Analytical Solution of the Tracer Equation for the Homogeneous Five-Spot Problem
- A.R. Almeida (Petrobras S/A) | R.M. Cotta (Universidade Federal do Rio de Janeiro)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 1996
- Document Type
- Journal Paper
- 31 - 38
- 1996. Society of Petroleum Engineers
- 5.5 reservoir Simulation
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The Generalized Integral Transform Technique (GITT) has been successfully employed in various heat and fluid flow problems. This paper describes the first application of the GITT in petroleum reservoir simulation problems.
The two-dimensional tracer equation for the five-spot pattern is solved analytically by means of this approach, with the usual assumptions of steady-state single-phase flow and unit mobility ratio. The solution is compared, in a wide range of situations, with alternative analytical (when possible) or numerical (finite differences) solutions, as well as with experimental data from other sources.
For the success of enhanced oil recovery (EOR) projects, the knowledge of the reservoir characteristics is essential. In accordance with Brigham and Abbaszadeh-Dehghani1, traditional studies like geology, geophysics, coring, well logging and well testing, help us to define the reservoir and their heterogeneities, but "for EOR evaluation thc most important data to be sought is the nature of the flow from well to well"1. For this purpose, there are two kinds of tests: well-to-well transient pressure tests and well-to-well tracer tests1. Brigham and Abbaszadeh-Dehghani1 state that the two tests are complementary because they measure different characteristics.
Since the beginning of this century, tracers have been used in underground porous media2. Several articles have been published, the great majority concerned with qualitative analysis of well-to-well tracer tests, involving important information like flow barriers, directional flow trends, cornmunication between reservoirs, and so on. However, some authors had dedicated their efforts in developing ways of extracting quantitative information from tracer tests. In order to achieve this goal, it is necessary to have a good model and, of course, a reliable way of solving it.
Tracer transport in a porous media is generally accepted to be subjected to two basic phenomena: convection and hydrodynamic dispersion3. This led to modelling tracer flow by the convection-diffusion (dispersion) equation. The particular features of this equation, associated with complex geometries involved in petroleum reservoirs, result in a difficult problem which can only be solved, by the well know conventional analytical methods, in simple cases. Even for numerical methods these equations are among the more difficult to solve since the behavior of the system ranges from parabolic to almost hyperbolic, depending on the ratio of convective to dispersive contributions to tracer transport4.
The recent developments on the so-called Generalized Integral Transform Technique (GITT)5 opened new perspectives of finding computationally exact (prescribed accuracy) solutions for a great variety of linear and nonlinear problems. This approach is now quite well established for different heat and fluid flow problems, like those governed by the boundary layer and Navier-Stokes equations5; this is, however, the first application in petroleum reservoir engineering.
Here, the solution through GITT of the two-dimensional tracer flow equation in fully developed five-spot patterns is described. The classical assumptions of horizontal, homogeneous and isotropic reservoir, with single phase steady state flow (unit mobility ratio), are adopted. Additionally, an ideal tracer is assumed, that is, with no adsorption, chemical reaction or radioactive decay. Although these assumptions restrict the application of the solution in the analysis of actual field situations, it will be instructive and useful both as a benchmark and as a design tool. This classical problem was chosen as a first step in the application of the GITT to reservoir problems, although there is no major drawback in removing any of the assumptions.
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