Comparing the performance of empirical black‐box models for river flow forecasting in the Heihe River Basin,Northwestern China

For many practical reasons, the empirical black‐box models have become an increasingly popular modelling tool for river flow forecasting, especially in mountainous areas where very few meteorological observatories exist. In this article, precipitation data are used as the only input to estimate river flow. Using five empirical black‐box models—the simple linear model, the linear perturbation model, the linearly varying gain factor model, the constrained nonlinear system model and the nonlinear perturbation model–antecedent precipitation index—modelling results are compared with actual results in three catchments within the Heihe River Basin. The linearly varying gain factor model and the nonlinear perturbation model yielded excellent predictions. For better simulation accuracy, a commonly used multilayer feed‐forward neural network model (NNM) was applied to incorporate the outputs of the individual models. Comparing the performance of these models, it was found that the best results were obtained from the NNM model. The results also suggest that more reliable and precise predictions of river flow can be obtained by using the NNM model while also incorporating the combined outputs of different empirical black‐box models. Copyright © 2012 John Wiley & Sons, Ltd.     

Matrix–fracture transfer shape factor for modeling flow of a compressible fluid in dual-porosity media

The matrix–fracture transfer shape factor is one of the important parameters in the modeling of fluid flow in fractured porous media using a dual-porosity concept. Warren and Root [36] introduced the dual-porosity concept and suggested a relation for the shape factor. There is no general relationship for determining the shape factor for a single-phase flow of slightly compressible fluids. Therefore, different studies reported different values for this parameter, as an input into the flow models. Several investigations have been reported on the shape factor for slightly compressible fluids. However, the case of compressible fluids has not been investigated in the past. The focus of this study is, therefore, to find the shape factor for the single-phase flow of compressible fluids (gases) in fractured porous media. In this study, a model for the determination of the shape factor for compressible fluids is presented; and, the solution of nonlinear gas diffusivity equation is used to derive the shape factor. The integral method and the method of moments are used to solve the nonlinear governing equation by considering the pressure dependency of the viscosity and isothermal compressibility of the fluid. The approximate semi-analytical model for the shape factor presented in this study is verified using single-porosity, fine-grid, numerical simulations. The dependency of the shape factor on the gas specific gravity, pressure and temperature are also investigated. The theoretical analysis presented improves our understanding of fluid flow in fractured porous media. In addition, the developed matrix–fracture transfer shape factor can be used as an input for modeling flow of compressible fluids in dual-porosity systems, such as naturally fractured gas reservoirs, coalbed methane reservoirs and fractured tight gas reservoirs.     

Forecasting monthly inflow with extreme seasonal variation using the hybrid SARIMA-ANN model

The optimal operation of dam reservoirs can be programmed and managed by predicting the inflow to these structures more accurately. To this end, there are various linear and nonlinear models. However, some hydrological problems like inflow with extreme seasonal variation are not purely linear or nonlinear. To improve the forecasting accuracy of this phenomenon, a linear Seasonal Auto Regressive Integrated Moving Average (SARIMA) model is combined with a nonlinear Artificial Neural Network (ANN) model. This new model is used to predict the monthly inflow to the Jamishan dam reservoir in West Iran. A comparison of the SARIMA and ANN models with the proposed hybrid model’s results is provided accordingly. More specifically, the models’ performance in forecasting base and flood flows is evaluated. The effect of changing the forecasting period length on the models’ accuracy is studied. The results of increasing the number of SARIMA model parameters up to five are investigated to achieve more accurate forecasting. The hybrid model predicts peak flood flows much better than the individual models, but SARIMA outperforms the other models in predicting base flow. The obtained results indicate that the hybrid model reduces the overall forecast error more than the ANN and SARIMA models. The coefficient of determination of the hybrid, ANN and SARIMA models were 0.72, 0.64 and 0.58, and the root mean squared error values were 1.02, 1.16 and 1.27 respectively, during the forecast period. Changing the forecasting length also indicated that these models can be used in the long term without increasing the forecast error.     

