Polynomial chaos expansion and response surface method for nonlinear modelling of reference evapotranspiration

The feasibility of polynomial chaos expansion (PCE) and response surface method (RSM) models is investigated for modelling reference evapotranspiration (ET₀). The modelling results of the proposed models are validated against the M5 model tree and multi-layer perceptron neural network (MLPNN) methods. Two meteorological stations, Isparta and Antalya, in the Mediterranean region of Turkey, are inspected. Various input combinations of daily air temperature, solar radiation, wind speed and relative humidity are constructed as input attributes for the ET₀. Generally, the modelling accuracy is increased by increasing the number of inputs. Including wind speed in the model inputs considerably increases their accuracy in modelling ET₀. Mean absolute error (MAE), root mean square error (RMSE), agreement index (d) and Nash-Sutcliffe efficiency (NSE) are used as comparison criteria. The PCE is the most accurate model in estimating daily ET₀, giving the lowest MAE (0.036 and 0.037 mm) and RMSE (0.047 and 0.050 mm) and the highest d (0.9998 and 0.9999) and NSE (0.9992 and 0.9996) with the four-input PCE models for Isparta and Antalya, respectively.     

Stochastic analysis of flux and head moments in a heterogeneous aquifer system

This study investigates the behavior of flux and head in a strongly heterogeneous three-dimensional aquifer system. The analyses relied on data from 520 slug tests together with 38,000 one-foot core intervals lithological data from the site of the General Separations Area in central Savannah River Site, South Carolina, USA. The skewness in the hydraulic conductivity histograms supported the geologic information for the top two aquifers, but revealed stronger clay content, than was reported for the bottom aquifer. The log-normal distribution model described adequately the hydraulic conductivity measurements for all three aquifers although, other distributions described equally well the bottom aquifer measurements. No apparent anisotropy on the horizontal plane was found for the three aquifers, but ratios of horizontal to vertical correlation lengths between 33 and 75 indicated a strong stratification at the site. Three-dimensional Monte Carlo stochastic simulations utilized a grid with larger elements than the support volume of measurements, but of sub-REV (representative elementary volume) dimensions. This necessitated, on one hand, the use of upscaled hydraulic conductivity expressions, but on the other hand did not allow for the use of anisotropic effective hydraulic conductivity expressions (Sarris and Paleologos in J Stoch Environ Res Risk Assess 18: 188–197, 2004). Flux mean and standard deviations components were evaluated on three vertical cross-sections. The mean and variance of the horizontal flux component normal to a no-flow boundary tended to zero at approximately two to three integral scales from that boundary. Close to a prescribed head boundary both the mean and variance of the horizontal flux component normal to the boundary increased from a stable value attained at a distance of about five integral scales from that boundary. The velocity field 〈q_x〉 was found to be mildly anisotropic in the top two aquifers, becoming highly anisotropic in the bottom aquifer; 〈q_y〉 was anisotropic in all three aquifers with directions of high continuity normal to those of the 〈q_x〉 field; finally, 〈q_z〉 was highly anisotropic in all three aquifers, with higher continuity along the east–west direction. The mean head field was found to be continuous, despite the high heterogeneity of the underlying hydraulic conductivity field. Directions of high continuity were in alignment with field boundaries and mean flow direction. Conditioning did not influence significantly the expected value of the flux terms, with more pronounced being the effect on the standard deviation of the flux vector components. Conditioning reduced the standard deviations of the horizontal flux components by as much as 50% in the bottom aquifer. Variability in the head cross-sections was affected only marginally, with an average 10% reduction in the respective standard deviation. Finally, the location of the conditioning data did not appear to have a significant effect on the surrounding area, with uniform reduction in standard deviations.     

Inverse analysis of stochastic moment equations for transient flow in randomly heterogeneous media

