A finite element model for simulating surface run‐off and unsaturated seepage flow in the shallow subsurface

This work introduces water–air two‐phase flow into integrated surface–subsurface flow by simulating rainfall infiltration and run‐off production on a soil slope with the finite element method. The numerical model is formulated by partial differential equations for hydrostatic shallow flow and water–air two‐phase flow in the shallow subsurface. Finite element computing formats and solution strategies are presented to obtain a numerical solution for the coupled model. An unsaturated seepage flow process is first simulated by water–air two‐phase flow under the atmospheric pressure boundary condition to obtain the rainfall infiltration rate. Then, the rainfall infiltration rate is used as an input parameter to solve the surface run‐off equations and determine the value of the surface run‐off depth. In the next iteration, the pressure boundary condition of unsaturated seepage flow is adjusted by the surface run‐off depth. The coupling process is achieved by updating the rainfall infiltration rate and surface run‐off depth sequentially until the convergence criteria are reached in a time step. A well‐conducted surface run‐off experiment and traditional surface–subsurface model are used to validate the new model. Comparisons with the traditional surface–subsurface model show that the initiation time of surface run‐off calculated by the proposed model is earlier and that the water depth is larger, thus providing values that are closer to the experimental results.     

A new modeling approach for simulating microtopography-dominated,discontinuous overland flow on infiltrating surfaces

Realistic modeling of discontinuous overland flow on irregular topographic surfaces has been proven to be a challenge. This study is aimed to develop a new modeling framework to simulate the discontinuous puddle-to-puddle (P2P) overland flow dynamics for infiltrating surfaces with various microtopographic characteristics. In the P2P model, puddles were integrated in a well-delineated, cascaded drainage system to facilitate explicit simulation of their dynamic behaviors and interactions. Overland flow and infiltration were respectively simulated by using the diffusion wave model and a modified Green–Ampt model for the DEM-derived flow drainage network that consisted of a series of puddle-based units (PBUs). The P2P model was tested by using a series of data from laboratory overland flow experiments for various microtopography, soil, and rainfall conditions. The modeling results indicated that the hierarchical relationships and microtopographic properties of puddles significantly affected their connectivity, filling–spilling dynamics, and the associated threshold flow. Surface microtopography and rainfall characteristics also exhibited strong influences on the spatio-temporal distributions of infiltration rates, runoff fluxes, and unsaturated flow. The model tests demonstrated its applicability in simulating microtopography-dominated overland flow on infiltrating surfaces.     

Adaptive multi-scale parameterization for one-dimensional flow in unsaturated porous media

In the analysis of the unsaturated zone, one of the most challenging problems is to use inverse theory in the search for an optimal parameterization of the porous media. Adaptative multi-scale parameterization consists in solving the problem through successive approximations by refining the parameter at the next finer scale all over the domain and stopping the process when the refinement does not induce significant decrease of the objective function any more. In this context, the refinement indicators algorithm provides an adaptive parameterization technique that opens the degrees of freedom in an iterative way driven at first order by the model to locate the discontinuities of the sought parameters. We present a refinement indicators algorithm for adaptive multi-scale parameterization that is applicable to the estimation of multi-dimensional hydraulic parameters in unsaturated soil water flow. Numerical examples are presented which show the efficiency of the algorithm in case of noisy data and missing data.     

Validity limits for the van Genuchten–Mualem model and implications for parameter estimation and numerical simulation

The calculation of the relative hydraulic conductivity function based on water retention data is an attractive and widely used approach, since direct measurements of unsaturated conductivities are difficult. We show theoretically under which conditions an air-entry value for water retention data is definitely required when using the statistical approach of Mualem. Moreover we rigorously specify the conditions for which the classical van Genuchten–Mualem model leads to wrong predictions of relative hydraulic conductivity and, hence, an alternative formulation including an air-entry value should be used. Significant consequences are demonstrated for the inverse parameter estimation based on multistep outflow experiments. Furthermore it is shown that the use of a physically correct formulation of the water retention curve including an air-entry value and the derived hydraulic conductivity function influences not only the stability of numerical simulations but also their final results. This is especially grave as simulations with van Genuchten–Mualem parameters are frequently used to compare experiments and simulations and to draw conclusions on the correctness of Richards’ equation.     

