In this thesis the development of a compressor for a gas network simulation will be discussed. The modelled compressor is a centrifugal compressor and this compressor is common to use when compressing gas. At first the equations to describe a centrifugal compressor and an ex- ample parameter set are given. These equations will be used in an algorithm to calculate either the suction pressure, the discharge pressure or the volume flow. A compressor can only operate when it is in a feasible area. This feasible area is defined by both a surge and choke line as well as the minimum and maximum rotation speed. The gas network considered consists of a compressor plus a pipeline in front and after the compressor. To model a pipeline the Weymouth pressure-drop equation is used. The pipe- compressor system can be described as a system of seven equations. There are seven unknown variables and two given variables. The given variables are the pressure at the beginning of the first pipeline and the pressure at the end of the second pipeline. To solve the system of equations a nonlinear solver is used. The Newton method is an ex- ample of such a nonlinear solver and will be discussed for scalar equations and a system of equations. Solving the system of equations as part of a simulation, a function in Python called fsolve will be used. This function uses a more robust algorithm to get the solution quickly. The numerical behaviour of the solver will be considered during multiple simulations. Fur- thermore a parameter study has been done to understand what kind of pipelines are needed for a certain pressure ratio. Finally two examples are discussed briefly.

There are several different methods to solve the steady-state flow problem of district heating networks and gas networks. In this thesis, the nodal method, the loop method, and the loop-node method are compared. This is done in order to determine for what type of network, with specific characteristics, which of these methods is preferred. In this comparison, the focus is on computer storage, sensibility to starting values, convergence properties, and computational time.

In light of our depleting fossil fuel reserves and the relatively `cheap' extraction of oil and in spite of the highly nonlinear nature of reservoirs, waterflooding has become big business. In recent times, the use of numerical reservoir simulation has not only become possible but has increasingly been used in the petroleum industry in the forecasting of output and money to be made. However, this numerical modelling and automated history matching is not without its problems. The inner workings of sophisticated commercial reservoir simulators are often taken for granted, i.e., ``black boxes''. These simulators are constructed around a numerical method, with its advantages and disadvantages. Herein, the input settings play a role in the stability and precision of results. For example, the chosen iteration method, grid spacing and time step size or even the choice of iteration parameters, all based on insufficient data, leads simulators to be unreliable and inefficient. Moreover, even under the assumption that such ``black boxes'' are able to produce a true prediction, this is entirely conditional on correctly establishing the current state and conditions. Thus, practitioners have many global settings given, many of which inaccurate. These many uncertainties can lead to costly mistakes. The aim of this study therefore: is to develop a tool that can quantify the uncertainty of a core flood model and a parameter estimation routine. Specifically, it investigates whether, given limited incoming data, an uncertain parameter can be estimated and then used to simulate and quantify the uncertainty of the water saturation and oil pressure in the core sample. In this context, we question if it be used to provide a history match of the core sample. To see how the uncertain input parameters are reflected in a model output, the tool, based on, the IMPES scheme simulates the two-phase flow, and the Ensemble Kalman filter (EnKF) to estimate the parameters. Building on the base twin experiment, a variety of twin experiments were performed to understand the parameter estimation, by presenting and visualizing the uncertainties in the data and states. We investigate the use of the EnKF for history matching and ways to improve are also explored. Based on the results of this study, it is concluded that the Ensemble Kalman Filter is capable of effective parameter estimation. With modification, it can also be used for history matching and uncertainty quantification. It clearly suggests that the utility of numerical modelling and automated history matching will continue to make contributions to the success of the exploration and extraction of fossil hydrocarbons.

This work is concerned with the simulation of gas networks using a hybrid modeling approach where different pipes are described withmodels belonging to the isothermal model hierarchy. It makes special emphasis in the coupling of algebraic and transient models and their effects in both steady-state and transient regime. The transient behavior of simple networks is studied in both non-dissipative and dissipative regime with a perturbation analysis. Physical effects concerning transmission and reflection of perturbations are studied both numerical and analytically. The transient response of the algebraic model is studied and compared with the results of numerical experiments. Finally a perturbative approach for computing the attenuation coefficient for dissipative transport is proposed and the analytical results are compared with the numerical experiments.

