Data integration with multiple-point geostatistics
Sebastien Strebelle, André G. Journel & Jef Caers<\/h2>
Stanford Center for Reservoir Forecasting,
Stanford University,
Stanford, CA 94305-2220, USA.
strebell@pangea.stanford.edu
Keywords: multiple-point statistics, geostatistics, data integration, seismic, channel systems
This paper introduces a new geostatistical approach to both data integration and the realistic modeling of geological information in reservoir models. Traditional geostatistics is limited by the variogram (direct variogram or cross-variogram), which is a two-point statistics. This presentation shows that a so-called multipoint geostatistics can integrate better ancillary geological information (from outcrops, analog models), than existing traditional models. Multiple-point geostatistics also allows integrating the actual physical relationship between petro-physical properties and the reservoir data, rather than relying on a mere statistical associations as is traditional to two-point geostatistics.
Abstract
The limitations of traditional geostatistics
Traditional geostatistics calls for the modeling of two-point correlations between petro-physical/facies properties at different locations in space. In geostatistics one traditionally uses the variogram as a structural model or measure of correlation. This structural model, which is obtained from well observations, allows us to establish a link between the data and the unknown, un-sampled locations in the reservoir. This seems a highly consistent and ive procedure: the variogram model, which conditions the patterns generated in the reservoir models, is obtained from the data of that reservoir. Nevertheless, the practice of geostatistics has clearly shown that 1) it is difficult to model the variogram model from sparse well data and the variogram needs to be constructed from ancillary information such as outcrops, 2) the variogram is only a very limited measure or means of quantification of actual spatial patterns occurring in the reservoir. The variogram does not include the full spatial structure; the variogram is only a limited summary of spatial continuity. Another tradition that has emerged in practicing Petroleum Geostatistics is the inclusion of so-called soft data (seismic data, production data) by modeling the two-point cross-correlation between facies/petro-physical properties and the soft data. The cross-variogram or co-variance is used as a structural model to represent the relation between facies and soft data. Again, since the cross-covariance is obtained from the facies observation at wells and the seismic over the entire reservoir, such procedure appears as ive. However in most cases one largely neglects the spatial pattern of the seismic itself because 1) the cross-variogram is only a limited two-point correlation measure and 2) predictions of facies or petro-physical properties are only based on co-located information. Seismic however may contain spatial patterns that are related to certain geological events (channeling, shale bodies).
Sequential simulation is often used as a tool to simulate reservoir models that reflect a certain structural model such as a variogram or honor the cross-variation between the simulated model and the soft data. Essentially, at each node to be simulated one estimates the conditional probability P(A|B) of
A = {facies category occurring} or A = {petro-physical property occurring}<\/h2>
and
B = {the well data and previously simulated nodes}<\/h2>
The local conditional distribution can be Gaussian (sequential Gaussian simulation), its mean and variance being depending on the covariance between A and each of the individual events in B. The term “individual” is important, since the variogram quantifies the correlation between two events at a time only.
The approach proposed in this paper attempts at modeling the conditional distribution P(A|B) by relating facies event A to the multiple-point event B, B now being considered as one single event not a set of separate single well observations or simulated nodes. This will allow to model geological structures richer than the traditional geostatistical approaches. In the next section we address the modeling of P(A|B)
The second improvement lies in the integration of soft data into the reservoir models. Traditionally, from well observations, one would attempt at modeling the conditional distribution P(A|C) where A is the same as above and
C = {co-located seismic datum, i.e. co-located with the event A}<\/h2>
Again, this restricts too much the information provided by the seismic and neglects largely the spatial patterns and physics of the seismic itself. In order to take into account the pattern variability of seismic one therefore needs to model the conditional distribution P(A|C) where
C = {window of co-located seismic, containing the local seismic pattern}
Finally, one needs to combine P(A|B) and P(A|C) into P(A|B,C) in order to simulate reservoir models that reflect a particular geological vision and is constrained to well and seismic data.
The new tools of multiple-point geostatistics<\/h1>
<\/h1>
In order to go beyond the traditional variogram model or two-point correlation measures one needs to construct a training image or training reservoir model. This training image is an analog of what one would like to observe, in term of geological structures, in the simulated reservoir models. Indeed, as it is already difficult to extract from wells a suitable variogram model it will be even more difficult to determine higher order statistics needed to determine the conditioning of event A on a multiple-point event B. Training images provide a dense dataset from which many multiple-point statistics including two-point statistics can be extracted. Training images are often difficult to come by and when available are deemed too specific, not representative for the actual reservoir, hence using them would lead to too subjective modeling of the reservoir. However, one should acknowledge that in the current state of the art of geostatistics, multiple-point information is never quantified and is always implicitly delivered in the reservoir model through the specific algorithm applied. It is not because these higher moments are not explicitly visible, they should be deemed more ive. In fact a training image allows a full quantification of all higher order moments or patterns before any geostatistical modeling is attempted. Geological interpretation is therefore best delivered in images, which can be obtained for example using a non-conditional Boolean technique, through intense outcrop sampling, or even simple geological re-constructions.
An algorithm termed snesim, which solves a single normal equation, has been developed that allows exporting the patterns in the training image to the actual subsurface reservoir model conditioned to local well data. The core idea of the snesim code is to model, from the training image, the conditional distribution P(A|B) for any possible A and B. Therefore, the snesim program allows, before the geostatistical simulation is started, to store all possible distribution values P(A|B).
Traditionally, one would model, using a scatter-plot, the conditional distribution of facies A, given the co-located seismic data at the wells. The idea is now to model the conditional distribution of a facies event A observed at the well and a co-located window of seismic data C. The window definition and geometry should depend on the particular application at hand. A neural network approach is proposed to automatically model that conditional distribution P(A|C).
