Uncertain Predictions of Flow and Transport in Random Porous Media: The Implications for Process Planning and Control

摘要

Traditional predictions of flow and transport in porous media are based on mass balance equations in the form of partial differential equations (PDEs), where the flux at every point is defined by Darcy's law, q=KVh, i.e., the flux is proportional to hydraulic head gradient, where K is the hydraulic conductivity of the medium (a tensor or a scalar; essentially, a material property); it is further assumed that Darcy's law applies to transient multiphase flow in three dimensions [14,26,36,55]. The solutions of these PDEs constitute groundwater models, oil reservoir simulators, geothermal models, and models of flow and transport in soils/vadosezone. Due to the similarity between the linear Darcy's law and Ohm's law in electricity, Fourier law in heat conduction, and Hooke's law in elasticity, such models (or PDE solutions) are similar and commonly interchangeable between these fields. Natural porous formations are heterogeneous, and display spatial variability of their geometric and hydraulic properties. This variability is of irregular and complex nature. It generally defies a precise quantitative description because of insufficient information on all relevant scales [9,18,26, 29,30,32,33,86,91]. In practice, only sparse measurements are available (limited by cost of drilling and monitoring). Under lack of exhaustive information, the higher the variability, the higher is the uncertainty. Geostatistics is commonly used to analyze and interpolate between measurements in mining and oil explorations, as well as hydrology and soil sciences, using methods such as "kriging", where the uncertainties in "krigged" values are also quantified [33,35,36,42,56-58,76,77,81,83,84,87, 104,105]. Frequently, these data are collected on different scales that may differ from the required scale of predictions. The task of quantitatively relating measurements and properties on different scales is difficult and intriguing [4,5,7,13,27,29, 30,38-40,46,59,78,86,91,101,108,109]. Lack of information in both observed results (output) and measured material properties (parameters) causes uncertain predictions. Spatial variability and uncertainty have lead engineers and geologists to use probabilistic theories that translate the uncertainty to a random space function (RSF) or a random field, consisting of an ensemble of (infinite number of) equally probable "realizations" of parameter values, all having the same spatial statistics, particularly correlation structure [107,76,77,23,33,35,36,42,56,58,83,84, 85,87,91]. Imbedded in this approach is a geostatistical model of an assumed joint pdf. In practice, only the first two moments are considered, with an underlying assumption of multivariate normal distribution; in particular, the theoretical semi-variogram (or simply, "variogram") - the reciprocal of the covariance function, and the mean and variance of the pdf. Since these joint moments are inferred from spatial data, the assumption of ergodicity (i.e., assuming that the ensemble and spatial statistics are identical - a theorem that cannot be proven on real data) must be invoked, which, in turn, implies some kind of stationarity (or statistical homogeneity) [33,35,36,49]. Further, in order to determine the variogram model from available spatial data, an inverse method has to be used to estimate the parameters of this variogram; sensitivity to data errors on one hand, and identifiability problems (of model parameters) on the other hand [81,87,104,105] lead to uncertainty in the geostatistical model itself, which is usually ignored (in fact, the common practice is to fit the variogram model to the experimental variogram by eye and by subjective judgement of model type, degree of stationarity (drift), and statistical anisotropy). Another ignored uncertainty is in the "measured" hydraulic conductivity value that are actually inferred from hydraulic tests interpreted by simplistic models that assume local homogeneity, which is somewhat inconsistent with the RSF app