Method to characterize aquitards above leaky aquifers with water supply wells

Leaky aquifers provide protected drinking water since the aquifer is overlain by an aquitard, and this study develops a method to estimate hydraulic properties of the latter deposit. Steady pumping, supply well shutdowns, and slug tests generate data in adjacent monitoring well clusters that characterize the aquitard. An existing steady model estimates a site-averaged value of the aquitard permeability k if its thickness is known, and this site-averaged estimate may be compared with local k estimates from conventional and extended slug tests. A shutdown attenuation model estimates a local value of the consolidation coefficient C _V, which combines with the local k value to specify the compressibility α of the aquitard. The method is illustrated for the Fowl Meadow Aquifer, a stratified drift deposit used as a drinking water supply in eastern Massachusetts (USA), with an overlying silt aquitard of 10 m thickness. Steady data and theory suggest a site-averaged k of 2.3?×?10^–17?m² for the aquitard, while the shutdown attenuation model generates local C _V values that vary from 10^–5 to 10^–3?m²/s. The slug tests yield a local k variation (10^–17–10^–13?m²) that brackets the site-averaged value, and an α range of 10^–9–10^–7?Pa^–1.     

A new computational strategy for solving two‐phase flow in strongly heterogeneous poroelastic media of evolving scales

We develop a new computational methodology for solving two‐phase flow in highly heterogeneous porous media incorporating geomechanical coupling subject to uncertainty in the poromechanical parameters. Within the framework of a staggered‐in‐time coupling algorithm, the numerical method proposed herein relies on a Petrov–Galerkin postprocessing approach projected on the Raviart–Thomas space to compute the Darcy velocity of the mixture in conjunction with a locally conservative higher order finite volume discretization of the nonlinear transport equation for the saturation and an operator splitting procedure based on the difference in the time‐scales of transport and geomechanics to compute the effects of transient porosity upon saturation. Notable features of the numerical modeling proposed herein are the local conservation properties inherited by the discrete fluxes that are crucial to correctly capture the fingering patterns arising from the interaction between heterogeneity and nonlinear viscous coupling. Water flooding in a poroelastic formation subject to an overburden is simulated with the geology characterized by multiscale self‐similar permeability and Young modulus random fields with power‐law covariance structure. Statistical moments of the poromechanical unknowns are computed within the framework of a high‐resolution Monte Carlo method. Numerical results illustrate the necessity of adopting locally conservative schemes to obtain reliable predictions of secondary recovery and finger growth in strongly heterogeneous deformable reservoirs. Copyright © 2011 John Wiley & Sons, Ltd.     

Modeling fractures as interfaces with nonmatching grids

We consider a model for fluid flow in a porous medium with a fracture. In this model, the fracture is treated as an interface between subdomains, on which specific equations have to be solved. In this article, we analyze the discrete problem, assuming that the fracture mesh and the subdomain meshes are completely independent, but that the geometry of the fracture is respected. We show that despite this nonconformity, first-order convergence is preserved with the lowest-order Raviart–Thomas(-Nedelec) mixed finite elements. Numerical simulations confirm this result.     

A penalty mechanism for discontinuous Galerkin processing of Darcy velocity

The Darcy velocity plays an important role in the flow in porous media, particularly when a miscible displacement is concerned. One major requirement when approximating this velocity is the continuity of its normal component. The discontinuous Galerkin methods, by nature, are not well designed for this challenge, since approximations are performed in space of totally discontinuous polynomials.We propose in such context a penalty approach, in order to enhance the continuity of the normal component of the Darcy velocity. The resulting formulation is shown to be stable whatever the origin of the pressure but requires the inversion of a global matrix. We then propose two modifications leading to the inversion of only local matrices. Error estimates are furnished and the analysis of the penalty parameter vis-a-vis the computed pressure is addressed. We show that the proposed reconstructions have better performance compared to the simple local differentiation of the computed pressure. Numerical tests are provided to illustrate the theoretical results.     

