Boundary dominated flow in low permeability reservoir with threshold pressure gradient

Due to the extremely complicated pore structures and strong fluid-rock interaction, fluid flow in low permeability reservoir does not obey Darcy's law. It is non-Darcy flow associated with threshold pressure gradient. Threshold pressure gradient (TPG) is the level of pressure gradient that has to be attained to enable fluid to overcome the viscous forces and start flowing. So, applying traditional well-testing theory in low permeability reservoir will lead to incorrect understanding of reservoir behavior; then, a new mathematical model for describing fluid flow in low permeability reservoir should be established. In non-Darcy flow in low permeability reservoirs, the fluid flow boundary is controlled by threshold pressure gradient and extended outward continuously as production goes on, while reservoir outside this boundary remains to original conditions. Once the moving boundary reaches physical reservoir boundary, it is called boundary dominated flow. This paper presents new mathematical models for boundary dominated flow under two different conditions: constant pressure boundary and closed boundary. Analytical solutions are obtained by using Greens' function with a numerical approximation. It is concluded that, during transient flow, the pressure derivative is not a horizontal line but a concave curve which goes upwards. The bigger threshold pressure gradient brings bigger flow resistance, so the slower pressure wave propagates, and the later boundary dominated flow starts. During boundary dominated flow, the pressure derivative is independent of threshold pressure gradient. A numerical simulation is carried out to validate the analytical solution and approves the validity of the analytical solution. The solution proposed in this paper provides a suggestive tool for welltesting in low permeability reservoir with threshold pressure gradient.