Effects of model sensitivity and nonlinearity on nonlinear regression of ground water flow

Nonlinear regression is increasingly applied to the calibration of hydrologic models through the use of perturbation methods to compute the Jacobian or sensitivity matrix required by the Gauss-Newton optimization method. Sensitivities obtained by perturbation methods can be less accurate than those obtained by direct differentiation, however, and concern has arisen that the optimal parameter values and the associated parameter covariance matrix computed by perturbation could also be less accurate. Sensitivities computed by both perturbation and direct differentiation were applied in nonlinear regression calibration of seven ground water flow models. The two methods gave virtually identical optimum parameter values and covariances for the three models that were relatively linear and two of the models that were relatively nonlinear, but gave widely differing results for two other nonlinear models. The perturbation method performed better than direct differentiation in some regressions with the nonlinear models, apparently because approximate sensitivities computed for an interval yielded better search directions than did more accurately computed sensitivities for a point. The method selected to avoid overshooting minima on the error surface when updating parameter values with the Gauss-Newton procedure appears for nonlinear models to be more important than the method of sensitivity calculation in controlling regression convergence.     

Nonlinear FE model updating and reconstruction of the response of an instrumented seismic isolated bridge to the 2010 Maule Chile earthquake

Nonlinear finite element (FE) modeling has been widely used to investigate the effects of seismic isolation on the response of bridges to earthquakes. However, most FE models of seismic isolated bridges (SIB) have used seismic isolator models calibrated from component test data, while the prediction accuracy of nonlinear FE models of SIB is rarely addressed by using data recorded from instrumented bridges. In this paper, the accuracy of a state‐of‐the‐art FE model is studied through nonlinear FE model updating (FEMU) of an existing instrumented SIB, the Marga‐Marga Bridge located in Viña del Mar, Chile. The seismic isolator models are updated in 2 phases: component‐wise and system‐wise FEMU. The isolator model parameters obtained from 23 isolator component tests show large scatter, and poor goodness of fit of the FE‐predicted bridge response to the 2010 Mw 8.8 Maule, Chile Earthquake is obtained when most of those parameter sets are used for the isolator elements of the bridge model. In contrast, good agreement is obtained between the FE‐predicted and measured bridge response when the isolator model parameters are calibrated using the bridge response data recorded during the mega‐earthquake. Nonlinear FEMU is conducted by solving single‐ and multiobjective optimization problems using high‐throughput cloud computing. The updated FE model is then used to reconstruct response quantities not recorded during the earthquake, gaining more insight into the effects of seismic isolation on the response of the bridge during the strong earthquake.     

Non-equilibrium interphase heat and mass transfer during two-phase flow in porous media—Theoretical considerations and modeling

Mass and heat transfer occurring across phase-interfaces in multi-phase flow in porous media are mostly approximated using equilibrium relationships or empirical kinetic models. However, when the characteristic time of flow is smaller than that of mass or heat transfer, non-equilibrium situations may arise. Commonly, empirical approaches are used in such cases. There are only few works in the literature that use physically-based models for these transfer terms. In fact, one would expect physical approaches to modeling kinetic interphase mass and heat transfer to contain the interfacial area between the phases as a variable. Recently, a two-phase flow and solute transport model was developed that included interfacial area as a state variable [36]. In that model, interphase mass transfer was modeled as a kinetic process.     

Recession curve estimation for storm event separations

An accurate recession curve model is important for separating individual flow events, which is especially difficult over catchments in regions with a maritime climate where frequent rainfall events cause the flows to rise before they reach the baseflow level. The traditional recession curve equations are based on static linear and nonlinear reservoir models. These models work quite well for ground water dominated recession curves, but not so well when the direct runoff is significant in the recession part. In this study, a new modelling methodology is explored based on self-adaptive parameters in the linear and nonlinear reservoir models. It has been found that the adaptive forms performed better than the static ones, especially when a window for the adaptive parameter estimation is properly selected. While the nonlinear adaptive model had better accuracy over the linear one, it could become unstable if its window is too narrow, indicating that more research work is needed to find an useful pattern for the window size. A comparison between the recession curve models and PDM model (a rainfall-runoff model) has shown that they agreed quite well in most winter events, but less so in the summer.     

