Pilot Block Method Methodology to Calibrate Stochastic Permeability Fields to Dynamic Data
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Pilot Block Method Methodology to Calibrate Stochastic Permeability Fields to Dynamic Data Mickaele Le Ravalec-Dupin
Received: 20 January 2009 / Accepted: 4 October 2009 / Published online: 22 October 2009 © International Association for Mathematical Geosciences 2009
Abstract In the present paper, a new geostatistical parameterization technique is introduced for solving inverse problems, either in groundwater hydrology or petroleum engineering. The purpose of this is to characterize permeability at the field scale from the available dynamic data, that is, data depending on fluid displacements. Thus, a permeability model is built, which yields numerical flow answers similar to the data collected. This problem is often defined as an objective function to be minimized. We are especially focused on the possibility to locally change the permeability model, so as to further reduce the objective function. This concern is of interest when dealing with 4D-seismic data. The calibration phase consists of selecting sub-domains or pilot blocks and of varying their log-permeability averages. The permeability model is then constrained to these fictitious block-data through simple cokriging. In addition, we estimate the prior probability density function relative to the pilot block values and incorporate this prior information into the objective function. Therefore, variations in block values are governed by the optimizer while accounting for nearby point and block-data. Pilot block based optimizations provide permeability models respecting point-data at their locations, spatial variability models inferred from point-data and dynamic data in a least squares sense. A synthetic example is presented to demonstrate the applicability of the proposed matching methodology. Keywords Flow calibration · Inverse modeling · Parameterization · Blocks · Pilot points 1 Introduction Many petroleum or groundwater applications call for the modeling of the spatial distribution of transport properties in subsurface formations. Among these, one of the M. Le Ravalec-Dupin () IFP, 1 & 4 Avenue de Bois Préau, 92852 Rueil-Malmaison Cedex, France e-mail: [email protected]
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Math Geosci (2010) 42: 165–185
most heterogeneous and influential is permeability. High permeability regions form preferential flow paths, while low permeability regions form barriers to flow. Characterization of permeability is the key to forecast fluid displacements in oil reservoirs, underground gas storage sites, or aquifers. However, there is never enough data available to describe its spatial distribution. Typically, spatial variability is inferred from well tests or cores extracted from sparsely distributed wells. The soinvestigated volume constitutes only a tiny fraction of the subsurface formation. In addition, permeability was shown to strongly vary in areas of limited extent. As an example, next to the Waste Isolation Pilot Plant, New Mexico (USA), built for nuclear waste storage, permeability of the regional aquifer varies by five orders of magnitude (Lavenue
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