Calculation of geological uncertainties associated with 3-D geological models

3-D geological models are built with data collected in the field such as boreholes, geophysical measurements, pilot shafts or geological mapping. Unfortunately, these data are always limited in number. It implies that geological information is sparse and subsurface models are thus always built of both subjective interpretation and mathematical interpolation/extrapolation techniques. These models are therefore uncertain and this uncertainty is rarely pointed out in a geological prognosis. Our study proposes to bring a new methodology for the evaluation of geological uncertainties related to 3-D subsurface models and to test its potential use. The methodology we propose is based on the 3-D subsurface model, which is here considered as the most probable prediction (notion of best guess). The various geological interfaces that compose the subsurface model are handled individually as Gaussian random fields. At each location of an interface, the random function Z(u) describing the position of this interface is composed of a deterministic part m(u) which represents the expected position, and a random part σ(u)ε(u) which describes fluctuations around the predicted position. Then, a model of spatial variability (a variogram function γ(h)) is proposed in order to condition the random field according to available observations. Several structural constraints, such as the shape of folds and the thickness of layers can also be accounted for in this model. At this point, we are able to estimate the local variance all over the study area by the application of the kriging technique. Finally, the variability is converted into three-dimensional information by calculating probabilities, this describes the occurrence of the various rock masses that are present in the study area. The probabilities are calculated according to intersection rules that govern the stratigraphic sequence of the subsurface model, and they allow us to build a probabilistic model of subsurface structures in the form of a three-dimensional probability field. All of this has been incorporated in a computer program.