A closed‐form analytical solution for thermal single‐well injection‐withdrawal tests

A closed-form analytical solution for thermal single-well injection-withdrawal tests Yoojin Jung 1 and Karsten Pruess 1 [ 1 ] Thermal single-well injection-withdrawal (SWIW) tests entail pumping cold water into a hot and usually fractured reservoir, and monitoring the temperature recovery during subsequent backflow. Such tests have been proposed as a potential means to characterize properties of enhanced geothermal systems (EGS), such as fracture spacing, connectivity, and porosity. In this paper we develop an analytical solution for thermal SWIW tests, using an idealized model of a single vertical fracture with linear flow geometry embedded in impermeable conductive wall rocks. The analytical solution shows that the time dependence of temperature recovery is dominated by the heat exchange between fracture and matrix rock, but strong thermal diffusivities of rocks as compared to typical solute diffusivities are not necessarily advantageous for characterizing fracture-matrix interactions. The effect of fracture aperture on temperature recovery during backflow is weak, particularly when the fracture aperture is smaller than 0.1 cm. The solution also shows that temperature recovery during backflow is independent of the applied injection and backflow rates. This surprising result implies that temperature recovery is independent of the height of the fracture, or the specific fracture-matrix interface areas per unit fracture length, suggesting that thermal SWIW tests will not be able to characterize fracture growth that may be achieved by stimulation treatments. 1. Introduction [ 2 ] Enhanced geothermal systems (EGS) are engineered reservoirs that may be developed to produce energy from hot rock formations that are otherwise not economically viable for heat mining [Ge´rard et al., 2006]. To achieve a practically useful production capacity, the development of EGS commonly requires stimulation treatments, which usually involve water injection under high pressure. By applying stimulation treatments, we expect to increase the aperture, permeability, and size of pre-existing fractures and make additional fractures accessible to the injected fluid. [ 3 ] While increased fracture permeability is advanta- geous for improving the production of thermal energy, rapid migration of the injected water through preferential paths with insufficient heat transfer from the rock may result in premature thermal breakthrough at production wells, which would reduce the lifetime of the geothermal reservoir. For both success and sustainability of EGS, it is critical to ascertain the effectiveness of stimulation treat- ments for enhancing the fracture-rock matrix interface area, reducing flow impedance in the reservoir, and increasing flow rates of production wells. Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, USA. [ 4 ] Tracer tests have been proposed as a means to esti- mate fracture-matrix interface areas. Such tests generally involve the injection of aqueous solutes into one or more injection wells, and monitoring of tracer returns in fluids produced from offset observation or production wells [Shook, 2001 ; Sanjuan et al., 2006]. Interdiffusion of sol- ute tracers between fractures and rock matrix produces characteristic tails in tracer breakthrough curves that may permit the determination of fracture-matrix interface areas [Pruess, 2002 ; Pruess et al., 2005 ; Shan and Pruess, 2005]. However, because tracer breakthrough at offset ob- servation wells may be weak and slow, interwell tracer tests (ITT) may require fluid sampling over extended time peri- ods of weeks or even months. Also, suitable observation wells may not always be available. [ 5 ] Single-well injection-withdrawal (SWIW) tests, variously referred to as ‘‘huff and puff,’’ ‘‘push-pull,’’ or ‘‘injection-backflow’’ tests, can be an alternative to ITT. During an SWIW test, fluid with tracers is injected into a well and, after some quiescent or rest period, is produced out of the same well [Kocabas and Horne, 1987 ; Haggerty et al., 2001 ; Nalla and Shook, 2005 ; Ghergut et al., 2006, 2009 ; Neretnieks, 2007]. SWIW tests typically require a much shorter test duration from hours to a few days as com- pared to weeks or months for ITT. This holds out the prom- ise of obtaining test results much more quickly, which would provide significant economic benefits. Another potential advantage is that SWIW tests are much less affected than ITT by heterogeneities in the fracture network, potentially providing a clearer signal of matrix properties 1 of 12