Upscaling unsaturated hydraulic parameters for flow through heterogeneous anisotropic sediments

We compare two methods for determining the upscaled water characteristics and saturation-dependent anisotropy in unsaturated hydraulic conductivity from a field-scale injection test. In both approaches an effective medium approximation is used to reduce a porous medium of M textures to an equivalent homogenous medium. The first approach is a phenomenological approach based on homogenization and assumes that moisture-based Richards’ equation can be treated like the convective–dispersive equation (CDE). The gravity term, dK_z(θ)/d(θ), analogous to the vertical convective velocity in the CDE, is determined from the temporal evolution of the plume centroid along the vertical coordinate allowing calculation of an upscaled K_z(θ). As with the dispersion tensor in the CDE, the rate of change of the second spatial moment in 3D space is used to calculate the water diffusivity tensor, D(θ), from which an upscaled K(θ) is calculated. The second approach uses the combined parameter scale inverse technique (CPSIT). Parameter scaling is used first to reduce the number of parameters to be estimated by a factor M. Upscaled parameters are then optimized by inverse modeling to produce an upscaled K(θ) characterized by a pore tortuosity–connectivity tensor, L. Parameters for individual textures are finally determined from the optimized parameters by inverse scaling using scale factors determined a priori. Both methods produced upscaled K(θ) that showed evidence of saturation dependent anisotropy. Flow predictions with the STOMP simulator, parameterized with upscaled parameters, were compared with field observations. Predictions based on the homogenization method were able to capture the mean plume behavior but could not reproduce the asymmetry caused by heterogeneity and lateral spreading. The CPSIT method captured the effects of heterogeneity and anisotropy and reduced the mean squared residual by nearly 90% compared to local-scale and upscaled parameters from the homogenization method. The Pacific Northwest National Laboratory is operated for the US Department of Energy by Battelle under Contract DE-AC05-76RL01830.     

Bias and uncertainty in regression-calibrated models of groundwater flow in heterogeneous media

Groundwater models need to account for detailed but generally unknown spatial variability (heterogeneity) of the hydrogeologic model inputs. To address this problem we replace the large, m-dimensional stochastic vector β that reflects both small and large scales of heterogeneity in the inputs by a lumped or smoothed m-dimensional approximation γθ_∗, where γ is an interpolation matrix and θ_∗ is a stochastic vector of parameters. Vector θ_∗ has small enough dimension to allow its estimation with the available data. The consequence of the replacement is that model function f(γθ_∗) written in terms of the approximate inputs is in error with respect to the same model function written in terms of β, f(β), which is assumed to be nearly exact. The difference f(β) − f(γθ_∗), termed model error, is spatially correlated, generates prediction biases, and causes standard confidence and prediction intervals to be too small. Model error is accounted for in the weighted nonlinear regression methodology developed to estimate θ_∗ and assess model uncertainties by incorporating the second-moment matrix of the model errors into the weight matrix. Techniques developed by statisticians to analyze classical nonlinear regression methods are extended to analyze the revised method. The analysis develops analytical expressions for bias terms reflecting the interaction of model nonlinearity and model error, for correction factors needed to adjust the sizes of confidence and prediction intervals for this interaction, and for correction factors needed to adjust the sizes of confidence and prediction intervals for possible use of a diagonal weight matrix in place of the correct one. If terms expressing the degree of intrinsic nonlinearity for f(β) and f(γθ_∗) are small, then most of the biases are small and the correction factors are reduced in magnitude. Biases, correction factors, and confidence and prediction intervals were obtained for a test problem for which model error is large to test robustness of the methodology. Numerical results conform with the theoretical analysis.     