Cyclic hysteretic flow in porous medium column: model, experiment, and simulations

A periodic vertical movement of the groundwater table results in a subsequent cyclic response of the water content and pressure profiles in the vadose zone. The sequence of periodic wetting and drying processes can be affected by hysteresis effects in this zone. A one-dimensional saturated/unsaturated flow model based on Richards’ equation and the Mualem (Soil Sci. 137 (1984) 283) hysteresis model is formulated which can take into account multi-cycle hysteresis effects in the relation between capillary pressure and water content. The numerical integration of the unsaturated flow equation is based on a Galerkin-type finite element method. The flow domain is discretised by finite elements with linear shape functions. Simulations start with static water content and pressure profiles, which correspond to either a boundary drying or wetting retention curve. To facilitate the numerical solution of the hysteretic case an implicit non-iterative procedure was chosen for the solution of the nonlinear differential equation. Laboratory experiments were performed with a vertical sand column by imposing a high frequency periodic pressure head at the lower end of the column. The total water volume in the column, and the periodic water content profile averaged over time were measured. The boundary drying and wetting curves of the relation between water content and capillary pressure were determined by independent experiments. The simulations of the experimental conditions show a clear effect of the hysteresis phenomenon on the water content profile. The simulations with hysteresis agree well with the measurements. Computed dimensionless water content profiles are presented for different oscillation frequencies with and without consideration of hysteresis.     

Inverse methods for subsurface flow: A critical review of stochastic techniques

The development of stochastic methods for groundwater flow representation has undergone enormous expansion in recent years. The calibration of groundwater models, the inverse problem, has lately received comparable attention especially and almost exclusively from the stochastic perspective. In this review we trace the evolution of the methods to date with a specific view toward identifying the most important issues involved in the usefulness of the approaches. The methods are critiqued regarding practical usefulness, and future directions for requisite study are discussed.     

A multiscale method for subsurface inverse modeling: Single-phase transient flow

High-resolution geologic models that incorporate observed state data are expected to effectively enhance the reliability of reservoir performance prediction. One of the major challenges faced is how to solve the large-scale inverse modeling problem, i.e., to infer high-resolution models from the given observations of state variables that are related to the model parameters according to some known physical rules, e.g., the flow and transport partial differential equations. There are typically two difficulties, one is the high-dimensional problem and the other is the inverse problem. A multiscale inverse method is presented in this work to attack these problems with the aid of a gradient-based optimization algorithm. In this method, the model responses (i.e., the simulated state data) can be efficiently computed from the high-resolution model using the multiscale finite-volume method. The mismatch between the observations and the multiscale solutions is then used to define a proper objective function, and the fine-scale sensitivity coefficients (i.e., the derivatives of the objective function with respect to each node’s attribute) are computed by a multiscale adjoint method for subsequent optimization. The difficult high-dimensional optimization problem is reduced to a one-dimensional one using the gradient-based gradual deformation method. A synthetic single-phase transient flow example problem is employed to illustrate the proposed method. Results demonstrate that the multiscale framework presented is not only computationally efficient but also can generate geologically consistent models. By preserving spatial structure for inverse modeling, the method presented overcomes the artifacts introduced by the multiscale simulation and may enhance the prediction ability of the inverse-conditional realizations generated.     

瞬变电磁波场转换算法的改进

地球物理电磁场数据与虚拟地震波场数据之间存在数学上的等效转换关系,通过这种等效转换,可有效提高地球物理电磁法对地下目标体分界面的辨识度.但是这种转换在数学上属于不适定问题,可采用奇异值分解法处理.由于大奇异值控制计算矩阵的主要信息,小的奇异值控制计算矩阵的次要信息,传统的截断奇异值分解法只保留大奇异值,而忽略小的奇异值,导致数值解不够精确.本文提出一种新的修正方案——改进截断奇异值法,采用岭估计方法计算由小的奇异值引起的虚拟波场.模型计算结果表明：改进截断奇异值法比传统的奇异值分解法得到的波场转换结果更好,对某煤矿采空区探测数据进行了处理,成功分辨出采空区分界面.

