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 non-linear
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 speed-up 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 open-source 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 high-fidelity responses taken over time at different parameter
points, and speed-up times of ≈ 500 are achieved, corresponding to speed-up
times found in the literature for linear problems. The purpose of the saturated
groundwater model is to demonstrate that a POD-based ROM can approximate the
full model response over the entire parameter domain, highlighting the excellent
interpolation qualities and speed-up 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
high-fidelity response. This did not yield in any speed-up times, since the full
matrices of non-linear 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 non-linear 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 speed-up times in the range of
10 to 100. The adaptation, notably, does not use any interpolation techniques,
favouring a code intrusive, but physics-based, 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 POD-based 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 POD-based ROM’s. A further
advantage of this optimisation approach is that the ROM is built on a set of
high-fidelity 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 POD-based
ROM’s. The speed up times and extrapolation properties of POD-based 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 nie-lineêre 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 Richards-vergelyking beskryf word. As gevolg
van die nie-lineêre 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, Suid-Afrika, 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 nie-lineêre 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 fisika-gebaseerde benadering, en maak nie gebruik van
interpolasie tegnieke nie. Die gebruik van ’n fisika-gebaseerde 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 fisika-gebaseerde 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 tyd-rowende 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.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/97116 |
| Date | 03 1900 |
| Creators | Wise, John Nathaniel |
| Contributors | Venter, Gerhard, Batton-Hubert, Mireille, Stellenbosch University. Faculty of Engineering. Dept. of Mechanical and Mechatronic Engineering. |
| Publisher | Stellenbosch : Stellenbosch University |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 155 pages : illustrations |
| Rights | Stellenbosch University |
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