Seismic Porosity Mapping in the Ekofisk Field Using a New Form of Collocated Cokriging

Abstract

Abstract An important practical problem in the geosciences is the integration of seismic attribute information in subsurface mapping applications. The aim is to utilize a more densely sampled secondary variable such as seismic impedance to guide the interpolation of a related primary variable such as porosity. The collocated cokriging technique was recently introduced to facilitate the integration process. Here we propose a simplified implementation of collocated cokriging based on a Bayesian updating rule. We demonstrate that the cokriging estimate at one point can be obtained by direct updating of the kriging estimate with the collocated secondary data. The linear update only requires knowledge of the kriging variance and the coefficient(s) of correlation between primary and secondary variables. No cokriging system need be solved and no reference to spatial cross-covariances is required. The new form of collocated cokriging is applied to predict the lateral variations of porosity in a reservoir layer of the Ekofisk Field, Norwegian North Sea. A cokriged porosity map is obtained by combining zone average porosity data at more than one hundred wells and acoustic impedance information extracted from a 3-D seismic survey. Utilization of the seismic information yields a more detailed and reliable image of the porosity distribution along the flanks of the producing structure. Introduction In recent years there has been considerable interest in the general problem of integrating different types of data in reservoir modelling applications. A typical example of data integration involves the utilization of seismic information to constrain the mapping of reservoir properties between wells. The aim is to use one or more densely sampled secondary attributes, such as acoustic impedance, amplitude or travel time extracted from 3-D seismic data to guide the interpolation of a related primary variable such as porosity, shale volume or depth. In a geostatistical framework, integration of secondary information is often achieved using some form of cokriging to account for the spatial cross-correlation between primary and secondary variables. The recently introduced collocated cokriging technique facilitates the integration of multiple secondary attributes by eliminating the burden of spatial cross-covariance modelling. Here, we propose a simplified implementation of collocated cokriging which uses a Bayesian updating rule to construct a Gaussian posterior distribution for the primary variable at each point. The posterior distribution is calculated by taking the product of a Gaussian kernel, obtained by simple kriging of the primary data, and a secondary Gaussian likelihood function. Through the application of this Bayesian rule, we show that cokriging estimates can be obtained directly from kriging estimates by adding a linear function of the collocated secondary data. The simplified implementation of collocated cokriging is particularly appealing as (i) it decouples the influence of the primary and secondary data, (ii) no cokriging system need be solved and (iii) no reference to spatial cross-covariance functions is required. The new technique is applied in the context of a case study of seismic porosity mapping in the Ekofisk Field. P. 21