Analytical Solution for Gas/Oil Displacement with Temperature Variation
Authors: Zhu Jichun , Jessen Kristian , Orr Franklin M. 
Published: December 2007

Abstract
Analytical solutions are available for the one-dimensional conservation equations that describe multicomponent gas displacement through porous media with components partitioning between oil and gas phases. Past research on such analytical solutions has been limited to isothermal displacements. In this paper, we relax this restriction and present solutions where the injection gas has different temperature than that of the initial reservoir. We limit this study to systems with two components, and use Peng-Robinson equation of state to represent the phase behavior. It is observed that in regions where temperature variations dominate, the temperature variations propagate more slowly due to heat exchange between the flowing phases and rock matrix. In regions where composition variation dominates, temperature variations remain extremely small. As a result, a solution consists of a slow moving temperature wave upstream that covers the majority of the temperature difference between the initial oil and injected gas, and a near-isothermal solution downstream. A careful examination of the effect of variable injection composition and rock heat capacity reveals that it is possible for the slow moving temperature waves to catch up and merge with the trailing composition waves and alter the solution structure under certain conditions without affecting the fast moving compositional waves downstream. Therefore, for the near-isothermal waves, constant temperature is a good assumption. For cases when trailing temperature waves do not interact with the near-isothermal waves downstream, the solution method can be extended to systems with more than two components with relative ease.
A one-dimensional upstream weighted finite-different scheme was also developed to simulate the gas injection processes with temperature variation. The numerical solutions confirm the analytical solutions, though some details of the displacement behavior are obscured by numerical dispersion. The analytical solutions presented allow assessment of the accuracy of numerical simulations of gas displacement processes with temperature variation.

Introduction
Gas injection has been widely adopted in the oil industry as an efficient method of enhanced oil recovery. The injected gases are compressed before arriving at the oil layers, often at a different temperature than that of the reservoir. The temperature difference is especially evident if the gas injection procedure is scheduled after waterflooding using cold sea water, as has been practiced in some of the North Sea and Alaskan oil fields. When the injected gas mixes with the reservoir fluids originally in place, two phases may form when local thermodynamic phase equilibrium is established. The reservoir fluids may exchange heat with the reservoir rock due to temperature difference and chemical species through adsorption and desorption at the reservoir rock surfaces. This process occurs repeatedly as the newly formed fluid phases propagate through the reservoir rocks at different rates and along different paths, according to the physical properties of the flowing phases and the heterogeneity of the reservoir rocks. Hence, the gas injection process is a problem of two-phase flow through a porous medium coupled with phase behavior of a multicomponent mixture.
Analytical solution methods can be used to isolate individual mechanisms that drive the gas injection process and provide a better understanding of the solution physics. Analytical solutions can also be used in combination with streamline methods for reservoir simulation to yield very fast computational tools for modeling three-dimensional displacement1. However, in order to eliminate some of the complexity of the general flow, it is often assumed that flow occurs at constant temperature. The objective of this study is to extend the analytical theory by including and studying the physical effects of variable temperature.