Modelling of solute transport in rivers under different flow rates: A case study without transient storage

A methodology to derive solute transport models at any flow rate is presented. The novelty of the proposed approach lies in the assessment of uncertainty of predictions that incorporate parameterisation based on flow rate. A simple treatment of uncertainty takes into account heteroscedastic modelling errors related to tracer experiments performed over a range of flow rates, as well as the uncertainty of the observed flow rates themselves. The proposed approach is illustrated using two models for the transport of a conservative solute: a physically based, deterministic, advection-dispersion model (ADE), and a stochastic, transfer function based, active mixing volume model (AMV). For both models the uncertainty of any parameter increases with increasing flow rate (reflecting the heteroscedastic treatment of modelling errors at different observed flow rates), but in contrast the uncertainty of travel time, computed from the predicted model parameters, was found to decrease with increasing flow rate.     

Effect of fracture pressure depletion regimes on the dual-porosity shape factor for flow of compressible fluids in fractured porous media

A precise value of the matrix-fracture transfer shape factor is essential for modeling fluid flow in fractured porous media by a dual-porosity approach. The slightly compressible fluid shape factor has been widely investigated in the literature. In a recent study, we have developed a transfer function for flow of a compressible fluid using a constant fracture pressure boundary condition [Ranjbar E, Hassanzadeh H, Matrix-fracture transfer shape factor for modeling flow of a compressible fluid in dual-porosity media. Adv Water Res 2011;34(5):627-39. doi:10.1016/j.advwatres.2011.02.012]. However, for a compressible fluid, the consequence of a pressure depletion boundary condition on the shape factor has not been investigated in the previous studies. The main purpose of this paper is, therefore, to investigate the effect of the fracture pressure depletion regime on the shape factor for single-phase flow of a compressible fluid. In the current study, a model for evaluation of the shape factor is derived using solutions of a nonlinear diffusivity equation subject to different pressure depletion regimes. A combination of the heat integral method, the method of moments and Duhamel’s theorem is used to solve this nonlinear equation. The developed solution is validated by fine-grid numerical simulations. The presented model can recover the shape factor of slightly compressible fluids reported in the literature. This study demonstrates that in the case of a single-phase flow of compressible fluid, the shape factor is a function of the imposed boundary condition in the fracture and its variability with time. It is shown that such dependence can be described by an exponentially declining fracture pressure with different decline exponents. These findings improve our understanding of fluid flow in fractured porous media.     

Nonlinear model reduction of unconfined groundwater flow using POD and DEIM

Nonlinear groundwater flow models have the propensity to be overly complex leading to burdensome computational demands. Reduced modeling techniques are used to develop an approximation of the original model that has smaller dimensionality and faster run times. The reduced model proposed is a combination of proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM). Solutions of the full model (snapshots) are collected to represent the physical dynamics of the system and Galerkin projection allows the formulation of a reduced model that lies in a subspace of the full model. Interpolation points are added through DEIM to eliminate the reduced model's dependence on the dimension of the full model. POD is shown to effectively reduce the dimension of the full model and DEIM is shown to speed up the solution by further reducing the dimension of the nonlinear calculations. To show the concept can work for unconfined groundwater flow model, with added nonlinear forcings, one-dimensional and two-dimensional test cases are constructed in MODFLOW-OWHM. POD and DEIM are added to MODFLOW as a modular package. Comparing the POD and the POD-DEIM reduced models, the experimental results indicate similar reduction in dimension size with additional computation speed up for the added interpolation. The hyper-reduction method presented is effective for models that have fine discretization in space and/or time as well as nonlinearities with respect to the state variable. The dual reduction approach ensures that, once constructed, the reduced model can be solved in an equation system that depends only on reduced dimensions.     

Experimental seismic investigation of Sefid‐rud concrete buttress dam model on shaking table

Owing to the devastating M7.6 earthquake of 20 June 1990 that occurred in the northern province of Iran, Sefid‐rud concrete buttress dam located near the epicenter was severely shaken. The crack penetrated throughout the dam thickness near slope discontinuity, causing severe leakage, but with no general failure. In this study, nonlinear seismic response of the highest monolith with empty reservoir is investigated experimentally through model testing. A geometric‐scaled model of 1:30 was tested on a shaking table with high‐frequency capability to study dynamic cracking of the model and serve as data for nonlinear computer model calibration. Three construction joints are set up in the model to simulate effects of construction aspects. The experimental results are then compared with smeared crack and damage mechanics finite‐element simulations using nonlinear concrete constitutive models based on fracture mechanics. The crack patterns obtained from numerical models are in good agreement with those obtained from shaking table tests for the case of including construction joint effects and rigid foundation. Copyright © 2008 John Wiley & Sons, Ltd.     

