W.A. van El. Modeling Dispersion and Mixing in EOR Processes

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1 W.A. van El Modeling Dispersion and Mixing in EOR Processes

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3 Title : Modeling Dispersion and Mixing in EOR Processes Author(s) : W.A. van El Date : June 015 Professor(s) : Prof. dr. W.R. Rossen Supervisor(s) : Prof. dr. W.R. Rossen Postal Address : Section for Petroleum Engineering Department of Geoscience & Engineering Delft University of Technology P.O. Box 508 The Netherlands Telephone : (31) (secretary) Telefax : (31) Copyright 008 Section for Petroleum Engineering All rights reserved. No parts of this publication may be reproduced, Stored in a retrieval system, or transmitted, In any form or by any means, electronic, Mechanical, photocopying, recording, or otherwise, Without the prior written permission of the Section for Petroleum Engineering

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5 Abstract Dispersion has two meanings in reservoir engineering: (1) the variety of arrival times at a production well for an injected tracer, which is mostly a function of heterogeneity of permeability in the reservoir, and () mixing of injected fluids in situ. It is very difficult to distinguish these two processes in situ, while the success of different processes depends on different aspects of dispersion. For instance: a geothermal producer depends on the arrival time of cold water, while low-salinity water and miscible-flood displacements are harmed by in situ mixing. Moreover, simulation studies need to contend with numerical dispersion, a purely artificial effect of numerical simulation. In this thesis we develop a first-order-accurate two-dimensional streamline-based simulator to model in-situ mixing of tracer particles. A coordinate transformation is performed from an (x,y) to an (x,ψ) domain, such that all streamlines are parallel, before applying the convection-diffusion equation. When using streamline methods, numerical dispersion mostly occurs in the longitudinal direction which effects particle distributions in the same way as longitudinal diffusion. Taylor (1953) described how the effect of transverse diffusion effects the distribution of particles more than longitudinal diffusion. The streamline model is used to simulate Taylor dispersion in a tube in order to study the effect of longitudinal numerical dispersion on the accuracy of the model. As predicted by (Taylor 1953), numerical dispersion is found to diminish in importance as higher transverse diffusion coefficients are introduced, increasing the accuracy of the model. This effect is further studied by comparing the first-order accurate model with a higher-order accurate model and analytical solutions. After establishing that the model works, it is compared to a particle-tracking simulator which is considered to be state-of-the-art. Particle-tracking simulators are free of numerical dispersion but cannot accurately model in-situ concentrations unless exceedingly large numbers of particles are included. Rather, they infer mixing has taken place from the variance of particles positions obtained from flow-reversal tests and assume part of this mixing has taken place in forward flow. The streamline-based model developed in this thesis shows similar results to the model developed by Berentsen et al. (007) in the Taylor limit, but slightly different results in the convective limit and intermediate scenarios. The potential of the streamline-based simulator is shown by applying the model to a realistic, although only two-dimensional, scenario of a heterogeneous reservoir. Especially in the convective limit does the streamline based model show more accurate results than a conventional simulator. Finally, we apply the model to a situation which includes adsorption of particles by the solid phase. The adsorption and flow depend on local concentrations; a situation a particle-tracking model cannot easily model.

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7 Acknowledgements First of all I would like to express my appreciation to the members of my assessment committee for sharing their knowledge and insight in this project: Prof. Dr. W.R. Rossen, Prof. Dr. Ir. T.J. Heimovaara and Dr. H. Hajibeygi. Special thanks should be given to Prof. Dr. W.R. Rossen for always being ready to help, his guidance and the very interesting discussions we ve had on the topic of this thesis. Furthermore I would like to express my gratitude to Prof. Dr. L.W. Lake, Dr. C.W.J. Berentsen and Dr. A.K. John for the fruitful discussions we ve had and sharing their expertise with me. W.A van El Delft, June 015

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9 Table of Contents 1 Introduction... 1 Concepts and Definitions Dispersion Numerical dispersion Streamline and streamtube modeling Previous work: particle tracking and flow reversal Taylor s dispersion in a tube Methodology Velocity-field generation Streamline tracing Corner point grid and streamtube determination Convection-diffusion Results and discussion Taylor dispersion in a tube Quantifying numerical dispersion Comparison with high-resolution model Comparison with particle-tracking model Comparison with Virtual Asset Sorption of particles Conclusion Conclusions Recommendations Nomenclature Appendix A Velocity field model Appendix B Streamline seeds Appendix C Convection-diffusion model Appendix D Quantification of numerical dispersion References... 51

