Worth of secondary data compared to piezometric data for the probabilistic assessment of radionuclide migration

A common approach for the performance assessment of radionuclide migration from a nuclear waste repository is by means of Monte-Carlo techniques. Multiple realizations of the parameters controlling radionuclide transport are generated and each one of these realizations is used in a numerical model to provide a transport prediction. The statistical analysis of all transport predictions is then used in performance assessment. In order to reduce the uncertainty on the predictions is necessary to incorporate as much information as possible in the generation of the parameter fields. In this regard, this paper focuses in the impact that conditioning the transmissivity fields to geophysical data and/or piezometric head data has on convective transport predictions in a two-dimensional heterogeneous formation. The Walker Lake data based is used to produce a heterogeneous log-transmissivity field with distinct non-Gaussian characteristics and a secondary variable that represents some geophysical attribute. In addition, the piezometric head field resulting from the steady-state solution of the groundwater flow equation is computed. These three reference fields are sampled to mimic a sampling campaign. Then, a series of Monte-Carlo exercises using different combinations of sampled data shows the relative worth of secondary data with respect to piezometric head data for transport predictions. The analysis shows that secondary data allows to reproduce the main spatial patterns of the reference transmissivity field and improves the mass transport predictions with respect to the case in which only transmissivity data is used. However, a few piezometric head measurements could be equally effective for the characterization of transport predictions.     

Worth of secondary data compared to piezometric data for the probabilistic assessment of radionuclide migration

A common approach for the performance assessment of radionuclide migration from a nuclear waste repository is by means of Monte-Carlo techniques. Multiple realizations of the parameters controlling radionuclide transport are generated and each one of these realizations is used in a numerical model to provide a transport prediction. The statistical analysis of all transport predictions is then used in performance assessment. In order to reduce the uncertainty on the predictions is necessary to incorporate as much information as possible in the generation of the parameter fields. In this regard, this paper focuses in the impact that conditioning the transmissivity fields to geophysical data and/or piezometric head data has on convective transport predictions in a two-dimensional heterogeneous formation. The Walker Lake data based is used to produce a heterogeneous log-transmissivity field with distinct non-Gaussian characteristics and a secondary variable that represents some geophysical attribute. In addition, the piezometric head field resulting from the steady-state solution of the groundwater flow equation is computed. These three reference fields are sampled to mimic a sampling campaign. Then, a series of Monte-Carlo exercises using different combinations of sampled data shows the relative worth of secondary data with respect to piezometric head data for transport predictions. The analysis shows that secondary data allows to reproduce the main spatial patterns of the reference transmissivity field and improves the mass transport predictions with respect to the case in which only transmissivity data is used. However, a few piezometric head measurements could be equally effective for the characterization of transport predictions.     

Geostatistics and Bayesian updating for transmissivity estimation in a multiaquifer system in Manitoba,Canada

In ground water flow and transport modeling, the heterogeneous nature of porous media has a considerable effect on the resulting flow and solute transport. Some method of generating the heterogeneous field from a limited dataset of uncertain measurements is required. Bayesian updating is one method that interpolates from an uncertain dataset using the statistics of the underlying probability distribution function. In this paper, Bayesian updating was used to determine the heterogeneous natural log transmissivity field for a carbonate and a sandstone aquifer in southern Manitoba. It was determined that the transmissivity in m²/sec followed a natural log normal distribution for both aquifers with a mean of -7.2 and - 8.0 for the carbonate and sandstone aquifers, respectively. The variograms were calculated using an estimator developed by Li and Lake (1994). Fractal nature was not evident in the variogram from either aquifer. The Bayesian updating heterogeneous field provided good results even in cases where little data was available. A large transmissivity zone in the sandstone aquifer was created by the Bayesian procedure, which is not a reflection of any deterministic consideration, but is a natural outcome of updating a prior probability distribution function with observations. The statistical model returns a result that is very reasonable; that is homogeneous in regions where little or no information is available to alter an initial state. No long range correlation trends or fractal behavior of the log-transmissivity field was observed in either aquifer over a distance of about 300 km.     

Effect of Conditioning Randomly Heterogeneous Transmissivity on Temporal Hydraulic Head Measurements in Transient Two-Dimensional Aquifer Flow

We investigated the effect of conditioning transient, two-dimensional groundwater flow simulations, where the transmissivity was a spatial random field, on time dependent head data. The random fields, representing perturbations in log transmissivity, were generated using a known covariance function and then conditioned to match head data by iteratively cokriging and solving the flow model numerically. A new approximation to the cross-covariance of log transmissivity perturbations with time dependent head data and head data at different times, that greatly increased the computational efficiency, was introduced. The most noticeable effect of head data on the estimation of head and log transmissivity perturbations occurred from conditioning only on spatially distributed head measurements during steady flow. The additional improvement in the estimation of the log transmissivity and head perturbations obtained by conditioning on time dependent head data was fairly small. On the other hand, conditioning on temporal head data had a significant effect on particle tracks and reduced the lateral spreading around the center of the paths.     

