Pressure Buildup in a Well Produced at Constant Pressure

Abstract

Summary Dimensionless analytic expressions of the well-pressure buildup resulting from shutting in a well after producing it at constant pressure were obtained, in both Laplace space and time, using different mathematical approaches to solve the pertinent differential equations, including the effects of both wellbore storage and skin damage. Such a pressure response is said to have been obtained at constant-pressure-buildup (CPBU) conditions. The well, represented by a cylindrical source of nonzero radius, produces a single compressible fluid from an infinite and homogeneous reservoir. For a vertical well, the reservoir flow is always radial and horizontal. For the first approach in Laplace space, the pressure distribution in the reservoir at the shut-in time was used as the initial condition to obtain the transient pressure response upon shutting in the well. In the second approach, the solution in Laplace space was inverted analytically, yielding an expression containing infinite integrals. Duhamel's principle was used in the third approach, yielding the same integrals as the second approach. Subject to the inherent numerical errors in the integration and inversion algorithms, all solutions gave the same numerical results over a wide range of the production period, the elapsed time since the well was shut in, the storage coefficients, and the skin damage. As required by material balance, all pressures are restored to the value of the initial reservoir pressure as the infinite reservoir eventually completely recharges the amount of fluid produced before the well being shut in. The pressure response divided by the production rate at the time the well is shut in is denoted as the specific pressure response and has the same units as the pressure response under constant rate buildup (CRBU) conditions. It is thus not surprising that the general trends of the specific pressure response and the specific pressure increment are somewhat similar for both producing conditions, and that a number of interpretation procedures developed for CRBU conditions using the exponential integrals are applicable under CPBU conditions, including use of Horner plots. The Laplace-space procedure used to obtain numerically inverted values of the pressure response under CPBU conditions can be formally extended to provide expressions for the shut-in pressure response for any rate or pressure history during the production period. Although the emphasis is on the description of the well pressure and the specific-well-pressure buildup after shut-in, and of their time derivatives, the paper also presents analytic Laplace-space expressions for the pressure and radial flow within the reservoir after the well is shut in.