New mathematical models for production performance of a well producing at constant bottomhole pressure

New mathematical models are developed in this paper to forecast the production performance of a well producing from the center of a circular closed-boundary reservoir at constant bottomhole pressure and the pressure buildup behavior after shut-in. Both transient flow and boundary dominated flow are studied. The models are based on single-phase fluid flow of constant compressibility, viscosity, and formation volume factor in homogeneous reservoir with uniform thickness. A fully analytical solution is obtained through combinations of Dirac delta function, Bessel functions, Laplace transform, Green's function, and inverse Laplace transform. Stehfest's method is used to convert the obtained solution from the Laplace transform domain into the real time domain. Gaussian quadrature is used to approximate the integral of a function. The complete procedure of governing equations is described in detail to allow verification. The proposed mathematical models in this paper are based on fully analytical solutions to diffusivity equations, and the solutions which are obtained by Green's function, Gaussian quadrature, and numerical inverse Laplace transform are efficient to forecast the production performance of a well producing at constant bottomhole pressure. A computer modelling group (CMG) simulation is run to verify the production decline and pressure buildup performance following constant bottomhole pressure production. The results of the simulation match well with those of the models. The proposed models in this paper are reliable and the solutions are with high order of accuracy; they are fast tools to forecast the production performance of a well producing at constant flowing bottomhole pressure.