Thermal Effects in Cyclic Operation of Storage Reservoirs

Abstract

Summary. This paper addresses the long-term temperature and pressure variations of natural gas that is stored cyclically in underground reservoirs to avoid the formation of gas hydrates. A detailed examination of the well and near-well reservoir emphasizes the cumulative importance of the Joule-Thomson effect over many years. Introduction Advance knowledge of gas temperatures during injection and withdrawal has generated much interest in the underground storage of natural gas over successive annual cycles. A complete annual cycle consists broadly of four parts:injection during the summer months,a brief shut-in or dormant period during the fall,withdrawal in the winter, anda second dormant period in the spring. A typical sequence, shown in Fig. 1, is then repeated over ensuing years. As the gas is stored and retrieved over a complete cycle, it interacts with its surroundings in a complex manner. To gain a complete understanding of the process, it is necessary to consider variations with time and position of gas temperature and pressure in the well, rock temperature in the formation surrounding the well, and gas temperature and pressure in the reservoir storage region. From injection to subsequent recovery, there is a loss of pressure and the Joule-Thomson effect tends to cool the gas. On the other hand, the gas tends to warm up because of the higher geothermal effect at increasing depths. Because of these two competing processes, the gas may be retrieved at a lower or higher temperature than it is injected. Consequently, there is then the possibility of hydrate formation, which would greatly reduce the deliverability of a well. Gas hydrates form at temperatures as high as 65 deg. F [18 deg. C] or higher when accompanied by large pressures (greater than 2,000 psia [greater than 13.8 MPa]) and produce solids, which decrease the available area for flow (see Fig. 2). An accurate knowledge of the temperature distribution is therefore necessary to prevent such occurrences. In earlier work, analytical solutions were predominant, while Ramey developed a calculational procedure that became the standard for many years. However, these analytical solutions used several simplifying assumptions, such as uniform rock properties, steady flow, and constant fluid properties. Subsequent calculations have continued to use analytical solutions, although fewer assumptions have been made and the emphasis is on speed and ease of computation. Numerical solutions are also available that introduce flexibility and added accuracy by allowing different flow options and even fewer simplifications. The missing premise, however, has been the effect that time has on the analysis. This paper therefore focuses on the unsteady-state solution of temperature in the well column and includes the temperature effects in the surrounding rock formation and the storage region, both of which have been previously ignored. Finite-element and finite-difference methods are used for added accuracy. Gas temperature and pressure profiles are shown as functions of time to indicate when hydrates may form for a given system, initial conditions, and an injection/withdrawal schedule. Results also show the approach of the system to equilibrium over several yearly operating cycles. Development The transient problem under consideration, shown in Fig. 3, involves the injection or withdrawal of natural gas in a well and its accompanying storage or depletion in the formation surrounding the bottom of the well. Assuming that the flow rate and temperature of the gas are known at the surface, the object is to determine the temperature and pressure of the gas as functions of position and time within the well column and in the immediately adjacent storage region. The model is divided into three parts to account additionally for the thermal effects of the geothermal gradient surrounding the well:the well and its related hardware, extending as far as the casing-cement/rock interface;the rock formation that extends-beyond the cement/rock interface; andthe storage region adjacent to the bottom of the well. In the following analysis, physical and thermal properties are not assumed to be constant and may be continuously updated throughout the simulation. However, short-term transient effects in the wellbore, which are attendant on any sudden changes in injection or withdrawal, are neglected. The Well. The procedure is to solve the energy and momentum equations simultaneously to yield both temperature and pressure distributions. The energy balance governs the variations of enthalpy with elevation in the well and is given by (1) where the sign on u(t) is plus for withdrawal and minus for injection. The momentum-energy balance governs the variation of pressure with elevation in the well: (2) where the plus/minus sign on the last term is for injection/ withdrawal. In all equations, it is understood that appropriate conversion factors, such as gc and J, will be added when ultimately working with specific units. Conditions in the well are assumed to reach equilibrium very quickly in comparison with the slower processes described below. Hence, an explicit time derivative is not needed in the well equations. Several additional relations are needed to solve Eqs. 1 and 2. First, the enthalpy is expressed in terms of both the temperature and the pressure, which automatically incorporates the Joule-Thomson coefficient: (3) The Joule-Thomson effect, as represented by (delta H/delta p)T, is negative for the temperature and pressure ranges found in storage and production of natural gas. SPERE P. 1295^