Effect of low velocity non-Darcy flow on pressure response in shale and tight oil reservoirs

Low velocity non-Darcy flow in shale and tight oil reservoirs is described by nonlinear or nonhomogeneous models. These models, especially for well shut-in period, are usually solved by numerical method, since the traditional pressure superposition principle is no longer applicable. The current paper presents a modified pressure superposition principle, accounting for the pseudo Threshold Pressure Gradient (TPG), and its mathematical proof. The proposed principle indicates that the total change of bottom hole pressure (BHP) in shut-in period is equal to the superposition of BHP change in a real well with pseudo TPG and that in a virtual well without pseudo TPG. The new principle is applied to the derivation of an analytical solution to the nonhomogeneous problem during the well shut-in period. Type curves calculated from the analytical solution show that the pseudo TPG leads to curve up-warping in switch-on period but down-warping in shut-in period, which agree with previous numerical results, and can be explained by the moving-boundary theory. Throughout the switch-on period, a closed moving-boundary is generated when the pressure gradient is less than the pseudo TPG. The boundary is closer to the well with higher pseudo TPG. However, during the shut-in period, a supply moving-boundary, which was generated during previous production or injection period, is earlier to be reached for virtual well with higher pseudo TPG. The flow is steady state afterwards. Matching of field data by the analytical solution results in the pseudo TPG in the investigation zone. The interpretation of the field case shows that pseudo TPG equals 0.104 MPa/m, generating a pressure drop as high as 6.35 MPa across the investigation zone during the well testing period.