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11 1 Introduction Dispersion may be seen as the difference in arrival times of particles that were initially uniformly injected at a producer. Any observed differences are caused by changes in particle velocities. By measuring arrival times, a basic understanding of reservoir heterogeneity can be obtained, as changes in permeability will change particle velocities. However, spreading and mixing of particles in-situ, causing additional dispersion, cannot be modeled based on arrival times only, while it can play a crucial role in enhanced oil recovery (EOR) techniques. Examples where in-situ mixing is important for the effectiveness of an EOR method include low-salinity water flooding, where in-situ mixing of injected water with connate water can strongly influence the process efficiency (Jerauld et al. 006) and miscible-flood displacements, where in-situ mixing degrades miscibility by lowering concentrations (Adepoju et al. 013). We look into accurately modeling true in-situ mixing based on streamline techniques. Streamline based models have been extensively used in the petroleum and ground-water industries to model flow through the subsurface for several decades (Thiele et al. 011, King and Datta-Gupta 1998). As reservoir models become increasingly detailed, requiring many orders of magnitude more grid cells, it is becoming ever more important to develop computationally efficient simulators. For this reason interest in streamline modeling has increased significantly during recent years as it is a particularly effective technique in solving large and geologically complex, heterogeneous systems (Datta-Gupta and King 1995, Samier et al. 001, Thiele et al. 1996). An added advantage of streamline modeling is that grid cells are aligned in the direction of flow, an important concept we will extensively use in our model as this restricts the primary numerical diffusion, often caused by a convective term in the transport equation, to be in the flow direction only. Underlying most streamline simulators, all flow is assumed to be in the direction of flow only. However, true in-situ mixing may not be modeled accurately if it is assumed there is no cross-flow between streamlines, as in-situ mixing depends on concentration gradients and not on velocity vectors. We extend the definition of streamlines such that convective processes occur in the flow direction only while diffusive dispersion occurs across streamlines. To model true in-situ mixing we create a two-dimensional model based on a streamline approach, using MATLAB 014b. The transport equation that we use is the convection-diffusion equation which will be modeled using an explicit scheme in time, an upwind scheme for the convective term and central differences for the diffusive term. We construct a grid based on streamlines to be used for transport calculations, defining streamtube boundaries and geometries. We then use the principle of the grid being aligned in the direction of flow to perform a coordinate transformation from an (x,y)-domain to a (x,ψ)-domain, where ψ represents the stream function. The convection-diffusion equation will be solved on the transformed domain. Unlike most streamline simulators, the model includes fluxes both parallel- (convective) and perpendicular (diffusive) to the flow direction such that we can model in-situ mixing accurately. Based on the analysis of dispersion by Taylor (1953) we hypothesize that for certain regimes the numerical dispersion in the longitudinal direction (parallel to the flow direction) is negligible compared to other sources of axial dispersion, specifically tranverse diffusion and velocity gradients, resulting in accurate simulation results. The model will first be applied to Taylor dispersion in a tube to study the effect of numerical diffusion on the simulation results and then compared to a more accurate model. Consequently we will compare our simulator with a state-of-the-art particle-tracking simulator, developed by Berentsen et al. (007), and study the results. 1

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13 Concepts and Definitions.1 Dispersion In this thesis we define dispersion as the mixing of miscible chemical components as they are transported through porous media (Lake 1986). The result can be different from what is observed from production data (Bear 1988). For example: a reservoir may consist of a white component while a black component is uniformly injected at an injector well. Assuming the two components are miscible, a change in color will be observed when the black component reaches the production well, reflecting reservoir heterogeneity, even if there is no mixing in sity. If the reservoir is entirely homogeneous, production will quite abruptly change from white component to black component when the black component reaches the producer. If the reservoir is heterogeneous, the mixture of produced components will initially be white and, over time, turn grey until finally production is entirely black. One definition of dispersion is difference in arrival times at the producer. The problem in defining dispersion based on production data is that we don t know if the mixing took place in the reservoir or in the production well. The definition we use is equal to in-situ mixing of the components. We will focus on the dispersion of the components before they reach a producer, representing the true mixing within the porous medium. To continue the previous example of black and white components; we focus on the color of the flowing phase along its path, from injection to production. The physical cause of dispersion of miscible chemical components is a combination of convective spreading due to different fluid velocities in heterogeneous porous media and molecular-diffusion driven by concentration gradients (Lake 1986). Dispersion is governed by conservation of mass and can be defined as t x x y C v C D C C L D T (1) where C is the injected chemical concentration, v superficial velocity in equation (1) assumed to be in the x- direction, φ porosity and D L and D T the longitudinal and transverse diffusion coefficients respectively. After dividing all terms by the porosity, the porosity term is often included in the diffusion coefficients instead. We define the longitudinal and transverse diffusion coefficients as D L D D, DT () where D is the actual diffusion coefficient. The combined effect of convection and diffusion causes and represents the in-situ mixing and spreading of injected chemicals.. Numerical dispersion When numerically modeling flow through a porous medium, it is not only diffusion and convection that cause dispersion of a solute. Discretizing the convection-diffusion equation (1) and solving the equation on a grid leads to a numerical artifact we refer to as numerical dispersion, which is not a physical dispersion. Numerical dispersion occurs through several mechanisms. The first and most important cause is because a grid cell, by definition, can only contain a single value for concentration. Therefore, as soon as mass enters a cell, the concentration of the entire cell is adjusted; mixing within the grid cell is instantaneous. As the concentration of the cell is considered to be uniform, some of the solute starts spilling to the next grid cell immediately. So for every timestep, some solute spreads to a new cell, dispersing the solute. Figure 1 shows an example of numerical dispersion due to instantaneous mixing within a cell (physical diffusion is excluded in this example). The top figure shows the initial condition, where a pulse with unit concentration is injected into the first grid cell. As it flows downstream, the solute is dispersed as shown in the lower figure. After only five timesteps, the 3