Effect of Conditioning Randomly Heterogeneous Transmissivity on Temporal Hydraulic Head Measurements in Transient Two-Dimensional Aquifer Flow

We investigated the effect of conditioning transient, two-dimensional groundwater flow simulations, where the transmissivity was a spatial random field, on time dependent head data. The random fields, representing perturbations in log transmissivity, were generated using a known covariance function and then conditioned to match head data by iteratively cokriging and solving the flow model numerically. A new approximation to the cross-covariance of log transmissivity perturbations with time dependent head data and head data at different times, that greatly increased the computational efficiency, was introduced. The most noticeable effect of head data on the estimation of head and log transmissivity perturbations occurred from conditioning only on spatially distributed head measurements during steady flow. The additional improvement in the estimation of the log transmissivity and head perturbations obtained by conditioning on time dependent head data was fairly small. On the other hand, conditioning on temporal head data had a significant effect on particle tracks and reduced the lateral spreading around the center of the paths.     

Monotonic solution of flow and transport problems in heterogeneous media using Delaunay unstructured triangular meshes

Transport problems occurring in porous media and including convection, diffusion and chemical reactions, can be well represented by systems of Partial Differential Equations. In this paper, a numerical procedure is proposed for the fast and robust solution of flow and transport problems in 2D heterogeneous saturated media. The governing equations are spatially discretized with unstructured triangular meshes that must satisfy the Delaunay condition. The solution of the flow problem is split from the solution of the transport problem and it is obtained with an approach similar to the Mixed Hybrid Finite Elements method, that always guarantees the M-property of the resulting linear system. The transport problem is solved applying a prediction/correction procedure. The prediction step analytically solves the convective/reactive components in the context of a MAST Finite Volume scheme. The correction step computes the anisotropic diffusive components in the context of a recently proposed Finite Elements scheme. Massa balance is locally and globally satisfied in all the solution steps. Convergence order and computational costs are investigated and model results are compared with literature ones.     

Transmissivity identification through multi-directional aquifer stimulation

This paper presents a geostatistical approach to multi-directional aquifer stimulation in order to better identify the transmissivity field. Hydraulic head measurements, taken at a few locations but under a number of different steady-state flow conditions, are used to estimate the transmissivity. Well installation is generally the most costly aspect of obtaining hydraulic head measurements. Therefore, it is advantageous to obtain as many informative measurements from each sampling location as possible. This can be achieved by hydraulically stimulating the aquifer through pumping, in order to set-up a variety of flow conditions. We illustrate the method by applying it to a synthetic aquifer. The simulations provide evidence that a few sampling locations may provide enough information to estimate the transmissivity field. Furthermore, the innovation of, or new information provided by, each measurement can be examined by looking at the corresponding spline and sensitivity matrix. Estimates from multi-directional stimulation are found to be clearly superior to estimates using data taken under one flow condition. We describe the geostatistical methodology for using data from multi-directional simulations and address computational issues.     

Transmissivity identification through multi-directional aquifer stimulation

This paper presents a geostatistical approach to multi-directional aquifer stimulation in order to better identify the transmissivity field. Hydraulic head measurements, taken at a few locations but under a number of different steady-state flow conditions, are used to estimate the transmissivity. Well installation is generally the most costly aspect of obtaining hydraulic head measurements. Therefore, it is advantageous to obtain as many informative measurements from each sampling location as possible. This can be achieved by hydraulically stimulating the aquifer through pumping, in order to set-up a variety of flow conditions. We illustrate the method by applying it to a synthetic aquifer. The simulations provide evidence that a few sampling locations may provide enough information to estimate the transmissivity field. Furthermore, the innovation of, or new information provided by, each measurement can be examined by looking at the corresponding spline and sensitivity matrix. Estimates from multi-directional stimulation are found to be clearly superior to estimates using data taken under one flow condition. We describe the geostatistical methodology for using data from multi-directional simulations and address computational issues.     

Stochastic analysis of spatial variability in unconfined groundwater flow

In this paper, spatial variability in steady one-dimensional unconfined groundwater flow in heterogeneous formations is investigated. An approach to deriving the variance of the hydraulic head is developed using the nonlinear filter theory. The nonlinear governing equation describing the one-dimensional unconfined groundwater flow is decomposed into three linear partial differential equations using the perturbation method. The linear and quadratic frequency response functions are obtained from the first- and second-order perturbation equations using the spectral method. Furthermore, under the assumption of the exponential covariance function of log hydraulic conductivity, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the linear system are derived. The results show that the variance derived herein is less than that of Gelhar (1977). The reason is that the log transmissivity is linearized in Gelhars work. In addition, the analytical solutions of both the spectrum and the variance of the hydraulic head produced from the quadratic system are derived as well. It is found that the correlation scale and the trend in mean of log hydraulic conductivity are important to the dimensionless variance ratio.     

