Using Well Windows in Full-Field Reservoir Simulation

Summary

This paper presents the general and practical applicability of the "windowing technique"¹ to model wells in full-field reservoir simulation. Windows modeling the near-wellbore area and the wellblock itself, constructed by the Perpendicular-Bisectional (PEBI) or k-orthogonal PEBI (k-PEBI) method, will be introduced for all wells of a reservoir or for selected "problem wells."

The windowing technique, introduced by Deimbacher and Heinemann, ¹ allows a time-dependent replacement of grids for a defined area during a simulation run. A window can represent any kind of well with a gridded wellbore and an appropriate grid pattern around the well. Such an approach makes the generally used Peaceman well model superfluous. Because the gridded wellbore and the gridblocks around it are small (some cubic feet), the computational stability requires small timesteps and a greater number of Newton-Raphson iterations. It is obvious that this is not feasible if the solution for the full-scale model and the well windows must be performed simultaneously. Therefore, in a first step, the fully implicit solution for the full-scale model will be calculated, but the inner blocks of the windows are solved for the pressure only, without updating the saturations and mole fractions. This solution provides the boundary influx for the windows. In a second step, the windows are calculated for the same overall timestep with up to 1,000 small local steps.

This paper presents the general and practical applicability of this method. Windows, constructed by the PEBI (k-PEBI) method, can be introduced automatically for all the wells (vertical, horizontal, and slanted) or for certain "problem wells."

For testing purposes, real case field models were used. It will be shown that the quality of the results obtained for the model calculated with integrated radial grids around the wells and small overall timesteps are equal to those obtained for the same model using the windowing and local timestepping techniques. Further, it will be shown that solving the window model with large timestep lengths for the first solution step and small lengths for the second solution step results in equal or smaller CPU times in comparison to the conventional model.

Introduction

Adequate well representation in hydrocarbon reservoir simulation has been one of the central problems in the petroleum industry and a topic of extensive research over the past decades.

In 1978, Peaceman² presented the concept of the equivalent wellblock radius, allowing us to relate the computed pressure of a perforated block to the well flowing pressure for different block geometries. This approach was frequently discussed and adapted to fit different requirements, such as arbitrary well locations in a square or rectangular block.³

In 1986, Pedrosa and Aziz⁴ introduced the concept of "hybrid grids" in a Cartesian block system, trying to overcome the insufficiencies of the Peaceman model. Gridding the near-wellbore area by hybrid grids honors the radial nature of flow, while small block sizes help in handling large saturation changes caused by high production rates. The paper showed that the solution of the equations in the radially gridded well region can be decoupled from the solution of the equations of the conventional grid system by applying either an iterative or a direct solution method. Pedrosa and Aziz gave preference to the direct approach, finding it able to converge faster.

In 1996, Ding⁵ proposed a numerical well model, applicable for numerous geometries of vertical, deviated, and horizontal wells in flexible grids. Heinemann et al.^1,6,7 introduced the windowing technique (i.e., a time-dependent replacement of grids or parameters in a limited area within the original block model, combined with local timestepping that is highly suitable to model the nearwellbore area). A direct solution approach is used but is fundamentally different from the approach used by Pedrosa and Aziz. A window can contain all kinds of wells, constructed by gridding techniques such as 3D or 3D k-PEBI gridding,⁸ and it can be solved with local timestep lengths different from the overall timestep length. Today, Peaceman's approach is still commonly used in reservoir simulation, while gridding the near-wellbore area is only applied to investigate single well problems, such as coning behavior. The applicability of the windowing and local timestepping techniques for well testing in full-field reservoir simulation was shown by Abdelmawla.⁹

The authors of this paper will extend the idea, presenting a procedure for modeling all production and injection wells, both vertical and horizontal, using the windowing and local timestepping techniques in full-field reservoir simulation. The method will be validated with a small-scale coning example and then applied to a real full-field case. The results will be compared with runs produced by a conventional simulation model.

The Windowing Technique

The windowing technique is a numerical multipurpose tool with advantages for a large number of simulation problems. It allows for the time-dependent replacement of grids and parameters during a simulation run. Using the local timestepping technique, the equations for the window area can be solved using different timestep lengths (local timesteps) than those used for the equations of the full-scale model (overall timesteps).

The window grid can be regular (refined Cartesian, hexagonal, etc.) or completely irregular. Window blocks are not part of the underlying basic grid model; they overlap the basic gridblocks or (to be more precise) replace them when the window is activated.

We use the term "window image blocks" when speaking of the basic gridblocks that surround a window and the term "window boundary blocks" when referring to the outermost blocks inside a window. For any grid used in the window area, it is essential to construct the window boundary blocks in such a way that their size will be approximately equal to the size of blocks in the basic grid.

To represent the window area, it will be sufficient to parameterize the window grid using the same rock data as for the basic gridblocks. A detailed description of the window grid parameterization can be found in Ref. 10. More complex algorithms to parameterize radial grids, taking upscaling into account, have been the topic of recent developments (Ref. 11).

It should be mentioned that the possibility of parameterizing the window grids differently from the basic grid could be used to automatically match stochastic geological maps to dynamic engineering data. This idea is beyond the scope of this work and will not be discussed in this paper. A window can be switched on (activated) and off (deactivated) at any time during a simulation run.