14 solute is smeared over several grid cells; the width of the pulse has increased from one grid block to several and the peak concentration is greatly reduced. 0 timesteps 5 timesteps Figure 1 Example of numerical dispersion due to instantaneous mixing of solute within a grid cell. The top figure shows the initial injection of a pulse with unit concentration. The lower figure shows the dispersed pulse after five time steps. The worst-case scenario for numerical dispersion is when the flow direction is diagonal to the grid. When this scenario occurs, the flow vector is separated into vectors which connect neighboring cell nodes. Flow then occurs between the cell nodes along these vectors with velocities v x and v y. The result can be illustrated using Pascal s triangle (see Figure ). Pascal s triangle is normally used to represent the binomial coefficients in a triangular array but it can be used to represent numerical dispersion in a clear way. The concentration of every cell can be obtained by dividing each element in a row of Pascal s triangle by the sum of the elements in that particular row. The sum of a row then equals unity which in this example represents the injected concentration flowing downstream. By numerically solving the convection-diffusion equation for this scenario the solute starts dispersing in the transverse direction. An example for the time to travel five grid blocks is shown in Figure. In this example, each successive diagonal row represents tracer concentration when it has travelled further. Numerical smearing now occurs through convection driven fluxes occurring diagonally to the grid. Each grid block receives flow from two grid blocks upstream and mixes them. Figure Numerical dispersion when the flow direction (arrow top left) is diagonal to the grid. Each successive diagonal row represents tracer concentration when it has travelled further. Solute spreads in the longitudinal direction as well as in the transverse direction even in scenarios without diffusion. Note that the color scale of each figure is the same. As we will be modeling diffusion as well as convection, even in scenarios where the grid direction equals the flow direction (as was used to illustrate numerical dispersion in Figure 1), numerical dispersion will occur in the transverse direction as well through modeling of transverse diffusion. The underlying mechanism causing numerical dispersion due to diffusion is the same as the mechanism causing numerical dispersion due to modeling of convection; as soon as solute enters a grid cell (in this case by diffusion), the particles instantaneously mix throughout the whole grid cell and start spilling to the next grid cell. To show the effect of 4

15 transverse numerical dispersion on simulation results, we show an example (Figure 3) where we compare numerical results with an analytical solution. Specifically, we how the spreading of particle concentrations in one dimension where initially half the domain is filled with concentration equal to unity; this situation is similar to heat conduction between two infinite plates initially at different temperatures. The analytical solution, in dimensionless form, is defined as (Lake 1986) C D 1 xd t D erfc td N Pe (3) where N Pe equals the inverse of the diffusion coefficient. Figure 3 below shows the comparison between the analytical solution (red dashed line) and second order accurate numerical results (blue line). The lines do not match exactly as a result of numerical dispersion. The effect however, is far smaller than the effect of numerical dispersion due to convection as was illustrated in Figure 1 and Figure. Figure 3 Example comparison between the analytical solution (red dashed line) as defined by equation (3) and numerical results (blue line). The result is shown at t D = 0.5 for a domain which is initially filled with concentration equal to unity for -0.5 < x D < 0. Numerical dispersion leads to inaccurate simulation results and can cause significant problems when modeling in-situ mixing. It is not dispersion of a physical nature and as such is an unwanted effect. We assume that numerical dispersion is in the longitudinal direction only when the grid is aligned with the flow direction (Figure 1), and has a smaller effect than when the grid is aligned diagonally to the flow direction. Moreover, when flow is aligned with the grid, numerical dispersion occurs in the longitudinal direction only and does not occur in the transverse direction..3 Streamline and streamtube modeling To reduce numerical dispersion, we use the method of streamline modeling. Streamlines are the basis for every streamline based-model, where streamlines are defined as lines tangent to the vectors of a potential field; streamlines are aligned in the flow direction. For reservoir engineering the potential field is defined as a velocity field based on pressure gradients. The velocity field may be obtained using conventional reservoir simulators based on a finite difference or finite volume method for single-phase flow. To construct streamlines, the bi-stream functions (ψ,χ) have to be solved, which are defined as (Bear 1988) v (4) 5

16 where the superficial velocity v is divergence free if the fluid is considered to be incompressible. The intersection between a constant value for ψ and a constant value for χ defines a streamline. Equation (4) applied to two dimensions can be done by setting χ = z and results in the simpler form (Datta-Gupta and King 1995) v x, y v y x (5) where equation (5) is written in component form. The solution to equation (5) can be obtained directly but is often determined by tracing a particle s flow path from seed (injector) to sink (producer) through the velocity field. By definition this path is, indeed, tangent to the velocity field, which is the principle of a streamline. The algorithm we apply to track streamline paths was first defined by Pollock (1988) and will be discussed in more detail in section 3.. In this work we use streamline modeling to get rid of transverse numerical dispersion as described in section., since streamlines can be used to align the grid in the local flow direction. Although diffusive fluxes also lead to transverse numerical dispersion in the same way convection leads to numerical dispersion when the grid is aligned to the flow direction, the effect is much smaller, as modeling diffusion can easily be done far more accurately. Thus, numerical dispersion occurs in the longitudinal direction mostly..4 Previous work: particle tracking and flow reversal Recent studies on modeling dispersion include the work of Berentsen and van Kruijsdijk (008) and John et al. (010). These studies apply random-walk particle-tracking methods to solve the convection-diffusion equation (1). The Particle-tracking methods used are considered to be free of numerical dispersion. The models describe particles flowing downstream tracing streamlines by using velocity vectors, representing convection, and introduce diffusion using a random-number-generator (John et al. 010, Berentsen et al. 007). The randomnumber-generator is presented by a Gaussian redistribution function and describes the chance a particle will jump to a nearby streamline. Therefore, the Gaussian redistribution function has to be related to the dispersion coefficients shown in equation (1). To measure dispersion, particle-tracking methods keep track of the variance of particles over time; the number of particles of every x-coordinate are measured every time-step and the spread of the particles is measured as the particle variance. Both works describe transverse diffusion using the idea of flow reversal to distinguish between dispersive effects due to convective spreading and molecular-diffusion. The principle is defined as follows: solute transport is allowed for a finite amount of time. After this time the velocity field is reversed and transport is allowed to continue for an equal amount of time, such that particles flow back towards the inlet. At the end of a flow reversal, convective spreading by itself results in zero dispersion; the concentration at the inlet reflects the concentration upon injection. The underlying assumption is that a particle flows downstream along a streamline and after flow reversal tracks the same streamline back to the point of origin. When molecular-diffusion is included in the process, the dispersion on flow reversal is not zero. Rather, dispersion continues to increase even after flow reversal. This effect is based on the principle that diffusion allows a particle to jump streamlines. After flow reversal the path of a particle cannot be retraced along a single streamline. Rather, after flow reversal the particle keeps jumping streamlines and increases dispersion. Dispersion due to diffusion is an effect based on time and concentration gradients regardless of flow direction. To make such a model, which includes diffusion, reversible requires a negative diffusion coefficient, which is not physical (Berentsen and van Kruijsdijk 008). The particle-tracking method is very powerful, as it is free of numerical dispersion (John et al. 010). But, the models can't show mixing in forward flow because it is obscured by the flow field. They can only infer the insitu mixing after flow is reversed to the starting point, from the variance in positions, and infer that a portion of that was going on in forward flow. Moreover, if transport depends on concentration of the solute (as in 6