Analytical solutions of one-dimensional macrodispersion in stratified porous media by the quadrupole method: convergence to an equivalent homogeneous porous medium

In this article, the quadrupole method is implemented in order to simulate the effects of heterogeneities on one dimensional advective and diffusive transport of a passive solute in porous media. Theoretical studies of dispersion in heterogeneous stratified media can bring insight into transport artefacts linked to scale effects and apparent dispersion coefficients. The quadrupole method is an efficient method for the calculation of transient response of linear systems. It is based here on the Laplace transform technique. The analytical solutions that can be derived by this method assists understanding of upscaled parameters relevant to heterogeneous porous media.First, the method is developed for an infinite homogeneous porous medium. Then, it is adapted to a stratified medium where the fluid flow is perpendicular to the interfaces. The first heterogeneous medium studied is composed of two semi-infinite layers perpendicular to the flow direction each having different transport properties. The concentration response of the medium to a Dirac injection is evaluated. The case studied emphasises the importance in the choice of the boundary conditions.In the case of a periodic heterogeneous porous medium, the concentration response of the medium is evaluated for different numbers of unit-cells. When the number of unit cells is great enough, depending on the transport properties of each layer in the unit cell, an equivalent homogeneous behaviour is reached. An exact determination of the transport properties (equivalent dispersion coefficient) of the equivalent homogeneous porous medium is given.     

Solving nonlinear water management models using a combined genetic algorithm and linear programming approach

Gradient-based nonlinear programming (NLP) methods can solve problems with smooth nonlinear objectives and constraints. However, in large and highly nonlinear models, these algorithms can fail to find feasible solutions, or converge to local solutions which are not global. Evolutionary search procedures in general, and genetic algorithms (GAs) specifically, are less susceptible to the presence of local solutions. However, they often exhibit slow convergence, especially when there are many variables, and have problems finding feasible solutions in constrained problems with “narrow” feasible regions. In this paper, we describe strategies for solving large nonlinear water resources models management, which combine GAs with linear programming. The key idea is to identify a set of complicating variables in the model which, when fixed, render the problem linear in the remaining variables. The complicating variables are then varied by a GA. This GA&LP approach is applied to two nonlinear models: a reservoir operation model with nonlinear hydropower generation equations and nonlinear reservoir topologic equations, and a long-term dynamic river basin planning model with a large number of nonlinear relationships. For smaller instances of the reservoir model, the CONOPT2 nonlinear solver is more accurate and faster, but for larger instances, the GA&LP approach finds solutions with significantly better objective values. The multiperiod river basin model is much too large to be solved in its entirety. The complicating variables are chosen here so that, when they are fixed, each period's model is linear, and these models can be solved sequentially. This approach allows sufficient model detail to be retained so that long-term sustainability issues can be explored.     

湍流输送的非线性热力学性质

湍流输送是一种热力学不可逆过程,本文利用非线性热力学研究了湍流输送的特征. 将热力学流对热力学力以平衡态作为参考态进行Taylor展开,可以得到湍流输送系数是系统宏观参量梯度的Taylor级数. 线性湍流输送系数是Reynolds湍流闭合方案的K闭合湍流输送系数;而湍流输送系数非线性项则是系统偏离热力学平衡态所造成的热力学非线性效应. 湍流输送系数这一热力学性质提供了一种热力学湍流闭合方案. 线性湍流输送系数是正定的,湍流输送只能使系统宏观参量均匀化;而在远离平衡态的热力学非线性区,可能导致湍流输送系数负黏性现象. 在最小熵产生态的条件下,热力学流对热力学力Taylor展开的各级系数间存在一种递推关系. 利用这种递推关系大大减少了由实验确定的Taylor级数的系数个数.     

A generalized transformation approach for simulating steady-state variably-saturated subsurface flow

A numerical method is proposed to accurately and efficiently compute a direct steady-state solution of the nonlinear Richards equation. In the proposed method, the Kirchhoff integral transformation and a complementary transformation are applied to the governing equation in order to separate the nonlinear hyperbolic characteristic from the linear parabolic part. The separation allows the transformed governing equation to be applied to partially- to fully-saturated systems with arbitrary constitutive relations between primary (pressure head) and secondary variables (relative permeability). The transformed governing equation is then discretized with control volume finite difference/finite element approximations, followed by inverse transformation. The approach is compared to analytical and other numerical approaches for variably-saturated flow in 1-D and 3-D domains. The results clearly demonstrate that the approach is not only more computationally efficient but also more accurate than traditional numerical solutions. The approach is also applied to an example flow problem involving a regional-scale variably-saturated heterogeneous system, where the vadose zone is up to 1 km thick. The performance, stability, and effectiveness of the transform approach is exemplified for this complex heterogeneous example, which is typical of many problems encountered in the field. It is shown that computational performance can be enhanced by several orders of magnitude with the described integral transformation approach.     

