Geological Modelling and Reservoir Simulation

Abstract

Summary.The main mathematical techniques used in building geological models for input to fluid flow simulation are reviewed. The subject matter concerns the entire geological and reservoir simulation modelling workflow relating to the subsurface. To provide a realistic illustration of a complete fluid flow model, a short outline of two-phase incompressible flow through porous media is given. The mathematics of model building is discussed in a context of seismic acquisition, processing and interpretation, well logging and geology. Grid generation, geometric modelling and spatial statistics are covered in considerable detail. A few new results in the area of geostatistics are proved. In particular the equivalence of radial basis functions, general forms of kriging and minimum curvature methods is shown. A Bayesian formulation of uncertainty assessment is outlined. The theory of inverse problems is discussed in a general way, from both deterministic and statistical points of view. There is a brief discussion of upscaling. A case for multiscale geological modelling is made and the outstanding research problems to be solved in building multiscale models from many types of data are discussed.

Hierarchical Nonlinear Approximation for Experimental Design and Statistical Data Fitting

“…To this end, we prefer to work with kriging, a well-established statistical method first introduced by Krige in the 1950's and then formalized as a fundamental geostatistical technique by Matheron [17] in 1963, and later utilized by Sacks, Welch, Mitchell, and Wynn [25] in computer experiments. We remark that kriging is under specific conditions equivalent to Bayesian methods (see [6] for details), and also to radial basis function interpolation (as recently shown in [9,32]).…”

Uncertainty quantification and optimal decisions

“…4 To determine an optimal policy we first need to define a function that measures the cost, in some sense, of implementing any given policy. This is another situation that calls for mathematical modelling: in this case driven by economic and statistical theory.…”

Covariance functions motivated by spatial random field models with local interactions

“…Finally, we define a generalized spectrum of correlation scales λ T HE theory of spatial random fields (SRFs) is a powerful mathematical framework for modelling spatial variability [1]- [3]. The SRF theory has a multidisciplinary scope of applications that include fluid mechanics [4], computational and probabilistic engineering mechanics [5], materials science [6]- [8], hydrological modeling [9]- [11], petroleum engineering [12]- [14], environmental monitoring [15]- [17], mining exploration and mineral reserves estimation [18], [19], environmental health [20], geophysical signal processing [21], image analysis [22], [23], machine learning [24] 1 statistical cosmology [25], [26], medical image registration [27], as well as in structural and functional mapping of the brain [28]- [30].…”

“…Proposition 2. The Bessel-Lommel covariance C BL xx (z; θ) as defined in (10) and (11) is given by means of the following tripartite sum, where z = k c r, d ≥ 2, and ν = d/2 − 1: I: Lommel functions S ν+2l,ν−1 (z) and S ν+2l+1,ν (z) for l = 0, 1, 2 used in C BL xx (z; θ) given by (14). The expressions are based on (12) where d ≥ 2 is the space dimension and ν = d/2 − 1.…”