Multi-resolution adaptive modeling of groundwater flow and transport problems

Many groundwater flow and transport problems, especially those with sharp fronts, narrow transition zones, layers and fingers, require extensive computational resources. In this paper, we present a novel multi-resolution adaptive Fup approach to solve the above mentioned problems. Our numerical procedure is the Adaptive Fup Collocation Method (AFCM), based on Fup basis functions and designed through a method of lines (MOL). Fup basis functions are localized and infinitely differentiable functions with compact support and are related to more standard choices such as splines or wavelets. This method enables the adaptive multi-resolution approach to solve problems with different spatial and temporal scales with a desired level of accuracy using the entire family of Fup basis functions. In addition, the utilized collocation algorithm enables the mesh free approach with consistent velocity approximation and flux continuity due to properties of the Fup basis functions. The introduced numerical procedure was tested and verified by a few characteristic groundwater flow and transport problems, the Buckley–Leverett multiphase flow problem, the 1-D vertical density driven problem and the standard 2-D seawater intrusion benchmark–Henry problem. The results demonstrate that the method is robust and efficient particularly when describing sharp fronts and narrow transition zones changing in space and time.     

On the tensorial nature of advective porosity

Field tracer tests indicate that advective porosity, the quantity relating advective velocity to Darcy flux, may exhibit directional dependence. Hydraulic anisotropy explains some but not all of the reported directional results. The present paper shows mathematically that directional variations in advective porosity may arise simply from incomplete mixing of an inert tracer between directional flow channels within a sampling (or support) volume ω of soil or rock that may be hydraulically isotropic or anisotropic. In the traditional fully homogenized case, our theory yields trivially a scalar advective porosity equal to the interconnected porosity ϕ, thus explaining neither the observed directional effects nor the widely reported experimental finding that advective porosity is generally smaller than ϕ. We consider incomplete mixing under conditions in which the characteristic time t_D of longitudinal diffusion along channels across ω is much shorter than the characteristic time t_H required for homogenization through transverse diffusion between channels. This may happen where flow takes place preferentially through relatively conductive channels and/or fractures of variable orientation separated by material that forms a partial barrier to diffusive transport. Our solution is valid for arbitrary channel Peclet numbers on a correspondingly wide range of time scales t_D ⩽ t ≪ t_H. It shows that the tracer center of mass is advected at a macroscopic velocity which is generally not collinear with the macroscopic Darcy flux and exceeds it in magnitude. These two vectors are related through a second-rank symmetric advective dispersivity tensor Φ. If the permeability k of ω is a symmetric positive-definite tensor, so is Φ. However, the principal directions and values of these two tensors are generally not the same; whereas those of k are a fixed property of the medium and the length-scale of ω, those of Φ depend additionally on the direction and magnitude of the applied hydraulic gradient. When the latter is large, diffusion has negligible effect on Φ and one may consider tracer mass to be distributed between channels in proportion to the magnitude of their Darcy flux. This is made intuitive through a simple example of an idealized fracture network. Our analytical formalism reveals the properties of Φ but is too schematic to allow predicting the latter accurately on the basis of realistic details about the void structure of ω and tracer mass distribution within it. Yet knowing the tensorial properties of Φ is sufficient to allow determining it indirectly on the basis of ω-scale hydraulic and tracer data, including concentrations that represent homogenized samples extracted from (or sensed externally across) an ω-scale plume.     

Stochastic modeling of Iranian earthquakes and estimation of ground motion for future earthquakes in Greater Tehran

Strong-motion data from eight significant well-documented earthquakes in Iran have been simulated using a stochastic modeling technique for finite faults proposed by Beresnev and Atkinson [Bull Seismol Soc Am 87 (1997) 67–84; Seism Res Lett 69 (1998) 27–32]. The database consists of 61 three-component records from eight earthquakes of magnitude ranging from M 6.3 to M 7.4, recorded at hypocentral distances up to 200 km. The model predictions are in good agreement with available Iranian strong-motion data as evidenced by near-zero average of differences between logarithms of the observed and predicted values for all frequencies. The strength factor, sfact, a quantity that controls the high-frequency radiation from the source is determined, on an event-by-event basis, by fitting simulated to observed response spectra.     

