Upscaling of fractured oil reservoirs using homogenization including non-equilibrium capillary pressure and relative permeability

Recovery from incompletely water-wet fractured reservoirs can be extremely low. A reason for the low recovery is related to wetting issues, whereas the reason for slow recovery can be the non-equilibrium behavior of capillary pressure. One of the non-equilibrium theories is developed by Barenblatt et al. and it modifies both capillary pressure and relative permeabilities. The other theory is developed by Hassanizadeh et al. and it only deals with non-equilibrium effects for capillary pressure. To incorporate non-equilibrium in larger-scale problems, we apply homogenization to derive an upscaled model for fractured reservoirs in which the non-equilibrium effects are included. We formulate a fully implicit three-dimensional upscaled numerical model. Furthermore, we develop a computationally efficient numerical approach to solve the upscaled model. We use simulations to determine the range of delay times and capillary-damping coefficients for which discernable effects occur in terms of oil recovery. It is shown that at low Peclet numbers, i.e., when the residence time of the fluids in the fracture is long with respect to the imbibition time, incorporation of delay times of the order of few months have no significant effect on the oil recovery. However, when the Peclet number is large, the delay times reduce the rate of oil recovery. We discuss for which values of the delay time (Barenblatt) and capillary-damping coefficient (Hassanizadeh), significant delays in oil production occur.