An analytical study on three‐dimensional versus two‐dimensional water table‐induced flow patterns in a Tóthian basin

Although it has been increasingly acknowledged that groundwater flow pattern is complicated in the three‐dimensional (3‐D) domain, two‐dimensional (2‐D) water table‐induced flow models are still widely used to delineate basin‐scale groundwater circulation. However, the validity of 2‐D cross‐sectional flow field induced by water table has been seldom examined. Here, we derive the analytical solution of 3‐D water table‐induced hydraulic head in a Tóthian basin and then examine the validity of 2‐D cross‐sectional models by comparing the flow fields of selected cross sections calculated by the 2‐D cross‐sectional model with those by the 3‐D model, which represents the “true” cases. For cross sections in the recharge or discharge area of the 3‐D basin, even if head difference is not significant, the 2‐D cross‐sectional models result in flow patterns absolutely different from the true ones. For the cross section following the principal direction of groundwater flow, although 2‐D cross‐sectional models would overestimate the penetrating depth of local flow systems and underestimate the recharge/discharge flux, the flow pattern from the cross‐sectional model is similar to the true one and could be close enough to the true one by adjusting the decay exponent and anisotropy ratio of permeability. Consequently, to determine whether a 2‐D cross‐sectional model is applicable, a comparison of hydraulic head difference between 2‐D and 3‐D solutions is not enough. Instead, the similarity of flow pattern should be considered to determine whether a cross‐sectional model is applicable. This study improves understanding of groundwater flow induced by more natural water table undulations in the 3‐D domain and the limitations of 2‐D models accounting for cross‐sectional water table undulation only.     

三维井间电磁场的正反演计算

提出了一种计算体积分方程的方法－改进型局域非线性迭代（MLNI），并用其对三维井间电磁场进行反演计算，该方法将井间大尺度散射体分为近场和远场区域两部分，它们的位置和尺寸均随场点位置的变化而改变，采用局域非线性近似计算近场区域的影响，将远场区域的影响作为外部激励源，采用迭代方法计算，该方法具有计算速度快，所需内存量少，收敛性好的优点，在反演中采用基于MLNI的非线性反演方法，利用这种方法可以将成像区域集中于一定范围内而不是整个三维空间，该反演方法由于考虑到了雅可比矩阵元素的非线性项，因而与传统的Born迭代反演方法相比更为精确和稳定，考虑到信息量和计算机内存的限制，第1次成像采用双重体元分割法进行较粗略的成像，然后缩小成像范围进行第2次较精确的成像，数值计算结果表明，MLNI是一种有效的计算井间大尺度异常体散射场的方法，将该方法用于反演过程能够得到较高分辨率的三维井间电导率图像。     

An efficient numerical model for multicomponent compressible flow in fractured porous media

An efficient and accurate numerical model for multicomponent compressible single-phase flow in fractured media is presented. The discrete-fracture approach is used to model the fractures where the fracture entities are described explicitly in the computational domain. We use the concept of cross flow equilibrium in the fractures. This will allow large matrix elements in the neighborhood of the fractures and considerable speed up of the algorithm. We use an implicit finite volume (FV) scheme to solve the species mass balance equation in the fractures. This step avoids the use of Courant–Freidricks–Levy (CFL) condition and contributes to significant speed up of the code. The hybrid mixed finite element method (MFE) is used to solve for the velocity in both the matrix and the fractures coupled with the discontinuous Galerkin (DG) method to solve the species transport equations in the matrix. Four numerical examples are presented to demonstrate the robustness and efficiency of the proposed model. We show that the combination of the fracture cross-flow equilibrium and the implicit composition calculation in the fractures increase the computational speed 20–130 times in 2D. In 3D, one may expect even a higher computational efficiency.     

3D Lagrangian modelling of saltating particles diffusion in turbulent water flow

A 3D Lagrangian model of the saltation of solid spherical particles on the bed of an open channel flow, accounting for turbulence-induced mechanisms, is proposed and employed as the key tool of the study. The differences between conventional 2D models and a proposed 3D saltation model are discussed and the advantages of the 3D model are highlighted. Particularly, the 3D model includes a special procedure allowing generation of 3D flow velocity fields. This procedure is based on the assumption that the spectra of streamwise, vertical and transverse velocity components are known at any distance from the bed. The 3D model was used to identify and quantify effects of turbulence on particle entrainment and saltation. The analysis of particle trajectories focused on their diffusive nature, clarifying: (i) the effect of particle mobility parameter; (ii) the effect of bed topography; and (iii) the effect of turbulence. Specifically, the results of numerical simulations describing the abovementioned effects on the change in time of the variance are presented. In addition, the change in time of the skewness and kurtosis, which are likely to reflect the turbulence influence on the spread of particles, are also shown. Two different diffusion regimes (local and intermediate) for each of the investigated flow conditions are confidently identified.     

