NIELSEN, SØREN B., Department of Earth Sciences, The University of Aarhus, KERRY GALLAGHER, Imperial College of Science, Technology and Medicine.

Abstract: Efficient Sampling of 3-D Basin Modelling Scenarios

Quantitative basin models contain many poorly constrained parameters which need to be adjusted in the process of data fitting and model calibration. Least squares inverse methods have been applied for this purpose by a number of authors (e.g. Lerche, 1988). A single solution is, however, not very informative because statistical data errors and uncertain a priori parameter values induce an infinite family of possible solutions. Many other fine solutions therefore exist which, unfortunately, may have very different implications regarding the hydrocarbon potential of an area.

This ambiguity problem has been considered in a steady state 1-D model (Gallagher and Sambridge, 1992) and in a transient 1-D model (Nielsen, 1996). Nielsen (1998) used repeated least squares inversion of statistically perturbed data and a priori information in a 2-D transient model to quantify the ambiguity induced in the thermal and maturation history. However, to achieve a truly representative sample of basin modelling scenarios statistical sampling techniques such as Markov Chain Monte Carlo (MCMC) must be used.

The Bayesian approach to the solution of inverse problems is deceivingly simple. It combines the a priori information, ? (m) on the model vector, m, with a model likelihood function, L(m), to achieve the a posteriori probability density function, s (m), representing the information we have on the system (Tarantola and Valette, 1982):

L(m) measures the fit between observed data and data predicted from the model, and k is a normalisation constant. Eq(1) solves the general non-linear inverse basin modelling problem because, in principle, s(m) can be mapped by evaluating the known right-hand side by forward runs of the basin model. However, this approach rapidly gets out of hand. If the model vector, m, contains 10 variable parameters each discretised by 10 values, exhaustive mapping of s(m) requires 10¹⁰ forward runs. On the passing we note that if ? (m) and L(m) are chosen to be Gaussian functions the least squares solution corresponds to a local maximum of s(m).

A more feasible approach to the exploration of s(m) is statistical sampling by MCMC.This technique allows for producing, say, 10³ realisations of the model vector, m, drawn from the a posteriori distribution, s(m). This is then a truly representative sample of the basin model scenarios. Unfortunately on the order of 10⁶ forward runs may be needed to obtain the 10³ random draws from s(m). For a general 3-D basin model incorporating fluid flow this may be computationally prohibitive. However, when considering thermal aspects of basin models such as heat flow, temperature history, maturation, and petroleum generation Fast Approximate Forward (FAF) procedures exist (Nielsen, 1998) which can generate on the order of 10 ⁶ 3-D forward runs fast enough to exploit the potential of MCMC.