A mathematical model is presented that describes the effect of heat losses on pressure behavior in falloff testing of steam-injection wells. A general solution is presented in the form of type curves (with the heat-loss factor as a parameter), along with the analytical solutions for the asymptotic cases. A new method of falloff data analysis also is proposed.

The effect of an electric field on the long wave motion of the surface of a conducting fluid layer is considered. A Korteweg–de Vries (KdV)-type equation with coefficients depending on the applied field is derived. The speed of the solitary wave on the fluid layer is seen to be reduced by the electric field. It is found that there are two critical values of the applied voltage that lead to (i) breaking up of the solitary waves and (ii) bifurcation of solutions of the governing equations. For a given value of the imposed potential, solitary waves of elevation and depression are possible depending on the value of surface tension.

Over the years, attempts have been made to include system heterogeneities into mathematical descriptions and to determine their effect on pressure response. In this study, a general case of a system with layers of non-uniform thicknesses is considered. The mathematical description is based on assuming radial flow within individual layers and pseudosteady-state cross flow between the adjacent layers. Since the angle of the interphasing plane may assume any angle between 0 and 90 degrees, layered (uniform thicknesses) and composite systems constitute the lim iting cases. The analytical solution is found in Laplace space and the numerical inversion to real space is obtained by using the Stehfest algorithm. The model is verified for several simpler subsystems and its sensitivity to reservoir parameters is investigated. Solutions are presented in a graphical form for a selected set of parameters. Based on this model a method of well test data analysis is proposed. Introduction Well testing is a valuable source of information on reservoir properties and well conditions. Test data are interpreted on the basis of pressure transient theory, which is deve1oped for various physical concepts of the reservoir flow conditions. In this study, modeling of complex reservoir systems is attempted in order to enhance the quality of the well test data analysis. It is well recognized that all petroleum reservoirs are heterogeneous to a certain degree. The heterogeneity manifests itself-in terms of variable rock and/or fluid properties, which occur due to either natural processes or human activities. Heterogeneities may be observed both in the radial and the vertical directions. Modelling of heterogeneous systems is usually approached by considering several subsystems each with properties lumped at some characteristic value. Such an approach has been used over the years and has given rise to the concepts of composite and the layered reservoir models. Literature Review In this section, only the key papers are mentioned. A complete review of the work done on the subject of multilayer pressure transient theory was published by Ehlig-Economides and Joseph(l) in 1985; the contributions made by individual authors are categorized according to the solution method, number of layers, existence of the formation crosstlow and types of the boundary conditions. The models describing the pressure response of layered reservoirs may be classified into two groups depending whether or not the interlayer flow is considered. One of the first descriptions of crossflow between two layers was introduced by Barrenblatt et al(2), who proposed the double porosity model for naturally fractured reservoirs. Such systems are featured by pressure imbalance between high diffusivity fractures and low diffusivity matrix blocks: the resultant fluid transfertends to equalize pressures in the two subsystems. Assuming the lump parameter concept, the flow between a matrix and a fracture is proportional to their pressure difference. This approach has been successfully used in simulating crossflow between layers of different properties.

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Composite reservoir systems are frequently encountered in production or injection operations. In most of the recent modelling efforts a set of concentric cylinders has been used to describe a composite nature of a reservoir. Some systems, however, exhibit nonisotropical rock heterogeneities. In this paper, an elliptical flow model accounting for the non isotropic reservoir properties is proposed. The model describes the effects of the wellbore storage the wellbore skin and the front skin. The model sensitivity to the storativity, F, [defined by equation (AJ)] and mobility ratios and the front skin was investigated as well as the effect of these parameter on the flow characteristics Introduction Oil and gas reservoirs are frequently featured by heterogeneous rock and fluid properties. Formation heterogeneities may either be of a natural origin, such as vugs, or man-made; for instance, fractures intercepting wells, injected water bubbles, the burned or steam swept regions in thermal oil recovery processes, etc. Formations which are featured by radial or quasi-radial discontinuities in terms of fluid mobilities and hydraulic diffusivities are referred to as composite systems. The majority of studies on composite reservoirs provide the system description in terms of the radial coordinates; rock isotropy is [hen a necessary requirement. However, several situations exist when elliptical flow may occur: a case of anisotropic systems; a fracture intercepting a well water influx from an aquifer into an elliptical reservoir and the gravity override and/or underride in case of thermal processes. In the recent past, various aspects of elliptical flow under the reservoir conditions were investigated by Kucuk and Brigham(l), Obut and Enekin(2) and Stanislav et al.(3). The objective of this study is to determine the system sensitivity to a variety of parameters under the line source inner boundary condition. Mathematical Model A two-zone elliptical composite system is considered. Each region is featured by different fluid and rock properties. A well is located at the origin of the coordinate system as shown in Figure I. The model is developed subject to the following assumptions: single phase fluid of constant compressibility is considered; the Formation is characterized by a constant thickness and homogeneous properties in each of the two regions; the two-dimensional elliptical flow takes place in both regions; the front (the interphase between the two regions) is stationary during the test period. Other features of the model will be introduced in connection with the formulation of the boundary conditions. The dimensionless form of the flow equation can be written as follows: region I: Equation (1) (Available in full paper) Equation (2) (Available in full paper) Equation (3) (Available in full paper) Equation (4) (Available in full paper) Equation (5) (Available in full paper) Equation (6) (Available in full paper) Equation (7) (Available in full paper) Equation (8) (Available in full paper) The set of equations, equations (I) to (8), represents a complete mathematical description of the problem considered. All dimensionless quantities are defined in Appendix A. FIGURE 1. Coordinate system FIGURE 2, Effects of CO and Sh on P∞D for a homogenous system.