A Markov Chain Model for Subsurface Characterization: Theory and Applications
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Letter to the Editor Comment on “A Markov Chain Model for Subsurface Characterization: Theory and Applications” by A. Elfeki and M. Dekking Elfeki and Dekking (2001) present a specific type of Markov chain model, for which they developed a simulation method that conditions model realizations to well data. On page 586 of their paper, the authors make the following statement about the conditioning of a different type of models, known as object-based models in reservoir engineering: “One of the advantages of this [i.e., the authors’] methodology is that, in conditioning to wells the geological features on a certain level (vertical coordinates) are kept at their level in the simulation. The objectbased models used by Chessa and Martinius (1992) and Chessa (1995) do not have this advantage.” The statement in the last sentence is incorrect. Although the statement refers to a different class of models than the model used by the authors, the practical implications of conditioning models of reservoir location and geometry to well data may be such that it seems reasonable to make some essential points about this problem. The purpose of this letter is to rephrase the authors’ remark correctly and to summarize some problems that play a crucial part in the correct conditioning of object-based models. At the same time, this letter gives a short summary of different conditional simulation approaches for these models. The cited study by Chessa and Martinius (1992) is dated, and more research about the conditioning problem for object-based models has been done afterwards. The mathematical model behind object-based models is a marked point process (Stoyan, Kendall, and Mecke, 1987; Daley and Vere-Jones, 1988). The ‘points’ of the ordinary (i.e., unmarked) point process are used to denote the reference locations of sandstone bodies, shales or faults, and the marks of these points define the geometry of such bodies (e.g., in terms of length, width, and thickness). Object-based models are based on this concept and were introduced by Haldorsen (see, e.g., Haldorsen and Lake, 1984).
Published online: 10 November 2006. 503 C 2006 International Association for Mathematical Geology 0882-8121/06/0500-0503/1
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Letter to the Editor
Haldorsen’s object-based simulation method does reproduce the vertical location (i.e., stratigraphic depth) and thickness of bodies intersected by wells, by simply assigning their observed values to intersected bodies. However, Berkhout, Chessa, and Martinius (1996) and Chessa (1995) correct Haldorsen’s method, since the latter does not produce the theoretically correct statistics about the horizontal location and the length of intersected bodies. This can be argued on the basis of the so-called waiting-time paradox in probability theory, which implies that well sands or shales are larger on average than interwell sands or shales. By simulating well and interwell bodies from the same length distribution, Haldorsen’s method may underestimate the average length of well sands or shales by a factor 2. The valid
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