17 adsorption or phase partitioning), particle tracking can't model it (unless it counts particles in grid blocks, which brings back numerical dispersion or computes the probability of particles occupying the same position). Our goal is to have a way to model dispersive mixing that can be applied in forward flow without particle tracking..5 Taylor s dispersion in a tube Taylor (1953) analyzed the effects of convection and diffusion for different time scales by injecting soluble components in solvent flowing through a capillary tube. For small times during the experiment, transport was determined to be entirely convection dominated; spreading occurs due to different velocities. As time increases, particularly diffusion transverse to the direction of flow becomes more dominant as tracer particles change streamlines due to concentration gradients; spreading then occurs mainly due to transverse diffusion combined with non-uniform velocity. When enough time passes, a tracer particle may sample all individual streamlines and approximate the average flow velocity. The net effect then appears to be plug flow where radial concentration variation is almost zero and the parabolic concentration profile is eventually entirely lost (Taylor 1953, Lake and Hirasaki 1981, Berentsen and van Kruijsdijk 008, John et al. 010). Figure 4 illustrates the analysis of Taylor for different time scales; initially flow is dominated by convection and for later times flow is dominated by convection and transverse diffusion. Figure 4 Schematic of Taylor dispersion in a tube by (John et al. 010). For early times transport is dominated by convection and the concentration profile reflects a parabola. For later times transport is dominated by convection and (transverse) diffusion and flow represents a plug. An important remark on the work by Taylor is his conclusion that longitudinal diffusion may have a negligible effect on dispersion for both small and large timescales; the total dispersion of the solute is dominated by convection, transporting the solute downstream, and transverse diffusion. In the long run, especially for scenarios with a large transverse dispersion coefficient, the effect of transverse diffusion is that it (partly) negates dispersion in the longitudinal direction. Through transverse movement of solute, particles obtain an average flow velocity very quickly if the transverse dispersion coefficient is large enough compared to the flow velocity. Particles will then flow downstream as a plug. The greater the transverse dispersion coefficient, the less time is required for particles to obtain the average flow velocity and thus the less wide the plug will be. Longitudinal diffusion has no such effect: whether particles change position in the longitudinal direction through diffusion has no effect on their velocity as they do not change streamline. Moreover, changes in the longitudinal direction will be counteracted by transverse diffusion and convection. Thus, the effect of spreading due to concentration gradients in the longitudinal direction is far smaller than the effect of diffusion in the transverse direction. This specific scenario occurs when the time necessary for a radial variation in concentration to decrease owing to radial diffusion is much shorter than the time necessary for an appreciable change in concentration to occur through longitudinal convection (Taylor 1954). This condition can be expressed as L v R 3.8 D (6) 7

18 where L is the length of the zone with appreciable concentration gradients (approximately the width of a pulse), R is the radius of the tube, and is the average velocity. Equation (6) was analytically deduced by Taylor (1953) such that radial variations in concentrations are less than 1/e of its initial value. In this scenario, diffusion acts as though it counteracts dispersion in the longitudinal direction due to convection. By aligning the grid in the flow direction using streamlines, the largest numerical dispersion occurs in the longitudinal direction. The effect of numerical dispersion is similar to longitudinal diffusion as it diffuses solute due to concentration differences and not due to differences in velocity, as will be discussed further in section 4.. However, dispersion due to diffusion in the longitudinal direction is negligible under certain conditions. The combination of aligning the grid with the local flow direction through streamlines and the findings of Taylor should allow us to accurately model the convection-diffusion equation and to get rid of most of the numerical dispersion. 8

