Reservoir simulation is an important tool for decision making and field development management. It enables reservoir engineers to predict reservoir production performance, update an existing model to reproduce monitoring data, assess alternative field development scenarios and design robust production optimization strategies by taking into account the existing uncertainties. A big obstacle in automating model calibration and production optimization approaches is the massive computation required to predict the response of real reservoirs under proposed changes in the model inputs. To speed up reservoir response predictions without compromising accuracy, fast surrogate models have been proposed. These models are either derived by preserving the physics of the involved processes (e.g. mass balance equations) to provide reliable long-range predictions or are developed based solely on statistical relations, in which case they can only provide short-range predictions due to the absence of the physical processes that govern the long-term behavior of the reservoir. We present an alternative solution that combines the advantages of both statistics-based and physics-based methods by deriving the flow predictions in complex two-dimensional models from one-dimensional flow network models. The existing injection/production wells in the original model form the nodes or vertices of the flow network. Each pair of wells (nodes) in the flow network is connected using a one-dimensional numerical simulation model; hence, the entire reservoir is reduced to a connected network of one-dimensional simulation models where the coupling between the individual one-dimensional models is enforced at the nodes where network edges intersect. The proposed flow network model provides a useful and fast tool for characterizing inter-well connectivity, estimating drainage volume between each pair of wells, and predicting reservoir production over an extended period of time for optimization purposes. We estimate the parameters of the flow network model using a robust training approach to ensure that the flow network model reproduces the response of the original full model under a wide range of development strategies. This step helps preserve the flow network model's predictive power during the production optimization when development strategies can change at different iterations. The robust networks training and the subsequent production optimization iterations are computationally efficient as they are performed with the faster flow network model. We demonstrate the effectiveness and applicability of our proposed flow network modeling approach to rapid production optimization using two-phase waterflooding simulations in synthetic and benchmark models.

Data generated by complex, computational models can provide highly accurate predictions of hydrological and hydrodynamic data in multiple dimensions. Unfortunately, however, for large data sets, running these models is often timeconsuming and computationally expensive. Thus, finding a way to reduce the running time of these models, while still producing comparable results, is of notable interest. The Inference Engine is a proposed system for doing just this. It takes previously generated model data and uses them to predict additional data. Its performance, both accuracy and running time, has been compared to the performance of the actual models, in increasingly difficult data prediction tasks, and it is able, with sufficient accuracy, to quickly predict unknown model data.

In this thesis we conduct a numerical study of the 1D viscous Burgers' equation and several Reduced Order Models (ROMs) over a range of parameter values. This study is motivated by the need for robust reduced order models that can be used both for design and control. Thus the model should first, allow for selection of optimal parameter values in a trade space and second, identify impacts from changes of parameter values that occur during development, production and sustainment of the designs. To facilitate this study we apply a Finite Element Method (FEM) and where applicable, the Group Finite Element Method (GFE) due its demonstrated stability and reduced complexity over the standard FEM. We also utilize Proper Orthogonal Decomposition (POD) as a model reduction technique and modifications of POD that include Global POD, and the sensitivity based modifications Extrapolated POD and Expanded POD. We then use a single baseline parameter in the parameter range to develop a ROM basis for each method above and investigate the error of each ROM method against a full order "truth" solution for the full parameter range. / Master of Science

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.

This thesis presents a study of reduced-order models based on proper orthogonal decomposition applied to non-isothermal transport phenomena in °uidized beds. A numerical °ow solver called Multiphase Flow with Interphase eXchanges (MFIX) was used to generate a database of solution snapshots for proper orthogonal decomposi- tion (POD). Using POD, time independent basis functions were extracted from the data and the governing equations of the numerical solver were projected onto the basis functions to generate reduced-order models. A reduced-order model was constructed that simulates multi-phase isothermal and non-isothermal °ow. In the reduced-order models (ROMs) the large number of partial di®erential equations were replaced by a much smaller number of ordinary di®erential equations. These reduced-order models were applied to two reference cases, a time extrapolation case and a time-dependent period boundary condition case. Three additional acceleration techniques were devel- oped to further improve computational e±ciency of the POD based ROM: 1) Database splitting, 2) Freezing the matrix of the linear system and 3) Time step adjustment. Detailed numerical analysis of both the full-order model, MFIX and the POD-based ROM, including estimating the number of operations and the CPU time per iteration, was performed as part of this study. The results of this investigation show that the reduced-order models are capable of producing qualitatively accurate results with less than 5% error with a two-order of magnitude reduction of computational costs.

This thesis presents advances in reduced-order modeling based on proper orthogonal decomposition (POD) for single and two-phase flows. Reduced-order models (ROMs) are generated for two-phase gas-solid flows. A multiphase numerical flow solver, MFIX, is used to generate a database of solution snapshots for proper orthogonal decomposition. Time-independent basis functions are extracted using POD from the data, and the governing equations of the MFIX are projected onto the basis functions to generate the multiphase POD-based ROMs. Reduced-order models are constructed to simulate multiphase two-dimensional non-isothermal flow and isothermal flow particle kinetics and three-dimensional isothermal flow. These reduced-order models are applied to three reference cases. The results of this investigation show that the two-dimensional reduced-order models are capable of producing qualitatively accurate results with less than 5 percent error with at least an order of magnitude reduction of computational costs. The three-dimensional ROM shows improvements in computational costs. This thesis also presents an algorithm based on mathematical morphology used to extract discontinuities present in quasi-steady and unsteady flows for POD basis augmentation. Both MFIX and a Reynolds Average Navier-Stokes (RANS) flow solver, UNS3D, are used to generate solution databases for feature extraction. The algorithm is applied to bubbling uidized beds, transonic airfoils, and turbomachinery seals. The results of this investigation show that all of the important features are extracted without loss in accuracy.