Since the 1950s, numerical models are widely used in reservoir simulators to predict and optimize oil recovery from petroleum reservoirs. Commercial simulators typically combine multiphase porous media flow models with a separate module for the transport of tracers. By decoupling flow and transport, the transport equation can be solved using a more accurate and efficient numerical scheme than the fully implicit first order scheme that is commonly used to solve the flow equations. The accuracy of the numerical transport scheme is of high importance in Enhanced Oil Recovery modelling in order to accurately predict the influence of EOR techniques on the oil production. Of all numerical methods considered in this research, the explicit high-resolution flux-limiter method with the van Leer limiter performed best in terms of accuracy and efficiency. The accuracy and monotonicity of the numerical transport scheme strongly depend on the underlying flowsolution. To ensure monotonicity of the scheme in all model situations considered, a partially implicit method is introduced that switches to the monotone first order implicit scheme where necessary. In addition, care must be taken when modelling influences of the polymer and surfactant concentration on the flow. Bad choices can lead to instabilities or cause deviations in the model outcome. The conventional approach for modelling the hydrodynamic acceleration of polymer via a constant velocity enhancement factor results in an ill-posed system. To obtain a well-posed system, some form of a saturation dependent velocity enhancement factor could be used instead. However, which physical and numerical model is best suited for modelling the hydrodynamic acceleration remains an open question. When modelling surfactant flooding, the implementation of a discontinuous transition in relative permeabilities around the surfactant front results in oscillations in the numerical solution over time. These oscillations can be resolved or at least diminished by applying some form of interpolation between the two sets of relative permeabilities. However, there is no guarantee that the oscillations disappear completely and the interpolation function has to be carefully selected as it can have a large impact on the solution profile.

In the past decades, several EOR (Enhanced Oil Recovery) techniques have been developed to increase the amount of oil recoverable from a reservoir. Among these techniques, polymer flooding consists in the injection of a solution of water and polymer into the reservoir, and it is performed in order to lower the mobility of injected water. A peculiarity of a water-polymer flow through a porous medium is the velocity enhancement, or hydrodynamic acceleration, that affects the polymer molecules. Experimentally, polymers are observed to travel faster than inert chemical species. Therefore, when simulating numerically a polymer flooding, a velocity enhancement effect has to be incorporated into the governing equations to accurately predict oil recovery. The simple model that introduces a constant enhancement factor, usually implemented in commercial simulators, leads to an ill-posed problem, causing stability issues in the simulations, which further leads to a monotonicity loss in the solution: the polymer is predicted to pile-up at the water-oil interface. While accumulation of polymer at the water front is not necessarily unphysical, it should not be the result of a mathematical ill-posed problem. Hence, an alternative model, employing a saturation dependent factor, has been proposed in the work of Bartelds et al. (G.A. Bartelds, J. Bruining, and J. Molenaar. The modeling of velocity enhancement in polymer flooding. Transport in Porous Media, 26(1):75–88, 1997), and has been extended by Hilden et al. (S.T. Hilden, H.M. Nilsen, and X. Raynaud. Study of the well-posedness of models for the inaccessible pore volume in polymer flooding. Transport in Porous Media, 114(1):65–86, 2016) to overcome some physical restrictions. In this thesis, these models are re-examined through a thorough analytical study in the one-dimensional case. An analytical solution, resulting in an acceleration of the polymer front, is computed for the well-posed model proposed by Bartelds, while the extended enhancement factor derived by Hilden is shown to lead again to an ill-posed problem. A study of the monotonicity of the numerical schemes reveals that accumulation of polymer at the water front is strictly related to ill-posedness. Obtaining accurate numerical solutions is not an easy task. Commercial simulators adopt fully implicit schemes to solve the multi-phase flow, and transport of chemical agents is solved through the development and coupling of separate modules. In this case, the use of high-resolution methods for the transport of polymer is not compatible with the first order solver for the underlying flow. To investigate the interaction of the enhancement model proposed by Bartelds with other physical phenomena, adsorption of polymer onto the reservoir rock is added to the governing equations. The problem maintains the well-posedness property, but because of a raise in equations complexity, only numerical results are presented. The model is then extended to the two-dimensional case and, relying on numerical solutions, similar conclusions as in the one-dimensional case are derived.

The transport of a solute dissolved in a fluid flowing trough porous media is, next to advection and diffusion, determined by hydrodynamic dispersion. This be- haviour is commonly characterized using the longitudinal and transverse dispersion coefficients. Laboratory and field measurements of these coefficients tend to differ, which might be attributed to heterogeneities found in field porous media. To investigate this, a stratified porous medium consisting of two layers is con- sidered. Each layer has different physical properties, resulting in a different average fluid velocity. As a consequence of the difference in velocity, transport of the solute occurs between the two layers. Under certain circumstances the layers start to be- have as one single layer, with one single effective dispersion coefficient, explaining the discrepancy between field and laboratory measurements. The two-layer stratified porous medium is characterized using a dimensionless number. It is investigated for which values of this number the porous medium acts as one single layer, and for which values the medium behaves as two separate layers. This is done by introducing an index, which effectively measures the behaviour of the medium in terms of these two limit cases. The calculation of the index is done using a numerical simulation of flow and dispersion in the stratified porous medium. It was found that the dimensionless number was in general a good predictor of the behaviour of the stratified porous medium. The system behaved as one single layer if the dimensionless number (after a correction with a certain factor) was much greater than unity. Similarly, the system behaved as two separate layers if the number was much less than unity. However, this number failed if the ratio of the two layer thicknesses was varied. A correction to the dimensionless number was suggested, taking the ratio into account.