Local problem-based <Emphasis Type="Italic">a posteriori</Emphasis> error estimators for discontinuous Galerkin approximations of reactive transport

We consider adaptive discontinuous Galerkin (DG) methods for solving reactive transport problems in porous media. To guide anisotropic and dynamic mesh adaptation, a posteriori error estimators based on solving local problems are established. These error estimators are efficient to compute and effective to capture local phenomena, and they apply to all the four primal DG schemes, namely, symmetric interior penalty Galerkin, nonsymmetric interior penalty Galerkin, incomplete interior penalty Galerkin, and the Oden–Babuška-Baumann version of DG. Numerical results are provided to illustrate the effectiveness of the proposed error estimators.     

A local discontinuous Galerkin scheme for Darcy flow with internal jumps

We present a new version of the local discontinuous Galerkin method which is capable of dealing with jump conditions along a submanifold Γ_LG (i.e., Henry’s Law) in instationary Darcy flow. Our analysis accounts for a spatially and temporally varying, non-linear permeability tensor in all estimates which is also allowed to have a jump at Γ_LG and gives a convergence order result for the primary and the flux unknowns. In addition to this, different approximation spaces for the primary and the flux unknowns are investigated. The results imply that the most efficient choice is to choose the degree of the approximation space for the flux unknowns one less than that of the primary unknown. The only stabilization in the proposed scheme is represented by a penalty term in the primary unknown.     

Fast linear solver for diffusion problems with applications to pressure computation in layered domains

Accurate simulation of fluid pressures in layered reservoirs with strong permeability contrasts is a challenging problem. For this purpose, the Discontinuous Galerkin (DG) method has become increasingly popular. Unfortunately, standard linear solvers are usually too inefficient for the aforementioned application. To increase the efficiency of the conjugate gradient (CG) method for linear systems resulting from symmetric interior penalty (discontinuous) Galerkin (SIPG) discretizations, we cast an existing two-level preconditioner into the deflation framework. The main idea is to use coarse corrections based on the DG solution with polynomial degree p = 0. This paper provides a numerical comparison of the performance of the original preconditioner and the resulting deflation variant in terms of scalability and overall efficiency. Furthermore, it studies the influence of the SIPG penalty parameter, weighted averages in the SIPG formulation (SWIP), the smoother, damping of the smoother, and the strategy for solving the coarse systems. We have found that the penalty parameter can best be chosen diffusion-dependent. In that case, both two-level methods yield fast and scalable convergence. Whether preconditioning or deflation is to be favored depends on the choice of the smoother and on the damping of the smoother. Altogether, both two-level methods can contribute to cheaper and more accurate fluid pressure simulations.     

Estimation of velocity function parameters in unconsolidated sands using semblance velocity analysis

To properly understand seismic wave propagation in unconsolidated sand layers, it is important to estimate the parameters of their continuous velocity–depth functions. This study proposes a procedure to estimate the V ₀ and k parameters of a specific velocity function, where V ₀ is the direct P-wave velocity at the ground surface and k is the velocity gradient. The V ₀ and k parameters are generally independent of each other. However, it is possible to relate them numerically because both depend strongly on the porosity (?) and water saturation (S _w). The proposed procedure starts by tabulating V ₀ and k for 0.05?≤???≤?0.5 sampled at Δ??=?0.05 and S _w?=?0.6, so that only V ₀ is needed for fitting. Then, time–distance (T-X) type curves of the direct arrival are calculated for the corresponding values of V ₀ and k parameters values. The type curves are fitted then to the observed shot gather through a modification of the classic semblance velocity analysis method. Once the best-fit V ₀ value is found, the corresponding k, ?, and S _w values are picked from a V ₀–k–? lookup table. The procedure is applied on synthetic shot gathers with various amounts of additive Gaussian random noise. Results show that the method is robust and tolerant to low to moderate amounts of noise.     

Analysis of a mixed discontinuous Galerkin method for instationary Darcy flow

We present an a priori stability and convergence analysis of a new mixed discontinuous Galerkin scheme applied to the instationary Darcy problem. The analysis accounts for a spatially and temporally varying permeability tensor in all estimates. The proposed method is stabilized using penalty terms in the primary and the flux unknowns.     

