Abstract
Many problems in reservoir characterization require the formulation and solution of an inverse problem. The definition of the inverse problem is the prediction of reservoir properties from measurements made on the reservoir. The most familiar example of an inverse problem is well log interpretation. Well logging involves the acquisition of various kinds of data for the purpose of predicting reservoir properties of the earth formations near the borehole.
Conventional inverse methods used in the industry today typically involve the constrained minimization of a weighted sum of the squared deviations between a set of measurements and a forward model equation or equations. The forward model equations are either empirically or theoretically derived equations. These equations relate the reservoir property to be predicted (e.g., saturation, permeability, etc.) to measurements (e.g., resistivity, T2 distributions, porosity, etc.). Adjustable parameters in the forward model equations are determined using a "calibration database" of laboratory measurements made on a suite of representative samples. This traditional methodology suffers from fundamental limitations that reservoir rocks and fluids are too complex to be accurately described by the simple idealized equations that are used today as forward models. Additionally, the forward models contain adjustable parameters which can vary over a wide range, thereby leading to additional inaccuracies in reservoir properties.
Our paper discusses a new model-independent inversion method that overcomes the limitations and inaccuracies of the conventional method. The new inversion method predicts reservoir properties without the need to use idealized model equations or to solve minimization problems. The new method divides the calibration database into input measurements (e.g., used for the predictions) and outputs (e.g., the reservoir properties to be predicted). The outputs are mapped to the inputs using model-independent mapping functions constructed from Gaussian radial basis functions (RBF). The RBF mapping functions are accurate representations of the functional relationship between the database inputs and outputs. The coefficients of the mapping function can be uniquely determined using the database. Once the coefficients are determined, there are no adjustable parameters.
The new method is applicable to a wide variety of reservoir characterization problems. The construction of the mapping functions is presented in detail. We apply the method to a challenging problem in nuclear well logging — accurate predictions of formation thermal neutron capture cross sections (sigma) from well logging data. Logs of formation sigma predicted using the conventional regression method and the new method are compared and discussed.