     

Efficient methods for solving water flow in variably saturated soils under prescribed flux infiltration

The non-linear solvers in numerical solutions of water flow in variably saturated soils are prone to convergence difficulties. Many aspects can give rise to such difficulties and in this paper we address the gravity term and the prescribed-flux boundary in the Picard iteration. The problem of the gravity term in the Picard iteration is iteration-to-iteration oscillation as the gravity term is treated, by analogy with the time-step advance technique, ‘explicitly’ in the iteration. The proposed method for the gravity term is an improvement of the ‘implicit’ approach of Zhang and Ewen [Water Resour. Res. 36 (2000) 2777] by extending it to heterogeneous soil and approximating the inter-nodal hydraulic conductivity in the diffusive term and the gravity term with the same scheme. The prescribed-flux boundary in traditional methods also gives rise to iteration-to-iteration oscillation because there is no feedback to the flux in the solution at the new iteration. To reduce such oscillation, a new method is proposed to provide such a feedback to the flux. Comparison with traditional Picard and Newton iteration methods for a wide range of problems show that a combination of these two proposed methods greatly improves the stability and consequently the computational efficiency, making the use of small time step and/or under-relaxation solely for convergence unnecessary.     

A smoothed particle hydrodynamics model for droplet and film flow on smooth and rough fracture surfaces

Flow on fracture surfaces has been identified by many authors as an important flow process in unsaturated fractured rock formations. Given the complexity of flow dynamics on such small scales, robust numerical methods have to be employed in order to capture the highly dynamic interfaces and flow intermittency. In this work we use a three-dimensional multiphase Smoothed Particle Hydrodynamics (SPH) model to simulate surface tension dominated flow on smooth fracture surfaces. We model droplet and film flow over a wide range of contact angles and Reynolds numbers encountered in such flows on rock surfaces. We validate our model via comparison with existing empirical and semi-analytical solutions for droplet flow. We use the SPH model to investigate the occurrence of adsorbed trailing films left behind droplets under various flow conditions and its importance for the flow dynamics when films and droplets coexist. It is shown that flow velocities are higher on prewetted surfaces covered by a thin film which is qualitatively attributed to the enhanced dynamic wetting and dewetting at the trailing and advancing contact lines. Finally, we demonstrate that the SPH model can be used to study flow on rough surfaces.     

A mixed-dimensional finite volume method for two-phase flow in fractured porous media

We present a vertex-centered finite volume method for the fully coupled, fully implicit discretization of two-phase flow in fractured porous media. Fractures are discretely modeled as lower dimensional elements. The method works on unstructured, locally refined grids and on parallel computers with distributed memory. An implicit time discretization is employed and the nonlinear systems of equations are solved with a parallel Newton-multigrid method. Results from two-dimensional and three-dimensional simulations are presented.     

Improved numerical solvers for implicit coupling of subsurface and overland flow

Due to complex dynamics inherent in the physical models, numerical formulation of subsurface and overland flow coupling can be challenging to solve. ParFlow is a subsurface flow code that utilizes a structured grid discretization in order to benefit from fast and efficient structured solvers. Implicit coupling between subsurface and overland flow modes in ParFlow is obtained by prescribing an overland boundary condition at the top surface of the computational domain. This form of implicit coupling leads to the activation and deactivation of the overland boundary condition during simulations where ponding or drying events occur. This results in a discontinuity in the discrete system that can be challenging to resolve. Furthermore, the coupling relies on unstructured connectivities between the subsurface and surface components of the discrete system, which makes it challenging to use structured solvers to effectively capture the dynamics of the coupled flow. We present a formulation of the discretized algebraic system that enables the use of an analytic form of the Jacobian for the Newton–Krylov solver, while preserving the structured properties of the discretization. An effective multigrid preconditioner is extracted from the analytic Jacobian and used to precondition the Jacobian linear system solver. We compare the performance of the new solver against one that uses a finite difference approximation to the Jacobian within the Newton–Krylov approach, previously used in the literature. Numerical results explores the effectiveness of using the analytic Jacobian for the Newton–Krylov solver, and highlights the performance of the new preconditioner and its cost. The results indicate that the new solver is robust and generally outperforms the solver that is based on the finite difference approximation to the Jacobian, for problems where the overland boundary condition is activated and deactivated during the simulation. A parallel weak scaling study highlights the efficiency of the new solver.     

