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1 
Soil moisture change due to variable water tableKamat, Madhusudan Sunil 27 May 2016 (has links)
The thesis numerically models and investigates the effect of a variable water table on the soil moisture content. The modelling is done using COMSOL and Richards' equation. The temporal variation plots can be used to find the capillarity of the soil and its impact on other phenomenon such as vapor intrusion and infiltration.

2 
Inverse modelling and optimisation in numerical groundwater flow models using proportional orthogonal decompositionWise, John Nathaniel 03 1900 (has links)
Thesis (PhD)Stellenbosch University, 2015. / ENGLISH ABSTRACT: Numerical simulations are widely used for predicting and optimising the
exploitation of aquifers. They are also used to determine certain physical parameters,
for example soil conductivity, by inverse calculations, where the model
parameters are changed until the model results correspond optimally to measurements
taken on site. The Richards’ equation describes the movement of an
unsaturated fluid through porous media, and is characterised as a nonlinear
partial differential equation. The equation is subject to a number of parameters
and is typically computationally expensive to solve. To determine the parameters
in the Richards’ equation, inverse modelling studies often need to be undertaken.
In these studies, the parameters of a numerical model are varied until
the numerical response matches a measured response. Inverse modelling studies
typically require 100’s of simulations, which implies that parameter optimisation
in unsaturated case studies is common only in small or 1D problems in the
literature.
As a solution to overcome the computational expense incurred in inverse
modelling, the use of Proper Orthogonal Decomposition (POD) as a Reduced
Order Modelling (ROM) method is proposed in this thesis to speedup individual
simulations. An explanation of the Finite Element Method (FEM) is given using
the Galerkin method, followed by a detailed explanation of the Galerkin POD
approach. In the development of the Galerkin POD approach, the method of
reducing matrices and vectors is shown, and the treatment of Neumann and
Dirichlet boundary values is explained.
The Galerkin POD method is applied to two case studies. The first case study
is the Kogelberg site in the Table Mountain Group near Cape Town in South Africa.
The response of the site is modelled at one well over the period of 2 years, and is
assumed to be governed by saturated flow, making it a linear problem. The site
is modelled as a 3D transient, homogeneous site, using 15 layers and ≈ 20000
nodes, using the FEM implemented on the opensource software FreeFem++.
The model takes the evapotranspiration of the fynbos vegetation at the site into
consideration, allowing the calculation of annual recharge into the aquifer. The
ROM is created from highfidelity responses taken over time at different parameter
points, and speedup times of ≈ 500 are achieved, corresponding to speedup
times found in the literature for linear problems. The purpose of the saturated
groundwater model is to demonstrate that a PODbased ROM can approximate the
full model response over the entire parameter domain, highlighting the excellent
interpolation qualities and speedup times of the Galerkin POD approach, when
applied to linear problems.
A second case study is undertaken on a synthetic unsaturated case study,
using the Richards’ equation to describe the water movement. The model is a 2D
transient model consisting of ≈ 5000 nodes, and is also created using FreeFem++.
The Galerkin POD method is applied to the case study in order to replicate the
highfidelity response. This did not yield in any speedup times, since the full
matrices of nonlinear problems need to be recreated at each time step in the
transient simulation.
Subsequently, a method is proposed in this thesis that adapts the Galerkin POD
method by linearising the nonlinear terms in the Richards’ equation, in a method
named the Linearised Galerkin POD (LGP) method. This method is applied to
the same 2D synthetic problem, and results in speedup times in the range of
10 to 100. The adaptation, notably, does not use any interpolation techniques,
favouring a code intrusive, but physicsbased, approach. While the use of an
intrusively linearised POD approach adds to the complexity of the ROM, it avoids
the problem of finding kernel parameters typically present in interpolative POD
approaches.
Furthermore, the interpolation and possible extrapolation properties inherent
to intrusive PODbased ROM’s are explored. The good extrapolation properties,
within predetermined bounds, of intrusive POD’s allows for the development of
an optimisation approach requiring a very small Design of Experiments (DOE)
sets (e.g. with improved Latin Hypercube sampling). The optimisation method
creates locally accurate models within the parameter space using Support Vector
Classification (SVC). The region inside of the parameter space in which the
optimiser is allowed to move is called the confidence region. This confidence
region is chosen as the parameter region in which the ROM meets certain accuracy
conditions. With the proposed optimisation technique, advantage is taken of the
good extrapolation characteristics of the intrusive PODbased ROM’s. A further
