Three‐dimensional unsaturated flow modeling using cellular automata

The parabolic partial differential equation describing fluid flow in partially saturated porous media, Richards' equation, is highly nonlinear due to pressure head dependencies in the specific soil moisture capacity and relative hydraulic conductivity terms. In order to solve Richards' equation several numerical techniques have been developed, which, starting from the discretization of the partial differential equation, produced even more accurate models, leading to complex and computationally expensive simulations for large‐scale systems. A three‐dimensional unsaturated flow modeling developed through a simulation environment based on cellular automata (CA) is described in this paper. The proposed model represents an extension of the original computational paradigm of cellular automata, because it uses a macroscopic CA approach where local laws with a clear physical meaning govern interactions among automata. This CA structure, aimed at simulating a large‐scale system, is based on functionalities capable of increasing its computational capacity, both in terms of working environment and in terms of the optimal number of processors available for parallel computing. The model has been validated with reference multidimensional solutions taken from benchmarks in literature, showing a good agreement even in the cases where nonlinearity is very marked. Furthermore, some analyses have been carried out considering quantization techniques aimed at transforming the CA model into an asynchronous structure. The use of these techniques in a three‐dimensional benchmark allowed a considerable reduction in the number of local interactions among adjacent automata without changing the efficiency of the model, especially when simulations are characterized by scarce mass exchanges. Finally, from a computational point of view the higher efficiency values were achieved running the model on a parallel architecture, obtaining a high speedup very close to the optimal with the maximum number of processors available.

[1]  John von Neumann,et al.  Theory Of Self Reproducing Automata , 1967 .

[2]  Y. Mualem,et al.  A conceptual model of hysteresis , 1974 .

[3]  A. R. Mitchell,et al.  The Finite Difference Method in Partial Differential Equations , 1980 .

[4]  P. Huyakorn,et al.  Techniques for Making Finite Elements Competitve in Modeling Flow in Variably Saturated Porous Media , 1984 .

[5]  D. R. Nielsen,et al.  On describing and predicting the hydraulic properties of unsaturated soils , 1985 .

[6]  G. Marsily.  Quantitative Hydrogeology: Groundwater Hydrology for Engineers , 1986 .

[7]  P. Huyakorn,et al.  A three‐dimensional finite‐element model for simulating water flow in variably saturated porous media , 1986 .

[8]  Tommaso Toffoli,et al.  Cellular Automata Machines , 1987, Complex Syst..

[9]  Pierre Lallemand,et al.  Lattice Gas Hydrodynamics in Two and Three Dimensions , 1987, Complex Syst..

[10]  Zanetti,et al.  Use of the Boltzmann equation to simulate lattice gas automata. , 1988, Physical review letters.

[11]  J. Jiménez,et al.  Boltzmann Approach to Lattice Gas Simulations , 1989 .

[12]  Eric F. Wood,et al.  NUMERICAL EVALUATION OF ITERATIVE AND NONITERATIVE METHODS FOR THE SOLUTION OF THE NONLINEAR RICHARDS EQUATION , 1991 .

[13]  Sauro Succi,et al.  The lattice Boltzmann equation: a new tool for computational fluid-dynamics , 1991 .

[14]  Y. Qian,et al.  Lattice BGK Models for Navier-Stokes Equation , 1992 .

[15]  Matthaeus,et al.  Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[16]  Gedeon Dagan,et al.  Upscaling of permeability of anisotropic heterogeneous formations: 1. The general framework , 1993 .

[17]  Michael A. Celia,et al.  MICROMODEL STUDIES OF THREE-FLUID POROUS MEDIA SYSTEMS : PORE-SCALE PROCESSES RELATING TO CAPILLARY PRESSURE-SATURATION RELATIONSHIPS , 1993 .

[18]  Mario Putti,et al.  A comparison of Picard and Newton iteration in the numerical solution of multidimensional variably saturated flow problems , 1994 .

[19]  Stephen Wolfram,et al.  Cellular Automata And Complexity , 1994 .