A method for incorporating equilibrium chemical reactions into multiphase flow models for CO2 storage

CO₂ injection and storage in deep saline aquifers involves many coupled processes, including multiphase flow, heat and mass transport, rock deformation and mineral precipitation and dissolution. Coupling is especially critical in carbonate aquifers, where minerals will tend to dissolve in response to the dissolution of CO₂ into the brine. The resulting neutralization will drive further dissolution of both CO₂ and calcite. This suggests that large cavities may be formed and that proper simulation may require full coupling of reactive transport and multiphase flow. We show that solving the latter may suffice whenever two requirements are met: (1) all reactions can be assumed to occur in equilibrium and (2) the chemical system can be calculated as a function of the state variables of the multiphase flow model (i.e., liquid and gas pressure, and temperature). We redefine the components of multiphase flow codes (traditionally, water and CO₂), so that they are conservative for all reactions of the chemical system. This requires modifying the traditional constitutive relationships of the multiphase flow codes, but yields the concentrations of all species and all reaction rates by simply performing speciation and mass balance calculations at the end of each time step. We applied this method to the H₂O–CO₂–Na–Cl–CaCO₃ system, so as to model CO₂ injection into a carbonate aquifer containing brine. Results were very similar to those obtained with traditional formulations, which implies that full coupling of reactive transport and multi-phase flow is not really needed for this kind of systems, but the resulting simplifications may make it advisable even for cases where the above requirements are not met. Regarding the behavior of carbonate rocks, we find that porosity development near the injection well is small because of the low solubility of calcite. Moreover, dissolution concentrates at the front of the advancing CO₂ plume because the brine below the plume tends to reach high CO₂ concentrations quite rapidly. We conclude that carbonate dissolution needs not to be feared.     

Coarsening of three-dimensional structured and unstructured grids for subsurface flow

We present a generic, semi-automated algorithm for generating non-uniform coarse grids for modeling subsurface flow. The method is applicable to arbitrary grids and does not impose smoothness constraints on the coarse grid. One therefore avoids conventional smoothing procedures that are commonly used to ensure that the grids obtained with standard coarsening procedures are not too rough. The coarsening algorithm is very simple and essentially involves only two parameters that specify the level of coarsening. Consequently the algorithm allows the user to specify the simulation grid dynamically to fit available computer resources, and, e.g., use the original geomodel as input for flow simulations. This is of great importance since coarse grid-generation is normally the most time-consuming part of an upscaling phase, and therefore the main obstacle that has prevented simulation workflows with user-defined resolution. We apply the coarsening algorithm to a series of two-phase flow problems on both structured (Cartesian) and unstructured grids. The numerical results demonstrate that one consistently obtains significantly more accurate results using the proposed non-uniform coarsening strategy than with corresponding uniform coarse grids with roughly the same number of cells.     

Curvilinear immersed boundary method for simulating coupled flow and bed morphodynamic interactions due to sediment transport phenomena

The fluid-structure interaction curvilinear immersed boundary (FSI-CURVIB) numerical method of Borazjani et al. [3] is extended to simulate coupled flow and sediment transport phenomena in turbulent open-channel flows. The mobile channel bed is discretized with an unstructured triangular mesh and is treated as a sharp-interface immersed boundary embedded in a background curvilinear mesh used to discretize the general channel outline. The unsteady Reynolds-averaged Navier-Stokes (URANS) equations closed with the k − ω turbulence model are solved numerically on a hybrid staggered/non-staggered grid using a second-order accurate fractional step method. The bed deformation is calculated by solving the sediment continuity equation in the bed-load layer using an unstructured, finite-volume formulation that is consistent with the CURVIB framework. Both the first-order upwind and the higher-order hybrid GAMMA schemes [12] are implemented to discretize the bed-load flux gradients and their relative accuracy is evaluated through a systematic grid refinement study. The GAMMA scheme is employed in conjunction with a sand-slide algorithm for limiting the bed slope at locations where the material angle of repose condition is violated. The flow and bed deformation equations are coupled using the partitioned loose-coupling FSI-CURVIB approach [3]. The hydrodynamic module of the method is validated by applying it to simulate the flow in an 180° open-channel bend with fixed bed. To demonstrate the ability of the model to simulate bed morphodynamics and evaluate its accuracy, we apply it to calculate turbulent flow through two mobile-bed open channels, with 90° and 135° bends, respectively, for which experimental measurements are available.     

