite element method. The threedimensional pore fluid diffusion in the reservoir and the induced stresses are modeled by the boundary integral equation method. The numerical results have been verified through comparing to the analytical solutions given in Nygren and Ghassemi (2006) for an infinite radial fracture problem. Thereafter, the numerical model is used to study the pore pressure and stress fields in the rock matrix resulting from injection/extraction in a circular fracture. INTRODUCTION The production of geothermal energy from low permeability reservoirs is achieved by water circulation in natural and/or man-made fractures, and is often referred to as enhanced or engineered geothermal systems (EGS). Cold water injection perturbs the in-situ stress state within the reservoir leading to fracture initiation and/or activation of discontinuities such as faults and joints which is often manifested as multiple microseismic events. Detection and interpretation of microseismic events using downhole receiver arrays (e.g., Brady et al., 1994; Warpinski et al., 1996) can be monitored and analyzed to provide useful information on the stimulated zone, fracture growth, and geometry of the geological structures and the in-situ stress state (Warpinski et al., 2001; Guiterez, 2003; Pine, 1984). Micro-seismic events are believed to be associated with rock failure in shear, and shear slip on new or pre-existing fracture planes (Peasron, 1981). Effective interpretation of micro-seismicity can benefit from the knowledge of the hydro-thermomechanical mechanisms associated with injection in the reservoir, and the resulting stress variations that play a key role in rock failure around the main hydraulic fracture. These include the stresses due to the opening of a hydraulic fracture, and thermoelastic and poroelastic stresses due to rock cooling and fluid leak-off into the rock mass. In general, an injection-induced fracture problem consists of (1) fluid flow and heat transport within the fracture, (2) fluid flow in the matrix, (3) conductive and advetcive heat transport in the matrix, and (4) fracture propagation. Some solutions for problems involving the first three parts have been presented (Ghassemi and Zhang, 2006; Nygren & Ghassemi, 2006; Nygren at al., 2005). Most of these as well as other studies of the subject were based on some simplifications such as uniform pressure in line fractures, special reservoir geometry and onedimensional fluid diffusivity in the reservoir, etc. Other 3D models use the finite element method (e.g., Kohl et al. 1995). A three-dimensional boundary element model has been developed by Ghassemi et al. (2007) to study the impact of thermal stresses on the reservoir matrix and the fracture without considering poroelastic effects. In this paper, a three-dimensional poroelastic model is developed to study the poroelastic response of the reservoir to fluid injection into an irregularly-shaped fracture or a fractured zone (see Fig. 1). The fluid flow in the fracture is assumed to be lubrication flow and is modeled using the finite element method. The three-dimensional pore fluid diffusion in the reservoir and the induced stresses is treated by the boundary integral equation method; and the displacement discontinuity boundary element method is used to model the fracture itself. The adoption of the boundary element method eliminates the need for discretizing the reservoir to account for the threedimensional effect of fluid diffusivity in the reservoir. Both the finite element method and the boundary element methods use the same mesh, and they are coupled. GOVERNING EQUATIONS The governing equations of the problem are briefly described below. Fluid flow in fracture The fracture is assumed to be flat (or with moderate curvature), of finite size but can be irregular in shape. The reservoir is assumed to be poroelastic and of infinite extent. In this paper, we assume that the flow in the fracture is laminar and governed by the lubrication flow equation: ( ) ( ) ( ) 2 3 12 , ,0, , , , p x y t x y t w x y μ ∇ = − q (1) where 2 ∇ is the gradient operator in the fracture plane, ( ) , , 0, p x y t the fluid pressure in the fracture, μ the fluid viscosity, ( ) , w x y the fracture width, ( ) , , x y t q the fluid discharge, and A is the fracture plane. Assuming the fluid to be incompressible and the fracture aperture does not vary with time, the fluid continuity equation could be written as: ( ) ( ) ( ) ( ) ( ) ( ) 2 , , 2 , , , , l i i i e e e x y t v x y t Q t x x y y Q t x x y y δ δ ∇ ⋅ = − + − − − − − q (2) where 2 ∇ ⋅ is the two-dimensional divergence operator, l v the leak-off fluid from one side of the fracture wall into reservoir, ( ) i Q t and ( ) e Q t respectively the fluid injection and extraction rates, ( ) , i i x y and ( ) , e e x y injection and extraction well locations, and δ the Dirac delta function. Using Eqs. (1) and (2) to eliminate q , yields ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 2 , , ,0, 24 , , , 12 , l