A Study of a Thermal Discontinuity in Well Test Analysis
Authors: Mangold Donald C. , Tsang Chin Fu , Lippmann Marcelo J. , Witherspoon Paul A. 
Published: December 2007

Summary
The presence of zones of different temperatures in nonisothermal reservoirs may resemble permeability boundaries during well testing. This investigation employed numerical modeling to examine such effects in drawdown, buildup, and injection well tests. The results indicate that nonisothermal influence can be detected and accounted for by tests of sufficient duration with suitably placed observation wells.

Introduction
A challenge to the interpretation of well test results from nonisothermal reservoirs is the effect on the pressure response due to fluid and rock temperature-dependent properties. Since zones of different temperatures in the reservoir may resemble permeability boundaries, care in interpretation of results is required. This study is an examination of these temperature effects on well test data where the producing well is completed in the center of a circular hot zone surrounded by a concentric cooler water region. The investigation was carried out with the intent of comparing the results of such tests with the classical Theis solution and discovering ways in which temperature differences can be accounted for in well test data analysis.Only recently have there been discussions of nonisothermal well testing in the literature. An analytical study by Tsang and Tsang motivated this study by giving semilog results to be expected from cold-water injection into a hot reservoir. By assuming a specific form for the variation of the permeability/viscosity ratio (k/mu) across the transition zone (from cold to hot water), they were able to derive solutions that follow the classic Theis curve for each region. From early- and later-time data, reservoir transmissivity (kh/mu) and storativity (phi ch) can be determined so that compressibility (c), reservoir permeability (k), porosity (phi), and thickness (h) can be evaluated. Their results prompted this numerical investigation of viscosity effects where production, buildup, and partial-penetration well tests are considered.

Background
The standard methods of well test data analysis in isothermal reservoirs have been documented thoroughly by Earlougher. The problems of well testing in geothermal reservoirs have been discussed by only a few authors in the past decade. Recently, Narasimhan and Witherspoon have reviewed the problems in carrying out and analyzing well tests in these systems. They mention that the conventional concept of transmissivity should be replaced by absolute permeability (k) or the product kh. In another recent study of reinjection at the East Mesa geothermal area, pressure distributions based on viscosity ratios of 1:1.9 between hot and cold water were described. The semilog plots show a distinctive change in slope at the cold front. The approach used, however, was a series of steady-state runs with the cold front at varying distances from the production well and with energy and mass transport equations decoupled. Rice used computer models to describe drawdown and buildup tests in single- and two-phase reservoirs under isothermal conditions. Earlier, Chappelear and Volek Modeled the injection of a hot liquid into a porous medium using temperature-dependent viscosity. They assumed that specific heat and density were independent of temperature, and they calculated temperature (not pressure) distributions within the reservoir, caprock, and bedrock.
JPT
P. 1095^