A Parallelepiped Model To Analyze The Pressure Behavior Of Geothermal Steam Wells Penetrating Vertical Fractures
Authors: Cinco-Ley Heber , Brigham W.E. , Economides M. , Miller F.G. , Ramey H.J. , Barelli A. , Manetti G. 
Published: December 2007

Abstract
Reservoir geometry is often the basis for development of models used to analyze field transient pressure data. The presence of a deep horizontal pressure data. The presence of a deep horizontal steamwater interface or boiling front first gave rise to the idea of a constant pressure boundary. Faults suggest vertical no-flow boundaries, and impermeable rocks over lying the steam zone indicate a no-flow cap.
In the past, parallelepiped models have been used to analyze the results of predesigned buildup and well interference tests. The model described herein is of more general use, and can be used to analyze long-term pressure-production well data. In this study, it is pressure-production well data. In this study, it is applied to analyze pressure data of a drawdown test for a geothermal reservoir.
As in earlier models, the boiling front is assumed to be a constant fluid-pressure boundary, and the side —as well as the top—are assumed to be impermeable boundaries. Equations of reservoir pressure behavior are derived using Green's functions and source functions. Graphs describing dimensionless pressure as a function of time and various reservoir parameters are provided. Partially penetrating fractures common to provided. Partially penetrating fractures common to many geothermal well systems are considered in the development of the model.

Introduction
Many geothermal areas are characterized by reservoirs whose dimensions are controlled by sealing fault or low permeability boundaries. Some of these reservoirs produce steam diluted with small quantities of noncondensable gases. There is often evidence of boiling water at a considerable distance from the producing horizon (presumably lying below the steam cap). In addition, the transient pressure behavior of these wells sometimes indicates they intersect large, high conductivity fractures. These are natural fractures; no hydraulic fracturing has been done in any of these wells.
This general description leads naturally to a model in the shape of a parallelepiped with closed boundaries on five sides and a constant pressure boundary on the bottom. The fracture intersecting the well can be simulated as a rectangular-shaped source or sink. Such a system was described by Atkinson et al. for a limited set of geometric conditions which approximated the producing system in the Travale area, in Italy. Because of the success of this approach, personnel at Stanford University and at ENEL, in Pisa, have developed general programs for generating long-term pressure-production forecasts for a variety of parallelepiped conditions and well-fracture geometries. This approach appears to be generally useful, for many geothermal systems worldwide will have fault-controlled geometry of the type described here.
The purpose of this work is to determine which unique characteristics of such systems can be identified. This, in turn, will lead to greater confidence in predictions of the long-range producing characteristics predictions of the long-range producing characteristics of these systems.

DESCRIPTION OF THE MODEL
Let us consider a well intersected by a vertical fracture in a parallelepiped reservoir, as shown in Fig. 1. The system is bounded laterally by vertical impermeable planes; the top of the reservoir is a horizontal no-flow boundary, and underlying the reservoir there is a constant pressure plane.
The general assumptions for this model are as follows:The well produces at a constant flowrate in an anisotropic, homogeneous reservoir of constant properties (k and phi are independent of pressure and properties (k and phi are independent of pressure and temperature).The reservoir contains a slightly compressible fluid of constant viscosity, mu, and compressibility, c. Although this assumption is not valid, there is considerable evidence that it is a good approximation for gaseous systems when the m(p) function is used.There are small pressure gradients and negligible gravity effects in the reservoir.