The analysis of flow-induced fracture (also known as hydraulic fracture) propagation is important in various applications, such as petroleum recovery [M. P. Cleary, SPE Distinguished Author Series--Journal of Petroleum Technology 13-21 (January 1988)], nuclear waste disposal, and any process that involves the separation of solid surfaces by the motion of fluids between them. The present work addresses a particular problem in hydraulic fracture, i.e. characterization of gravity-driven motion of multiple stages of immiscible fluids within a narrow fracture cavity. Among capabilities developed to study the present problem [M. P. Cleary and Amaury F. Junior, SPE Journal 24825 (October, 1992)], a suite of numerical algorithms, named PARFES (acronym for PARallel Finite Element Solvers)--designed specially to take advantage of highly parallel computer environments--is introduced here. PARFES is composed of three modules:
PARFESl--tracks the interface motion and mesh nodal distribution of a given fluid stage; PARFESAX--models the axisymmetrical multiple stage flow problem; PARFES2--a nonlinear Newton-Raphson algorithm to determine nonsymmetrical motions. Current designs for (expensive) commercial oil and gas fracture treatments tend to greatly overestimate the effective area filled by proppant particles (carried by fluid stages) within the cavity, consequently obtaining less than desirable post-fracture-treatment oil/gas production rates. The PARFES algorithms help to better track the movement of proppant-laden fluid stages within the fracture cavity [Amaury F. Junior, Ph.D. Thesis, MIT (1992)], as well as fluid losses to the adjacent strata.