Scale Effects on the Application of Saturation-Height Functions to Reservoir Petrofacies Units

Summary

An internally self-consistent reservoir model has been constructed to investigate the effects of scale on the numerical form and application of four saturation-height functions and on the resulting predictions of reservoir fluid saturations. The treatment uses reservoir partitioning into strongly characterized, physically distinct litho-units, or petrofacies, for purposes of petrophysical interpretation, and a separate, correlative reservoir zonation for subsequent parameter mapping and volumetrics. The context is therefore that of a deterministic reservoir model.

The numerical difference between a petrofacies-specific saturation-height function calibrated at the well-log scale (<1 m) and the same type of function generated at the reservoir zonal scale (>10 m) varies with the function used and the physical character of the particular petrofacies unit. The application of saturation-height functions established separately at these two different scales has allowed the departures introduced through scale transgression to be assessed quantitatively. This has been done by comparing the predicted zonal hydrocarbon saturations generated for reservoir mapping with benchmark values of the same parameter. For this deterministic approach to reservoir description, an extended algorithm that uses bulk volume water as predictand provides the most accurate evaluation of water saturation at the reservoir zonal scale, which must be used in volumetrics applications. In contrast, an exponential form of the saturation-height function shows relatively large errors at both scales.

The significance and variability of scale effects have required the formulation of a structured approach to the optimum application of saturation-height functions in deterministic reservoir description. Through this facility, it is possible to contain the uncertainty associated with the distribution of hydrocarbon saturation at the reservoir mapping and volumetrics stages.

Introduction

The application of saturation-height functions constitutes a primary link between the reservoir properties inferred at wells through core and log analysis and the distribution of those same properties within a reservoir model for purposes of evaluating hydrocarbons in place. This linkage is founded on reservoir mapping, which requires that lithologically descriptive reservoir parameters be quantified at a regular array of grid nodes. The mapping process draws upon stratigraphy in one or more of its several forms. For purposes of reservoir geology, stratigraphy is expressed in terms of reservoir zonation. The stratigraphic zones comprise the reservoir architecture, the geological configuration within which reservoir properties are distributed and mapped.

There is an immediate scale disparity between the petrophysical interpretation of core and well-log data, which are at the centimeter-to-meter scale, and the mapping of reservoir properties through correlatable reservoir zones, which have thicknesses that can exceed the decameter scale. This disparity has important implications for the generation of saturation-height functions. Like most other petrophysical algorithms, these functions are empirical; therefore, they are applicable only over the ranges of the independent and dependent variables used to establish them. Further, empirical algorithms are strictly valid only at the scale at which they have been established because the constituent coefficients, exponents, and intercepts delivered by regression analysis are known to be scale-dependent; there have been clear indications of this dependence in the broader context of petrophysics.¹

Saturation-height functions should therefore be established at the scale at which they are to be applied. Because the scale of application is that of geological mapping, saturation-height functions should be established at the reservoir zonal scale. However, the source data for these functions are core measurements and well logs. For this reason, many saturation-height functions founded on core and log data have been applied indiscriminately at the reservoir zonal scale, a practice that mixes the scales of establishment and application and thereby degrades the prediction of water saturation (S_w).

The purpose of this paper is to examine the effects of a scale transgression of this kind in the generation and use of saturation-height functions. Such an exercise requires an exact solution against which to judge the predictive performance of an algorithm. This means that actual field data cannot be used because they are affected by too many uncertainties. For this reason, the analysis draws upon a reservoir model that has been designed to map exactly reservoir properties between two wells, specifically onto an intermediate grid node at which water saturations can be predicted and compared with benchmark values. The design and building of an exact reservoir model is not a trivial exercise, and special consideration has been given to the requirements of the model in the context of the stated objectives.

This paper describes the construction of a 2D, internally consistent reservoir model, in which every parameter that is used in the prediction of water saturation is known everywhere. Moreover, the depth variation of water saturation itself has been quantified at the two control wells and at one point of prediction, the hypothetical grid node, where benchmark values of zonally averaged water saturation can be used to judge the predictive performance of saturation-height functions. The functions themselves are established using data from the control wells. These functions are generated in two ways: first, by regressing input data at the well-log scale (<1 m), and then by regressing data that have been averaged over physical units of the reservoir and therefore relate to the reservoir zonal scale (>10 m). The differences between the results generated by separately applying both groups of algorithms provide a quantitative guide to the effects of scale on the use of saturation-height functions.