Reservoir performance during waterflooding is important to reservoir engineers. Analytical and semi-analytical flow models with different assumptions have been presented and used widely to describe the flow dynamics of such a process. Most assume that one of the terms, typically capillary forces, can be neglected or considered constant diffusion coefficients, so the simplified diffusive-convective type of flow equations can be solved analytically or by numerical methods. Obtaining analytical or semi-analytical solutions to non-linear diffusive-convective flow equations, including capillary, gravity and viscous forces simultaneously, has been a challenge.
This paper presents a theoretical study of the effects that controlling flow parameters have on saturation profiles and breakthrough time during oil recovery by waterflooding. A mathematical non-linear diffusive-convective type model for immiscible oil-water displacement in one-dimensional vertical homogeneous porous media considering the three chief forces (capillary, gravity and viscous) is derived and solved numerically by using a finite-difference formulation with fully implicit scheme in time and central differences in space.
Dimensionless equations are written so that any of the three forces can be investigated independently; capillary and gravity forces can be "turned on or off." The effects of varying fluid viscosity, injection flow rate, system length or wettability, for both displacing and displaced fluids, can be understood thoroughly. The flow model is versatile enough that it allows for variations of the shape of the relative permeability and capillary pressure functions. The impact of these functions in the driving forces and on oil recovery is analyzed.
The contribution of each of the forces to the dimensionless water velocity and its impact on oil recovery was studied in four flow cases: viscous, viscous-gravity, viscous-capillary and viscous-gravity-capillary; all possible flow cases in water injection problems were considered. Graphical results are discussed.