Randal J. Barnes

High-order line elements in modeling two-dimensional groundwater flow

This paper presents a new Analytic Element formulation for high-order line elements in modeling two-dimensional groundwater flow. These elements are line-doublets, line-dipoles and line-sinks. The jump functions for line elements are expressed as Chebyshev series. The unknown coefficients are computed by applying the principle of overspecification to the boundary conditions. The use of the high-order elements and the principle of overspecification have resulted in high precision and significant improvements in computational efficiency compared to the existing collocation-based formulation. The new formulation is currently being used in the development of the Metropolitan Area Groundwater Model for Twin Cities, Minnesota, USA and for enhancements of the next release of National Groundwater Model for The Netherlands.

Two-dimensional flow through large numbers of circular inhomogeneities

An implicit analytic solution is presented for two-dimensional groundwater flow through a large number of non-intersecting circular inhomogeneities in the hydraulic conductivity. The locations, sizes and conductivity of the inhomogeneities may be arbitrarily selected. The influence of each inhomogeneity is expanded in a series that satisfies the Laplace equation exactly. The unknown coefficients in this expansion are related to the coefficients in the expansion of the combined discharge potential from all other elements. Using a least squares formulation for the boundary conditions and an iterative algorithm, solutions can be obtained for a very large number of inhomogeneities (e.g. 10,000) on a personal computer to any desired precision, up to the machines limit. Such precision and speed allows the development of a numerical laboratory for investigating two-dimensional flow and convective transport.

Three-dimensional flow through large numbers of spheroidal inhomogeneities

An implicit analytic solution is presented for three-dimensional (3D) groundwater flow through a large number of non-intersecting spheroidal inhomogeneities in the hydraulic conductivity. The locations, dimensions, and conductivity of the inhomogeneities may be arbitrarily selected. The specific discharge potential due to each inhomogeneity is expanded in a series that satisfies the Laplace equation exactly. The unknown coefficients in this expansion are related to the coefficients in the expansion of the combined specific discharge potential from all other elements. Using a least-squares formulation for the boundary conditions, a superblock approach, and an iterative algorithm, solutions can be obtained for a very large number of inhomogeneities (e.g. 10,000) on a personal computer to any desired precision, up to the machines limit. Such speed and precision allows the development of a numerical laboratory for investigating 3D flow and convective transport.

The Variogram Sill and the Sample Variance

The relationship between the sill of the variogram and the sample variance is explored. The common practice of using the sample variance as an estimate of the variogram sill is questioned, and a conceptual framework for determining the appropriateness of this heuristic is constructed.

Infill sampling criteria to locate extremes

Three problem-dependent meanings for engineering “extremes” are motivated, established, and translated into formal geostatistical (model-based) criteria for designing infill sample networks. (1) Locate an area within the domain of interest where a specified threshold is exceeded, if such areas exist. (2) Locate the maximum value in the domain of interest. (3) Minimize the chance of areas where values are significantly different from predicted values. An example application on a simulated dataset demonstrates how such purposive design criteria might affect practice.

The superblock approach for the analytic element method

An approach, the superblock approach, is presented for increasing computational efficiency of analytic element models. The approach is based on computing the combined effect of functions using both asymptotic expansions and Taylor Series expansions. The superblocks are used to reduce both the computational effort required to determine the coefficients in the analytic element model and to reduce the effort expended in generating contour plots and streamlines. An application of flow is presented in an aquifer with one hundred thousand circular impermeable objects. The errors in the simulation are within the machine accuracy.

Modeling secondary oil migration with core-scale data: Viking Formation, Alberta basin

The Viking Formation in the Alberta basin contains approximately 88.7 x 106 m3 (5.579 x 108 bbl) of recoverable oil, which migrated more than 200 km, as indicated by oil-source rock correlation. Simulating the mechanisms controlling secondary oil migration (hydrodynamics, buoyancy, and permeability heterogeneity) is beneficial for exploration, but it remains extremely difficult to predict oil occurrences. Although core-scale petrophysical data for the Viking Formation are abundant (> 69,000 core plugs), the extent of fracture permeability and permeability alteration due to diagenesis are unknown. Moreover, sampling bias may affect the permeability distribution in unpredictable ways. Numerical simulations of oil migration were conducted using the highest core-plug measurement of permeability from each borehole to obtain an upper bound on oil migration velocities. This permeability model is not appropriate for simulating stratigraphic entrapment of oil, but it does reveal that core-scale data are in the appropriate range of magnitude to have allowed significant oil migration. Regional groundwater flow was essential for charging several of the largest and most distant oil fields in the Viking Formation. Maximum core-plug permeability data are useful for modeling the extent of secondary oil migration and may have applications to fluid flow and transport modeling in other foreland settings.

Infill-sampling design and the Cost of classification errors

The criterion used to select infill sample locations should depend on the sampling objective. Minimizing the global estimation variance is the most widely used criterion and is suitable for many problems. However, when the objective of the sampling program is to partition an area of interest into zones of high values and zones of low values, minimizing the expected cost of classification errors is a more appropriate criterion. Unlike the global estimation variance, the cost of classification errors incorporates both the sample locations and the sample values into an objective infill-sampling design criterion.

Bounding the required sample size for geologic site characterization

The proposed objective of limited sample geologic site characterization is to minimize the chance of unknown and unexpected extremes. This problem proves to be extremely difficult when the data are spatially correlated. A generalization of the classical one-sided nonparametric tolerance interval, based upon the statistical concept of associated random variables, establishes a rigorous, almost distribution-free, tool for computing the minimum required sample size for site characterization. An upper bound on the required number of samples follows from a heuristic measure for the quantity of information in a spatially dependent sample; the measure presented is the equivalent number of uncorrelated samples and is calculated using an estimated variogram. An empirical check of the upper and lower bounds, using more than 2 million simulations and seven real data sets produces a heuristic rule for quantifying the required number of samples.

Adding bounds to kriging

The ordinary kriging interpolation algorithm is extended by the inclusion of explicit lower and upper bounds on the estimate. The associated estimation variance is written as the ordinary kriging variance plus a non-negative correction term.