Tentative study of flow patterns in the North Aegean Sea using NOAA-AVHRR images and 2D model simulation

A statistical technique for image processing, the maximum cross correlation (MCC) method, was utilized on sequences of NOAA-AVHRR thermal data in order to explore the surface advective current dynamics at the discharge region of the Hellespont in the North Aegean Sea. A 2D numerical flow model was also used in order to simulate the barotropic flow pattern of the surface water layer. The model was forced with diurnal wind fields obtained for the same period as the satellite infrared images. The currents (magnitude and direction) derived from the two methods compare satisfactorily despite the fact that some model simplifications were made.     

Computer modeling of geoelectromagnetic fields in three-dimensional media by the finite element method

This work describes the basic approaches to the solution of three-dimensional (3D) problems of geoelectromagnetism with the use of the finite element method and the possibilities of the GeoEM program complex for its implementation. The methods of modeling geoelectromagnetic fields for the most widely known types of controllable sources and the mathematical apparatus for the solution of problems of magnetotelluric soundings are considered. Examples of the calculations of 3D fields for the horizontal and vertical electric lines in planning electric exploration works on the shelf are presented, as well as an example of the 3D interpretation of array transient electromagnetic sounding data during the search for deep-seated target objects in conditions of a mostly heterogeneous upper part of the section.     

Coordinate mapping of analytical contaminant transport solutions to non-uniform flow fields

Existing analytical solutions to 2D and 3D contaminant transport problems are limited by the mathematically convenient assumption of uniform flow. An approximate method is developed herein for coordinate mapping of 2D (vertically-averaged) transport solutions to non-uniform steady-state irrotational and divergence-free flow fields in single-layer aquifers. The method enables existing analytical transport solutions to be applied to aquifer systems with wells, non-uniform saturated thickness, surface water features, and (to a limited degree) heterogeneous hydraulic conductivity and recharge. This mass-conservative coordinate mapping approach is inexact in its approximation of the dispersion process but is still sufficiently accurate for many simple flow systems. The degree of model error is directly proportional to the variation of velocity magnitude within the domain. These mapped analytical solutions are compared to numerical simulation results and the coordinate mapping errors are investigated. The methods described herein may be used in the traditional capacity of analytical transport models, i.e., screening and preliminary site assessment, without sacrificing accuracy by assuming locally uniform flow conditions or applying an ad-hoc coordinate transformation. The solutions benefit from the traditional advantages of analytical methods, particularly the removal of artifacts due to spatial and temporal discretization: no time-stepping or numerical discretization is required.     

Δx≠Δy in 3D depth migration via McClellan transformations

An intuitive method is presented for extending Hale–McClellan migration operators to handle surveys binned and stacked with in‐line and cross‐line spacings unequal. This avoids re‐interpolating the input to 3D migration, either externally or internally. The algorithm does not require an integer or rational ratio between the in‐line and cross‐line spacings in order to be applied nor does it deliberately introduce artificial aliasing by interleaving subsampled grids along the more finely sampled axis. Examples of its impulse response and application to the SEG–EAGE salt model are shown.     

Inversion of dynamic production data for permeability: fast streamline-based computation of sensitivity coefficients of fractional flow rate

Generation of permeability field in a reservoir model that matchs historical dynamic production data requires an inverse calculation. A gradient method is typically used to solve the inverse minimization problem and requires sensitivity coefficients of reservoir responses, e.g. fractional flow rate or pressure, with respect to the change in the permeability. This paper presents a novel semi-analytical streamline-based method for computing such sensitivity coefficients under the framework of two-phase (oil-water) flow conditions. This method is shown to be significantly faster and generate permeability fields with lower objective function than the traditional perturbation method. The method decomposes the multiple-dimensional full flow problem into multiple 1D problems along streamlines. The sensitivity of fractional flow rate at the production well is directly related to the sensitivity of time-of-flight (TOF) along each individual streamline and the sensitivity of pressure at grid cells along the streamline. The sensitivity of TOF of a streamline can be obtained analytically. The sensitivity of pressure is obtained as part of a fast single phase flow simulation. The proposed method is implemented in a geostatistically based inverse technique, called the sequential self-calibration (SSC) method. Results for fractional flow rate sensitivities are presented and compared with the traditional perturbation method. This new method can be easily extended to compute sensitivity coefficients of saturation (concentration) data.     

