This work shows that the traditional version of geological models of oil and gas fields obtained by a computer approach is not the only possible one and it prevents the development of modeling as a whole, since it is not truly mathematical.
Given that computers do not work with images, but with numbers, a novel approach is presented for the construction of truly mathematical geological models. The proposed model has an unusual appearance and is not intended for visual analysis, but it is more effective for forecasting. The mathematical basis of the novel approach is the cascades of fuzzy-logical matrices, which are formed from spatial coordinates and considered geological parameters.
Suppose that for each point in the geological grid there is a coordinate vector, in the simplest case these are the lateral coordinates X and Y, as well as the vertical coordinate Z. There is also a set of points (wells) at which the specified coordinates and the values of considered geological parameter, for example, porosity or oil saturation are determined. If some seismic parameter is added to them, which can be taken from grids constructed according to seismic data at the points of the wells, then four coordinates become available.
Preliminary, all considered geological parameters should be normalized in the range from -1.0 to + 1.0 in order to standardize and equalize them.
Four coordinates give six independent pairs. A matrix is constructed for each of these pairs. The matrix size can be different - from 100 per 100 to 1000 per 1000.
Next, the values of the considered geological parameter at the well points determined by four coordinates are applied to these matrices. Certainly, such points are much smaller than the points in the matrix, therefore, to fill the entire polygon of the matrix, the interpolation method is used, based on the idea of the lattice Boltzmann equations.
The number of fuzzy-logical matrices in one geological model can reach several hundreds.
Using the obtained matrices, one can construct membership functions and predict the values of the selected geological parameters, as well as the distribution of initial hydrocarbon reserves or the effectiveness of new drilling at the field.
The novel approach to geological modeling based on the cascades of fuzzy-logical matrices may seem complicated. However, the calculation of these cascades is carried out completely automatically, since they are the truly mathematical functions, and not the illustrations of the geological structure of the filed, and they are directly used in forecasting calculations.
The cascades of fuzzy-logical matrices can be considered as a new form of machine learning algorithms, for which it is advisable to use big data sets. It opens up the additional possibilities for the application of machine learning methods in geological modeling of oil and gas fields with conventional and unconventional reserves.