HIGH ORDER BOUND-PRESERVING DISCONTINUOUS GALERKIN METHODS AND THEIR APPLICATIONS IN PETROLEUM ENGINEERING

This report contains researches in the theory of high-order bound-preserving (BP) discontinuous Galerkin (DG) method and their applications in petroleum engineering. It contains both theoretical analysis and numerical experiments. The compressible miscible displacements and wormhole propagation problem, arising in petroleum engineering, is used to describe the evolution of the pressure and concentrations of different components of fluid in porous media. The important physical features of concentration and porosity include their boundedness between 0 and 1, as well as the monotone increasing for porosity in wormhole propagation model. How to keep these properties in the simulation is crucial to the robustness of the numerical algorithm. In the first project, we develop high-order bound-preserving discontinuous Galerkin methods for the coupled system of compressible miscible displacements on triangular meshes. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the jth component,cj, should be between 0 and 1. The main idea is stated as follows. First, we apply the second-order positivity-preserving techniques to all concentrations c′ js and enforce P jcj= 1 simultaneously to obtain physically relevant boundedness for every components. Then, based on the second-order BP schemes, we use the second-order numerical fluxes as the lower order one to combine with high-order numerical fluxes to achieve the high-order accuracy. Finally, since the classical slope limiter cannot be applied to polynomial upper bounds, we introduce a new limiter to our algorithm. Numerical experiments are given to demonstrate the high-order accuracy and good performance of the numerical technique. In our second project, we propose high-order bound-preserving discontinuous Galerkin methods to keep the boundedness for the porosity and concentration of acid, as well as the monotone increasing for porosity. The main technique is to introduce a new variable r to replace the original acid concentration and use a consistent flux pair to deduce a ghost equation such that the positive-preserving technique can be applied on both original and deduced equations. A high-order slope limiter is used to keep a polynomial upper bound which changes over time for r. Moreover, the high-order accuracy is attained by the flux limiter. Numerical examples are given to demonstrate the high-order accuracy and bound-preserving property of the numerical technique

Simplifying reservoir models by flow regime

This study focuses on the interaction between geological heterogeneity and the reservoir processes which govern fluid flow in porous media. We have developed and tested a measure of heterogeneity which uses the coefficient of variation of the vorticity of the flow field to quantify the impact of geological uncertainty on oil recovery. We go on to explore the vorticity formulation of the equations of motion in porous media as a basis for understanding reservoir dynamics, particularly in the presence of heterogeneity and density differences. We derive dimensionless numbers to quantify the relative importance of viscosity and density differences, molecular diffusion, dispersion, and permeability heterogeneity on reservoir flow behaviour. This approach is used to develop an objective measure of the impact of permeability heterogeneity on reservoir performance, which we have compared with traditional heterogeneity indices and shown how it may be used for realistic 2D and 3D geological models. We have used our heterogeneity index, and the dimensionless numbers to analyse the impact of heterogeneity, buoyancy effects, mobility ratio and dispersion on breakthrough time and recovery for first contact miscible gas injection processes using geologically realistic reservoir models. We find that the new heterogeneity number, in conjunction with these dimensionless numbers, provides meaningful results for real non-linear reservoir flows. We present phase diagrams which show how reservoir performance depends on mobility ratio, viscous-gravity ratio, and heterogeneity. We have proposed that the phase diagram, and a comparison of these dimensionless numbers can be used to identify the key factors which control recovery, thus assisting the engineer in determining appropriate enhanced oil recovery (EOR) techniques, without resort to detailed flow simulation. This will enable a quick, and more robust, evaluation of the impact of geological uncertainty in the field

