标题：Problem 4: some consideration provided by some students in Chongqing University, China
Attached are some consideration about Problem 4 "On a Mathematical Model and an Automated System for Geologic Zonation", which are provided by some students in Chongqing University, China.
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This is interesting. My guess is that the workflow for a problem of this kind is:
Solve the local inverse problem to compute the electrical properties of the rocks. There are, of course many ways of doing this but they may already have solved that problem. (Is the problem to solve the electrical inverse problem, or is it to build a geological model?)Then, if you are lucky they will have cored the wells and have some real information about the flow properties of the rocks, such as water saturation (that's what they try to infer from resistivity logs via 'Archie's Law' - see Wikipedia) and the permeability and porosity. Then, you have to decide where layers are located in each of the wells. If there is seismic data then the topology of the stratigraphy will be known, more or less, which is a great help. If there is no seismic, the uncertainty is all the greater. One important piece of information is the opinion of a geologist as to the depositional environment of the sediments. That is, are they desert sands (eolian), river sands (fluvial, deltaic) or of various oceanic types (turbidite, and so on) (to give you a taste of the jargon)?
Then you make a big assumption. Assume that the rock properties are a realisation of some stochastic process. The easiest model (this is what we did for the Brewery problem at the Hong Kong staudy group last year) is to assume that the log of the property is a Gaussian random field (noting that permeability often has a log-normal distribution). They usually build a stratigraphic grid, ie a geometric model of the reservoir and then treat the properties of each layer as a two dimensional function, which is interpolated from the wells via a system of 'stratigraphic coordinates'. That is, you have to map the reservoir into a cube of cubes, and perform the interpolation in the transformed space. (There is an alternative method that I have been working on that avoids this, but it is not accepted by the industry even though it would be more convenient.)
From information about the deposotional environment you can guess the correlation function (the jargon is the 'variogram' which is related, and equal to 0.5<(k(x) - k(x+y))^2 >, which is a linear transformation of the correlation function). The most important parameter is the correlation length, which can be much greater than the well separation, if you are lucky. Then, the most probable interpolant of the data uses this variogram in a process known as 'Kriging'. I attach a review of mine which gives some of the background theory. There is much commercial software for performing this task. In particular the 'Petrel' package from Schlumberger. There are also some academic software packages around. Of particular interest is the GSLIB package from Stanford, which is available as a CD in the back of an Oxford University Press Book. The authors are Deutsch and Journel. Come to think of it, the book 'Geostatistical Reservoir Modelling', OUP 2002 by Clayton Deutsch is a good background reference. What usually catches everyone out is the presence of fractures, or other high permeability features (such as fossilised river beds).
Perhaps this is enough for now, as I may have completely misunderstood the real problem.
Regards,
Chris