Abstract
A mathematical model is developed and tested for the production of natural gas with water encroachment and gas entrapment. The model is built on the material and volumetric balance relations, the Schilthuis water drive model, and a gas entrapment mechanism which assumes that the rate of gas entrapment is proportional to the volumetric rate of water influx. This model represents an alternative to the large grid models because of its low computer, maintenance, and manpower costs.
Introduction
Reservoir simulation can be considered to have started in 1936 when Schilthuis first stated the material balance equation for petroleum reservoirs. This led to various attempts to simulate the reservoir for zero-dimensional through two-dimensional cases, including the use of electrolytic and electrical analogs and the use of simplified equations and simple geometries solved analytically. Numerical methods awaited the development of the digital computer; the landmark paper here, of course, was that of Bruce et al. in 1953. Much research in numerical analysis was stimulated by the attempts to solve the mathematical equations on computers efficiently and accurately. The reservoir model presented here approaches a natural gas reservoir from a different point of view. It is a zero-dimensional model which does not deal with the detailed interior (microscopic) structure of the reservoir. It assumes that the reservoir consists of an isothermal mass of gas at temperature Ti in porous rock surrounded by an infinite aquifer. As gas is withdrawn from the reservoir, the resulting pressure gradient (approximated by the pressure difference) allows water to infiltrate at a rate proportional to the pressure difference (Schilthuis model). As the water infiltrates, it traps gas at a volumetric rate proportional to the volumetric infiltration rate. Although this is a very simple reservoir model, results show that it is quite realistic. It can be useful in modeling pressure/production relationships for policy and planning purposes. Of course, it does not model the detailed internal behavior of the reservoir needed for certain engineering purposes.
Lumped Parameter Model of Natural Gas Reservoir
By a lumped parameter model we mean one in which the distributed variables, which normally vary with position, are replaced by a single, effective quantity. For example, in electronic circuit theory, it is usually sufficient to consider a point or "lumped" resistance, capacitance, and inductance in place of distributed quantities in a circuit. Likewise in the oil or gas reservoir, for many cases of interest we may replace the pressure and density with average or "lumped" quantities. This gives a zero-dimensional model in that the space dependence of these quantities need not be calculated explicitly. For example, if we consider an ideal gas with uniform pressure, we may relate the pressure p, the volume V, and the total number of moles of gas G by the relation
(1)
where R is the universal gas constant and Ti is the initial temperature of the reservoir, which is assumed to be isothermal.
A New Lumped Parameter Model for Natural Gas Pipelines in State Space