Coupled flow and geomechanics modeling for fractured poroelastic reservoirs

textTight gas and shale oil play an important role in energy security and in meeting an increasing energy demand. Hydraulic fracturing is a widely used technology for recovering these resources. The design and evaluation of hydraulic fracture operation is critical for efficient production from tight gas and shale plays. The efficiency of fracturing jobs depends on the interaction between hydraulic (induced) and naturally occurring discrete fractures. In this work, a coupled reservoir-fracture flow model is described which accounts for varying reservoir geometries and complexities including non-planar fractures. Different flow models such as Darcy flow and Reynold's lubrication equation for fractures and reservoir, respectively are utilized to capture flow physics accurately. Furthermore, the geomechanics effects have been included by considering a multiphase Biot's model. An accurate modeling of solid deformations necessitates a better estimation of fluid pressure inside the fracture. The fractures and reservoir are modeled explicitly allowing accurate representation of contrasting physical descriptions associated with each of the two. The approach presented here is in contrast with existing averaging approaches such as dual and discrete-dual porosity models where the effects of fractures are averaged out. A fracture connected to an injection well shows significant width variations as compared to natural fractures where these changes are negligible. The capillary pressure contrast between the fracture and the reservoir is accounted for by utilizing different capillary pressure curves for the two features. Additionally, a quantitative assessment of hydraulic fracturing jobs relies upon accurate predictions of fracture growth during slick water injection for single and multistage fracturing scenarios. It is also important to consistently model the underlying physical processes from hydraulic fracturing to long-term production. A recently introduced thermodynamically consistent phase-field approach for pressurized fractures in porous medium is utilized which captures several characteristic features of crack propagation such as joining, branching and non-planar propagation in heterogeneous porous media. The phase-field approach captures both the fracture-width evolution and the fracture-length propagation. In this work, the phase-field fracture propagation model is briefly discussed followed by a technique for coupling this to a fractured poroelastic reservoir simulator. We also present a general compositional formulation using multipoint flux mixed finite element (MFMFE) method on general hexahedral grids with a future prospect of treating energized fractures. The mixed finite element framework allows for local mass conservation, accurate flux approximation and a more general treatment of boundary conditions. The multipoint flux inherent in MFMFE scheme allows the usage of a full permeability tensor. An accurate treatment of diffusive/dispersive fluxes owing to additional velocity degrees of freedom is also presented. The applications areas of interest include gas flooding, CO₂ sequestration, contaminant removal and groundwater remediation.Petroleum and Geosystems Engineerin

Large scale cavity dissolution: From the physical problem to its numerical solution

Dissolution of underground cavities by ground water (or solutions) may cause environmental problems and geological hazards. Efficient modeling and numerical solving of such phenomena are critical for risk analysis. To solve the cavity dissolution problems, we propose to use a porous medium based local non-equilibrium diffuse interface method (DIM) which does not need to track the dissolution fronts explicitly as the sharp front methods (such as ALE). To reduce the grid blocks when using the DIM method, an adaptive mesh refinement (AMR) method is used to have higher resolutions following the moving fronts. An efficient fully implicit scheme is used by taking care of the velocities across the gridblock interfaces on the AMR grid. Numerical examples of salt dissolution under different flow conditions were performed to validate the modeling and numerical solving. Core-scale and reservoir-scale cases were carried out to study the mass transport and the evolution of the profiles of the dissolution fronts. Gravity-driven physical instabilities are found to be more strong in the infinite channel with upper and lower planes than in the 3D tube configuration under the same condition. The implementations with the AMR method also showed a very good computational efficiency, while obtaining good agreement with the finest-grid solutions

A finite volume approach for the numerical analysis and solution of the Buckley-Leverett equation including capillary pressure

The study of petroleum recovery is significant for reservoir engineers. Mathematical models of the immiscible displacement process contain various assumptions and parameters, resulting in nonlinear governing equations which are tough to solve. The Buckley-Leverett equation is one such model, where controlling forces like gravity and capillary forces directly act on saturation profiles. These saturation profiles have important features during oil recovery. In this thesis, the Buckley-Leverett equation is solved through a finite volume scheme, and capillary forces are considered during this calculation. The detailed derivation and calculation are also illustrated here. First, the method of characteristics is used to calculate the shock speed and characteristics curve behaviour of the Buckley-Leverett equation without capillary forces. After that, the local Lax-Friedrichs finite-volume scheme is applied to the governing equation (assuming there are no capillary and gravity forces). This mathematical formulation is used for the next calculation, where the cell-centred finite volume scheme is applied to the Buckley- Leverett equation including capillary forces. All calculations are performed in MATLAB. The fidelity is also checked when the finite-volume scheme is computed in the case where an analytical solution is known. Without capillary pressure, all numerical solutions are calculated using explicit methods and smaller time steps are used for stability. Later, the fixed-point iteration method is followed to enable the stability of the local Lax-Friedrichs and Cell-centred finite volume schemes using an implicit formulation. Here, we capture the number of iterations per time-steps (including maximum and average iterations per time-step) to get the solution of water saturation for a new time-step and obtain the saturation profile. The cumulative oil production is calculated for this study and illustrates capillary effects. The influence of viscosity ratio and permeability in capillary effects is also tested in this study. Finally, we run a case study with valid field data and check every calculation to highlight that our proposed numerical schemes can capture capillary pressure effects by generating shock waves and providing single-valued saturation at each position. These saturation profiles help find the amount of water needed in an injection well to displace oil through a production well and obtains good recovery using the water flooding technique