Stochastic analysis of bounded unsaturated flow in heterogeneous aquifers: Spectral/perturbation approach

This paper describes a stochastic analysis of steady state flow in a bounded, partially saturated heterogeneous porous medium subject to distributed infiltration. The presence of boundary conditions leads to non-uniformity in the mean unsaturated flow, which in turn causes non-stationarity in the statistics of velocity fields. Motivated by this, our aim is to investigate the impact of boundary conditions on the behavior of field-scale unsaturated flow. Within the framework of spectral theory based on Fourier–Stieltjes representations for the perturbed quantities, the general expressions for the pressure head variance, variance of log unsaturated hydraulic conductivity and variance of the specific discharge are presented in the wave number domain. Closed-form expressions are developed for the simplified case of statistical isotropy of the log hydraulic conductivity field with a constant soil pore-size distribution parameter. These expressions allow us to investigate the impact of the boundary conditions, namely the vertical infiltration from the soil surface and a prescribed pressure head at a certain depth below the soil surface. It is found that the boundary conditions are critical in predicting uncertainty in bounded unsaturated flow. Our analytical expression for the pressure head variance in a one-dimensional, heterogeneous flow domain, developed using a nonstationary spectral representation approach [Li S-G, McLaughlin D. A nonstationary spectral method for solving stochastic groundwater problems: unconditional analysis. Water Resour Res 1991;27(7):1589–605; Li S-G, McLaughlin D. Using the nonstationary spectral method to analyze flow through heterogeneous trending media. Water Resour Res 1995; 31(3):541–51], is precisely equivalent to the published result of Lu et al. [Lu Z, Zhang D. Analytical solutions to steady state unsaturated flow in layered, randomly heterogeneous soils via Kirchhoff transformation. Adv Water Resour 2004;27:775–84].     

A Finite Layer Formulation for Groundwater Flow to Horizontal Wells

A finite layer approach for the general problem of three‐dimensional (3D) flow to horizontal wells in multilayered aquifer systems is presented, in which the unconfined flow can be taken into account. The flow is approximated by an integration of the standard finite element method in vertical direction and the analytical techniques in the other spatial directions. Because only the vertical discretization is involved, the horizontal wells can be completely contained in one specific nodal plane without discretization. Moreover, due to the analytical eigenfunctions introduced in the formulation, the weighted residual equations can be decoupled, and the formulas for the global matrices and flow vector corresponding to horizontal wells can be obtained explicitly. Consequently, the bandwidth of the global matrices and computational cost rising from 3D analysis can be significantly reduced. Two comparisons to the existing solutions are made to verify the validity of the formulation, including transient flow to horizontal wells in confined and unconfined aquifers. Furthermore, an additional numerical application to horizontal wells in three‐layered systems is presented to demonstrate the applicability of the present method in modeling flow in more complex aquifer systems.     

可控源电磁场三维自适应矢量有限元正演模拟

本文实现了可控源电磁(CSEM)场三维自适应矢量有限元正演算法,该算法采用非结构四面体单元进行三维网格剖分,能够真实模拟地形起伏和复杂电性异常体.采用一次场和二次场分离的方式计算电磁场响应,能够有效解决有限元模拟中的源点奇异性,提高场源附近电磁场数值精度,其中一次场利用CSEM一维正演算法解析求得,二次场采用矢量有限元方法求得.并利用基于后验误差估计的自适应网格细化算法指导网格细化,以减少人为设计网格导致的误差.通过一维和三维模型的数值模拟,验证了本文算法的有效性:一维模型有限元解与解析解吻合得很好,电磁场振幅相对误差在1%左右,相位差整体小于1°;三维模型有限元解与有限体积解吻合得也很好.模拟了一个含三维倾斜板状异常体的可控源电磁响应,表明了本文算法模拟复杂地电结构电磁场的能力和有效性.     

