Description Of The Three-Dimensional Two-Phase Simulator Shaft78 For Use In Geothermal Reservoir Studies
Authors: Pruess Karsten , Schroeder Ron C. , Witherspoon Paul A. 
Published: December 2007

The algorithm of SHAFT78 is based on mass and energy-balance equations for two-phase flow in a porous medium. We convert the basic differential porous medium. We convert the basic differential equations into integrated finite-difference form. This allows regular and irregular reservoir shapes to be handled with the same ease. The equations are solved semi-implicitly for discrete time steps. The present version of SHAFT78 treats the coupling between present version of SHAFT78 treats the coupling between mass and energy flow in a noniterative way. Special provisions are made to compute phase transitions with provisions are made to compute phase transitions with high accuracy.

We have verified the program by computing a number of sample problems that previously had been investigated by other authors. Our model studies reveal that two-phase geothermal reservoirs are capable of a great variety of pressure responses upon production. We show that the pressure responses upon production. We show that the standard technique of estimating reserves by extrapolating a plot of p/Z vs cumulative production is not applicable to two-phase geothermal reservoirs. We develop a bulk model (Lumped parameters) for a two-phase reservoir that admits an analytical solution for pressure decline upon production. From this we conclude that in many cases production. From this we conclude that in many cases pressure will be a linear function of cumulative production, pressure will be a linear function of cumulative production, with the slope allowing an estimate of total reservoir volume. Reserve assessment requires knowledge of average porosity and vapor saturation, which cannot be obtained porosity and vapor saturation, which cannot be obtained from pressure decline curves.

Introduction

Geothermal reservoirs are distinguished by the fact that, in contrast to oil and gas reservoirs, fluid flow, in general, is not isothermal. The processes of water boiling and steam condensation involve exchange of large quantities of heat between the fluid and the rock matrix. The flow of steam and water alters both the distribution of mass and energy in the reservoir. A theoretical description of two-phase geothermal reservoirs, then, must consider both mass and energy transport.

In recent years, a number of numerical simulators for geothermal reservoirs have been developed. The general purpose of these is to aid reservoir engineers in (1) purpose of these is to aid reservoir engineers in (1) determining characteristic parameters of reservoirs (most important among those being the reserves of fluid and heat) and (2) simulating the performance of reservoirs upon production and injection. production and injection. The various simulators differ in the approximations made in the underlying physical model (e.g., dependence of rock and fluid properties on thermo-dynamic variables), in the geometrical definition of the reservoir (one-, two-, or three-dimensional, regular or irregular shape), in the choice of thermodynamic variables, and in the mathematical techniques used for solving the coupled mass and energy transport equations.

Criteria for desirable performance of numerical simulators depend in part on the particular problems to be investigated. Different problems often will differ in the required level of detail to be resolved, and in the optimum balance of speed and accuracy of computation. Much can be learned about two-phase flow in porous media from model studies for idealized systems. Such studies can be performed with less-than-three-dimensional models and algorithms that are based on regular grid spacings. For modeling natural geothermal reservoirs, on the other hand, it is important that irregular three-dimensional geometries may be handled easily.

In comparison with other two-phase simulators that have been discussed in the literature, the main distinctive feature of SHAFT78 is that it uses an integrated finite-difference method (IFD). We solve finite-difference equations that are obtained by formally integrating the basic partial differential equations for mass and energy flow over arbitrary polyhedral volume elements. This method is as easily applicable to irregular geometries of actual reservoirs as it is to idealized regular geometries; yet the relative simplicity of the finite-difference method is retained in the theory and algorithms.