Method to Improve Well Model Efficiency in Reservoir Simulation

2011 ◽  
Author(s):  
David Alan Edwards ◽  
Baris Guyaguler
Keyword(s):  
Reservoir Simulation ◽  
Well Model ◽  
Model Efficiency

Representing Wells in Numerical Reservoir Simulation: Part 1 - Theory

1981 ◽  
Vol 21 (03) ◽  
pp. 323-338 ◽  
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＾

A Fully-Implicit Fully-Coupled Well Model for Parallel Mega-Cell Reservoir Simulation

2005 ◽  
Author(s):  
Larry S.K. Fung ◽  
HoJeen Su ◽  
Cheng T. Tan ◽  
Kesavalu Hemanthkumar ◽  
Jorge A. Pita
Keyword(s):  
Reservoir Simulation ◽  
Fully Implicit ◽  
Well Model ◽  
Fully Coupled

A Generalized 3D Well Model for Reservoir Simulation

SPE Journal ◽  
1996 ◽  
Vol 1 (04) ◽  
pp. 437-450 ◽  
Author(s):  
Y. Ding
Keyword(s):  
Reservoir Simulation ◽  
Well Model

Representation of Wells in Numerical Reservoir Simulation

1998 ◽  
Vol 1 (01) ◽  
pp. 18-23 ◽  
Author(s):  
Yu Ding ◽  
Gerard Renard ◽  
Luce Weill
Keyword(s):  
Finite Difference ◽  
Reservoir Simulation ◽  
Radial Flow ◽  
Numerical Models ◽  
Vertical Flow ◽  
Vertical Wells ◽  
New Approach ◽  
Well Model ◽  
Wellbore Pressure ◽  
Standard Models

Summary In reservoir simulation, linear approximations generally are used for well modeling. However, these types of approximations can be inaccurate for fluid-flow calculation in the vicinity of wells, leading to incorrect well-performance predictions. To overcome such problems, a new well representation1 has been proposed that uses a "logarithmic" type of approximation for vertical wells. In this paper, we show how the new well model can be implemented easily in existing simulators through the conventional productivity index (PI). We discuss the relationship between wellbore pressure, wellblock pressure, and flow rate in more detail, especially for the definition of wellblock pressure. We present an extension of the new approach to off-center wells and to flexible grids. Through this extension, the equivalence of various gridding techniques for the well model is emphasized. The key element is the accurate calculation of flow components in the vicinity of wells. Introduction The well model plays an important role in reservoir simulation because the precision of calculation in well-production rate or bottomhole pressure is directly related to this well model. The main difficulty of well modeling is the problem of singularity because of the difference in scale between the small wellbore diameter (less than 0.3 m) and the large wellblock grid dimensions used in the simulation (from tens to hundreds of meters), and to the radial nature of the flow around the well (i.e., nonlinear but logarithmic variation of the pressure away from the well). Thus, the wellblock pressure calculated by standard finite-difference methods is not the wellbore pressure. Peaceman2,3 first demonstrated that wellblock pressure calculated by finite difference in a uniform grid corresponds to the pressure at an equivalent wellblock radius, r0, related to gridblock dimensions. Assuming a radial flow around the well, he demons-trated that this radius could be used to relate the wellblock pressure to the wellbore pressure. However, there are problems with this approach in many practical reservoir simulation studies:For routinely used nonuniform Cartesian grids,4 there is no easy means to determine an r0 value.In three-dimensional (3D) cases with non-fully-penetrating wells, the basic radial flow assumption does not apply,5 whereas vertical flow effects must be included.6Off-center wells are not correctly treated.7,8Treatment of the well model is much more complicated with non Cartesian or flexible grids.9–11 The aim of this paper is to show that the new well representation1 proposed in a previous paper can handle these problems accurately. Wellblock Pressure Calculation A previous paper1 presented a new approach particularly well-suited to nonuniform grids for the modeling of vertical wells in numerical simulation. The principle of this new approach, which is based on a finite-volume method, is to calculate new interblock distances that improve the modeling of flow in the vicinity of wells. Because the new approach was originally presented for two-dimensional (2D)-XY problems, it was shown that for such problems the wellbore pressure could be calculated without both the intermediate computation of the wellblock pressure and introduction of an equivalent wellblock radius. However, for at least two reasons, it is convenient to keep this standard method commonly used in numerical models, which consists of relating the wellbore pressure and wellblock pressure through the use of a numerical PI and equivalent wellblock radius. One reason is practical. To implement the new approach more easily into standard numerical models, it is better to keep their internal structure unchanged. The other reason is dictated by the necessity of having a wellblock pressure in particular 3D simulation studies. When a well partially penetrates the reservoir or when there is communication between different layers, there is a vertical flow component in the vicinity of the well that necessitates that the wellblock pressure be calculated. How should the new approach be implemented in standard reservoir simulators- In these simulators, a numerical PI is used in the well model to relate the wellbore pressure, pw, to the wellblock pressure, p0. Usually, this PI is written as where r0 is the equivalent wellblock radius at which the pressure is equal to p0. Within the new well representation,1 to obtain a pressure p0 corresponding to a radius r0, it is sufficient to use equivalent wellblock transmissibilities relating p0 to the pressures of adjacent blocks through equivalent interblock distances, Leq, i (Fig. 1: where ?x0, ?y0 are the wellblock dimensions. For instance, in the x+ direction, Leq,1 is written where ?1+2 arctg (?y0 /?x0) is the angle formed by the right wellblock interface seen from the well. Because wellblock transmissibilities in standard models are conventionally expressed by the new approach can be implemented easily in standard models multiplying the conventional wellblock transmissibilities by constant factors. For instance, in the x+ direction, this factor is By use of equivalent transmissibilities, the calculated wellblock pressure, p0, should correspond to the equivalent wellblock radius, r0, which is involved in transmissibility calculations (Eq. 3). Then, the wellblock pressure can be related to the wellbore pressure with the conventional PI (Eq. 1).