One‐dimensional and two‐dimensional hydrodynamic modeling derived flow properties: impacts on aquatic habitat quality predictions

Studies of the effects of hydrodynamic model dimensionality on simulated flow properties and derived quantities such as aquatic habitat quality are limited. It is important to close this knowledge gap especially now that entire river networks can be mapped at the microhabitat scale due to the advent of point‐cloud techniques. This study compares flow properties, such as depth and velocity, and aquatic habitat quality predicted from pseudo‐2D and fully 2D hydrodynamic modeling. The models are supported by high‐resolution, point‐cloud derived bathymetries, from which close‐spaced cross‐sections were extracted for the 1D modeling, of three morphologically and hydraulically different river systems. These systems range from small low‐gradient meandering pool–riffle to large steep confined plane‐bed rivers. We test the effects of 1D and 2D models on predicted hydraulic variables at cross‐sections and over the full bathymetry to quantify the differences due to model dimensionality and those from interpolation. Results show that streambed features, whose size is smaller than cross‐sectional spacing, chiefly determine the different results of 1D and 2D modeling whereas flow discharge, stream size, morphological complexity and model grid sizes have secondary effects on flow properties and habitat quality for a given species and life stage predicted from 1D and 2D modeling. In general, the differences in hydraulic variables are larger in the bathymetric than in the cross‐sectional analysis, which suggests that some errors are introduced from interpolation of spatially disaggregated simulated variables with a 1D model, instead of model dimensionality 1D or 2D. Flow property differences are larger for velocity than for water surface elevation and depth. Differences in weighted usable area (WUA) derived from 1D and 2D modeling are relatively small for low‐gradient meandering pool–riffle systems, but the differences in the spatial distribution of microhabitats can be considerable although clusters of same habitat quality are spatially comparable. Copyright © 2014 John Wiley & Sons, Ltd.     

Cross-matching with interpreted warping of 3D streamer and 3D ocean-bottom-cable data at Valhall for time-lapse assessment

Legacy streamer data and newer 3D ocean‐bottom‐cable data are cross‐matched and analysed for time‐lapse analysis of geomechanical changes due to production in the Valhall Field. The issues relating to time‐lapse analysis using two such distinctly different data sets are addressed to provide an optimal cross‐matching workflow that includes 3D warping. Additionally an assessment of the differences between the imaging using single‐azimuth streamer and multi‐azimuth ocean‐bottom‐cable data is provided. The 3D warping utilized in the cross‐matching procedure is sensitive to acquisition and processing differences but is also found to provide valuable insight into the geometrical changes that occur in the subsurface due to production. As such, this work also provides a demonstration of the use of high‐resolution 3D interpreted warping to resolve the 3D heterogeneity of the compaction and subsidence. This is an important tool for Valhall, and possibly other fields, where compaction and subsidence (and monitoring thereof) are key factors in the reservoir management since the predominant observed production‐induced changes are compaction of the soft, high‐porosity chalk reservoir, due to pore‐pressure reduction, and the resultant overburden subsidence. Such reservoir compaction could have significant implications for production by changing permeabilities and production rates. Furthermore the subsidence effects could impact upon subsea installations and well‐bore stability. Geomechanical studies that have previously been used to model such subsidence and compaction are only constrained by observed surface displacements and measured reservoir pressure changes, with the geological overburden being largely neglected. The approaches suggested herein provide the potential for monitoring and assessment in three dimensions, including the probable heterogeneity and shearing, that is needed for full understanding of reservoir compaction and the resultant effects on the overburden to, for example, mitigate well‐bore failures.     

Efficient scheme for the shallow water equations on unstructured grids with application to the Continental Shelf

In this paper, a shallow-water flow solver is presented, based on the finite-volume method on unstructured grids The method is suitable for flows that occur in rivers, channels, sewer systems (1D), shallow seas, rivers, overland flow (2D), and estuaries, lakes and shelf breaks (3D). We present an outline of the numerical approach and show three 2D test cases and an application of tidal propagation on the Continental Shelf. The benefits of applying an unstructured grid were explored by creating an efficient model network that aims at keeping the number of grid cells per wavelength constant. The computational speed of our method was compared with that of WAQUA/TRIWAQ and Delft3D (the commonly used structured shallow-flow solvers in The Netherlands), and comparable performance was found.     