Parametric uncertainty is one of many possible causes of divergence for the Kalman filter. Frequently, state estimation errors caused by imperfect model parameters are reduced by including the uncertain parameters as states (i.e., augmenting the state vector). For many situations, this not only improves the state estimates, but also improves the accuracy and precision of the parameters themselves. Unfortunately, not all filters benefit from this augmentation due to computational restrictions or because the parameters are poorly observable. A parameter with low observability (e.g., a set of high order gravity coefficients, a set of camera offsets, lens calibration controls, etc.) may not acquire enough measurements along a particular trajectory to improve the parameter's accuracy, which can cause detrimental effects in the performance of the augmented filter. The problem is then how to reduce the dimension of the augmented state vector while minimizing information loss. This dissertation explored possible implementations of reduced-order filters which decrease computational loads while also minimizing state estimation errors. A theoretically rigorous approach using the ?consider" methodology was taken at discrete time intervals were explored for linear systems. The continuous dynamics, discretely measured (continuous-discrete) model was also expanded for use with nonlinear systems. Additional techniques for reduced-order filtering are presented including the use of additive process noise, an alternative consider derivation, and the minimum variance reduced-order filter. Multiple simulation examples are provided to help explain critical concepts. Finally, two hardware applications are also included to show the validity of the theory for real world applications. It was shown that the minimum variance consider Kalman filter (MVCKF) is the best reduced-order filter to date both theoretically and through hardware and software applications. The consider method of estimation provides a compromise between ignoring parameter error and completely accounting for it in a probabilistic sense. Based on multiple measures of optimality, the consider filtering framework can be used to account for parameter error without directly estimating any or all of the parameters. Furthermore, by accounting for the parameter error, the consider approach provides a rigorous path to improve state estimation through the reduction of both state estimation error and with a consistent variance estimate. While using the augmented state vector to estimate both states and parameters may further improve those estimates, the consider estimation framework is an attractive alternative for complex and computationally intensive systems, and provides a well justified path for parameter order reduction.

The primary objective of this work is to develop an approach for evaluating GHG mitigation strategies that considers the detailed operation of the electricity system in question and to ascertain whether considering the detailed operation of the electricity system materially affects the assessment. A secondary objective is to evalute the potential benefit of flexible CO2 capture and storage. An electricity system simlator is developed based upon a deregulated electricity system containing markets for both real and reserve power. Using the IEEE RTS ???96 as a test case, the performance of the electricity system is benchmarked with GHG regulation. Two different implementations of CO2 capture are added to the electricity system ??? fixed CO2 capture and flexible CO2 capture ??? and the impact of having CCS is assessed. The results indicate that: - the assessment of GHG mtigation strategies for the electricity generation subsector should consider the detailed operation of the electricity system in question, - cost of generation alone is not necessarily a good indicator of the economic impact of GHG regulation or the deployment of a GHG mitigation strategy, - adding CCS, at even a single generating unit, can significantly reduce GHG emissions and moderate the ecnomic impact of GHG regulation relative to the cases where CCS is not present, and - a generating unit with a flexible CCS processes participates preferentially in the reserve market enabling it to increase its net energy benefit. It is conclued that there is a significant potential advantage to generating units with flexible CCS processes. The flexibiity of existing and novel CCS process should be an assessment and design criterion, respectively, and the development of novel CCS processes with optimial operability is a suggested area of future research activity. A reduced-order model of a coal-fired generating unit with flexible CO2 capture is developed and integrated into the MINLP formulation of an economic dispatch model. Both of these efforts, not observed previously in the literature, constitute an important contribution of the work as the methodology provides a template for future assessmments of CCS and other electricity mitigation strategies in the electricity generation subsector.

abstract: This thesis focuses on the continued extension, validation, and application of combined thermal-structural reduced order models for nonlinear geometric problems. The first part of the thesis focuses on the determination of the temperature distribution and structural response induced by an oscillating flux on the top surface of a flat panel. This flux is introduced here as a simplified representation of the thermal effects of an oscillating shock on a panel of a supersonic/hypersonic vehicle. Accordingly, a random acoustic excitation is also considered to act on the panel and the level of the thermo-acoustic excitation is assumed to be large enough to induce a nonlinear geometric response of the panel. Both temperature distribution and structural response are determined using recently proposed reduced order models and a complete one way, thermal-structural, coupling is enforced. A steady-state analysis of the thermal problem is first carried out that is then utilized in the structural reduced order model governing equations with and without the acoustic excitation. A detailed validation of the reduced order models is carried out by comparison with a few full finite element (Nastran) computations. The computational expedience of the reduced order models allows a detailed parametric study of the response as a function of the frequency of the oscillating flux. The nature of the corresponding structural ROM equations is seen to be of a Mathieu-type with Duffing nonlinearity (originating from the nonlinear geometric effects) with external harmonic excitation (associated with the thermal moments terms on the panel). A dominant resonance is observed and explained. The second part of the thesis is focused on extending the formulation of the combined thermal-structural reduced order modeling method to include temperature dependent structural properties, more specifically of the elasticity tensor and the coefficient of thermal expansion. These properties were assumed to vary linearly with local temperature and it was found that the linear stiffness coefficients and the "thermal moment" terms then are cubic functions of the temperature generalized coordinates while the quadratic and cubic stiffness coefficients were only linear functions of these coordinates. A first validation of this reduced order modeling strategy was successfully carried out. / Dissertation/Thesis / M.S. Aerospace Engineering 2011

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