advantage of this optimisation approach is that the ROM is built on a set of
highfidelity responses obtained prior to the inverse modelling study, avoiding
the need for full simulations during the inverse modelling study.
In the methodologies and case studies presented in this thesis, initially infeasible
inverse modelling problems are made possible by the use of the PODbased
ROM’s. The speed up times and extrapolation properties of PODbased ROM’s
are also shown to be favourable.
In this research, the use of POD as a groundwater management tool for saturated and unsaturated sites is evident, and allows for the quick evaluation of
different scenarios that would otherwise not be possible. It is proposed that a form
of POD be implemented in conventional groundwater software to significantly
reduce the time required for inverse modelling studies, thereby allowing for more
effective groundwater management. / AFRIKAANSE OPSOMMING: Die Richards vergelyking beskryf die beweging van ’n vloeistof deur ’n onversadigde
poreuse media, en word gekenmerk as ’n nielineêre parsiële differensiaalvergelyking.
Die vergelyking is onderhewig aan ’n aantal parameters en
is tipies berekeningsintensief om op te los. Om die parameters in die Richards
vergelyking te bepaal, moet parameter optimering studies dikwels onderneem
word. In hierdie studies, word die parameters van ’n numeriese model verander
totdat die numeriese resultate die gemete resultate pas. Parameter optimering
studies vereis in die orde van honderde simulasies, wat beteken dat studies wat
gebruik maak van die Richards vergelyking net algemeen is in 1D probleme in
die literatuur.
As ’n oplossing vir die berekingskoste wat vereis word in parameter optimering
studies, is die gebruik van Eie Ortogonale Ontbinding (POD) as ’n Verminderde
Orde Model (ROM) in hierdie tesis voorgestel om individuele simulasies te versnel
in die optimering konteks. Die Galerkin POD benadering is aanvanklik ondersoek
en toegepas op die Richards vergelyking, en daarna is die tegniek getoets op
verskeie gevallestudies.
Die Galerkin POD metode word gedemonstreer op ’n hipotetiese gevallestudie
waarin water beweging deur die Richardsvergelyking beskryf word. As gevolg
van die nielineêre aard van die Richards vergelyking, het die Galerkin POD
metode nie gelei tot beduidende vermindering in die berekeningskoste per simulasie
nie. ’n Verdere gevallestudie word gedoen op ’n ware grootskaalse terrein in
die Tafelberg Groep naby Kaapstad, SuidAfrika, waar die grondwater beweging
as versadig beskou word. Weens die lineêre aard van die vergelyking wat die
beweging van versadigde water beskryf, is merkwaardige versnellings van > 500
in die ROM waargeneem in hierdie gevallestudie.
Daarna was die die Galerkin POD metode aangepas deur die nielineêre terme
in die Richards vergelyking te lineariseer. Die tegniek word die geLineariserde
Galerkin POD (LGP) tegniek genoem. Die aanpassing het goeie resultate getoon,
met versnellings groter as 50 keer wanneer die ROM met die oorspronklike simulasie
vergelyk word. Al maak die tegniek gebruik van verder lineariseering, is
die metode nogsteeds ’n fisikagebaseerde benadering, en maak nie gebruik van
interpolasie tegnieke nie. Die gebruik van ’n fisikagebaseerde POD benaderings
dra by tot die kompleksiteit van ’n volledige numeriese model, maar die
kompleksiteit is geregverdig deur die merkwaardige versnellings in parameter
optimerings studies.
Verder word die interpolasie eienskappe, en moontlike ekstrapolasie eienskappe,
inherent aan fisikagebaseerde POD ROM tegnieke ondersoek in die
navorsing. In die navorsing word ’n tegniek voorgestel waarin hierdie inherente
eienskappe gebruik word om plaaslik akkurate modelle binne die parameter
ruimte te skep. Die voorgestelde tegniek maak gebruik van ondersteunende vektor
klassifikasie. Die grense van die plaaslik akkurate model word ’n vertrouens
gebeid genoem. Hierdie vertrouens gebied is gekies as die parameter ruimte
waarin die ROM voldoen aan vooraf uitgekiesde akkuraatheidsvereistes. Die
optimeeringsbenadering vermy ook die uitvoer van volledige simulasies tydens
die parameter optimering, deur gebruik te maak van ’n ROM wat gebaseer is op
die resultate van ’n stel volledige simulasies, voordat die parameter optimering
studie gedoen word. Die volledige simulasies word tipies uitgevoer op parameter
punte wat gekies word deur ’n proses wat genoem word die ontwerp van
eksperimente.
Verdere hipotetiese grondwater gevallestudies is onderneem om die LGP en
die plaaslik akkurate tegnieke te toets. In hierdie gevallestudies is die grondwater
beweging weereens beskryf deur die Richards vergelyking. In die gevalle studie
word komplekse en tydrowende modellerings probleme vervang deur ’n POD
gebaseerde ROM, waarin individuele simulasies merkwaardig vinniger is. Die
spoed en interpolasie/ekstrapolasie eienskappe blyk baie gunstig te wees.
In hierdie navorsing is die gebruik van verminderde orde modelle as ’n grondwaterbestuursinstrument
duidelik getoon, waarin voorsiening geskep word vir
die vinnige evaluering van verskillende modellering situasies, wat andersins
nie moontlik is nie. Daar word voorgestel dat ’n vorm van POD in konvensionele
grondwater sagteware geïmplementeer word om aansienlike versnellings
in parameter studies moontlik te maak, wat na meer effektiewe bestuur van
grondwater sal lei.

