The problem is that of modelling and characterizing heterogeneity
in reservoirs in the shallow subsurface of the earth. The
classical approach here is to use variograms which are
mathematically equivalent to autocovariance functions. However,
for an engineer conditional probabilities are easier to interpret
than autocovariance functions, which makes modelling by Markov
chains attractive. In the simplest model the states of the chain
represent the different lithologies like sand, shale and rock. The
reservoir is discretised to a certain scale into square regions
and the transition matrix of the chain describes the probabilities
of transition from one lithology to another. There is a major
problem here: Markov chains describe a one-dimensional chain of
states, but for reservoirs one would of course prefer a 2d
(sections) or 3d description. Since the beginning of the seventies
a nice mathematical theory has been developed of objects that
generalise Markov chains to higher dimensions - regretfully the
objects go under the somewhat confusing name Markov random fields.
Even more regretfully engineers do often not like to use Markov
random fields. The main reason for this is the problem known as
``the intractability of the normalising constant'', which makes
simulations expensive and likelihoods difficult to work with. In
Elfeki's model two ordinary Markov chains, one for the horizontal
direction and one for the vertical direction, are what he calls
``coupled''^{3} to generate a two-dimensional
image of a section of a reservoir.

0.6Fig1.eps

Figure 1. *Simulation of a reservoir with 5 types of
lithology.*

The image, built out of rows of little squares called pixels, is generated from top to bottom and from left to right by specifying the states in an initial top row and left column, and then making transitions according to the following rule: the state of pixel is deduced from the states of pixels and by

Here and are the two ordinary Markov chains
representing the horizontal, respectively the vertical direction,
and is a normalising constant which arises by forcing
transitions in the and chain to the *same *
state. Hence, writing for the set of all states,

Actually the process

See Figure 1 for a simulation of a section of a field with 5 different types of lithology. The vertical transition probabilities can be estimated from information from well logs, and the horizontal transition matrix from geological surveys.