Analytical modeling of one-dimensional diffusion in layered systems with position-dependent diffusion coefficients

Diffusion in stratified porous media is common in the natural environment. The objective of this study is to develop analytical solutions for describing the diffusion in layered porous media with a position-dependent diffusion coefficient within each layer. The orthogonal expansion technique was used to solve a one-dimensional multi-layer diffusion equation in which the diffusion coefficient is expressed as a segmented linear function of positions in the porous media. The behavior of the solutions is illustrated using several examples of a three-layer system, with constant diffusion coefficient α₁ in layer 1 (0 < x < l₁), α₃ in layer 3(l₂ < x < l₃), and a linearly position-dependent diffusion coefficient α₁(1 + Δ(x − l₁)/(l₂ − l₁)) in the center layer (Δ = α₃/α₁ − 1). Because of the asymmetry of the layered system, the diffusion and related concentration distributions are also asymmetrical. For a given Δ value, the smaller the value of (l₂ − l₁)/l₃, the more significant the accumulation of concentration in the middle transition zone (l₁ < x < l₂), the sharper the change in the concentration profile of spatial distribution. Therefore, transition between two layers has significant effects on diffusion.     

Conditions when anisotropy is negligible for solute transfer in sediment beds of lakes or streams

To reduce the complexity and save computation time, an isotropic and a scalar dispersion model are explored and compared to the anisotropic advection/dispersion model to study the interstitial flow in a stream and lake sediment induced by a periodic pressure wave. In these systems, the solute transport is controlled by the ratio (R = a/(LS)) of the pressure wave steepness (a/L) to the stream slope (S), and the dispersivity ratio (λ = α_L/L) that measures the longitudinal dispersivity (α_L) relative to the pressure wave length (L). Through a series of numerical experiments, the conclusion is reached that a scalar dispersion model can be applied with satisfactory results for advection-dominated transport, i.e. when R ?  0.1 and λ ? 0.01, or λ ? 0.0001, i.e. Peclet number (Pe) ? 10000; an isotropic dispersion model can be applied when R ? 10 or λ ? 0.001, and the full anisotropic advection/dispersion model has to be applied when R > 10 and λ > 0.001.     

Evolution of droplets in subsea oil and gas blowouts: Development and validation of the numerical model VDROP-J

The droplet size distribution of dispersed phase (oil and/or gas) in submerged buoyant jets was addressed in this work using a numerical model, VDROP-J. A brief literature review on jets and plumes allows the development of average equations for the change of jet velocity, dilution, and mixing energy as function of distance from the orifice. The model VDROP-J was then calibrated to jets emanating from orifices ranging in diameter, D, from 0.5 mm to 0.12 m, and in cross-section average jet velocity at the orifice ranging from 1.5 m/s to 27 m/s. The d₅₀/D obtained from the model (where d₅₀ is the volume median diameter of droplets) correlated very well with data, with an R² = 0.99. Finally, the VDROP-J model was used to predict the droplet size distribution from Deepwater Horizon blowouts. The droplet size distribution from the blowout is of great importance to the fate and transport of the spilled oil in marine environment.     

A marching in space and time (MAST) solver of the shallow water equations. Part I: The 1D model

A new approach is presented for the numerical solution of the complete 1D Saint-Venant equations. At each time step, the governing system of partial differential equations (PDEs) is split, using a fractional time step methodology, into a convective prediction system and a diffusive correction system. Convective prediction system is further split into a convective prediction and a convective correction system, according to a specified approximated potential. If a scalar exact potential of the flow field exists, correction vanishes and the solution of the convective correction system is the same solution of the prediction system. Both convective prediction and correction systems are shown to have at each x − t point a single characteristic line, and a corresponding eigenvalue equal to the local velocity. A marching in space and time (MAST) technique is used for the solution of the two systems. MAST solves a system of two ordinary differential equations (ODEs) in each computational cell, using for the time discretization a self-adjusting fraction of the original time step. The computational cells are ordered and solved according to the decreasing value of the potential in the convective prediction step and to the increasing value of the same potential in the convective correction step. The diffusive correction system is solved using an implicit scheme, that leads to the solution of a large linear system, with the same order of the cell number, but sparse, symmetric and well conditioned. The numerical model shows unconditional stability with regard of the Courant–Friedrichs–Levi (CFL) number, requires no special treatment of the source terms and a computational effort almost proportional to the cell number. Several tests have been carried out and results of the proposed scheme are in good agreement with analytical solutions, as well as with experimental data.     