Pore network modelling of wettability effects on waterflood oil recovery from Agbada sandstone formation in the Niger Delta, Nigeria

A thesis Submitted to the School of Chemical and Metallurgical Engineering, Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfillment of the requirements for the degree of Doctor of Philosophy Johannesburg, 2016Wettability of a porous reservoir rock is an important factor that affects oil recovery during waterflooding. It is recognized as being important for multiphase properties. Understanding the variation of these properties in the field, due to wettability trends and different pore structures, is very critical for designing efficient and reliable processes and projects for enhanced hydrocarbon recovery. After primary drainage the reservoir wettability changes: if it was oil-wet initially, it gradually changes to water-wet during waterflooding. This change in reservoir wettability towards water-wet will reduce the residual oil saturation and improve the oil displacement efficiency. However, knowledge of the constitutive relationship between the pore scale descriptors of transport in the porous system is required to adequately describe wettability trend and its impact on oil recovery, particularly during waterflooding. In this work, the petrophysical properties that define fluid flow in the Agbada, Nigeria sandstone reservoir were determined using conventional experimental and x-ray CT scanning methods. Experimentally measured average porosity is 0.28, average permeability is 1699 mD, while the initial and irreducible water saturation is 0.22. Permeability in the x, y and z directions, ranging from 50 to 200 mD, were calculated from the pore network extracted from the Agbada sandstone rock. Results obtained from the Amott-Harvey wettability measurement method indicate that the reservoir is strongly water-wet, with Amott-Harvey index of about 0.9. The cross-over between the water and oil relative permeabilities occurred at saturations of the samples above 0.5, giving an indication of strong water-wetness. The work summarizes the mechanism of wettability alteration and characterizes the performance of the reservoir during waterflooding from injecting water, and relates the residual oil saturation, relative permeability and volumes of water injected to wettability and its effects on oil recovery. Waterflood oil recovery is computed using the Buckley-Leverett method based on the reservoir rock and fluid properties. Computed waterflood oil recovery using this method was about 60% of the oil initially in place. Plots of spontaneous imbibition rate show that the injection rate for optimal oil recovery is 40 bbls of injected water per day. At this rate, both the mobility and shock front mobility ratios are less than 1, leading to a stable flood front and absence of viscous fingering. Waterflooding is by far the most widely applied method of improved oil recovery over the years with good results in conventional and unconventional (tight oil) reservoirs It is relatively simple and cost effective: abundance and availability of water. Waterflood oil recovery factor is affected by internal and external factors. The placement of the injection and production wells, for example, impacts on the effectiveness of the waterflooding process. I considered the placement of the wells in a five-spot pattern as elements of an unbounded double periodic array of wells and assumed the reservoir to be homogeneous, infinite and isotropic, with constant porosity and permeability. Both fluids are treated as having slight but constant compressibility and their flow governed by Darcy’s law. The average pressure in the reservoir satisfies quasi-static flow or diffusion equation. I then assumed piston-like displacement of oil by injected water that takes account of viscosity diffence between both fluids and proposed a model based on the theory of elliptic functions, in particular Weierstrass p-functions functions. Oil-water contact movement, dimensionless time for water breakthrough at the production well, areal sweep and average reservoir pressures were modeled. The model was tested using Wolfram Mathematica 10 software and the results are promising. The thesis has therefore established that the Agbada sandstone reservoir is strongly water-wet and that waterflooding is a viable option for enhanced oil recovery from the reservoir.MT201