No abstract
The paper describes an approximate method to compute effective relative permeabilities and capillary pressure for 3D heterogeneous porous media. The method is based on the self-consistent approximation, and works in the capillary (quasi-static, low velocity) limit. The method needs as input data for the individual rock types making up the heterogeneous medium, their relative occurrence, and information related to the ratio between vertical and horizontal length scales of the heterogeneities. No actual realization of the heterogeneous medium is needed. The method is computationally very efficient. Example effective properties are given, demonstrating capillary trapping, and the tensorial nature of effective relative permeabilities. The accuracy of the self-consistent approximation is known to be quite satisfactory for most heterogeneity types. We have also studied the applicability of the capillary limit, using numerical simulation. At typical reservoir rates, the capillary limit appears to be valid only for heterogeneities at the sub-meter scale. Introduction Porous media have heterogeneities down to at least the cm-scale. It is inconceivable that a given reservoir can be described to such a level of detail. Even if it could, it would be practically impossible to honour the smallest scale individual heterogeneities in full field reservoir simulation. Fortunately, this is not necessary or even preferable, since only their average effect is of practical interest. Determination of the large scale effect of small scale heterogeneities is the upscaling problem. The reservoir upscaling problem has been a topic of active research for at least a decade. Several different concepts are in use. Under the proper separation of scales and associated conditions, one can define an upscaled fluid flow problem which is governed by partial differential equations of exactly the same form as the original small scale problem. The parameters entering the upscaled equations, in particular absolute and relative permeabilities and capillary pressure, are referred to as the effective properties of the heterogeneous medium. Effective properties are physical properties of the heterogeneous medium, valid on a larger spatial scale. The concept is defined without any reference to simulation or numerical methods. Its relation to the concept of dynamic pseudofunctions is discussed in Refs. 2 and 4. Effective properties are most conveniently defined via spatial averaging. The separation of scales conditions essentially state that the spatial scale of the heterogeneities must be much smaller than that over which spatial averaging takes place, and the scale of averaging must be much smaller than that over which averaged variables change significantly. In two-phase problems, even the large scale saturation must be approximately constant over the scale of averaging. These conditions appear to limit somewhat the applicability of the concept of effective properties. Problems containing pronounced saturation fronts would seem to be inadmissible, for instance. Fortunately, the literature suggests that the separation of scales conditions are overly severe, and that effective properties are applicable in a wide range of practical situations. Given a statistical description of the heterogeneous medium at hand, two contrasting strategies exist to compute its effective properties. Consider the one-phase case, for simplicity. In what one might term the indirect approach, an actual realization of the reservoir is constructed. One-phase fluid flow problems are solved accurately in that realization, to produce spatial pressure distributions, whereupon proper averaging delivers the effective absolute permeability. P. 119
While a number of methods, including semi-log/linear plot of water cut versus cumulative oil and water-oil ratio versus cumulative oil, have been published for extrapolating water cut during oil decline, the challenges of internal consistency, accuracy and limited areas of application have remained unresolved. These make the application of the methods threatening to the associated estimates of oil reserves and investments on water-handling facilities. In this paper, based on an investigation of internal/practical consistency and the physics of fluid flow, a didactic analysis of water cut models derivable from popular oil rate models of Arps, Li and Horne, as well as the Flowing Material Balance is carried out. This culminated in the development of an internally consistent water cut model, applicable during exponential decline of oil production. The development supports an exponential function of oil-cut vs production time. Using both hypothetic and field examples, applicability and performance of the derived model are demonstrated, and compared with six of the popular water cut models, including those of Ershagi and Omoregie, Liu, Warren and Purvis. Besides its simplicity, the proposed model consistently performs better and shows robustness. Although the form of the proposed model is here limited to exponential oil rate decline, the principles can easily be extended to other oil production decline trends. Introduction Besides the traditional focus of applying water cut data for reserves estimation, reservoir surveillance and management, the need to comply with increasingly stringent regulatory requirements for disposal of produced water, is fast becoming another key driver for reliable forecasts of water production. Clearly, for making robust investment decisions on water-handling facilities, it is critical to have reliable methods of predicting associated water production. There are several correlations in the literature and commercial packages dedicated to fitting water cut trends during oil decline. In general, these correlations can be divided into three main classes:using fractional flow theory, in which relative permeability functions are approximated, to establish water cut (or water-oil ratio) variation with oil recovery (1–4),using Arps' model and its modifications, for example semi-log water cut versus oil recovery, andobserved trends, for example linear water cut versus oil recovery. While these methods have been applied extensively, none has been found to be sufficiently robust (5), and for those not based on the fractional flow theory, besides the problem of internal consistency, curve-fitting by simple polynomial approximation do not result in satisfactory answers in most cases (6). Assumption of constant gross liquid production is also an inherent limitation of the three classes.
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