Impacts of Scale-up on Fluid Flow from Plug to Gridblock Scale in Reservoir Rock
Authors: Kumar Arun T.A. , Jerauld G.R. 
Published: December 2007

Abstract
Computational tools were developed to scale up petrophysical properties in a spatially periodic, mixed-wet, heterogeneous rock and applied to realistic reservoir descriptions comprising more than 106 fine-grid data points from a sandstone oil reservoir. Two-phase effective relative permeability and capillary pressure were obtained by scale-up from plug to gridblock scale. The steady-state technique employed is significantly faster than unsteady-state scale-up methods that require extensive fine-grid simulation. The results illustrate differences between scaled-up horizontal and vertical relative-permeability curves and rock curves. Differences between using periodic and no-flow boundary conditions were assessed. Effective properties are compared using direct pressure solution and averaging techniques for the limiting cases of low and high capillary number. The effect of wettability variations on the scaled-up relative permeabilities is illustrated. At low capillary number, considerable saturation variation was observed in the ratio of the total mobility in the vertical and horizontal directions. For the descriptions tested, the calculated capillary-number, dependent effective horizontal relative-permeability curves were close to the high-capillary-number limit at realistic reservoir flow rates.

Introduction
Modeling of fluid flow through sedimentary units is of great importance in assessing the performance of both hydrocarbon reservoirs and aquifers. On the one hand, most sedimentary rocks display structure from the mm or cm scale upwards. The permeability contrast and thickness of these structures vary considerably and depend on the depositional conditions, but some degree of small-scale permeability variation is always present. On the other hand, the reservoirs of economic importance involve huge volumes of rock that far exceed the limits of current computers if detailed descriptions are included. It is therefore necessary to treat simulation of such processes in stages and perform some sort of scale-up procedure before performing simulations to compute the overall performance of a reservoir.
Scale-up in this context refers to a technique of extrapolating results obtained at one scale (size), i.e., "fine-grid" to another larger scale, i.e., "coarse-grid". All properties (e.g., permeability, relative permeability, and capillary pressure) defined at a fine-grid system are transformed into equivalent properties defined at a coarse-grid system such that the two systems act similarly. The input properties at the coarser scale take into account the flow effects of the smaller-scale heterogeneity structure. Using a scale-up technique, we generate a representative average or effective property or function (for example permeability, relative permeability, capillary pressure, etc.) for a coarser grid, which is the gridblock scale in our case. The methods for calculating effective relative permeabilities include those of Jacks et al., Kyte and Berry, Stone, King et al. and Pickup and Sorbie.
Solving a single-phase flow problem is a fundamental step towards solving multi-phase problems. On the whole, the literature suggests that the single-phase scale-up is largely understood. There are a variety of techniques available and well-known limits on the accuracy of methods. Simple averages are quick and economical but are often not very accurate, while detailed numerical simulations generally give more accurate results but can be computationally intensive. For the size of problems currently undertaken in typical reservoir simulations, the computational effort necessary to perform rigorous single-phase scale-up is moderate and such approaches are rapidly becoming the norm. Multiphase scale-up is, however, much more difficult, and no consensus exists on the best methods to use except in a few situations.
The single-phase tensor technique forms the basis for the development of the multiphase tensor method. To date, two extreme cases, capillary-dominated and viscous-dominated flows, have been considered in the absence of gravity. The first step of the technique is to carry out a fine-scale multiphase simulation to calculate fine-scale properties, e.g. saturation, fractional flow, and total mobility. The next step is to use the single-phase periodic boundary condition methods to calculate the effective total mobility tensor at the coarse-scale. Lastly, the effective relative-permeability tensor is computed from the effective total mobility tensor and the total flow-weighted fractional flow. Thus far, this method has been tested only in a homogeneous field. The fine-scale multiphase simulation necessary before scale-up and the time-dependent nature of the effective phase-permeability tensor obtained cause the technique to be computationally expensive.
To understand the problem fully, one must consider two important aspects. First, the appropriate scale-up technique to use depends on the scales one is trying to account for as well as the dimensionless groups that control the scales. These have been reviewed in a recent paper. The basic idea is that one should consider the ratios of various forces controlling flow: gravitational, capillary, and viscous forces.
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