We present a nonlinear stochastic inverse algorithm that allows conditioning estimates of transient hydraulic heads, fluxes and their associated uncertainty on information about hydraulic conductivity (K) and hydraulic head (h  ) data collected in a randomly heterogeneous confined aquifer. Our algorithm is based on Laplace-transformed recursive finite-element approximations of exact nonlocal first and second conditional stochastic moment equations of transient flow. It makes it possible to estimate jointly spatial variations in natural log-conductivity (Y=lnK)$(Y=\ln K)$, the parameters of its underlying variogram, and the variance–covariance of these estimates. Log-conductivity is parameterized geostatistically based on measured values at discrete locations and unknown values at discrete “pilot points”. Whereas prior values of Y at pilot point are obtained by generalized kriging, posterior estimates at pilot points are obtained through a maximum likelihood fit of computed and measured transient heads. These posterior estimates are then projected onto the computational grid by kriging. Optionally, the maximum likelihood function may include a regularization term reflecting prior information about Y. The relative weight assigned to this term is evaluated separately from other model parameters to avoid bias and instability. We illustrate and explore our algorithm by means of a synthetic example involving a pumping well. We find that whereas Y and h can be reproduced quite well with parameters estimated on the basis of zero-order mean flow equations, all model quality criteria identify the second-order results as being superior to zero-order results. Identifying the weight of the regularization term and variogram parameters can be done with much lesser ambiguity based on second- than on zero-order results. A second-order model is required to compute predictive error variances of hydraulic head (and flux) a posteriori. Conditioning the inversion jointly on conductivity and hydraulic head data results in lesser predictive uncertainty than conditioning on conductivity or head data alone.     

Global sensitivity analyses for a complex hydrological model applied in an Alpine watershed

The increasing complexity of hydrological models results in a large number of parameters to be estimated. In order to better understand how these complex models work, efficient screening methods are required in order to identify the most important parameters. This is of particular importance for models that are used within an operational real‐time forecasting chain such as HQsim. The objectives of this investigation are to (i) identify the most sensitive parameters of the complex HQsim model applied in the Alpine Lech catchment and (ii) compare model parameter sensitivity rankings attained from three global sensitivity analysis techniques. The techniques presented are the (i) regional sensitivity analysis, (ii) Morris analysis and (iii) state‐dependent parameter modelling. The results indicate that parameters affecting snow melt as well as processes in the unsaturated soil zone reveal high significance in the analysed catchment. The snow melt parameters show clear temporal patterns in the sensitivity whereas most of the parameters affecting processes in the unsaturated soil zone do not vary in importance across the year. Overall, the maximum degree day factor (meltfunc_max) has been identified to play a key role within the HQsim model. Although the parameter sensitivity rankings are equivalent between methods for a number of parameters, for several key parameters differing results were obtained. An uncertainty analysis demonstrates that a parameter ranking attained from only one method is subjected to large uncertainty. Copyright © 2012 John Wiley & Sons, Ltd.     

Simulation of liquid waste buoyancy in a deep heterogeneous aquifer

The results of 3D numerical simulation of coupled flow and transport processes of light-waste migration in heterogeneous aquifer with brine are described. The study is based on the geological data and monitoring of deep-well injection of liquid waste at Chepetsk Mechanical Plant (ChMZ, Glazov). The buoyancy of light waste is simulated for the post-injection period of 300 years using different conceptual models of heterogeneity. The procedure of 3D geostatistical simulation based on transition probability analysis with TSIM code of the reservoir heterogeneity is presented. The results of the ascending waste migration simulation using SEAWAT2000 code indicate that both the variable fluid density and the hydraulic heterogeneity must be considered in the forecast models for determining the possible waste distribution during the injection and the post injection periods.     

Field study of hydrogeologic characterization methods in a heterogeneous aquifer

Hydraulic conductivity (K) and specific storage (S(s)) are required parameters when designing transient groundwater flow models. The purpose of this study was to evaluate the ability of commonly used hydrogeologic characterization approaches to accurately delineate the distribution of hydraulic properties in a highly heterogeneous glaciofluvial deposit. The metric used to compare the various approaches was the prediction of drawdown responses from three separate pumping tests. The study was conducted at a field site, where a 15 m × 15 m area was instrumented with four 18-m deep Continuous Multichannel Tubing (CMT) wells. Each CMT well contained seven 17 cm × 1.9 cm monitoring ports equally spaced every 2 m down each CMT system. An 18-m deep pumping well with eight separate 1-m long screens spaced every 2 m was also placed in the center of the square pattern. In each of these boreholes, cores were collected and characterized using the Unified Soil Classification System, grain size analysis, and permeameter tests. To date, 471 K estimates have been obtained through permeameter analyses and 270 K estimates from empirical relationships. Geostatistical analysis of the small-scale K data yielded strongly heterogeneous K fields in three-dimensions. Additional K estimates were obtained through slug tests in 28 ports of the four CMT wells. Several pumping tests were conducted using the multiscreen and CMT wells to obtain larger scale estimates of both K and S(s). The various K and S(s) estimates were then quantitatively evaluated by simulating transient drawdown data from three pumping tests using a 3D forward numerical model constructed using HydroGeoSphere (Therrien et al. 2005). Results showed that, while drawdown predictions generally improved as more complexity was introduced into the model, the ability to make accurate drawdown predictions at all CMT ports was inconsistent.     