[20]  Stéphane Zaleski,et al.  Modeling water infiltration in unsaturated porous media by interacting lattice gas‐cellular automata , 1994 .

[21]  Interacting Lattice Gas Automaton Study of Liquid-Gas Properties in Porous Media , 1996 .

[22]  Chen,et al.  Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann method. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[23]  L. Luo,et al.  Lattice Boltzmann Model for the Incompressible Navier–Stokes Equation , 1997 .

[24]  C. Mattiussi.  An Analysis of Finite Volume, Finite Element, and Finite Difference Methods Using Some Concepts from Algebraic Topology , 1997 .

[25]  W. E. Soll,et al.  The influence of coatings and fills on flow in fractured, unsaturated tuff porous media systems , 1998 .

[26]  P. Lallemand,et al.  Lattice-gas cellular automata, simple models of complex hydrodynamics , 1998 .

[27]  James P. Crutchfield,et al.  The Evolutionary Design of Collective Computation in Cellular Automata , 1998, adap-org/9809001.

[28]  Roberto Serra,et al.  Applying Cellular Automata to Complex Environmental Problems: The Simulation of the Bioremediation of Contaminated Soils , 1999, Theor. Comput. Sci..

[29]  Roberto Serra,et al.  An empirical method for modelling and simulating some complex macroscopic phenomena by cellular automata , 1999, Future Gener. Comput. Syst..

[30]  S. Orlandini.  Two-Layer Model of Near-Surface Soil Drying for Time-Continuous Hydrologic Simulations , 1999 .

[31]  Bastien Chopard,et al.  Lattice Boltzmann Computations and Applications to Physics , 1999, Theor. Comput. Sci..

[32]  Giandomenico Spezzano,et al.  Scalability Analysis and Performance Prediction for Cellular Programs on Parallel Computers , 2000, ACRI.

[33]  W. E. Soll,et al.  Pore scale study of flow in porous media: Scale dependency, REV, and statistical REV , 2000 .

[34]  Enzo Tonti,et al.  A Direct Discrete Formulation of Field Laws: The Cell Method , 2001 .

[35]  Albert J. Valocchi,et al.  Pore‐scale modeling of dissolution from variably distributed nonaqueous phase liquid blobs , 2001 .

[36]  William G. Gray,et al.  A Boltzmann-based mesoscopic model for contaminant transport in flow systems , 2001 .

[37]  J. L. Snowdon,et al.  CELL-DEVS QUANTIZATION TECHNIQUES IN A FIRE SPREADING APPLICATION , 2002 .

[38]  John W. Crawford,et al.  A novel three‐dimensional lattice Boltzmann model for solute transport in variably saturated porous media , 2002 .

[39]  Gabriel A. Wainer,et al.  Cell-DEVS quantization techniques in a fire spreading application , 2002, Proceedings of the Winter Simulation Conference.

[40]  John W. Crawford,et al.  A lattice BGK model for advection and anisotropic dispersion equation , 2002 .

[41]  Giandomenico Spezzano,et al.  Simulation of a cellular landslide model with CAMELOT on high performance computers , 2003, Parallel Comput..

[42]  G. M. Crisci,et al.  An extended notion of Cellular Automata for surface flows modelling , 2003 .

[43]  Dani Or,et al.  Lattice Boltzmann method for modeling liquid‐vapor interface configurations in porous media , 2004 .

[44]  Gianmarco Manzini,et al.  Mass-conservative finite volume methods on 2-D unstructured grids for the Richards’ equation , 2004 .

[45]  I. Ginzburg.  Equilibrium-type and link-type lattice Boltzmann models for generic advection and anisotropic-dispersion equation , 2005 .

[46]  John W. Crawford,et al.  Determination of soil hydraulic conductivity with the lattice Boltzmann method and soil thin-section technique , 2005 .

[47]  Dani Or,et al.  Simulation of gaseous diffusion in partially saturated porous media under variable gravity with lattice Boltzmann methods , 2005, Water resources research.