3 
Solving Partial Differential Equations Using Artificial Neural NetworksRudd, Keith January 2013 (has links)
<p>This thesis presents a method for solving partial differential equations (PDEs) using articial neural networks. The method uses a constrained backpropagation (CPROP) approach for preserving prior knowledge during incremental training for solving nonlinear elliptic and parabolic PDEs adaptively, in nonstationary environments. Compared to previous methods that use penalty functions or Lagrange multipliers,</p><p>CPROP reduces the dimensionality of the optimization problem by using direct elimination, while satisfying the equality constraints associated with the boundary and initial conditions exactly, at every iteration of the algorithm. The effectiveness of this method is demonstrated through several examples, including nonlinear elliptic</p><p>and parabolic PDEs with changing parameters and nonhomogeneous terms. The computational complexity analysis shows that CPROP compares favorably to existing methods of solution, and that it leads to considerable computational savings when subject to nonstationary environments.</p><p>The CPROP based approach is extended to a constrained integration (CINT) method for solving initial boundary value partial differential equations (PDEs). The CINT method combines classical Galerkin methods with CPROP in order to constrain the ANN to approximately satisfy the boundary condition at each stage of integration. The advantage of the CINT method is that it is readily applicable to PDEs in irregular domains and requires no special modification for domains with complex geometries. Furthermore, the CINT method provides a semianalytical solution that is infinitely differentiable. The CINT method is demonstrated on two hyperbolic and one parabolic initial boundary value problems (IBVPs). These IBVPs are widely used and have known analytical solutions. When compared with Matlab's nite element (FE) method, the CINT method is shown to achieve significant improvements both in terms of computational time and accuracy. The CINT method is applied to a distributed optimal control (DOC) problem of computing optimal state and control trajectories for a multiscale dynamical system comprised of many interacting dynamical systems, or agents. A generalized reduced gradient (GRG) approach is presented in which the agent dynamics are described by a small system of stochastic dierential equations (SDEs). A set of optimality conditions is derived using calculus of variations, and used to compute the optimal macroscopic state and microscopic control laws. An indirect GRG approach is used to solve the optimality conditions numerically for large systems of agents. By assuming a parametric control law obtained from the superposition of linear basis functions, the agent control laws can be determined via setpoint regulation, such</p><p>that the macroscopic behavior of the agents is optimized over time, based on multiple, interactive navigation objectives.</p><p>Lastly, the CINT method is used to identify optimal root profiles in water limited ecosystems. Knowledge of root depths and distributions is vital in order to accurately model and predict hydrological ecosystem dynamics. Therefore, there is interest in accurately predicting distributions for various vegetation types, soils, and climates. Numerical experiments were were performed that identify root profiles that maximize transpiration over a 10 year period across a transect of the Kalahari. Storm types were varied to show the dependence of the optimal profile on storm frequency and intensity. It is shown that more deeply distributed roots are optimal for regions where</p><p>storms are more intense and less frequent, and shallower roots are advantageous in regions where storms are less intense and more frequent.</p> / Dissertation

