Modeling Shallow Water Flows on General Terrains

A formulation of the shallow water equations adapted to general complex terrains is proposed. Its derivation starts from the observation that the typical approach of depth integrating the Navier-Stokes equations along the direction of gravity forces is not exact in the general case of a tilted curved bottom. We claim that an integration path that better adapts to the shallow water hypotheses follows the "cross-flow" surface, i.e., a surface that is normal to the velocity field at any point of the domain. Because of the implicitness of this definition, we approximate this "cross-flow" path by performing depth integration along a local direction normal to the bottom surface, and propose a rigorous derivation of this approximation and its numerical solution as an essential step for the future development of the full "cross-flow" integration procedure. We start by defining a local coordinate system, anchored on the bottom surface to derive a covariant form of the Navier-Stokes equations. Depth integration along the local normals yields a covariant version of the shallow water equations, which is characterized by flux functions and source terms that vary in space because of the surface metric coefficients and related derivatives. The proposed model is discretized with a first order FORCE-type Godunov Finite Volume scheme that allows implementation of spatially variable fluxes. We investigate the validity of our SW model and the effects of the bottom geometry by means of three synthetic test cases that exhibit non negligible slopes and surface curvatures. The results show the importance of taking into consideration bottom geometry even for relatively mild and slowly varying curvatures

Examination of the seepage face boundary condition in subsurface and coupled surface/subsurface hydrological models

A seepage face is a nonlinear dynamic boundary that strongly affects pressure head distributions, water table fluctuations, and flow patterns. Its handling in hydrological models, especially under complex conditions such as heterogeneity and coupled surface/subsurface flow, has not been extensively studied. In this paper, we compare the treatment of the seepage face as a static (Dirichlet) versus dynamic boundary condition, we assess its resolution under conditions of layered heterogeneity, we examine its interaction with a catchment outlet boundary, and we investigate the effects of surface/subsurface exchanges on seepage faces forming at the land surface. The analyses are carried out with an integrated catchment hydrological model. Numerical simulations are performed for a synthetic rectangular sloping aquifer and for an experimental hillslope from the Landscape Evolution Observatory. The results show that the static boundary condition is not always an adequate stand-in for a dynamic seepage face boundary condition, especially under conditions of high rainfall, steep slope, or heterogeneity; that hillslopes with layered heterogeneity give rise to multiple seepage faces that can be highly dynamic; that seepage face and outlet boundaries can coexist in an integrated hydrological model and both play an important role; and that seepage faces at the land surface are not always controlled by subsurface flow. The paper also presents a generalized algorithm for resolving seepage face outflow that handles heterogeneity in a simple way, is applicable to unstructured grids, and is shown experimentally to be equivalent to the treatment of atmospheric boundary conditions in subsurface flow models

Simulazione numerica del flusso e trasporto di contaminanti in mezzi porosi a saturazione e densità variabile

La legislazione riguardante la salvaguardia e la tutela delle risorse idriche, e tra queste le acque sotterranee, è in continua crescita in tutti i paesi industrializzati. La protezione delle acque di falda dal sovrasfruttamento e dalla contaminazione di origine diversa (rifiuti urbani e industriali, pesticidi e fertilizzanti, scorie nucleari, ecc...) richiede la previsione degli effetti indotti dalle attività umane sulla quantità e qualità delle risorse sotterranee, previsione che si può conseguire solo attraverso l'impiego di idonei modelli matematico-numerici. Un problema di stringente attualità in tutti i paesi che si affacciano sul Mediterraneo è l'inquinamento degli acquiferi costieri per intrusione di acqua di mare. La simulazione della penetrazione del cuneo salino comporta lo sviluppo di modelli accoppiati di flusso e trasporto che possono essere accuratamente ed efficientemente risolti col metodo degli elementi finiti (FEM) che viene qui implementato in un mezzo poroso tridimensionale a saturazione variabile, e che è quindi in grado di simulare sia la zona insatura (suoli superficiali) che quella satura (falde in pressione). Le non linearità che scaturiscono dall'accoppiamento e dalle leggi costitutive della permeabilità e del coefficiente di immagazzinamento nella zona insatura sono risolte con le tecniche di Picard e di Newton parziale. I modelli discreti finali linearizzati sono trattati col metodo dei gradienti coniugati opportunamente precondizionati per le matrici simmetriche di flusso (PGC) e quelle non simmetriche di trasporto (GMRES, Bi-CGSTAB, TFQMR). Le procedure descritte sono implementate nel codice FEM CODESA-3D (COupled variable DEnsity and SAturation) di cui è offerto un esempio applicativo.In the industrialized countries subsurface water resources are increasingly subject to regulations for protection from over-exploitation and from contamination arising from urban, industrial, nuclear, military, and agricultural activities. Prediction of the effects of anthropogenic impacts on water quantity and quality is an important part of proper aquifer management, and can be achieved through the use of mathematical models. As an example, seawater intrusion in coastal aquifers represents a serious environmental problem, especially in the countries of the Mediterranean basin, and can be simulated using coupled models of water flow and solute transport. Sophisticated groundwater models such as these can be accurately and effciently solved numerically via finite element discretizations of the three-dimensional porous medium. Both saturated (groundwater) and unsaturated (soil water) zones can be represented, and nonlinearities arising from storage-pressure head and conductivity-pressure head dependencies in the unsaturated zone and from coupling of the two equations can be resolved using Picard, Newton, or partial Newton methods. The resulting linearized systems of equations can be solved using a variety of preconditioned conjugate gradient-like methods applicable to symmetric and non-symmetric systems. The mathematical formulation and numerical procedures to be described form the basis of the CODESA-3D (COupled variable DEnsity and SAturation) model