岩石滞后非线性弹性响应的物理模型

介绍了岩石滞后非线性弹性的几个物理模型。赫兹颗粒接触模型是具有多尺度和滞后特性的经典模型,它预测了岩石中强烈的非线性;软的粘结系统几乎决定了岩石的力学性质,粘结系统中的流体对非线性响应的贡献特别显著,但是目前还没有搞清楚粘结系统和孔隙流体究竟是如何影响非线性响应的;GL模型是一个基于金属位错的物理模型,这是滞后动力行为方面一个开拓性的微观模型;PM模型是一个基于岩石细观尺度的唯象模型,它对理解岩石滞后非线性的机制和尺度是很重要的。     

Nonlinear dynamics of long-wave perturbations of the Kolmogorov flow for large Reynolds numbers

The nonlinear dynamics of long-wave perturbations of the inviscid Kolmogorov flow, which models periodically varying in the horizontal direction oceanic currents, is studied. To describe this dynamics, the Galerkin method with basis functions representing the first three terms in the expansion of spatially periodic perturbations in the trigonometric series is used. The orthogonality conditions for these functions formulate a nonlinear system of partial differential equations for the expansion coefficients. Based on the asymptotic solutions of this system, a linear, quasilinear, and nonlinear stage of perturbation dynamics is identified. It is shown that the time-dependent growth of perturbations during the first two stages is succeeded by the stage of stable nonlinear oscillations. The corresponding oscillations are described by the oscillator equation containing a cubic nonlinearity, which is integrated in terms of elliptic functions. An analytical formula for the period of oscillations is obtained, which determines its dependence on the amplitude of the initial perturbation. Structural features of the field of the stream function of the perturbed flow are described, associated with the formation of closed vortex cells and meandering flow between them. As a supplement, an asymptotic analysis of nonlinear dynamics of long-wave perturbations superimposed on a damped by small viscosity Kolmogorov flow (very large, but finite Reynolds numbers) is made. It is strictly shown that all velocity components of the perturbed flow remain bounded in this case.     

Copula entropy coupled with artificial neural network for rainfall–runoff simulation

The rainfall–runoff relationship is not only nonlinear and complex but also difficult to model. Artificial neural network (ANN), as a data-driven technique, has gained significant attention in recent years and has been shown to be an efficient alternative to traditional methods for hydrological modeling. However, for different input combinations, ANN models can yield different results. Therefore, input variables and ANN types need to be carefully considered, when using an ANN model for stream flow forecasting. This study proposes the copula-entropy (CE) theory to identify the inputs of an ANN model. The CE theory permits to calculate mutual information (MI) and partial MI directly which avoids calculating the marginal and joint probability distributions. Three different ANN models, namely multi-layer feed (MLF) forward networks, radial basis function networks and general regression neural network, were applied to predict stream flow of Jinsha River, China. Results showed that the inputs selected by the CE method were better than those by the traditional linear correlation analysis, and the MLF ANN model with the inputs selected by CE method obtained the best predicted results for the Jinsha River at Pingshan gauging station.     

Effectiveness of BIOECODS for peak flow attenuation: an appraisal using InfoWorks SD

The Bio-Ecological Drainage System (BIOECODS) is a sustainable drainage system, which adopts a “control at source” approach for urban storm water management in Malaysia. This study attempts to model a small-scale BIOECODS using InfoWorks SD. New modelling techniques are used to fully integrate the surface and on-line subsurface conveyance system, in which overland flow routing is described by a storm water management model that uses a nonlinear reservoir method and the kinematic wave approximation of the St Venant equation, and subsurface flow is described by the Horton method in conjunction with the Soil Conservation Service (SCS) curve number (CN) method. The observed water levels at primary outlets are compared with those obtained from model simulation. The modelling approach has been proven successful as the hydrographs (predicted and observed) match each other closely, with a mean error in the range of 4.58–7.32%. Results from the model showed that the BIOECODS is able to attenuate peak flow by 60–75%, and increase the lag time by 20 min within an area of <28?300 m² when compared with a traditional drainage system.     

