TY - JOUR

T1 - Mathematical framework of the well productivity index for fast forchheimer (non-darcy) flows in porous media

AU - Aulisa, Eugenio

AU - Ibragimov, Akif

AU - Valko, Peter

AU - Walton, Jay

N1 - Funding Information:
The research of the ¯rst author was partially supported by the NSF Grant DMS-0813825 and by the ARP Grant 0212-44-C399. The research of the fourth author was supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).

PY - 2009/8

Y1 - 2009/8

N2 - Motivated by the reservoir engineering concept of the well Productivity Index, we introduced and analyzed a functional, denoted as "diffusive capacity", for the solution of the initial-boundary value problem (IBVP) for a linear parabolic equation.21 This IBVP described laminar (linear) Darcy flow in porous media; the considered boundary conditions corresponded to different regimes of the well production. The diffusive capacities were then computed as steady state invariants of the solutions to the corresponding time-dependent boundary value problem. Here similar features for fast or turbulent nonlinear flows subjected to the Forchheimer equations are analyzed. It is shown that under some hydrodynamic and thermodynamic constraints, there exists a so-called pseudo steady state regime for the Forchheimer flows in porous media. In other words, under some assumptions there exists a steady state invariant over a certain class of solutions to the transient IBVP modeling the Forchheimer flow for slightly compressible fluid. This invariant is the diffusive capacity, which serves as the mathematical representation of the so-called well Productivity Index. The obtained results enable computation of the well Productivity Index by resolving a single steady state boundary value problem for a second-order quasilinear elliptic equation. Analytical and numerical studies highlight some new relations for the well Productivity Index in linear and nonlinear cases. The obtained analytical formulas can be potentially used for the numerical well block model as an analog of Piecemann.

AB - Motivated by the reservoir engineering concept of the well Productivity Index, we introduced and analyzed a functional, denoted as "diffusive capacity", for the solution of the initial-boundary value problem (IBVP) for a linear parabolic equation.21 This IBVP described laminar (linear) Darcy flow in porous media; the considered boundary conditions corresponded to different regimes of the well production. The diffusive capacities were then computed as steady state invariants of the solutions to the corresponding time-dependent boundary value problem. Here similar features for fast or turbulent nonlinear flows subjected to the Forchheimer equations are analyzed. It is shown that under some hydrodynamic and thermodynamic constraints, there exists a so-called pseudo steady state regime for the Forchheimer flows in porous media. In other words, under some assumptions there exists a steady state invariant over a certain class of solutions to the transient IBVP modeling the Forchheimer flow for slightly compressible fluid. This invariant is the diffusive capacity, which serves as the mathematical representation of the so-called well Productivity Index. The obtained results enable computation of the well Productivity Index by resolving a single steady state boundary value problem for a second-order quasilinear elliptic equation. Analytical and numerical studies highlight some new relations for the well Productivity Index in linear and nonlinear cases. The obtained analytical formulas can be potentially used for the numerical well block model as an analog of Piecemann.

KW - Diffusive capacity

KW - Nonlinear Darcy-Forchheimer flow

KW - Porous media

UR - http://www.scopus.com/inward/record.url?scp=70349404426&partnerID=8YFLogxK

U2 - 10.1142/S0218202509003772

DO - 10.1142/S0218202509003772

M3 - Article

AN - SCOPUS:70349404426

VL - 19

SP - 1241

EP - 1275

JO - Mathematical Models and Methods in Applied Sciences

JF - Mathematical Models and Methods in Applied Sciences

SN - 0218-2025

IS - 8

ER -