Deterministic sensitivity analysis for a model for flow in porous media

A deterministic method for sensitivity analysis is developed and applied to a mathematical model for the simulation of flow in porous media. The method is based on the singular value decomposition (SVD) of the Jacobian matrix of the model. It is a local approach to sensitivity analysis providing a hierarchical classification of the directions in both the input space and of those in the output space reflecting the degree of sensitiveness of the latter to the former. Its low computational cost, in comparison with that of statistical approaches, allows the study of the variability of the results of the sensitivity analysis due to the variations of the input parameters of the model, and thus it can provide a quality criterion for the validity of more classical probabilistic global approaches. For the example treated here, however, this variability is weak, and deterministic and statistical methods yield similar sensitivity results.     

Comparison of formation and fluid-column logs in a heterogeneous basalt aquifer

Deep observation boreholes in the vicinity of active production wells in Honolulu, Hawaii, exhibit the anomalous condition that fluid-column electrical conductivity logs and apparent profiles of pore-water electrical conductivity derived from induction conductivity logs are nearly identical if a formation factor of 12.5 is assumed. This condition is documented in three boreholes where fluid-column logs clearly indicate the presence of strong borehole flow induced by withdrawal from partially penetrating water-supply wells. This result appears to contradict the basic principles of conductivity-log interpretation. Flow conditions in one of these boreholes was investigated in detail by obtaining flow profiles under two water production conditions using the electromagnetic flowmeter. The flow-log interpretation demonstrates that the fluid-column log resembles the induction log because the amount of inflow to the borehole increases systematically upward through the transition zone between deeper salt water and shallower fresh water. This condition allows the properties of the fluid column to approximate the properties of water entering the borehole as soon as the upflow stream encounters that producing zone. Because this condition occurs in all three boreholes investigated, the similarity of induction and fluid-column logs is probably not a coincidence, and may relate to aquifer response under the influence of pumping from production wells.     

Calibration framework for a Kalman filter applied to a groundwater model

The paper presents a novel approach to the setup of a Kalman filter by using an automatic calibration framework for estimation of the covariance matrices. The calibration consists of two sequential steps: (1) Automatic calibration of a set of covariance parameters to optimize the performance of the system and (2) adjustment of the model and observation variance to provide an uncertainty analysis relying on the data instead of ad-hoc covariance values. The method is applied to a twin-test experiment with a groundwater model and a colored noise Kalman filter. The filter is implemented in an ensemble framework. It is demonstrated that lattice sampling is preferable to the usual Monte Carlo simulation because its ability to preserve the theoretical mean reduces the size of the ensemble needed. The resulting Kalman filter proves to be efficient in correcting dynamic error and bias over the whole domain studied. The uncertainty analysis provides a reliable estimate of the error in the neighborhood of assimilation points but the simplicity of the covariance models leads to underestimation of the errors far from assimilation points.     

Estimates of effective permeability for non-Newtonian fluid flow in randomly heterogeneous porous media

An understanding of the interplay between non-Newtonian effects in porous media flow and field-scale domain heterogeneity is of great importance in several engineering and geological applications. Here we present a simplified approach to the derivation of an effective permeability for flow of a purely viscous power–law fluid with flow behavior index n in a randomly heterogeneous porous domain subject to a uniform pressure gradient. A standard form of the flow law generalizing the Darcy’s law to non-Newtonian fluids is adopted, with the permeability coefficient being the only source of randomness. The natural logarithm of the permeability is considered a spatially homogeneous and correlated Gaussian random field. Under the ergodic hypothesis, an effective permeability is first derived for two limit 1-D flow geometries: flow parallel to permeability variation (serial-type layers), and flow transverse to permeability variation (parallel-type layers). The effective permeability of a 2-D or 3-D isotropic domain is conjectured to be a power average of 1-D results, generalizing results valid for Newtonian fluids under the validity of Darcy’s law; the conjecture is validated comparing our results with previous literature findings. The conjecture is then extended, allowing the exponents of the power averaging to be functions of the flow behavior index. For Newtonian flow, novel expressions for the effective permeability reduce to those derived in the past. The effective permeability is shown to be a function of flow dimensionality, domain heterogeneity, and flow behavior index. The impact of heterogeneity is significant, especially for shear-thinning fluids with a low flow behavior index, which tend to exhibit channeling behavior.     