On the topographic coupling at the core-mantle interface

The mean tangential stresses at a corrugated interface between a solid, electrically insulating mantle and a liquid core of magnetic diffusivity λ are calculated for uniform rotation of both mantle and core at an angular velocity Ω in the presence of a corotating magnetic field B. The core and mantle are assumed to extend indefinitely in the horizontal plane. The interface has the form z = η(x, y), where z is the upward vertical distance and x, y are the zonal and latitudinal distances respectively. The function η(x, y) has a planetary horizontal length scale (i.e. of the order of the radius of the Earth) and small amplitude and vertical gradient. The liquid core flows with uniform mean zonal velocity U₀ relative to the mantle. Ω and B possess vertical and horizontal components.The vertical (poloidal) component B_p is uniform and has a value of 5 G while the horizontal (toroidal) field B_T = B_pαz, where α is a constant. When |α| ? 1, the mean horizontal stresses are found to have the same order of magnitude (10^?2 N m^?2) as those inferred from variations in the decade fluctuations in the length of the day, although the exact numerical values depend on the orientation of Ω as well as on the wavenumbers in the zonal and latitudinal directions.The influence of the steepness (as measured by α) of the toroidal field on the stresses is investigated to examine whether the constraint that the mean horizontal stresses at the core-mantle interface be of the order of 10^?2 N m^?2 might provide a selection mechanism for the behaviour of the toroidal field in the upper reaches of the outer core of the Earth. The results indicate that the restriction imposed on α is related to the value assigned to the toroidal field deep into the core. For example, if |α| ? 1 then the tangential stresses are of the right order of magnitude only if the toroidal field is comparable with the poloidal field deep in the core.     

Error analysis of finite difference methods for two-dimensional advection–dispersion–reaction equation

In this paper, the numerical errors associated with the finite difference solutions of two-dimensional advection–dispersion equation with linear sorption are obtained from a Taylor analysis and are removed from numerical solution. The error expressions are based on a general form of the corresponding difference equation. The variation of these numerical truncation errors is presented as a function of Peclet and Courant numbers in X and Y direction, a Sink/Source dimensionless number and new form of Peclet and Courant numbers in X–Y plane. It is shown that the Crank–Nicolson method is the most accurate scheme based on the truncation error analysis. The effects of these truncation errors on the numerical solution of a two-dimensional advection–dispersion equation with a first-order reaction or degradation are demonstrated by comparison with an analytical solution for predicting contaminant plume distribution in uniform flow field. Considering computational efficiency, an alternating direction implicit method is used for the numerical solution of governing equation. The results show that removing these errors improves numerical result and reduces differences between numerical and analytical solution.     

An efficient,high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media

In this study, a probabilistic collocation method (PCM) on sparse grids is used to solve stochastic equations describing flow and transport in three-dimensional, saturated, randomly heterogeneous porous media. The Karhunen–Loève decomposition is used to represent log hydraulic conductivity Y=ln_Ks$Y=\ln K_s$. The hydraulic head h   and average pore-velocity v$\mathbf{v}$ are obtained by solving the continuity equation coupled with Darcy’s law with random hydraulic conductivity field. The concentration is computed by solving a stochastic advection–dispersion equation with stochastic average pore-velocity v$\mathbf{v}$ computed from Darcy’s law. The PCM approach is an extension of the generalized polynomial chaos (gPC) that couples gPC with probabilistic collocation. By using sparse grid points in sample space rather than standard grids based on full tensor products, the PCM approach becomes much more efficient when applied to random processes with a large number of random dimensions. Monte Carlo (MC) simulations have also been conducted to verify accuracy of the PCM approach and to demonstrate that the PCM approach is computationally more efficient than MC simulations. The numerical examples demonstrate that the PCM approach on sparse grids can efficiently simulate solute transport in randomly heterogeneous porous media with large variances.     