A two‐dimensional inverse model to identify transmissivity in an anisotropic aquifer

This study suggested a numerical model using the Tabu search algorithm along with the Adjoint State method to identify the hydrogeological characteristics of an anisotropic groundwater aquifer. The Tabu search algorithm was applied to identify the anisotropic transmissivity components to avoid a local optimum. Then, the Adjoint State method was used to calculate the sensitivity of the parameters in order to increase the efficiency of the optimization. For an anisotropic and homogeneous aquifer, results showed that the optimal procedure presented combining the Tabu search algorithm and the Adjoint State method might successfully identify the values of the transmissivity components. If the duration of the pumping test was long enough (12‐h pumping test), the value of the transmissivity components could be optimized with type‐curve, straight‐line, and Tabu search methods, along with the Adjoint State methods. If the duration of the pumping test was short (0·5‐h pumping test), the Tabu search method, along with the Adjoint State method proposed herein, might successfully optimize the transmissivity components. For an anisotropic but heterogeneous aquifer, results showed that the suggested optimal procedure still successfully identified the values of the transmissivity components. Copyright © 2008 John Wiley & Sons, Ltd.     

Coherence estimation algorithm using Kendall’s concordance measurement on seismic data

The coherence method is always used to describe the discontinuity and heterogeneity of seismic data. In traditional coherence methods, a linear correlation coefficient is always used to measure the relationship between two random variables (i.e., between two seismic traces). However, mathematically speaking, a linear correlation coefficient cannot be applied to describe nonlinear relationships between variables. In order to overcome this limitation of liner correlation coefficient. We proposed an improved concordance measurement algorithm based on Kendall’s tau. That mainly concern the sensitivity of the liner correlation coefficient and concordance measurements on the waveform. Using two designed numerical models tests sensitivity of waveform similarity affected by these two factors. The analysis of both the numerical model results and real seismic data processing suggest that the proposed method, combining information divergence measurement, can not only precisely characterize the variations of waveform and the heterogeneity of an underground geological body, but also does so with high resolution. In addition, we verified its effectiveness by the actual application of real seismic data from the north of China.     

Characterization of a hydraulically induced crystalline‐rock fracture

Hydraulic fracturing has become an important technique for enhancing the permeability of hydrocarbon source rocks and increasing aquifer transmissivity in many hard rock environments where natural fractures are limited, yet little is known about the nature or behaviour of these hydraulically induced fractures as conduits to flow and transport. We propose that these fractures tend to be smooth based on observed hydraulic and transport behaviour. In this investigation a multi‐faceted approach was used to quantify the properties and characteristics of an isolated hydraulically induced fracture in crystalline rocks. Packers were used to isolate the fracture that is penetrated by two separate observation wells located approximately 33 m apart. A series of aquifer tests and an induced gradient tracer test were performed to better understand the nature of this fracture. Aquifer test results indicate that full recovery is slow because of the overall low permeability of the crystalline rocks. Drawdown tests indicate that the fracture has a transmissivity of 1–2 m²/day and a specific storage on the order of 2–9 × 10^?7/m. Analysis of a potassium–bromide tracer test break through curve shows classic Fickian behaviour with minimal tailing analogous to parallel plate flow. Virtually all of the tracer was recovered, and the breakthrough curve dilution indicates that the swept area is only about 11% of a radial flow field and the estimated aperture is ≤0.5 mm, which implies a narrow linear flow region. These outcomes suggest that transport within these hydraulically induced ‘smooth’ fractures in crystalline rocks is rapid with minimal mixing, small local velocity fluctuations and no apparent diffusion into the host rock or secondary fractures. Copyright © 2016 John Wiley & Sons, Ltd.     

Combining geostatistics and flow simulators to identify transmissivity

The reconstruction of the transmissivity field from the more numerous experimental hydraulic head data, an inverse problem, remains the focus of continuing stochastic-based research. The difficulty of this problem arises not only from the complexity of the diffusion equation that links the two variables, but also from taking into account the physical aspects of the site under study; e.g. the boundary conditions, the effective recharge, and the geology. In practical applications, the validity of purely analytical techniques proposed to date is limited by certain simplifying assumptions, like the linearization of the flow equation, made in order to obtain a solution. For this reason, a hybrid methodology combining geostatistical techniques with deterministic numerical flow simulators is proposed. This combination allows the numerical calculation of the direct and cross covariances needed to cokrige the transmissivity from both the transmissivity and hydraulic head data. The flexibility of numerical flow simulators takes away the need for the simplifying assumptions of analytical techniques to apply the proposed methodology.