Evaluation of confidence intervals for a steady-state leaky aquifer model

The fact that dependent variables of groundwater models are generally nonlinear functions of model parameters is shown to be a potentially significant factor in calculating accurate confidence intervals for both model parameters and functions of the parameters, such as the values of dependent variables calculated by the model. The Lagrangian method of Vecchia and Cooley [Vecchia, A.V. & Cooley, R.L., Water Resources Research, 1987, 23(7), 1237–1250] was used to calculate nonlinear Scheffé-type confidence intervals for the parameters and the simulated heads of a steady-state groundwater flow model covering 450 km² of a leaky aquifer. The nonlinear confidence intervals are compared to corresponding linear intervals. As suggested by the significant nonlinearity of the regression model, linear confidence intervals are often not accurate. The commonly made assumption that widths of linear confidence intervals always underestimate the actual (nonlinear) widths was not correct. Results show that nonlinear effects can cause the nonlinear intervals to be asymmetric and either larger or smaller than the linear approximations. Prior information on transmissivities helps reduce the size of the confidence intervals, with the most notable effects occurring for the parameters on which there is prior information and for head values in parameter zones for which there is prior information on the parameters.     

Development of a surrogate model and sensitivity analysis for spatio-temporal numerical simulators

Stochastic Environmental Research and Risk Assessment - To evaluate the consequences on human health of radionuclide releases in the environment, numerical simulators are used to model the...     

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.     

Multicriteria sensitivity analysis as a diagnostic tool for understanding model behaviour and characterizing model uncertainty

Complex hydrological models are being increasingly used nowadays for many purposes such as studying the impact of climate and land‐use change on water resources. However, building a high‐fidelity model, particularly at large scales, remains a challenging task, due to complexities in model functioning and behaviour and uncertainties in model structure, parameterization, and data. Global sensitivity analysis (GSA), which characterizes how the variation in the model response is attributed to variations in its input factors (e.g., parameters and forcing data), provides an opportunity to enhance the development and application of these complex models. In this paper, we advocate using GSA as an integral part of the modelling process by discussing its capabilities as a tool for diagnosing model structure and detecting potential defects, identifying influential factors, characterizing uncertainty, and selecting calibration parameters. Accordingly, we conduct a comprehensive GSA of a complex land surface–hydrology model, Modélisation Environmentale–Surface et Hydrologie (MESH), which combines the Canadian land surface scheme with a hydrological routing component, WATROUTE. Various GSA experiments are carried out using a new technique, called Variogram Analysis of Response Surfaces, for alternative hydroclimatic conditions in Canada using multiple criteria, various model configurations, and a full set of model parameters. Results from this study reveal that, in addition to different hydroclimatic conditions and SA criteria, model configurations can also have a major impact on the assessment of sensitivity. GSA can identify aspects of the model internal functioning that are counter‐intuitive and thus help the modeller to diagnose possible model deficiencies and make recommendations for improving development and application of the model. As a specific outcome of this work, a list of the most influential parameters for the MESH model is developed. This list, along with some specific recommendations, is expected to assist the wide community of MESH and Canadian land surface scheme users, to enhance their modelling applications.     

A double-porosity model for a fractured aquifer with non-Darcian flow in fractures

Abstract

Non-Darcian flow in a finite fractured confined aquifer is studied. A stream bounds the aquifer at one side and an impervious stratum at the other. The aquifer consists of fractures capable of transmitting water rapidly, and porous blocks which mainly store water. Unsteady flow in the aquifer due to a sudden rise in the stream level is analysed by the double-porosity conceptual model. Governing equations for the flow in fractures and blocks are developed using the continuity equation. The fluid velocity in fractures is often too high for the linear Darcian flow so that the governing equation for fracture flow is modified by Forcheimer's equation, which incorporates a nonlinear term. Governing equations are coupled by an interaction term that controls the quasi-steady-state fracture—block interflow. Governing equations are solved numerically by the Crank-Nicolson implicit scheme. The numerical results are compared to the analytical results for the same problem which assumes Darcian flow in both fractures and blocks. Numerical and analytical solutions give the same results when the Reynolds number is less than 0.1. The effect of nonlinearity on the flow appears when the Reynolds number is greater than 0.1. The higher the rate of flow from the stream to the aquifer, the higher the degree of nonlinearity. The effect of aquifer parameters on the flow is also investigated. The proposed model and its numerical solution provide a useful application of nonlinear flow models to fractured aquifers. It is possible to extend the model to different types of aquifer, as well as boundary conditions at the stream side. Time-dependent flow rates in the analysis of recession hydrographs could also be evaluated by this model.     

