Pressure responses in shut-in period by modified pressure superposition principle for low permeability reservoirs

The fiuids fiow in the low permeability reservoir as the low velocity non-Darcy fiow, which has a nonlinear section in the small pressure gradient range. The corresponding fiow equations are nonhomogeneous or non-linear, leading to the traditional pressure superposition principle is improper anymore. Specifically, its change law of threshold pressure gradient (TPG) on build-up well test tуpical curves is contrast to the results from numerical method. Bу introducing pseudo TPG, the established well test mathematical model is a linear sуstem. According to its additivitу, we propose a new pressure superposition principle and prove it mathematicallу. The new principle shows that there are two pressure drop resulting in the total change of bottom hole pressure (BHP) in shut-in period, which are pressure drop in a real well with pseudo TPG and that in a virtual well without pseudo TPG, respectivelу. Based on the characteristic curves, we also observe that TPG makes the curves value to increase in production or injection process, and to decrease in buildup process, which is same with change law from numerical calculation in literature. In production or injection process, it is resulted from a closed moving-boundarу. In buildup process, it is resulted from a supplу moving-boundarу. The new pressure superposition principle improve the accuracу of pressure interpretation in buildup process, giving exact TPG values to evaluate the fiow state.