Fracture-matrix interface area contacted by injected fluid as a function of average saturation, mechanical aperture and counter-current imbibition

Capillary fluid transfer between fractures and water-wet siliclastic rock is a vital contributor to achieving significant recovery from waterfloods in naturally fractured reservoirs where the rock matrix contains most of the hydrocarbon volume, but the fractures provide for the fluid pathways. This transfer, however, is restricted to the subset of fractures that is actually reached by the injected water. Using discrete fracture network models and control volume finite element simulation, we find that the fracture surface area contacted by the injected water, Af,sw, is a function not only of the average saturation but strongly depends on rock properties and flow velocity. Conventional dualcontinua approaches fail to account for this non-linear behavior, thus introducing an error to the quantification of fluid exchange. Introduction Conventional thinking on numerical simulation of naturally fractured reservoirs (NFR) is subdued by multi-continua approaches. Fractures and matrix are treated as independent entities with their interaction being captured by so called transfer functions. Flow due to viscous pressure gradients may either occur in both, matrix and fractures (dual permeability), or only within the fractured domain (dual porosity). Based on a statistical network description, usually restricted to density and permeability of fractures, an effective permeability is averaged across grid blocks and a storage capacity is assigned (Warren and Root, 1963; Kazemi et al., 1976). This averaging of permeability over cells with an extent of up to a couple of hundred meters in diameter implicitly assumes the applicability of a representative elementary volume (REV). Recent literature, however, strongly indicates that such a REV with respect to flow behavior does not exist (Bonnet et al., 2001; Paluszny et al., 2007; Matthäi et al, 2009). The non-uniform but much rather fractal nature of fractures rock matrix with respect to conductivity, geometry/orientation and density along with the tendency to occur in sets demands for different means of accommodation. In Nelson Type II fractured reservoirs, the porous rock matrix acts as hydrocarbon storage whereas the fracture network provides for phase transport due to a viscous gradient (Nelson, 1982). Recovery in such a reservoir is controlled by the ability to drain oil to the fracture flow system from its adjacent matrix by means of capillary forces (fracture-matrix transfer, FMT). This exchange of phases is caused by the preference of the matrix to the water phase as opposed to the oil phase, being initially saturated with the latter. In case of water wet rock, the injected wetting fluid will imbibe the matrix through the fracture surface area. This exchange between phases is conventionally accommodated for by means of the above-mentioned Philipp LANG, Stephan MATTHÄI 2 transfer function. Naturally, this transfer only occurs at that fraction of the surface which actually is in contact with the injected fluid. This contacted area, however, is assumed to be constant by the dual-porosity approach. We developed a CSMP++ based numerical simulator using a control volume finite element scheme (Geiger et al., 2003; Paluszny et al., 2007; Matthäi et al., 2009) to take a closer look at the nature of the portion of total fracture area that is contacted by injected fluid, Af,sw. It is shown that it strongly depends on matrix properties and fracture flow velocity, obeying a non-linear behavior. This dependence reveals a significant simplification in dual continua models, where the average grid block saturation is considered to be the lone factor of how much contact area allows for phase transfer. As a result, the total volume exchanged between fracture and rock domain is liekly to be underestimated. The numerical reservoir simulator is presented which allows us to run discrete fracture network (DFN) models by using an exponential transfer function (Unsal et al., 2009). Fracture traits are computed accounting for geometry and subsurface stress regime. Simulation setups reflect quarters of a five-spot well pattern and fractured reservoir geometries of meso-scale size (i.e. up to 300m in diameter). Well pairs with constant rate and pressure condition, for the injector and producer respectively, cover a range of flow velocities. The effects of fracture-matrix transfer on flow within single fractures and the ensemble fracture networks are illustrated. Methodology Using the CSMP++ library, a control volume finite element (CVFE, Geiger et al., 2003) based simulator within the C++ programming language was developed. Flow phenomena accounted for in this simulator include viscous displacement, capillary spreading within the fractures and transfer of phases between fracture and adjacent matrix. Multiphase flow is restricted to a wateroil system as found in undersaturated reservoirs. The numerical solution technique of choice was IMPIMS, referring to the sequential implicit solution of both pressure and saturation. Simulation models base on DFN geometries and a finite element discretization. Each fracture’s permeability is computed accounting for internal fluid pressure and relative stress state. While flow through the network is simulated using the above approach, an exchange of phases between the fractures and what we call the ‘virtual matrix’ is imposed by means of a non-linear transfer function that acts as source/sink in the transport equation (Unsal et al., 2009). This is done in an effort to account for counter-current imbibition during e.g. waterflooding of an oil saturated water-wet fractured reservoir. The area of fracture elements in contact with the displacing phase is calculated and output versus the amount of injected volume. Parameter Unit Value Discretization a (fracture aperture) m 5.0 x 10 to 1.0 x 10 Piecewise constant k (fracture permeability) m 1.0 x 10 to 1.0 x 10 Piecewise constant km (matrix permeability) m 2 1.0 x 10 to 1.0 x 10 Piecewise constant kr,w (fracture relative permeability water) 0.0 to 1.0 x 10 10 Piecewise constant kr,nw (fracture relative permeability oil) 0.0 to 1.0 x 10 10 Piecewise constant Sw (water saturation) 0.0 to 1.0 Nodal property μo (oil density) Kg m -3 800.0 Nodal property μw (water density) Kg m -3 1000.0 Nodal property Fracture-Matrix Interface Area Contacted by Injected Fluid 3 ρo (oil viscosity) Pa sec 0.005 Nodal property ρw (water viscosity) Pa sec 0.001 Nodal property So,r (oil residual saturation) 0.0 Piecewise constant Sw,r (water residual saturation) 0.0 Piecewise constant So.i (oil initial saturation) 1.0 Piecewise constant pf (fracture fluid pressure) Pa 0.0 to 1.0 x 10 8 Nodal property σn (fracture element normal stress) N m -2 tbd Piecewise constant τ (fracture element shear stress) N m tbd Piecewise constant qfmt (fracture matrix transfer source) m 3 m sec tbd Piecewise constant fw (fractional flow water) 0.0 to 1.0 x 10 10 NA λo (oil phase mobility) tbd Piecewise constant λw (water phase mobility) tbd Piecewise constant λt (total mobility) tbd Piecewise constant pc m (matrix capillary pressure) Pa tbd Piecewise constant pc f (fracture capillary pressure) Pa tbd Piecewise constant Af (fracture surface area) m 2 tbd NA Af,sw (Af contacted by injected fluid) % tbd NA Table 1: Simulation variables as used in the CSMP++ CVFE DFN simulator for a two phase system (wetting phase water, non-wetting phase oil)