Analytic solutions to transient groundwater flow under time-dependent sources in a heterogeneous aquifer bounded by fluctuating river stage

Analytical solutions for the water table and lateral discharge in a heterogeneous unconfined aquifer with time-dependent source and fluctuating river stage were derived and compared with those in an equivalent homogeneous aquifer. The heterogeneous aquifer considered consists of a number of sections of different hydraulic conductivity values. The source term and river stage were assumed to be time-dependent but spatially uniform. The solutions derived is useful in studying various groundwater flow problems in a horizontally heterogeneous aquifer since the spatially piecewise-constant hydraulic conductivity and temporally piecewise-constant recharge and lateral discharge can be used to quantify variations in these processes commonly observed in reality. Applying the solutions derived to an aquifer of three sections of different hydraulic conductivity values shown that (1) the aquifer heterogeneity significantly increases the spatial variation of the water table and thus its gradient but it has little effect on lateral discharge in the case of temporally and spatially uniform recharge, (2) the time-dependent but spatially uniform recharge increases the temporal variation of groundwater table over the entire aquifer but its effect on lateral discharge is limited in the zone near the river, and (3) the effect of river stage fluctuation on the water table and lateral discharge is limited in the zone near the river and the effect of the heterogeneity is to increase lateral discharge to or recharge from the river.     

Development of probability distributions for urban hydrologic model parameters and a Monte Carlo analysis of model sensitivity

This paper proposes an approach to estimating the uncertainty related to EPA Storm Water Management Model model parameters, percentage routed (P_R) and saturated hydraulic conductivity (K_sat), which are used to calculate stormwater runoff volumes. The methodology proposed in this paper addresses uncertainty through the development of probability distributions for urban hydrologic parameters through extensive calibration to observed flow data in the Philadelphia collection system. The established probability distributions are then applied to the Philadelphia Southeast district model through a Monte Carlo approach to estimate the uncertainty in prediction of combined sewer overflow volumes as related to hydrologic model parameter estimation. Understanding urban hydrology is critical to defining urban water resource problems. A variety of land use types within Philadelphia coupled with a history of cut and fill have resulted in a patchwork of urban fill and native soils. The complexity of urban hydrology can make model parameter estimation and defining model uncertainty a difficult task. The development of probability distributions for hydrologic parameters applied through Monte Carlo simulations provided a significant improvement in estimating model uncertainty over traditional model sensitivity analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

A multiscale adjoint method to compute sensitivity coefficients for flow in heterogeneous porous media

A multiscale adjoint (MSADJ) method is developed to compute high-resolution sensitivity coefficients for subsurface flow in large-scale heterogeneous geologic formations. In this method, the original fine-scale problem is partitioned into a set of coupled subgrid problems, such that the global adjoint problem can be efficiently solved on a coarse grid. Then, the coarse-scale sensitivities are interpolated to the local fine grid by reconstructing the local variability of the model parameters with the aid of solving embedded adjoint subproblems. The approach employs the multiscale finite-volume (MSFV) formulation to accurately and efficiently solve the highly detailed flow problem. The MSFV method couples a global coarse-scale solution with local fine-scale reconstruction operators, hence yielding model responses that are quite accurate at both scales. The MSADJ method is equally efficient in computing the gradient of the objective function with respect to model parameters. Several examples demonstrate that the approach is accurate and computationally efficient. The accuracy of our multiscale method for inverse problems is twofold: the sensitivity coefficients computed by this approach are more accurate than the traditional finite-difference-based numerical method for computing derivatives, and the calibrated models after history matching honor the available dynamic data on the fine scale. In other words, the multiscale based adjoint scheme can be used to history match fine-scale models quite effectively.