Summary The Johnson-Bossler-Naumann (JBN) and related methods for interpreting displacement experiments provide explicit, or pointwise, estimates of relative permeability that must be interpolated or smoothed when entire relative permeability functions are desired. In this paper, we compare relative permeability functions obtained with an implicit (regression-based) method with those estimated with explicit values. We show that the functions obtained with the regression-based method are most consistent with the measured data.
Introduction The petroleum industry relies extensively on numerical reservoir simulation to predict how a reservoir will respond to different operating conditions. Those predictions are used to make important production decisions in the design of subsequent oil recovery operations. Accurate estimates of reservoir-rock properties, such as relative permeability curves, are desired to obtain reliable predictions of reservoir behavior. Accurate estimates of relative permeability curves are also desired for laboratory studies assessing various conditions, such as wettability and surface tension. Relative permeability curves normally are obtained from dynamic fluid-displacement experiments on reservoir core samples. They are not measured directly. Instead, they are inferred or estimated from measurements made during fluid-displacement experiments. We refer to the process of estimating relative permeabilities from measured data as an interpretive procedure. There are two distinct approaches for interpreting unsteady-state coreflood data to obtain relative permeability estimates. The JBN and related methods are explicit methods because relative permeability values are computed explicitly from coreflood data. With implicit methods, relative permeability curves are adjusted until values computed with a mathematical simulation of the laboratory experiment match, in some sense, the measured data. Interpretive procedures require some mathematical representation or model of the observed physical phenomena within which the properties of interest may be related to observed or measured quantities. The validity of the estimates is limited by the suitability of the model to represent the physical phenomena occurring during the experiment. Implicit procedures are not restricted to a specific mathematical representation of the experiment, whereas the explicit procedures are based on the Buckley-Leverett representation of the displacement process. One notable omission in the Buckley-Leverett representation is capillary pressure effects. When those effects are present in the experiment but not accounted for in the interpretive procedure, one may expect erroneous estimates of the relative permeability values. Displacement experiments normally are conducted at a high flow velocity to negate capillary pressure effects so that explicit methods may be used, although two methods have been proposed that attempt to mitigate capillary effects in explicit calculations of relative permeability values. The effects of capillary pressure can he included when data are interpreted with implicit methods. A key feature of explicit methods is that point values of relative permeabilities are computed; i.e., relative permeability estimates are obtained as values corresponding to some finite set of saturation values. In such a form, the relative permeability estimates generally are not suitable for entry into reservoir simulators or other subsequent analyses. Instead, an entire functional representation is desired-i.e., one should provide for the calculation of relative permeability values at any saturation value. This can be accomplished by fitting the relative permeability values by interpolation or smoothing and extrapolating those curves across the saturation region corresponding to the Buckley-Leverett flood front. Although, in principle, adequate smoothing or interpolation procedures may he devised, there is generally no guarantee that the resultant curves provide accurate predictions of the measured data when entered into a mathematical simulation of the experiment. If the curves do not provide accurate predictions of the experiments, they are probably not accurate representations of the relative permeability functions. To the best of our knowledge, the process of constructing and validating functional representations of explicitly calculated relative permeability values has not been addressed in the literature. The estimates provided by implicit methods (by contrast) are entire relative permeability curves or functions. Consequently, estimates are provided for the entire saturation range, so a separate data-filling step is not required. The curves obtained for the chosen functions are those that are most consistent with the measured data. In this paper, we compare and contrast explicit and implicit methods for estimating relative permeability curves. To compare the two approaches directly, we use the Buckley-Leverett representation in the implicit analysis so that the computational processes can he compared under conditions of the same physical description of the displacement process. The Buckley-Leverett representation is most suitable for analysis of a stable displacement of incompressible fluids in a (macroscopically) homogeneous medium, under conditions for which capillary effects are negligible. This representation, however, might also be useful for an approximate analysis of more complex situations for which heterogeneities or displacement instabilities give rise to fluid distributions that are not represented accurately with the ID Buckley-Leverett formulation. Note that the features investigated here would also be factors in comparing explicit and implicit methods that include capillary effects, although other features may also he important. We address, for the first time, the process of constructing estimates of relative permeability curves from explicitly calculated relative permeability values and validating the estimates with measured data.