Elliptic Functions in Modeling of Oil Recovery

The objective of this paper is to investigate the inflow performance of multiple vertical wells producing from and injecting into a closed reservoir of constant thickness under pseudosteady-state conditions. For this case we represent, like the method of imaginary sources, a closed reservoir as an element of unbounded doubly periodic array of wells and use the elliptic Weierstrass zeta- and sigma-functions to describe this inflow performance. This approach allow us: • to find the pressure distribution and the field of fluid velocities in the reservoir; • to calculate the productivity index (PI) and the Dietz’s shape factor for any shape of reservoirs, not only, like Dietz (1965), for rectangular and triangular ones; • to analyze the influence of a reservoir shape on the Dietz’s shape factor; • to establish, like Valko, Doublet and Blasingame (2000) for a rectangular reservoir, the influence matrix (IM) and the multiwell productivity matrix (MPM) for any shape of reservoirs; • to introduce the multiwell productivity index (MPI) as a norm of MPM and to find the optimal placement of producing wells in a closed reservoir, based on the maximum MPI condition; • to introduce, like Kaviani and Valko (2010) MPI-based method for a rectangular reservoir, the connectivity matrix (CM) for any shape of reservoirs and to evaluate on the base of the CM-approach the interwell connectivity of injector/producer wells in waterflooding of any shape of reservoirs.