A discontinuous Galerkin method for two-phase flow in a porous medium enforcing H(div) velocityand continuous capillary pressure

We consider the slightly compressible two-phase flow problem in a porous medium with capillary pressure. The problem is solved using the implicit pressure, explicit saturation (IMPES) method, and the convergence is accelerated with iterative coupling of the equations. We use discontinuous Galerkin to discretize both the pressure and saturation equations. We apply two improvements, which are projecting the flux to the mass conservative H(div)-space and penalizing the jump in capillary pressure in the saturation equation. We also discuss the need and use of slope limiters and the choice of primary variables in discretization. The methods are verified with two- and three-dimensional numerical examples. The results show that the modifications stabilize the method and improve the solution.     

Finite Element Solvers for Biot’s Poroelasticity Equations in Porous Media

We study and compare five different combinations of finite element spaces for approximating the coupled flow and solid deformation system, so-called Biot’s equations. The permeability and porosity fields are heterogeneous and depend on solid displacement and fluid pressure. We provide detailed comparisons among the continuous Galerkin, discontinuous Galerkin, enriched Galerkin, and two types of mixed finite element methods. Several advantages and disadvantages for each of the above techniques are investigated by comparing local mass conservation properties, the accuracy of the flux approximation, number of degrees of freedom (DOF), and wall and CPU times. Three-field formulation methods with fluid velocity as an additional primary variable generally require a larger number of DOF, longer wall and CPU times, and a greater number of iterations in the linear solver in order to converge. The two-field formulation, a combination of continuous and enriched Galerkin function space, requires the fewest DOF among the methods that conserve local mass. Moreover, our results illustrate that three out of the five methods conserve local mass and produce similar flux approximations when conductivity alteration is included. These comparisons of the key performance indicators of different combinations of finite element methods can be utilized to choose the preferred method based on the required accuracy and the available computational resources.

     

Trend surface analysis as a special case of IRF-k kriging

This short note establishes the equivalence between trend surface analysis with polynomials of orderk and IRF-k (intrinsic random function of orderk) kriging with a nugget effect covariance model.     

Kappa (k) derived from accelerograms recorded in the 2008 Wenchuan mainshock,Sichuan, China

High-frequency spectral decay factor, kappa (k), in the accelerograms of the Wenchuan mainshock was measured using strong motion data from 52 stations within 311 km of the epicenter. The derived k range from 0.0034 s to 0.0468 s. The correlation of k versus fault distance was given, which is k = 0.01288 + 5.9068 × 10^–5 R for the N-S component, k = 0.01881 + 1.4219 × 10^–5 R for the E-W component, and k = 0.00855 + 5.6086 × 10^–5 R for the U-D component. The analysis on the spatial variation of k demonstrates that k relates to source effect and propagation effect besides local site effect. Ground motions for the 52 stations were simulated using derived k and compared to actual recordings in terms of waveforms, amplitude spectra and response spectra. The results show agreement at shorter periods (<1 s), but a slight overestimation at longer periods (1–7 s).     

Part II. Discontinuous Galerkin method applied to a single phase flow in porous media

Discontinuous Galerkin numerical simulations of single phase flow problem are described in this paper. The simulations show the advantages of using discontinuous approximation spaces. hp convergence results are obtained for smooth solutions. Unstructured meshes and unsmooth solutions are also considered.     

Analysis of fracture propagation in a rock mass surrounding a tunnel under high internal pressure by the element-free Galerkin method

Fractures developed around high pressurized gas or air storage tunnels can progressively extend to the ground surface, eventually leading to an uplift failure. A tool reasonably reproducing the failure patterns is necessary for stability assessment. In this study, a numerical method based on the element-free Galerkin (EFG) method with a cohesive crack model is developed to simulate fracture propagation patterns in the rock mass around a tunnel under high internal pressure. A series of physical model tests was also conducted to validate the reliability of the developed method. A qualitative agreement between physical model tests and numerical results can be obtained. The in situ stress ratio, k, has a strong influence on both the position of crack initiation and the propagation direction. The numerical analyses were extended to full-scale problems. Numerical tests were performed to investigate the prime influencing factors on the failure patterns of a high pressurized gas circular tunnel with varying parameters. The results suggest that initial in situ stress conditions with a high k (larger than 1) is favorable for construction of pressurized gas or air storage tunnels.     

