A Model for Spontaneous Imbibition as a Mechanism for Oil Recovery in Fractured Reservoirs

The flow of oil and water in naturally fractured reservoirs (NFR) can be highly complex and a simplified model is presented to illustrate some main features of this flow system. NFRs typically consist of low-permeable matrix rock containing a high-permeable fracture network. The effect of this network is that the advective flow bypasses the main portions of the reservoir where the oil is contained. Instead capillary forces and gravity forces are important for recovering the oil from these sections. We consider a linear fracture which is symmetrically surrounded by porous matrix. Advective flow occurs only along the fracture, while capillary driven flow occurs only along the axis of the matrix normal to the fracture. For a given set of relative permeability and capillary pressure curves, the behavior of the system is completely determined by the choice of two dimensionless parameters: (i) the ratio of time scales for advective flow in fracture to capillary flow in matrix ; (ii) the ratio of pore volumes in matrix and fracture . A characteristic property of the flow in the coupled fracture-matrix medium is the linear recovery curve (before water breakthrough) which has been referred to as the "filling fracture" regime Rangel-German and Kovscek (J Pet Sci Eng 36:45-60, 2002), followed by a nonlinear period, referred to as the "instantly filled" regime, where the rate is approximately linear with the square root of time. We derive an analytical solution for the limiting case where the time scale of the matrix imbibition becomes small relative to the time scale of the fracture flow (i.e., ), and verify by numerical experiments that the model will converge to this limit as becomes large. The model provides insight into the role played by parameters like saturation functions, injection rate, volume of fractures versus volume of matrix, different viscosity relations, and strength of capillary forces versus injection rate. Especially, a scaling number is suggested that seems to incorporate variations in these parameters. An interesting observation is that at there is little to gain in efficiency by reducing the injection rate. The model can be used as a tool for interpretation of laboratory experiments involving fracture-matrix flow as well as a tool for testing different transfer functions that have been suggested to use in reservoir simulators.