19 3 Methodology In this dissertation we simulate incompressible tracer flow through a two-dimensional incompressible porous medium by applying a streamline method in MATLAB. We use a streamline approach because it defines a grid aligned in the flow direction. As the convective term we use is the least-accurate of the terms in the convectiondiffusion equation, the largest truncation error will occur in the direction of flow. The model will first be used to simulate Taylor dispersion. Because analysis by Taylor (1953) suggests that diffusion in the longitudinal direction is, for scenarios with a large dispersion coefficient, negligible, the effect of the truncation error due discretization of the convective term is reduced. We apply a streamtube approach where we keep track of streamline coordinates between which cross-flow may occur. We define streamtubes to be the area bounded exactly between two adjacent streamlines, i.e. through which given flow Q (same for all streamtubes) passes. Thus that we can easily determine the streamtube geometries. Once streamtubes are constructed and information on their geometry stored, we transform the streamtubes from an (x,y)-domain to an (x,ψ)-domain (Figure 5). We then apply a finite-volume method, using the stored geometries, to numerically solve the convection-diffusion equation for the tracer. We assume this tracer is injected uniformly over one boundary and is carried through the porous medium by a single-phase flow of carrier fluid towards the opposite boundary. This setup is such that we can simulate flow in a tube such as the experiments performed by Taylor (1953). We assume the tracer is fully miscible and has equal mobility to the solvent, such that we don t have to update streamlines over time. Note that velocity v x is not uniform in the (x,ψ) domain; it is inversely proportional to the width of the given streamtube in the (x,y) domain. Likewise, the effective transverse diffusion coefficient is inversely proportional to the width of the streamtube, reflecting the distance for transverse dispersion in the (x,y) domain. Figure 5 Example of coordinate transformation from the (x,y) domain to an (x,ψ) domain. 3.1 Velocity-field generation As discussed in section.3, streamlines are defined based on a potential field. In case of flow through porous medium this potential field is based on gradients in pressure resulting in velocity vectors. To construct the velocity field we assume single phase incompressible flow in a non-deformable permeable medium and no gravity contribution. The velocity field, derived from the pressure, can be obtained from Darcy s law q P (7) where q is the flow rate, the mobility defined as in which is the absolute permeability and the fluid viscosity, and the pressure gradient. To solve equation (7) numerically we apply a finite-volume method, similar to a conventional reservoir simulator, on a Cartesian grid. The discretization of equation (7) and further details on the numerical model can be found in 9

20 Appendix A Velocity field model. For now it is important to note that the numerical model solves the superficial velocity on the cell interfaces. In all cases we apply a constant pressure as boundary conditions at the inlet and outlet and no-flow at all other boundaries. Constant pressure allows determining the correct allocation of streamline seeds where constant injection velocity as boundary condition does not. Further details are discussed in Appendix B Streamline seeds. 3. Streamline tracing To construct streamlines we have to create curves that are tangent to the velocity field at every position. The tracing concept is simple: particles are traced through a flow field by calculating time steps required to reach cell boundaries. The particle is then moved to a new location determined by multiplying the velocity components with this time step to obtain the incremental changes in the particle s coordinates over that time interval (Pollock 1988). The equation we use to describe the Lagrange streamfunction ψ(x,y) is shown as equation (5) in section.3. Figure 6 Schematic representation of a particle path through a two-dimensional cell by Pollock (1988). This example shows a particle entering cell (i,j) through the left boundary (x 1) and two potential exit faces x and y being the right and top cell faces respectively. Cell face y 1 is not a potential exit as the velocity vector v y1 points inwards. To solve equations (5) for ψ(x,y), the equations have to be integrated over the velocity field and by definition we obtain (Hewett and Yamada 1997) y x x y (8) 0 0 ( x, y) v ( x, y ') dy ' v ( x', y) dx ' In our model equation (8) has to be solved numerically. We apply a tracing algorithm similar to the one described by Pollock (1988), where the velocity field is based on Cartesian grid. To compute streamline paths we first assume the velocity everywhere within the flow field as a piece-wise linear interpolation; the velocity gradient within every cell is assumed to be constant. As the velocities were solved on the cell interfaces, we can now define the velocity at any position within a grid cell (i,j). The linear interpolation in two dimensions can be described as 10

21 v v A ( x x ) x x1 x 1 v v A ( y y ) y y1 y 1 (9) In this equation x 1 and y 1 denote the left and bottom cell faces respectively (see Figure 6). A x and A y are defined as the velocity gradients within the cell (i,j) in the x- and y-direction respectively as / A v v x x x x x1 1 1 / 1 A v v y y y y y (10) Subscript defines the cell face opposite the cell face denoted 1 (Figure 6). Pollock (1988) has shown that linear interpolation of the cell face velocities satisfies the mass balance equation at every point inside that cell, which is important to remain mass conservative. To calculate the time required to reach each cell interface from the particle s current position (as an example we assume this is x 1), which we define as Δt e,x, Δt e,y1, Δt e,y, we first describe the movement of a particle through a cell as function of time. The change in velocity in the x-direction of particle P over time may be defined as dv dt x p dv dx dx dt x (11) p An equal expression can be described for the y-direction. Inserting equation (10) into equation (11), rephrasing the particle s velocity as (dx/dt) p = v xp and rearranging the terms yields the differential equation 1 dv v xp xp A dt (1) x Integrating equation (1) gives an equation for the time required for a particle to travel from its current position to a cell interface through which the particle potentially exits the cell. For example, if a particle enters a cell through face x 1 it can leave through faces x, y 1 and y depending on the direction of their respective velocity vectors (Figure 6). The integration yields, for a particle traveling from its current position to x 1 v ln x tex, A vxp (13) where subscript e denotes exit. If the velocity gradient is equal to zero, the calculation for the exit time in that direction is simplified to e, x p / xp 11 t x x v (14) Equation (13) and (14) give an exact solution such that no numerical error is introduced by this calculation. This derivation can be readily extended to two- or three-dimensions. In the model the time required to reach a cell interface only has to be calculated if that interface is a potential exit; only if the velocity vector at that interface is pointing outward can a particle travel towards that interface (Figure 6). The cell interface through which the particle P eventually exits the cell is the one that is reached first: the exit cell interface can be determined by taking the minimum time required to reach any of the cell boundaries: min(δt e,x, Δt e,y1, Δt e,y). The minimum time step then denotes which interface particle P reaches first. The final step of the algorithm is to determine the exact new coordinates of particle P on the exit cell-interface. To obtain an equation describing the exit coordinates, equation (9) is substituted in equation (13) for v xp. Rewriting the equation for the x-coordinate and substituting the exit time step Δt e for the time step Δt yields