Dynamic passive pressure from c–ϕ soil backfills

This technical note presents an analytical expression for the total passive pressure on a retaining wall from the c–? soil backfill subjected to both horizontal and vertical seismic inertial forces. The developed expression has been analysed for the special cases, and the results have been found identical to those proposed by earlier researchers on the subject. A numerical example, presented to illustrate the steps for the calculation of total dynamic passive pressure using the developed general expression, shows that the design value of total dynamic passive pressure as a resistance to the retaining wall movement should be obtained with upward vertical seismic inertial force in combination with the direction of horizontal seismic force towards the backfill.     

Optimal weighting in the finite difference solution of the convection-dispersion equation

Oscillation and numerical dispersion limit the reliability of numerical solutions of the convection-dispersion equation when finite difference methods are used. To eliminate oscillation and reduce the numerical dispersion, an optimal upstream weighting with finite differences is proposed. The optimal values of upstream weighting coefficients numerically obtained are a function of the mesh Peclet number used. The accuracy of the proposed numerical method is tested against two classical problems for which analytical solutions exist. The comparison of the numerical results obtained with different numerical schemes and those obtained by the analytical solutions demonstrates the possibility of a real gain in precision using the proposed optimal weighting method. This gain in precision is verified by interpreting a tracer experiment performed in a laboratory column.     

Numerical modelling-based comparison of longitudinal dispersion coefficient formulas for solute transport in rivers

River water quality models usually apply the Fischer equation to determine the longitudinal dispersion coefficient (D_x) in solving the advection–dispersion equation (ADE). Recently, more accurate formulas have been introduced to determine D_x in rivers, which could strongly affect the accuracy of the ADE results. A numerical modelling-based approach is presented to evaluate the performance of various D_x formulas using the ADE. This approach consists of a finite difference approximation of the ADE, a MATLAB code and a MS Excel interface; it was tested against the analytical ADE solution and demonstrated using eight well-known D_x formulas and tracer study data for the Chattahoochee River (USA), the Severn (UK) and the Athabasca (Canada). The results show that D_x has an important effect on tracer concentrations simulated with the ADE. Comparison between the simulated and measured concentrations confirms the appropriate performance of Zeng and Huai’s formula for D_x estimation. Use of the newly proposed equations for D_x estimation could enhance the accuracy of solving the ADE.     

Effective conductivity of periodic media with cuboid inclusions

This paper presents a numerical solution for the effective conductivity of a periodic binary medium with cuboid inclusions located on an octahedral lattice. The problem is defined by five dimensionless geometric parameters and one dimensionless conductivity contrast parameter. The effective conductivity is determined by considering the flow through the “elementary flow domain” (EFD), which is an octant of the unitary domain of the periodic media. We derive practical bounds of interest for the six-dimensional parameter space of the EFD and numerically compute solutions at regular intervals throughout the entire bounded parameter space. A continuous solution of the effective conductivity within the limits of the simulated parameter space is then obtained via interpolation of the numerical results. Comparison to effective conductivities derived for random heterogeneous media demonstrate similarities and differences in the behavior of the effective conductivity in regular periodic (low entropy) vs. random (high entropy) media. The results define the low entropy bounds of effective conductivity in natural media, which is neither completely random nor completely periodic, over a large range of structural geometries. For aniso-probable inclusion spacing, the absolute bounds of K_eff for isotropic inclusions are the Wiener bounds, not the Hashin-Shtrikman bounds. For isotropic inclusion and isoprobable conditions well below the percolation threshold, the results are in agreement with the self-consistent approach. For anisotropic cuboid inclusions, or at relatively close spacing in at least one direction (p > 0.2) (aniso-probable conditions), the effective conductivity of the periodic media is significantly different from that found in anisotropic random binary or Gaussian media.