4 
A Discontinuous Galerkin Finite Element Method Solution of OneDimensional Richards’ EquationXiao, Yilong 30 August 2016 (has links)
No description available.

5 
DualScale Modeling of TwoPhase Fluid Transport in Fibrous Porous MediaAshari, Alireza 23 November 2010 (has links)
The primary objective of this research is to develop a mathematical framework that could be used to model or predict the rate of fluid absorption and release in fibrous sheets made up of solid or porous fibers. In the first step, a twoscale twophase modeling methodology is developed for studying fluid release from saturated/unsaturated thin fibrous media made up of solid fibers when brought in contact with a moving solid surface. Our macroscale model is based on the Richards’ equation for twophase fluid transport in porous media. The required constitutive relationships, capillary pressure and relative permeability as functions of the medium’s saturation, are obtained through microscale modeling. Here, a mass convection boundary condition is considered to model the fluid transport at the boundary in contact with the target surface. The mass convection coefficient plays a significant role in determining the release rate of fluid. Moreover the release rate depends on the properties of the fluid, fibrous sheet, the target surface as well as the speed of the relative motion, and remains to be determined experimentally. Obtaining functional relationships for relative permeability and capillary pressure is only possible through experimentation or expensive microscale simulations, and needs to be repeated for different media having different fiber diameters, thicknesses, or porosities. In this concern, we conducted series of 3D microscale simulations in order to investigate the effect of the aforementioned parameters on the relative permeability and capillary pressure of fibrous porous sheets. The results of our parameter study are utilized to develop general expressions for kr(S) and Pc(S). Furthermore, these general expressions can be easily included in macroscale fluid transport equations to predict the rate of fluid release from partially saturated fibrous sheets in a time and costeffective manner. Moreover, the ability of the model has been extended to simulate the radial spreading of liquids in thin fibrous sheets. By simulating different fibrous sheets with identical parameters but different inplane fiber orientations has revealed that the rate of fluid spread increases with increasing the inplane alignment of the fibers. Additionally, we have developed a semianalytical modeling approach that can be used to predict the fluid absorption and release characteristics of multilayered composite fabric made up of porous (swelling) and soild (nonswelling) fibrous sheets. The sheets capillary pressure and relative permeability are obtained via a combination of numerical simulations and experiment. In particular, the capillary pressure for swelling media is obtained via height rise experiments. The relative permeability expressions are obtained from the analytical expressions previously developed with the 3D microscale simulations, which are also in agreement with experimental correlations from the literature. To extend the ability of the model, we have developed a diffusioncontrolled boundary treatment to simulate fluid release from partiallysaturated fabrics onto surfaces with different hydrophilicy. Using a custom made test rig, experimental data is obtained for the release of liquid from partially saturated PET and Rayon nonwoven sheets at different speeds, and on two different surfaces. It is demonstrated that the new semiempirical model redeveloped in this work can predict the rate of fluid release from wet nonwoven sheets as a function of time.