Towards a Stationary Monge--Kantorovich Dynamics: The Physarum Polycephalum Experience

In this work we propose an extension to the continuous setting of a model describing the dynamics of slime mold, Physarum Polycephalum (PP), which was proposed to simulate the ability of PP to find the shortest path connecting two food sources in a maze. The original model describes the dynamics of the slime mold on a finite-dimensional planar graph using a pipe-flow analogy whereby mass transfer occurs because of pressure differences with a conductivity coefficient that varies with the flow intensity. This model has been shown to be equivalent to a problem of \u201coptimal transportation\u201d on graphs. We propose an extension that abandons the graph structure and moves to a continuous domain. The new model couples an elliptic diffusion equation enforcing PP density balance with an ordinary differential equation governing the flow dynamics. We conjecture that the new system of equations presents a time-asymptotic equilibrium and that such an equilibrium point is precisely the solution of Monge--Kantorovich partial differential equations governing optimal transportation problems. To support this conjecture, we analyze the proposed model by recasting it into an infinite-dimensional dynamical system. We are then able to show well-posedness of the proposed model for sufficiently small times under the hypotheses of H\uf6lder continuous diffusion coefficients and essentially bounded forcing functions. Numerical results obtained with a simple fixed-point iteration combining P_1 / P_0 finite elements with backward Euler time stepping show that the approximate solution of our formulation of the transportation problem converges at large times to an equilibrium configuration that well compares with the numerical solution of the Monge--Kantorovich equations

Rapporto di ricerca bibliografica (Stato dell'arte dei modelli di flusso e trasporto in mezzi porosi)

In questo primo rapporto vengono presentate le equazioni fondamentali che reggono i fenomeni di flusso e trasporto in mezzi porosi insieme ai metodi numerici che verranno utilizzati per la soluzione di tali equazioni. I modelli matematici in questione sono basati su equazioni differenziali a derivate parziali che impongono il bilancio di massa sia per il fluido che per il soluto (inquinante disciolto in acqua). Queste equazioni vengono scritte in forma generale per un mezzo poroso tridimensionale; in dipendenza dal tipo di applicazione è possibile adottare modelli mono o bidimensionali che portano a semplificazioni notevoli. L'equazione di flusso è sviluppata per il caso di mezzi porosi a saturazione variabile e può essere quindi utilizzata contemporaneamente nella zona insatura (suoli superficiali) e satura (falde freatiche e artesiane). Nell'equazione di trasporto si considerano i processi di dispersione, diffusione e avvezione, insieme ad alcune fenomenologie di interazione chimico-fisica tra il soluto e la matrice porosa. Accanto a queste equazione, si descrive anche un modello, a scala di bacino, di afflussi-deflussi superficiali accoppiato con un modello di infiltrazione. Questo approccio viene tiene conto di fenomeni importanti qualora vi sia una stretta correlazione tra il moto dell'acqua in superficie e il moto dell'acqua nella zona insatura

Coupled surface runoff and subsurface flow model for catchment simulations

A distributed catchment scale numerical model for the simulation of coupled surface runoff and subsurface flow is presented. Starting from rainfall (evaporation) records, the model first determines the infiltration (exfiltration) rates in the soil, by evaluation of the soil field capacity at the specific conditions as calculated from the three-dimensional solution of the variably-saturated groundwater flow model (Richards’ equation). The flow rate that remains or returns to the surface, the so called overland flow, is then routed via a diffusion wave surface runoff model based on a Muskingum-Cunge scheme with variable parameters. Both hillslope and channel flow are described, and a special algorithm is used for the simulation of pools/lakes effects on storm-flow response. The importance of including detailed subsurface flow description in catchment simulations is shown on a simple testcase characterized by the presence of a central depression.583-59