Coupled Aquifer-Borehole Simulation

A model coupling fluid hydraulics in a borehole with fluid flow in an aquifer is developed in this paper. Conservation of momentum is used to create a one-dimensional steady-state model of vertical flow in an open borehole combined with radially symmetric flow in an aquifer and with inflow to the well through the wellbore screen. Both laminar and turbulent wellbore conditions are treated. The influence of inflow through the wellbore screen on vertical flow in the wellbore is included, using a relation developed by Siwoń (1987) . The influence of inflow reduces the predicted vertical variation in head up to 15% compared to a calculation of head losses due to fluid acceleration and the conventional Colebrook-White formulation of friction losses in a circular pipe. The wellbore flow model is embedded into the MODFLOW-2000 ground water flow code. The nonlinear conservation of momentum equations are iteratively linearized to calculate the conductance terms for vertical flow in the wellbore. The resulting simulations agree favorably with previously published results when the model is adjusted to meet the assumptions of the previous coupled models.     

Analytical solution of the solute transport equation for the binary homovalent ion exchange in groundwater

A binary homovalent ion exchange transport model governed by local chemical equilibrium is considered for a one-dimensional, steady flow in a homogeneous soil column. An analytical solution of the aqueous concentration distribution for the convex exchange is obtained by applying nonlinear shock wave theory. The main nonlinear feature is the breaking of fronts into shock waves. The corresponding mathematical theory is the method of characteristics with a special treatment of shock waves. The wave velocity and front thickness are also obtained to illustrate the front propagation and structure. The derivation of the solution presented may offer a wide range of application opportunities and may also provide a good approach for solving the binary heterovalent exchange transport model.     

地震活动性研究中的非线性动力学模型

本文利用非线性动力学模型研究了断层带的地震活动特性.断层系由多个耦合非线性单元模拟,每个基本单元由Maxwell体和刚性滑块组成.滑块的运动满足静、动摩擦强度本构关系.理论模拟的地震活动在短时间内是随机的,但当时间尺度足够大时,断层带的总体活动表现出某种规律性.多数实际观测到的地震活动现象均在这些非线性动力学模型中出现. 本文提出转换概率的观点来度量断层间和孕震区间的地震转移.模拟中发现活动地震带中存在条件安全区,这可能对地震预报的实践有重要的意义.     

A FEniCS-based programming framework for modeling turbulent flow by the Reynolds-averaged Navier-Stokes equations

Finding an appropriate turbulence model for a given flow case usually calls for extensive experimentation with both models and numerical solution methods. This work presents the design and implementation of a flexible, programmable software framework for assisting with numerical experiments in computational turbulence. The framework targets Reynolds-averaged Navier-Stokes models, discretized by finite element methods. The novel implementation makes use of Python and the FEniCS package, the combination of which leads to compact and reusable code, where model- and solver-specific code resemble closely the mathematical formulation of equations and algorithms. The presented ideas and programming techniques are also applicable to other fields that involve systems of nonlinear partial differential equations. We demonstrate the framework in two applications and investigate the impact of various linearizations on the convergence properties of nonlinear solvers for a Reynolds-averaged Navier-Stokes model.     

Evaluation of Bias Associated with Capture Maps Derived from Nonlinear Groundwater Flow Models

The impact of groundwater withdrawal on surface water is a concern of water users and water managers, particularly in the arid western United States. Capture maps are useful tools to spatially assess the impact of groundwater pumping on water sources (e.g., streamflow depletion) and are being used more frequently for conjunctive management of surface water and groundwater. Capture maps have been derived using linear groundwater flow models and rely on the principle of superposition to demonstrate the effects of pumping in various locations on resources of interest. However, nonlinear models are often necessary to simulate head‐dependent boundary conditions and unconfined aquifers. Capture maps developed using nonlinear models with the principle of superposition may over‐ or underestimate capture magnitude and spatial extent. This paper presents new methods for generating capture difference maps, which assess spatial effects of model nonlinearity on capture fraction sensitivity to pumping rate, and for calculating the bias associated with capture maps. The sensitivity of capture map bias to selected parameters related to model design and conceptualization for the arid western United States is explored. This study finds that the simulation of stream continuity, pumping rates, stream incision, well proximity to capture sources, aquifer hydraulic conductivity, and groundwater evapotranspiration extinction depth substantially affect capture map bias. Capture difference maps demonstrate that regions with large capture fraction differences are indicative of greater potential capture map bias. Understanding both spatial and temporal bias in capture maps derived from nonlinear groundwater flow models improves their utility and defensibility as conjunctive‐use management tools.