Stream tube integration method for analysing subsurface fluid flow

A stream tube integration method is introduced to solve transient subsurface fluid flow problems. The method combines a geometry-embedded form of Darcy's Law and the notion of location of average. Two types of problems, transient radial flow to a well of finite radius in an areally infinite aquifer and in a double porosity system, are solved by the stream tube integration method and the integral finite difference method. Results of the solutions show that the stream tube integration method, with fixed coarse mesh, are more accurate and better behaved than the integral finite difference method, with fine mesh. The fixed mesh stream tube integration method is readily extended to the moving mesh method. With much coarse mesh, the moving mesh technique can obtain the same accurate results as the fixed mesh stream tube integration method. It is suggested that the stream tube integration method is a viable way to state, solve, interpret and verify numerical solutions. The method provides efficient computation and improved accuracy for analysing subsurface fluid flow. © 1998 John Wiley & Sons, Ltd.     

Analytical approach for sheet flow transport in purely acceleration-skewed oscillatory flow

An instantaneous analytical approach is developed to predict sheet flow transport in purely acceleration-skewed oscillatory flow. The approach is derived from exponential approximations of velocity and concentration profiles above a mobile seabed, and it particularly considers factors of phase lead; phase lag (i.e. phase residual and phase shift); acceleration modification; and asymmetries in shear stress, roughness height, and boundary layer development. The approach can predict net boundary layer flow above a mobile seabed, and can revert to the classical bedload model. Instantaneous and net sediment transport rates are studied using the approach. The instantaneous sediment transport rate in an onshore flow stage can be approximated by a power function of velocity in which the exponent is confirmed to range between 1 and 5 with a decrease in the phase residual. The net sediment transport rate predicted using the approach is validated using a considerable amount of measured data, and compared with existing instantaneous and half-period type models that consider the phase lag or acceleration modification. For the net sediment transport rate in purely acceleration-skewed oscillatory flow, the phase residual is less important than the acceleration-skewed boundary layer difference between onshore and offshore acceleration stages.     

磁层顶边界层剪切流引起的MHD波模转化

采用非本征模方法（non-modal analysis）研究了磁层顶边界层中剪切流导致的MHD波模转化及其与背景流场的能量交换过程.发现在分别代表磁层顶内、外边界层及过渡区的均匀剪切流场中,初始设定的Alfvén波扰动可部分转化为快、慢磁声波.而且,在不同区域,根据等离子体参数的不同,发生的波模转化过程也不相同.在外边界,Alfvén波主要转化为慢磁声波;在内边界,Alfvén波则主要转化为快磁声波;而在二者之间的过渡区中,Alfvén波可同时转化为两种类型的磁声波.我们还发现,含有较强快波分量的扰动可从磁层顶剪切流场中获得能量而得到线性放大.上述物理过程可能对解释磁鞘至磁层的能量及动量异常输运现象有所帮助.     

A network theory approach for a better understanding of overland flow connectivity

Hydrological connectivity describes the physical coupling (linkages) of different elements within a landscape regarding (sub‐) surface flows. A firm understanding of hydrological connectivity is important for catchment management applications, for example, habitat and species protection, and for flood resistance and resilience improvement. Thinking about (geomorphological) systems as networks can lead to new insights, which has also been recognized within the scientific community, seeing the recent increase in the use of network (graph) theory within the geosciences. Network theory supports the analysis and understanding of complex systems by providing data structures for modelling objects and their linkages, and a versatile toolbox to quantitatively appraise network structure and properties. The objective of this study was to characterize and quantify overland flow connectivity dynamics on hillslopes in a humid sub‐Mediterranean environment by using a combination of high‐resolution digital‐terrain models, overland flow sensors and a network approach. Results showed that there are significant differences between overland flow connectivity on agricultural areas and semi‐natural shrubs areas. Significant positive correlations between connectivity and precipitation characteristics were found. Significant negative correlations between connectivity and soil moisture were found, most likely because of soil water repellency and/or soil surface crusting. The combination of structural networks and dynamic networks for determining potential connectivity and actual connectivity proved a powerful tool for analysing overland flow connectivity. Copyright © 2016 John Wiley & Sons, Ltd.