Numerical Simulation of Hysteretic Flow in Porous Media
Authors: Evrenos Attila I. , Comer A.G. 
Published: December 2007

This paper was prepared for the 44th Annual Fall Meeting of the Society of Petroleum Engineers of AIME, to be held in Denver, Colo., Sept. 28-Oct. 1, 1969. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in the JOURNAL paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines.

Abstract
This paper is concerned with the simulation of multi-cycle drainage-imbibition processes in hydrophilic rock. The theory processes in hydrophilic rock. The theory of two-phase immiscible displacement is extended to cover the phenomena of dynamic hysteresis in residual phase saturations, relative permeability, and capillarity. Semi-empirical equations are given for the loci of the dynamic hysteresis envelopes and scanning loops spanning the domains of relative permeability and capillarity. An algorithm is presented to track the saturation history and to select the appropriate subset from the set of semi-empirical equations describing hysteretic relative permeability and capillarity. The application of the computational techniques is demonstrated by simulation of a semi-hypothetical reservoir.

Introduction
The purpose of this paper is to report a numerical simulator for the multi-dimensional, hysteretic flow of two compressible, immiscible fluids in hydrocarbon reservoirs. The simulator was developed as a performance evaluation-prediction tool for performance evaluation-prediction tool for cyclic gas movement in "aquifer storage" and, in particular, estimation of residual gas remaining in an aquifer-reservoir complex at abandonment.
The movement of natural gas in contact with underground water is a special case of multi-phase immiscible displacement in porous media. Hence, the central problem of porous media. Hence, the central problem of this paper is the extension of the theory of immiscible displacement to include multi-cycle drainage-imbibition processes involving two compressible fluids, and to develop a mathematical simulator of the flow system. Such a simulator is also applicable to oil-water systems in the vicinity and above the bubble-point, and to gas-oil systems subject to sufficiently small pressure gradients.

STATEMENT OF PROBLEM
The popular theory of immiscible displacement in porous media is based on the fundamental concepts of Darcy's Law, capillarity, relative permeability, equation of state, and material balance. Multi-cycle drainage-imbibition processes differ from the non-cyclic displacement in that dynamic hysteresis exhibited by phase relative permeability and capillarity must be considered. permeability and capillarity must be considered.