Thermal Recovery Well Test Design and Interpretation

Abstract

Summary Because a reservoir undergoing a thermal recovery process is typically idealized as a composite reservoir, this study develops new design and interpretation equations by investigating pressure-derivative behavior of a well in a two-zone, radial, infinite or finite composite reservoir. Accurate design equations help establish the test duration required to observe a particular feature in well test data and thus the applicability of an interpretation method to determine front radius, or swept volume. This study shows that the dimensionless time to the end of the first semilog line (deviation time) based on front radius is a constant. The deviation-time method may be used if wellbore storage does not mask the first semilog line. Design equations for the time of the beginning of the second semilog line and the time to observe outer-boundary effects show that the intersection-time method is not suitable for thermal recovery well test analysis. Correlations developed for the end of pseudosteady-state behavior of the swept region should help select the correct Cartesian line to calculate swept volume. This paper also presents derivative type curves applicable for all front radii, with mobility and storativity ratios as parameters for infinitely large composite reservoirs. For closed and constant-pressure outer boundaries, the ratio of outer boundary to front radius is the third parameter. Introduction The behavior of composite reservoirs has recently attracted much attention, and many studies have appeared on this subject. A composite system can occur naturally or may be artificially created. Aquifers with two different permeabilities forming two regions, oil and water or gas and oil zones, with different properties in a reservoir, and a finite-thickness skin region are examples of naturally occurring two-zone composite systems. Secondary or tertiary recovery projects-such as waterflooding, polymer floods, gas injection, in-situ combustion, steamdrive, and CO2 miscible flooding-artificially create conditions wherein the reservoir can be viewed as consisting of two zones with different rock and/or fluid properties. Geothermal reservoir well tests can often be represented by a composite reservoir model. The pressure behavior of composite reservoirs has been considered extensively. Recently, Brown investigated pressure-derivative behavior of composite reservoirs but limited his study to mobility and storativity ratios on the order of 0.4 to 2.0 and 0.3 to 30, respectively. Such mobility and storativity ratios are typical of cases with finite-thickness skin regions around the wellbore. Wattenbarger and Rainey modeled a finite-thickness skin region as a composite system and obtained pressure-transient behavior for such systems using finite-difference techniques. Their solutions correspond to a range of mobility ratio from 0. 1 to 3.6, which is the range Brown used. In this study, we consider mobility and storativity ratio contrasts typical of thermal recovery processes and de-velop new design and interpretation equations by investigating pressure-derivative behavior. The Eggenschwiler et al. analytical solution was used for a two-zone, infinitely large composite reservoir. A constant rate was specified at the inner boundary. The Home et al. solution was used for finite composite reservoirs. The outer boundary for a finite reservoir can be either closed or at a constant pressure. Wellbore storage was neglected in this study. Implications of this study on different methods to calculate front radius or inner-zone volume are discussed. Pressure-Derivative Approach. Whenever a straight line is sought on a graph of pressure vs. a function of time, we seek a constant slope. Thus, pressure derivatives can be used directly. Also, a pressure derivative can enhance a pressure signal and may be more sensitive to disturbances in reservoir conditions. In general, a pressure-derivative graph shows the start and end of a given flow regime later and earlier, respectively, than is indicated on a pressure graph at a specified accuracy. This is illustrated for simple cases in Ref. 19. The examples in Ref. 19 suggest that there is often more detail evident in a derivative graph than is apparent in a pressure graph. Times of specific flow events determined from pressure-derivative analysis, however, can often be different from those determined from pressure analysis. Because the duration of a flow regime appears shorter on a derivative graph, care is required for well test analysis by a specialized method to ensure that pressure data are in the correct time range. Because of detail enhancement in a derivative graph, however, better type-curve matching may be possible with a derivative type curve. To use pressure derivatives, design equations and type curves based on pressure derivatives for the system under consideration are necessary. We consider composite reservoirs in this study. Fig. 1 is a schematic of a composite reservoir. Inner-and outer-region properties are assumed to be different. The distance R is the front radius, which is an important parameter sought in thermal recovery well test analysis. Strictly speaking, fronts in thermal recovery usually are not cylindrical owing to gravity effects. Thus, the front radius exists only in some average sense. It is perhaps better to speak of the volume of the inner region, especially when pseudosteady data are available. "Front radius" is used here, however, as a matter of convenience. This term has been used in most of the previous studies of well test analysis applied to thermal systems. Methods To Estimate Front Radius. Different methods have been proposed to estimate a front radius from pressure-vs.-time data:deviation-time method,intersection-time method,type-curve-matching methods, andpseudosteady-state method. The deviation-tune method uses the time at the end of the semilog pressure-vs.-time line corresponding to the inner region to calculate a front radius on the basis of a theoretical dimensionless deviation time. The intersection-time method uses the intersection time of two semilog lines corresponding to the mobilities of the inner and outer regions to calculate a front radius, again with a theoretical dimensionless intersection time as the basis. A type-curve-matching method was proposed by Bixel and van Poollen. Barua and Horne used automated type-curve matching to analyze thermal recovery well tests. Eggenschwiler et al. proposed a pseudosteady-state method for large mobility and stora-tivity contrasts. They observed that the swept region could behave like a closed system for a short duration after the end of the semi-log line corresponding to the inner-region mobility. During this time, a pseudosteady Cartesian line may develop with a slope that can be related to the swept volume. This method should be independent of the geometry of the swept region and has been applied by several investigators to field and simulated cases with apparent success. Infinite Outer-Boundary Results We now consider pressure-derivative behavior for a well producing (or injecting) at a constant rate in a two-zone composite, infinitely large reservoir. The derivative of well pressure with respect to the base 10 logarithm of time is (1)