The widely used Cartesian coordinate grid has some disadvantages in the description of boundaries, faults and discontinuities. In addition, a five-point scheme can cause significant grid orientation effects. A ninepoint scheme reduces this effect but makes the treatment of boundaries and heterogeneities more difficult.Orthogonal curvilinear coordinate systems have improved the modeling of reservoir shape and flow geometry. Mathematically they are based on a coordinate transformation and discretization of the transformed equations in the usual manner. Due to the lack of orthogonality additional mixed derivative terms are introduced which are difficult to discretize and therefore are usually neglected. However, because grid lines have to be coordinate lines, strictly orthogonal systems are not flexible enough to describe reservoirs of complicated shape. This paper describes a practical method for using irregular or locally irregular grids in reservoir simulation with the advantages of flexible approximation of reservoir geometry, simple treatment of boundary conditions and reduced grid orientation effects.Finite difference equations are set up by the so-called balance method. This method uses an integral formulation of the reservoir model equations equivalent to the commonly used differential equations. Integrating over grid blocks results in "balance equations" for each References and illustrations at end of paper 37 block. This can be done for various types of networks formed by triangles, convex quadrangles, polar meshes, curvilinear or locally refined grids. HEINRICHS 1 has proved consistence for this discretization scheme. The mesh can be refined locally. Well locations can be selected as mesh points to ensure that they are situated in the center of grid blocks. For triangular grids, the more isotropic distribution of grid points diminishes the orientation effect significantly.Numerical examples are presented comparing the proposed difference scheme with a nine-point Cartesian scheme . The performance of the method is illustrated by symmetry elements and complex simulation problems.

The simulation of the performance of a horizontal well has generated certain new and important challenges. These include the partial penetration of the well in the horizontal direction within the allocated drainage area, the positioning of the well between the vertical boundaries, the distance from the parallel horizontal boundaries, and the permeability anisotropy. In addition, there are special problems in the simulation of the response of fractures (natural and induced) in regard to their contact with the well (longitudinal or transverse), their conductivity, and the conductivity distribution along the fracture.We developed new numerical techniques to facilitate the simulation of these diverse problems. We use a locally refined perpendicular bisection grid to describe the horizontal (or deviated) wellbore. The grid is strictly orthogonal for the anisotropic case, and wellbore blocks are almost regular octagonal prisms. The transition to the coarse Cartesian grid is also orthogonal. The fully implicit formulation ensures the stability of the numerical solution. Our results are found to be in excellent agreement with published analytical or semianalytical approximations. In addition, the results offer flexibility that is not possible with analytical solutions. The grid system used is particularly amenable to handling practical problems with real reservoir geometries and configurations.This paper presents a comprehensive numerical simulation of problems associated with horizontal wells, including the arbitrary positioning of the well within a fully anisotropic medium. Hydraulic or natural fractures that intersect the well in the longitudinal or transverse direction are simulated for both infinite and finite conductivities.

No abstract

This paper describes a practical method in which irregular or locally irregular grids are used in reservoir simulation with the advantages of flexible approximation of reservoir geometry and reduced grid-orientation effects. Finite-difference equations are derived from an integral formulation of the reservoir model equations equivalent to the commonly used differential equations. Integrating over gridblocks results in material-balance equations for each block. This leads to a finite-volume method that combines the advantages of finite-element methods (flexible grids) with those of finite-difference methods (intuitive interpretation of flow terms). Grid-orientation effects are investigated. For grids based on triangular elements, the more isotropic distribution of gridpoints diminishes the orientation effect significantly. Numerical examples show that the regions of interest in a reservoir can be simulated efficiently and that well flow can be represented accurately.

The windowing technique, introduced by Deimbacher and Heinemann (Ref. 3), 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. As the gridded wellbore and the grid blocks around it are small (some cu-ft) 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 time step with up to 1000 small local steps. This paper presents the general and practical applicability of this method. Windows, constructed by the PEBI (k-PEBI) method, are introduced automatically for all of the wells (vertical, horizontal and slanted). For testing purposes a real case field model was 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 technique. Further it will be shown that solving the window model with large timestep lengths for the first and small lengths for the second solution step results in equal or smaller CPU times and less NR iterations 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 topic of extensive research over the past decades. In 1978 Peaceman (Ref. 6) presented the concept of the equivalent wellblock radius, allowing to relate the computed pressure of a perforated block with the well flowing pressure for different block geometries. This approach was frequently discussed and adapted to fit different requirements, such as deliberate locations in a square rectangular block (Ref. 8). In 1986 Aziz and Pedrosa introduced the concept of "hybrid grids" in a Cartesian block system (Ref. 1), trying to overcome the insufficiencies of the Peaceman model. Gridding the near wellbore area by hybrid grids honors the radial nature of flow, small block sizes help 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 either applying an iterative or a direct solution method. Aziz and Pedrosa gave preference to the direct approach, finding it to converge faster. In 1996 Ding (Ref. 7) proposed a numerical well model, applicable for numerous geometries of vertical, deviated and horizontal wells in flexible grids. Heinemann et al. 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 (Ref. 3, 4) that is highly suitable to model the near wellbore area. A direct solution approach is used, but 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 (Ref. 11), and can be solved using local timestep lengths different to the overall timestep length. Today the Peaceman 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.