22 1 x x v ( t )exp A t v A (15) e 1 xp p x e x1 x Repeating the derivation in the y-direction gives the particle P s new y-coordinate on the exit interface 1 y y v ( t )exp A t v A e 1 yp p y e y1 y (16) In equations (15) and (16) x e and y e denote the exit coordinates. Repeating these steps for every cell from source to sink and keeping track of the exit coordinates results in all the coordinates we use to define a stream function ψ(x,y). The algorithm for tracing a particle through a grid cell is repeated from a source (inlet) until a particle reaches a boundary or a sink (outlet). Although we use all the particle s coordinates to trace streamlines, we only store exit coordinates which line-up with Cartesian x-grid coordinates, or the vertical mesh. This is done to allow for transverse diffusion and easy determination of streamtubes and streamtube geometry, as will be discussed in the section Corner point grid and streamtube determination Once the stream function has been determined, we form a grid representing streamtubes bounded between every pair of consecutive streamlines (Figure 7). Streamtubes have a volume inversely related to the local velocity; for high local velocities the local streamline density is high and thus streamtube have a small local volume and vice versa for low local velocities (Thiele et al. 011). As such, our definition of a streamtube concurs with those set for streamlines. To define the geometry of streamtubes we generate a corner-point grid using the streamline coordinates and a vertical mesh. Figure 7A shows a schematic of two streamlines (ψ 1 and ψ ) overlying the Cartesian grid (dashed lines) used to generate the velocity field and streamlines. Figure 7B shows the resulting corner point grid which we use to determine streamtube geometries to be used for transport simulations. A B C Figure 7 A Streamlines (black lines denoted ψ 1 and ψ ) overlying the Cartesian grid used to generate the velocity field. Black dots represent stored coordinates used to generate a corner-point grid. The blue area represents a single streamtube bounded by two streamlines. B Example of the resulting corner-point grid (dashed lines) based on streamlines. C Example of the transformed grid on the (x,ψ)-domain used to solve the convection-diffusion equation. 1

23 Once we have determined and stored the geometry of every streamtube, a coordinate transformation is performed such that all streamlines are horizontal, parallel and equidistant as shown in Figure 7C; we transform streamlines from (x,y) coordinates to (x,ψ) coordinates. Essentially step B is only used to determine streamtube geometries such as volume, cell interfaces, distance between streamtube nodes and path lengths. We will then apply a Finite Volume Method (FVM) for modeling transport on the transformed domain of (x,ψ), where fluxes are appropriately scaled using the predetermined streamtube geometries. 3.4 Convection-diffusion As stated before, we assume a fully miscible tracer is carried through a two-dimensional porous medium by a single-phase flow of solvent. The convection-diffusion model describes dispersion of the tracer as it flows downstream, where we assume the longitudinal diffusion is negligible. The convective part describes dispersion of the tracer due to differences in flow velocity while the diffusive part of the equation describes dispersive effects due to concentration gradients. Solving for transport is done in the (x,ψ)-domain where all streamlines are parallel (Figure 5), instead of on the original (x,y)-domain. The convection-diffusion equation we apply is similar to the one described by Berentsen et al. (007), where we assume zero dispersion due to longitudinal diffusion t x C C C vx, D x, (17) where D τ is the transverse dispersion coefficient on the (x,ψ)-domain including the porosity term φ as shown in equation (1). Although we assume the value of the dispersion coefficient D T in equation (1) to be constant on the (x,y)-domain, the dispersion coefficient D τ(x,ψ) is now a variable; a smaller distance between streamlines on the (x,y)-domain results in a higher dispersion coefficient on the (x,ψ)-domain, while a larger distance between streamlines reduces the value for the dispersion coefficient such that the initial distance between streamlines is incorporated on the transformed domain. The transformation and scaling of the dispersion coefficient is discussed in further detail in Appendix C Convection-diffusion model. In this thesis we assume the physical dispersion coefficient in equation (1) is a positive constant, where it usually is a function of rock properties or flow velocity. For our purposes of studying the in-situ mixing and the effect of the dispersion coefficients, it is more useful to use constants as we can adjust their values and analyze the effects more easily. To discretize equation (17) we apply explicit time integration, a second order accurate finite difference scheme to the diffusion term and a first order accurate upwind scheme to model convection. Although applying a second order finite difference scheme to the convective term would be more accurate, this often leads to oscillations when applied to sharp shock fronts (LeVeque 199). Appendix C Convection-diffusion modeldescribes the full discretization and application of equation (17) in more detail. For the stability we apply the von Neumann stability analysis, which for the diffusion term yields 1 ( s) t min DT (18) where Δs equals the distance between streamtube nodes. For the convective term Neumann stability analysis results in x t (19) max( v ) x 13

24 By applying a CFL condition to the minimum timestep determined by both stability criteria, (18) and (19), we ensure the model is stable. The boundary conditions we apply in our model are that a tracer pulse is injected uniformly over the left-hand boundary for a single time step C( x0, y, t 0) 1 (0) As boundary conditions at the top and bottom boundary we assume zero, as was used to construct the velocity field and streamlines, and a constant pressure at the inlet and outlet. 14