Representing Wells in Numerical Reservoir Simulation: Part 2 - Implementation

1981 ◽  
Vol 21 (03) ◽  
pp. 339-344 ◽  
Author(s):  
John E. Chappelear ◽  
Alexander S. Williamson
Keyword(s):  
Relative Permeability ◽  
Reservoir Simulation ◽  
Model Development ◽  
Preceding Paper ◽  
Fluid Saturation ◽  
Well Model ◽  
Grid Block ◽  
The Difference ◽  
Reservoir Simulator ◽  
Water Block

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. This paper presents a succinct but comprehensive description of the installation of a well model in a simulator, including problems which may be encountered and possible remedies. This and the preceding paper, SPE 7697, present possible remedies. This and the preceding paper, SPE 7697, present a unified viewpoint of material, some of which may be already familiar to simulator developers. Introduction Our concern in this paper is the inclusion of a well model and well boundary conditions in a reservoir simulator. The source representation and the wellbore flow model are the basic components of the well model. The usefulness of the working version finally installed in a reservoir simulator depends greatly on the numerical implementation. We accordingly discuss numerical aspects of the well model for black-oil, compositional, and thermal well models.We have omitted a discussion of the incorporation into well models of surface gathering facilities and what could be called "well group constraints" such as lease, platform or pipeline constraints. These subjects easily could be the topics of several other papers.A satisfactory well model is frequently a key to successful simulation. Many of the details of well model development have not appeared in the petroleum literature. It is our hope that this paper may provide a basis for further work and discussion of this paper may provide a basis for further work and discussion of this essential topic. Implementation We shall discuss the implementation of the following equation (developed in Part 1) for the flow of each phase per completion interval. (1) Here, p is the phase (either oil, water, or gas). We note here certain aspects of this well model.1. The rates are in standard units.2. The relative permeability is calculated using the grid-block (average) fluid saturation from a well (i.e., not necessarily the grid-block) relative permeability table. It is at this point that the saturation boundary condition is imposed.3. The oil pressure is used to calculate the potential for all phases. Thus, capillarity, is not treated (i.e., no capillary end effect or water block). Also, the difference in phase pressures within a grid block due to gravity segregation is ignored.4. Zk is the vertical distance from the center of the kth completion interval to the center of the (k + 1)th completion interval (positive downward).5. The viscosity, formation volume factor, solution GOR, and density are calculated at gridblock pressure. Only the grid block for the completion interval is used.6. The skin and well radius are the same for every completion interval for each well.7. The external radius re of the grid block is a function of the grid-block, geometry. JPT P. 339

Sign in / Sign up

Export Citation Format

Share Document