A continuous/discontinuous Galerkin framework for modeling coupled subsurface and surface water flow

We consider conjunctive surface-subsurface flow modeling, where surface water flow is described by the shallow water equations and ground water flow by Richards’ equation for the vadose zone. Coupling between the models is based on the continuity of flux and water pressure. Numerical approximation of the coupled model using the framework of discontinuous Galerkin (DG) methods is formulated. In the subsurface, the local discontinuous Galerkin (LDG) method is used to approximate ground water velocity and hydraulic head; a DG method is also used to approximate surface water velocity and elevation. This approach allows for a weak coupling of the models and the use of different approximating spaces and/or meshes within each regime. A simplified LDG method based on continuous approximations to water head is also described. Numerical results that investigate physical and numerical aspects of surface–subsurface flow modeling are presented. This work was supported by National Science Foundation grant DMS-0411413.     

A time‐discontinuous Galerkin method for the dynamical analysis of porous media

We present a time‐discontinuous Galerkin method (DGT) for the dynamic analysis of fully saturated porous media. The numerical method consists of a finite element discretization in space and time. The discrete basis functions are continuous in space and discontinuous in time. The continuity across the time interval is weakly enforced by a flux function. Two applications and several numerical investigations confirm the quality of the proposed space–time finite element scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

Probabilistic Marcinkiewicz–Zygmund Inequalities on the Rotation Group

Approximation problems on the rotation group SO(3) naturally arise in various fields, like crystallography, chemistry, and biology. For example, in crystallographic texture analysis one is confronted with the problem of evaluating so-called orientation density functions (ODFs). In many situations one only has a finite number of measurements at scattered sampling nodes. In order to reconstruct ODFs over all rotations, so-called Marcinkiewicz–Zygmund inequalities on the rotation group are an important tool. These inequalities provide norm equivalences between polynomials on SO(3) and their sample values. Recently shown equivalences depend on a density parameter of the sampling set and the proven inequalities hold true for polynomials on SO(3) whose degree does not exceed an upper bound which is determined by this density parameter. In this paper, we show that we can enlarge this upper bound for the polynomial degree significantly if we are satisfied by such norm equivalences that hold with a given probability only. Moreover, we show that there are fixed sampling sets for which we get probabilistic Marcinkiewicz–Zygmund inequalities that hold for polynomials on SO(3) of all degrees.     

Macrofossil and pollen evidence for full-glacial steppe within an ecological mosaic along the Bluefish River,eastern Beringia

Plant and insect macrofossil assemblages dating to the full-glacial (late Wisconsinan) are rare from eastern Beringia. Here we present an assemblage of fossil pollen, insect and plant macrofossils recovered from alluvium at the Bluefish Exposure, northern Yukon Territory. Nine AMS radiocarbon ages place these data between ca. 18,880–16,440 ¹⁴C yr BP (22,313–19,597 cal. yr BP). These data indicate that xeric steppe, rich in bunchgrasses Poa and Elymus, Artemisia frigida and diverse forbs was interspersed within a mosaic of local vegetation types, including mid-rich fens, mesic graminoid meadows, steppe-tundra and herb-tundra. Macrofossils and minor pollen of tundra forbs suggest steppe-tundra plant associations within midslope elevations and discontinuous herb-tundra on high elevation uplands on exposed bedrock ridges. The composition and distribution of local vegetation was dependent on available moisture, drainage, aspect and elevation. Compositional and physiognomic similarities can be made with extrazonal steppe-dominated dry slopes and high elevation steppe-tundra ecotones in central Alaska and Yukon Territory. Our paleoecological data reflect environments inhabited by the diverse late Pleistocene Bluefish Caves fauna, including woolly mammoth, horse, steppe bison, and saiga antelope.     

Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I

Three Galerkin methods using discontinuous approximation spaces are introduced to solve elliptic problems. The underlying bilinear form for all three methods is the same and is nonsymmetric. In one case, a penalty is added to the form and in another, a constraint on jumps on each face of the triangulation. All three methods are locally conservative and the third one is not restricted. Optimal a priori hp error estimates are derived for all three procedures.