Scaleup techniques are commonly used to coarsen detailed geological models to scales more appropriate for reservoir simulation. These techniques typically generate upscaled permeabilities through solution of the local pressure equation, subject to some type of linear boundary condition. In the near-well region, coarse scale permeabilities computed in this manner may not adequately capture key features of the fine scale flow. In this article, we present a general method for computing accurate coarse scale wellblock transmissibilities and well indexes for vertical wells in two and three dimensions. The method is based on the solution of local well-driven flow problems subject to generic boundary conditions. Simulation results for two-dimensional systems indicate that the near-well treatment always provides improved predictions compared to those obtained using standard methods. In some parameter ranges, this improvement is quite substantial. Several three-dimensional models with vertical wells are also considered and improvement relative to the standard approach is achieved in all cases. Extension of the method to accommodate more general well configurations could result in significantly improved coarse scale simulations of nonconventional wells.
The highly detailed nature of geologic descriptions (currently on the order of 107 cells or more) significantly exceeds the capabilities of current serial reservoir simulators (which can handle on the order of 105 cells). This disparity has motivated the development and use of a variety of techniques for coarsening, or "upscaling," the geological model to a manageable level of detail. Scaleup techniques are becoming increasingly sophisticated and are now applied routinely in many reservoir simulation studies. For recent reviews of scaleup literature, see Refs. 1 through 3.
Virtually all existing scaleup techniques provide coarse scale effective (or, more properly, equivalent gridblock) permeabilities. Although some techniques additionally generate upscaled relative permeability functions (i.e., pseudorelative permeabilities), the computed coarse scale permeabilities are clearly one of the primary outputs of the upscaling process. Most existing techniques for the calculation of coarse scale permeability tensors are based on the (often implicit) assumption that the local pressure field is in some sense "slowly varying." This means that, in computing the permeability tensor for the coarse gridblock, flow through the corresponding fine scale region is typically driven by some type of "linear" pressure field. This linear field may be specified in one of several ways (e.g., periodic boundary conditions, linearly varying pressure boundary conditions, pressure-no flow boundary conditions) and there are important differences between the various methods. Nonetheless, the standard methods generally assume that the pressure field is slowly varying.
In the near-well region, the pressure field cannot be expected to be either linear or slowly varying. Further, the effects of heterogeneity in the wellblock or in neighboring blocks can significantly affect the pressure and flow fields. Many of the existing upscaling methods, therefore, may not be appropriate for flow in the immediate vicinity of a flowing well. The purpose of this article is to present and apply a general approach applicable for scaleup in the region near vertical wells. This approach will then be used to scaleup several two-dimensional and three-dimensional flow problems and will additionally be introduced into our general nonuniform coarsening scaleup methodology.4–6
Several previous investigators have considered the problem of scaleup in the near-well region7–9 and the current work builds directly on their findings and methods. Ding7 developed an approach that entails the solution of a single phase, well-driven flow problem. This solution is then used to determine wellblock transmissibilities in addition to the well index. Ding demonstrated a clear improvement with his technique over standard approaches. Lin8 developed well indexes for partially penetrating wells operating under "quasisteady state" conditions. He computed solutions for a variety of layered and heterogeneous flow problems. Soeriawinata et al.9 presented an analytical technique for the calculation of effective permeability for a coarse scale wellblock. They also demonstrated a significant improvement over standard approaches.
The approach presented in this article is most closely related to the method introduced by Ding.7 Consistent with his approach, we solve a fine grid problem and then compute upscaled wellblock transmissibilities and the well index such that the coarse grid problem provides a flow field in close agreement to the reference fine grid solution. In this work, we identify and solve specific local problems that appear to capture the key aspects of the flow field. In addition, we consider three-dimensional systems, whereas previous work was mainly concerned with two-dimensional problems. Further, we introduce some enhancements to the general approach which are capable of providing more accurate coarse scale descriptions in some cases.
Following the development of the general method, we present systematic results for single phase flow in two-dimensional heterogeneous permeability fields. This allows us to quantify the error in the standard approach, demonstrate the improvement offered by the near-well scaleup, and identify the parameter ranges in which the near-well modifications are necessary. Our findings on the performance of standard methods are consistent with the recent work of Aziz et al.10 and of Yamada and Hewett,11 who showed that standard approaches, applied in conjunction with a uniform coarsening of the fine scale description, can be quite inaccurate for horizontal well problems. We next present results for three-dimensional geostatistical models of several actual reservoirs. In these examples, the near-well scaleup is used with our general nonuniform coarsening scaleup methodology.4–6 In all cases considered our results represent an improvement over the standard approach. In some parameter ranges this improvement is relatively slight while in others it is dramatic.
We consider incompressible single phase flow in three-dimensional heterogeneous systems. Flow is driven by injection and production wells. Our interest is in computing upscaled properties such that a well in the coarse scale model will generate a flow field very close to the fine scale result. Specifically, for a fixed bottomhole pressure, we require that the well flow rates for the coarse model be very close to those in the fine scale model. Analogously, for a fixed flow rate, bottomhole pressures should be as close as possible for both.