25 4 Results and discussion To verify the correct functioning of our model we first study simulation results obtained using a parabolic velocity field to model and describe Taylor dispersion in a tube. We will study the effect of changing the transverse- and longitudinal dispersion coefficient of the model and then compare results obtained from our model with results described by Berentsen and van Kruijsdijk (008). Finally we will study the effect and importance of numerical smearing by comparing our results with those obtained from a more accurate numerical algorithm. We will show that for scenarios where longitudinal diffusion can be neglected, numerical dispersion has little effect on the results. 4.1 Taylor dispersion in a tube To study Taylor dispersion, the two-dimensional model has to be adjusted such that it incorporates the threedimensional geometry of a tube. Firstly, the velocity profile has to be set to be parabolic (Bird et al. 007) v( r) v 1 r R max (1) where v max is the maximum velocity, r the radial coordinate and R the radius of the tube. To obtain this velocity profile we simply put a permeability field into the model with an equal parabolic profile as described by equation (1). Through Darcy s law, shown in equation (7), the velocity field will assume the same profile. Secondly, the flow rate q has to be described in three dimensions using radial coordinates q( r) v 4 r r 4R max r 0 () Equation () is used to determine convective fluxes (flow rates) of every grid cell as if they cover a grid cell of half a disk (Figure 8). The flow area (light blue) represents the area of a grid cell on a longitudinal cross-section of the tube (dashed line). Likewise, the interface between grid cells in the radial direction A r, to be used for transverse diffusion, is obtained using the equation for the circumference of a circle divided by two Ar r (3) By substituting the transverse flow area shown in equation (3) and the flow rates obtained using equation () into the two-dimensional simulator, the three-dimensional properties of a tube are preserved. Figure 8 Figure showing half a disc used to determine flow surface of grid cells representing the threedimensional geometry of a tube, in a two-dimensional simulator (cross-section through the tube such as the dashed line). Equation () is used to determine the actual flow rate through such grid interfaces, applying the three-dimensional geometry. The blue area represents the flow area of an example grid block on the cross-section of half a disk. 15

26 Finally, the convection diffusion equation has to be transformed to radial coordinates. To describe dispersion of a concentration C, equation (1) is rephrased as (neglecting longitudinal diffusion) C v C D C Tr t x r r (4) Applying equation (1), (), (3) and (4) in our model, we are able to accurately describe convection and diffusion in a tube to run simulations representing Taylor dispersion. With a no-slip condition at the boundaries the resulting velocity field is parabolic as described by equation (1) and shown in Figure 9. Differences in velocity will lead to dispersion in the axial direction due to convection. When we allow for diffusion to take place, especially transverse diffusion, the tracer flow will become plug-like for longer time scales. The effect of diffusion for long time scales, where flow becomes plug-like, may also be obtained by increasing the diffusion coefficients which has the unit [L /t]; increasing the diffusion coefficient increases dispersion per unit time. By doing so the effect of diffusion may already be observed for relatively small periods of time. We apply different values for the diffusion coefficient to look at the effect of transverse diffusion and analyze the results for forward flow and flow-reversal tests as described in section.1. A B C Figure 9 A: Example of the permeability field used to model Taylor dispersion. B: Example of the velocity profile generated based on the permeability field shown in A. C: Example of streamlines generated based on the velocity field shown in B. Figure 9 shows the application of a parabolic permeability field (A) and the resulting velocity profile (B), as described by equation (1). Figure 9C shows the resulting streamlines. The boundary conditions we apply are no flow at the top and bottom boundaries and constant pressure at the boundaries in the flow direction. Tracer will be injected uniformly at t = 0 at x = 0 for a single timestep. Both D L and D T, as applied in equation (17) are initially set equal to zero such that we have convective flow only. Then D T will be exponentially increased from 10-7 to 10-5 to allow for diffusion. The values of other parameters that are used to model Taylor dispersion are shown in Table 1. It may be noted that in reality D L should equal a value close to the value of D T as was discussed in section.1. However, we assume diffusion in the longitudinal direction is negligible. This assumption is explained in section.5 and will be further discussed in sections 4. and

27 Table 1 Values for parameters used for modeling Taylor dispersion. All parameters are in SI units unless otherwise specified. Parameter Value L x 000 m N x 000 L y 1 m N y 50 N stream 50 T 100,090 s Δt 1 s D T A: = 0, B: 10-7, C: 10-6, D: 10-5 m /s D L 0 m /s P entry 0.0 Pa 0.0 Pa P exit Figure 10 shows simulation results (A, B, C and D) for different values of the transverse diffusion coefficient D T while all other parameters are as shown in Table 1. By changing the value for D T we show the idea put forward by Taylor (1953). The left column of Figure 10 (A1, A3, B1, B3, C1, C3, D1 and D3) shows simulation results after running simulations for a finite amount of time T (see Table 1) for a tracer pulse being injected uniformly into a porous medium, at x = 0. The value for T was chosen such that we inject exactly half of a pore volume. The x- and y-axis are in dimensionless forms where Δx D = 1/L x and Δy D = 1/L y. The right column of Figure 10 (A, A4, B, B4, C, C4, D and D4) shows the results of an echo test, after flow has been reversed for an equal amount of time T. For convenience the x-axis is extended by half a pore volume (PV) before the inlet and after the outlet, such that tracer particles do not exit the porous medium upon flow reversal. Below each in-situ concentration profile, two curves are shown, representing the concentration profile at a cross-section of the tube in the longitudinal direction. The red curve is a cross-section through the center of the streamtube closest to the boundary (at y = x) while the blue curve shows the concentration profile of the streamtube closest to the center of the tube (at y = 0.5 x). We first focus on results in the left column of Figure 10. From the results we observe that, as the value for the transverse dispersion coefficient increases, the pulses start to smear in the longitudinal direction; the concentration profile in the longitudinal direction becomes wider (scenarios B and C). When transverse diffusion becomes more and more dominant, the parabolic velocity profile is lost (scenarios C and D) and flow eventually becomes almost plug-like. In the article of Taylor (1953) he describes two limiting factors: 1. Changes in concentration due to convective processes along a tube take place in a time which is so short that the effect of molecular diffusion may be neglected.. The time necessary for appreciable effects to appear, owing to convective transport, is long compared with the time of decay during which radial variations of concentration are reduced to a fraction of their initial value through the action of molecular diffusion. We represent the first limit by setting the diffusion coefficient D T equal to zero, such that no diffusion takes place (scenario A of Figure 10). As the first limit describes, we observe changes in the longitudinal direction which are caused only by convection. To calculate the order of magnitude required for D T to reach the second limit, representing plug flow, we use (Taylor 1953) 17

