The Pressure Derivative Revisited—Improved Formulations and Applications

Abstract The proposed work provides a new definition of the pressure-derivative function [i.e., the β-derivative function, Δpβd(t)], which is defined as: Δpβd(t)=dln(Δp)dln(t)=1ΔptdΔpdt=Δpd(t)Δp (Δpd(t) is the "Bourdet" well testing derivative) This formulation is based on the "power-law" concept (i.e., the derivative of the logarithm of pressure drop with respect to the logarithm of time) — this is not a trivial definition, but rather a definition that provides a unique characterization of "power-law" flow regimes. The "power-law" flow regimes uniquely defined by the Δpβd(t) function are: [i.e., a constant Δpβd(t) behavior] CaseΔpβd(t)• Wellbore storage domination:1• Reservoir boundaries:— Closed reservoir (circle, rectangle, etc.).1— 2-Parallel faults (large time).1/2— 3-Perpendicular faults (large time).1/2• Fractured wells:— Infinite conductivity vertical fracture.1/2— Finite conductivity vertical fracture.1/4• Horizontal wells:— Formation linear flow.1/2 In addition, the Δpβd(t) function provides unique characteristic responses for cases of dual porosity (naturally-fractured) reservoirs. The Δpβd(t) function represents a new application of the traditional pressure derivative function, the "power-law" differentia-tion method (i.e., computing the d ln(Δp)/d ln(t) derivative) pro-vides an accurate and consistent mechanism for computing the primary pressure derivative (i.e., the Cartesian derivative, dΔp/dt) as well as the "Bourdet" well testing derivative [i.e., the "semilog" derivative, Δpd(t)=dΔp/d ln(t)]. The Cartesian and semilog derivatives can be extracted directly from the power-law derivative (and vice-versa) using the definition given above. Objectives The following objectives are proposed for this work: