Wellbore Heat Transmission and Pressure Drop for Steam/Water Injection and Geothermal Production: A Simple Solution Technique

Abstract

Summary This paper presents a straightforward iterative procedure for the wellbore heat transmission problem during upward or downward flow of a steam/water mixture. The mathematical model is taken directly from the literature and is based on material and momentum balances in the wellbore and a heat balance on the entire system including the surrounding media. The transient heat conduction equation is solved analytically by the application of successive Fourier and Laplace transforms. A simple superpositioning in the time domain permits a matching procedure similar to multiphase flow calculations in pipelines. This is in contrast to standard numerical schemes that involve the direct solution of a set of algebraic and ordinary and partial differential equations typical of reservoir stimulation. The pressure-drop calculations in the wellbore account for the slip concept and the prevailing flow regimes by means of standard two-phase correlations. The validity of the method is demonstrated by comparison with results of other numerical simulation studies and actual field data for both steam injection and geothermal production. Introduction Dafter estimated that 914,700 B/D [145 426 m3/d] oil was recovered in the noncommunist world by various EOR processes from 1979 through 1981. Thermal methods, of which steam injection (steam soak and steam- drive) is a major contributor, accounted for 73%. Thermal methods are based on the principle that an increase in temperature causes a dramatic reduction in oil viscosity. These methods are usually applied to the recovery of heavy oil or tar. As wet fluid is injected either down the wellbore or down the tubing/casing annulus to the formation being flooded., heat is transferred from or to the surrounding earth as a result of the difference in geothermal and injected fluid temperature. To evaluate the feasibility of an injection project, a reasonable estimate of the effective amount of heat carried by the fluid, its temperature, sandface pressure, and quality is important. Steam injection processes impose severe operating conditions on injection wells. Gates and Holmes have presented an excellent reference on field experience that presented an excellent reference on field experience that involves well-completion methods. Casing failures usually occur at the couplings because of thermal stresses. Well completion equipment constitutes a significant proportion of the cost of a thermal project. Economics require that the completion design should provide for minimum heat loss in the wellbore equipment, where the temperature must be kept low to prevent damage. The key to proper stress analysis in thermal recovery installations is accurate knowledge of temperatures involved. Hence, the problem to be solved can be stated simply: For a given mass, flow rate, completion design, surface temperature, quality, and geothermal gradient, what is the temperature of the injected fluid as a function of depth and time? By solving this problem, we can answer how much heat is lost to the surroundings and, thus, what the increase in wellbore equipment temperature will be. Review of the Mathematical Model The earliest work was based on analytic solutions with the line source concept-that is, the oil well was considered to be a cylinder of infinite length in an infinite medium. Moss and White derived an expression for the calculation of the temperature of water during hot water injection as a function of time, t. They assumed the following. 1. The physical properties of the fluid and the formation were independent of the depth and temperature. 2. The heat transfer factors for the completion were ignored. 3. The frictional losses and the kinetic energy effects were negligible. 4. The heat transfer in the wellbore was considered steady state. SPERE p. 148