6 
Implementing and Evaluating Variable Soil Thickness in the Community Land Model, Version 4.5 (CLM4.5)Brunke, Michael A., Broxton, Patrick, Pelletier, Jon, Gochis, David, Hazenberg, Pieter, Lawrence, David M., Leung, L. Ruby, Niu, GuoYue, Troch, Peter A., Zeng, Xubin 05 1900 (has links)
One of the recognized weaknesses of land surface models as used in weather and climate models is the assumption of constant soil thickness because of the lack of global estimates of bedrock depth. Using a 30arcs global dataset for the thickness of relatively porous, unconsolidated sediments over bedrock, spatial variation in soil thickness is included here in version 4.5 of the Community Land Model (CLM4.5). The number of soil layers for each grid cell is determined from the average soil depth for each 0.9 degrees latitude x 1.25 degrees longitude grid cell. The greatest changes in the simulation with variable soil thickness are to baseflow, with the annual minimum generally occurring earlier. Smaller changes are seen in latent heat flux and surface runoff primarily as a result of an increase in the annual cycle amplitude. These changes are related to soil moisture changes that are most substantial in locations with shallow bedrock. Total water storage (TWS) anomalies are not strongly affected over most river basins since most basins contain mostly deep soils, but TWS anomalies are substantially different for a river basin with more mountainous terrain. Additionally, the annual cycle in soil temperature is partially affected by including realistic soil thicknesses resulting from changes in the vertical profile of heat capacity and thermal conductivity. However, the largest changes to soil temperature are introduced by the soil moisture changes in the variable soil thickness simulation. This implementation of variable soil thickness represents a step forward in land surface model development.

7 
A Local Discontinuous Galerkin DualTime Richards' Equation Solution and Analysis on DualTime Stability and ConvergenceXiao, Yilong January 2021 (has links)
No description available.

8 
Construção de método de solução funcional para problemas de fluxo em meios porosos não saturadosFurtado, Igor da Cunha January 2017 (has links)
Neste estudo, consideramos um problema transiente de fluxo unidimensional vertical de água em meio poroso insaturado, modelado pela equação Richards nãolinear. As reações constitutivas de Van Genuchten são empregadas para representar a capacidade hidráulica e a condutividade. A fórmula da solução é otimizada e avaliada usando a equação governante em um critério de autoconsciente. Os resultados são apresentados para alguns tipos de solo e seus parâmetros relacionados, que são mencionados em literatura. / In this study, we consider a transiente vertical onedimensional flow problem of water in unsaturated porus media, modelled by the nonlinear Richards equation. Constitutive relations of Van Genutchten are employed to represent the hydraulic capacity and conductivity. The solution formula is optimized and evaluated using to governing equation for a selfconsistency criterion. The results are presented for some oil types and its related soil parameters, that are reported in the literature.

9 
Construção de método de solução funcional para problemas de fluxo em meios porosos não saturadosFurtado, Igor da Cunha January 2017 (has links)
Neste estudo, consideramos um problema transiente de fluxo unidimensional vertical de água em meio poroso insaturado, modelado pela equação Richards nãolinear. As reações constitutivas de Van Genuchten são empregadas para representar a capacidade hidráulica e a condutividade. A fórmula da solução é otimizada e avaliada usando a equação governante em um critério de autoconsciente. Os resultados são apresentados para alguns tipos de solo e seus parâmetros relacionados, que são mencionados em literatura. / In this study, we consider a transiente vertical onedimensional flow problem of water in unsaturated porus media, modelled by the nonlinear Richards equation. Constitutive relations of Van Genutchten are employed to represent the hydraulic capacity and conductivity. The solution formula is optimized and evaluated using to governing equation for a selfconsistency criterion. The results are presented for some oil types and its related soil parameters, that are reported in the literature.

10 
Construção de método de solução funcional para problemas de fluxo em meios porosos não saturadosFurtado, Igor da Cunha January 2017 (has links)
Neste estudo, consideramos um problema transiente de fluxo unidimensional vertical de água em meio poroso insaturado, modelado pela equação Richards nãolinear. As reações constitutivas de Van Genuchten são empregadas para representar a capacidade hidráulica e a condutividade. A fórmula da solução é otimizada e avaliada usando a equação governante em um critério de autoconsciente. Os resultados são apresentados para alguns tipos de solo e seus parâmetros relacionados, que são mencionados em literatura. / In this study, we consider a transiente vertical onedimensional flow problem of water in unsaturated porus media, modelled by the nonlinear Richards equation. Constitutive relations of Van Genutchten are employed to represent the hydraulic capacity and conductivity. The solution formula is optimized and evaluated using to governing equation for a selfconsistency criterion. The results are presented for some oil types and its related soil parameters, that are reported in the literature.

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