Two-dimensional structure of mountain wave observed by aircraft during the PYREX experiment

This study presents an experimental analysis from aircraft measurements above the Pyrenees chain during the PYREX experiment. The Pyrenees chain, roughly WE oriented, is a major barrier for northerly and southerly airflows. We present a case of southerly flow (15 October 1990) and three successive cases of northerly flows above the Pyrenees (14, 15 and 16 November 1990) documented by two aircraft. The aircraft have described a vertical cross section perpendicular to the Pyrenean ridge. This area is described via the thermodynamical and dynamical fields which have a horizontal resolution of 10 km. Three methods for computing the vertical velocity of the air are presented. The horizontal advection terms which play a role in the budget equations are also evaluated. The altitude turbulence zone of 15 October are shown via turbulent fluxes, turbulent kinetic energy (TKE), dissipation rate of TKE and inertial length-scale. A comparison of results obtained by eddy-correlation and inertial-dissi-pation method is presented. The experimental results show a warm and dry downdraft for the southerly flow with large values for advection terms. All the mountain wave cases are also shown to present an important dynamical perturbation just above the Pyrenees at upper altitudes.     

Finite layer method for flow in layered radial two-zone aquifer systems

A new finite layer method (FLM) is presented in this paper for transient flow analysis in layered radial two-zone aquifer systems. A radial two-zone system is an aquifer configuration in which a circular aquifer with finite radius is surrounded by a matrix possessing different permeability and storage properties. The aquifers can be pumped from fully or partially penetrating wells of infinitesimal radius. The trial function for drawdown is obtained through the use of piecewise linear correction functions in the present method. The trial function can satisfy the continuity conditions of flow and possess an appropriate continuity of C(0) at the two-zone interface. On the basis of Galerkin's method and the continuity condition of flow, the finite layer formulation is derived. The proposed method can cope with the anisotropy and layered heterogeneity in radial two-zone aquifer systems. Several numerical examples are presented to verify the validity of the present method through comparison with the analytical solution and the numerical results based on the finite difference method, in which a test of three-dimensional (3D) flow to a partially penetrating well in anisotropic two-zone aquifers is included. Furthermore, an additional application in simulating the two-zone flow in aquitard-aquifer systems is presented to demonstrate the applicability of FLM in modeling flow in more complex aquifer systems.     

Computational issues and applications of line-elements to model subsurface flow governed by the modified Helmholtz equation

Two new approaches are presented for the accurate computation of the potential due to line elements that satisfy the modified Helmholtz equation with complex parameters. The first approach is based on fundamental solutions in elliptical coordinates and results in products of Mathieu functions. The second approach is based on the integration of modified Bessel functions. Both approaches allow evaluation of the potential at any distance from the element. The computational approaches are applied to model transient flow with the Laplace transform analytic element method. The Laplace domain solution is computed using a combination of point elements and the presented line elements. The time domain solution is obtained through a numerical inversion. Two applications are presented to transient flow fields, which could not be modeled with the Laplace transform analytic element method prior to this work. The first application concerns transient single-aquifer flow to wells near impermeable walls modeled with line-doublets. The second application concerns transient two-aquifer flow to a well near a stream modeled with line-sinks.     

Optimal removal of topological artefacts in microtomographic images of porous materials

The analysis of flow at the pore scale in porous media has been facilitated with the use of microtomography. A powerful tool for quantifying the fluid structure using these tomographic 3D reconstructions is skeletonisation, but the significant disadvantage of this method is its sensitivity to noise, resulting in artefacts in the skeleton. A pre-processing of the 3D image is therefore required, but no method has yet proven to completely solve this problem. By developing a new procedure that, by construction, directly identifies the voxels and only those that are responsible for topological artefacts in the skeleton, we are able to remove all artefacts, and furthermore can prove that we do so by modifying a minimal amount of voxels in the segmented 3D image (i.e. the tomographic image in which each voxel has been assigned to either the porous or the solid phase). This is possible by identifying the three fundamental types of artefacts that can arise in a 3D skeleton, and dealing with each appropriately. Application to a microtomographic image of a sintered glass powder is presented. Impact of the different processing methods on the flow within its porosity is measured through the computed permeability deviations.