Three-Dimensional Well Model Pre-Processors for Reservoir Simulation With Horizontal and Curved Inclined Wells

Representing Wells in Numerical Reservoir Simulation: Part 1 - Theory

Society of Petroleum Engineers Journal ◽

10.2118/7697-pa ◽

1981 ◽

Vol 21 (03) ◽

pp. 323-338 ◽

Cited By ~ 18

Author(s):

Alexander S. Williamson ◽

John E. Chappelear

Keyword(s):

Boundary Condition ◽

Reservoir Simulation ◽

Singular Solution ◽

Numerical Solutions ◽

Applied Mathematics ◽

Internal Boundary ◽

Local Flow ◽

Well Model ◽

Grid Block ◽

Wellbore Flow

Abstract A reservoir simulation system uses an analytical model to represent flow within a grid block as it enters or leaves a well. This model is called a well model. We give a description here of the theoretical background of a well model, including how the sandface pressure and saturation boundary conditions can be calculated and how the well boundary itself can be replaced (approximately) by a source function. This paper and the following companion paper, SPE 9770, present a unified viewpoint of material, some of which may be already familiar to simulator developers. Introduction Our concern in this paper is the theory of representation of wells and the well boundary condition in a reservoir simulator.It frequently has been noted that, except in the case of a central well in a problem involving cylindrical coordinates, it is impractical to represent a well with an internal boundary. The ratio of well radius to desired grid-block length can be of order 0.001 or less. In such cases, an alternative procedure has evolved in which the well is represented by a source. The relationship between the source strength, the wellbore flow, and the flow in the surrounding grid blocks composes an essential part of the well model. Even when the grid around a well is sufficiently fine to represent the well as an internal boundary, other features such as partial perforation, partial penetration, or skin may be important to the local flow but extend over a "small" interval in relation to the appropriate grid-block dimension. Here also, a suitable source representation is advantageous. We shall develop the source representation of a well for a variety of circumstances.The well boundary condition generally involves the sandface pressure and flow rate. However, these quantities also must be consistent with the requirements of wellbore flow - i.e., reservoir and wellbore flows are coupled, and a wellbore flow model is required. We describe a means of treating a wide variety of wellbore flows without creating a numerically cumbersome simulator. We hope that this paper may provide a basis for further work and discussion of this essential topic. Review of Literature The source representation of a well can be described as a local, approximate, steady, singular solution of the flow equations. The idea of separating a singularity of this type for special treatment is an old idea in applied mathematics. In series solutions to certain elliptic and parabolic equations, it was found that the convergence of the series could be improved considerably by first extracting the singular part. In these cases the singular solution extended through the entire domain. The analogous approach using numerical methods in place of the series solution is also well known. The use of singular solutions in a purely local role in numerical solutions was introduced before the general use of digital computers. Woods' use of a local logarithmic expression in a solution of Poisson's equation by relaxation methods corresponds closely to the source representation of a well recently proposed by Peaceman. SPEJ P. 323＾

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Gravity Segregation in Two-Phase Displacement Processes

Society of Petroleum Engineers Journal ◽

10.2118/4630-pa ◽

1974 ◽

Vol 14 (06) ◽

pp. 619-632 ◽

Cited By ~ 14

Author(s):

Allan Spivak

Keyword(s):

Reservoir Simulation ◽

Three Dimensional ◽

Mobility Ratio ◽

Two Dimensional ◽

Two Phase ◽

Gravity Segregation ◽

Reservoir Performance ◽

Phase Displacement ◽

Gravity Effects ◽

Displacement Processes

Spivak, Allan,* Member SPE-AIME, Chevron Oil Field Research Co., La Habra, Calif. Abstract This paper describes a study of gravity segregation (underrun or override of injected fluids) in two-phase, secondary recovery displacement processes. Reservoir simulation was used to investigate the factors that influence gravity segregation and the magnitude of gravity effects for both water floods and gas floods. The degree of segregation for a given set of conditions was determined bycomparing the results of two-dimensional cross-sectional with one-dimensional horizontal calculations, andcomparing the results of three-dimensional vs two-dimensional a real calculations. The degree of segregation is quantitatively described by the dimensionless number E defined as (Rbt)no gravity -(Rbt)gravity E =, (Rbt)no gravity where Rbt is recovery at breakthrough. Gravity segregation effects in two-phase displacement processes were found to increase withincreasing processes were found to increase withincreasing permeability (either horizontal or vertical),permeability (either horizontal or vertical),increasing density difference,increasing mobility ratio,decreasing production rates. anddecreasing level of viscosity for a fixed viscosity ratio. A series of calculations was made in which the parameters that affect gravity segregation were varied. The effect of each parameter on the degree of gravity segregation was observed. The effect of both stratified and random heterogeneity was also studied. A correlation between the degree of gravity segregation and the dimensionless groups G = 0.00633 and M = mobility ratio was established. This correlation is based on the results of the simulator calculations and a detailed analysis of the equations for three-dimensional, two-phase. immiscible, incompressible flow. The correlation can be used to determine qualitatively whether gravity segregation will be a significant factor in a given flooding process. It can also be used to determine whether the assumption of vertical equilibrium is valid in the simulation by a two-dimensional a real simulation model of reservoirs where fluid saturations vary significantly in the vertical direction. Introduction For many years, gravity segregation in reservoir processes was not accounted for because it could processes was not accounted for because it could not be adequately handled in reservoir engineering calculations. The advent of reservoir simulation provided the capability to handle gravity, and it provided the capability to handle gravity, and it became apparent that gravity effects could significantly affect reservoir performance. This paper describes a study in which a three-dimensional paper describes a study in which a three-dimensional (3-D), two-phase, incompressible simulator was used to look at gravity effects in displacement processes. The objectives of this study wereto processes. The objectives of this study wereto determine what factors influence gravity segregation and in what way;to look at the magnitude of gravity segregation effects;to compare reservoir performance calculations with and without gravity performance calculations with and without gravity effects. PREVIOUS WORK ON GRAVITY PREVIOUS WORK ON GRAVITY SEGREGATION IN DISPLACEMENT PROCESSES Craig et al. did experimental work to study the effects of gravity segregation during water, gas, and solvent flooding. As a result of their studies, they concluded that segregation of fluids due to gravity effects could result in oil recoveries at breakthrough as low as 20 percent of those otherwise expected. They also concluded that performance may in some cases be influenced to a greater degree by heterogeneity than by gravity effects. They were not able to directly compare performance under a given set of conditions with and without gravity. SPEJ p. 619

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