28 D T v L R 3.8 max (5) Parameter L in equation (5) represents the length where the greater part of the change in concentration occurs. More specifically, for calculations we will use for L the distance between the points where the dimensionless cumulative concentrations equals 0.1 and 0.9. Using equation (5) allows us to calculate which value for the transverse dispersion coefficient is required to reduce the radial variations of concentration to less than 1/e of their initial value. Applying the values from Table 1 to equation (5), noting that the boundary conditions for pressure and size of the domain lead to a maximum velocity of 0.01 m/s, leads to D T >> Results D as shown in Figure 10 corresponds to this statement for the value of D T assuming that a ratio of at least 10 is a large enough such that the second Taylor limit is reached. We also observe, especially from results for scenarios B, C and D of Figure 10, that with increasing values for the transverse diffusion, the concentration profile becomes less wide and concentrations higher, as diffusion acts to prevent dispersion due to convection. 18

29 Figure 10 Example of the effect of dispersion for flow in a tube (Taylor dispersion). From top to bottom the transverse diffusion is set to A: D T = 0, B: D T = 10-7, C: D T = 10-6 and D: D T = 10-5 (axial diffusion is excluded). Shown are in-situ concentration profiles and concentration profiles of cross-sections through the center of the tube, y = 0.5 (blue), and through the center of the streamtube closest to the boundary at y = (red). Injection occurs at x = 0 for t = 0 for the duration of a single timestep. Note that the color scale varies among cases. 19

30 Next we focus on the results of the right column of Figure 10 (A, A4, B, B4, C, C4, D and D4). These results show the concentration profiles of a flow reversal test similar to the method described by Berentsen and van Kruijsdijk (008) and John et al. (010). A flow reversal test is used to distinguish between dispersion due to diffusion and dispersion due to convection (see section.1). During a flow reversal test the velocity profile is reversed (v x = -v x) when the time reaches the value described in Table 1, and flow is then continued for an equal amount of time. If flow was purely convective, tracer particles would retrace their exact paths back to their origin; the concentration profile after flow reversal would be a copy of the initial conditions. Dispersion due to diffusion is not reversible but increases with time regardless of the flow direction as diffusion is driven by concentration gradients, not velocity. To simulate a flow reversal tests one could reverse the velocity field as described by Berentsen et al. (007) and John et al. (010), which is very easy to do for particle tracking. However, when applying an upwind scheme to the convection-diffusion method simply reversing the flow does not work. The upwind scheme will become a downwind scheme if not adjusted properly. To prevent changing the scheme we apply a different approach: after flow occurs in the flow direction for a set amount of time we mirror the concentration profile in the longitudinal direction. We then continue flow for an equal amount of time and mirror the concentration profile again to obtain the echo test result. The velocity field is not altered during the simulation. Figure 11 shows an example of an echo test for a purely convective scenario. Figure 11A shows the initial condition at injection and figure 11B shows the in-situ concentration profile after transmission at t = T, at which point we mirror the concentration. This yields the in-situ concentration profile shown in figure 11C, we then continue flow for an equal amount of time T. After mirroring the result again we obtain figure 11D. Figure 11 Example of a flow reversal test where the concentration. First flow occurs in the x-direction (A B). Then the concentration is mirrored in the longitudinal direction (B C) at the time of flow reversal. Flow is then continued for an equal amount of time and the result is mirrored again when the echo test is finished, yielding result D. The result is equal to reversing the flow but without altering the velocity field. The transverse diffusion coefficient is set equal to zero. The results shown in Figure 10 for scenario B, C and D show that after flow reversal, spreading of the concentration continues. The pulse after flow reversal does not reflect the initial condition, which is a vertical line at x = 0 with a width of Δx and unit concentration. Instead, the tracer has spread more than before flow reversal; concentrations are lower and profiles wider. Although represented in a different way, this behavior is the same as was observed by Berentsen and van Kruijsdijk (008) and John et al. (010). This will be discussed further in section 4.4. The results for scenario A show a distinct difference. This scenario should result in a concentration profile reflecting the initial conditions, but it does not exactly match. This is due to longitudinal numerical dispersion as described in section. and will be discussed in more detail in the next sections. 0

31 4. Quantifying numerical dispersion From Figure 10 A we determined that our model is not entirely reversible for a purely convective scenario, as it should be. If the model excludes any diffusion, every particle should trace its path back to its origin. However, we do not obtain the initial conditions after flow reversal (Figure 10 A). Instead, there is dispersion of the concentration to be noted. The reason for this result is not of physical nature but due to numerical dispersion (sometimes called numerical diffusion or numerical smearing (Lantz 1971, Moghanloo 011)) as described in section.. Numerical dispersion results from numerical approximations of the concentration and the fact that every grid cell can only contain a single value for concentration for a given time. Thus as particles enter a grid cell, they are instantly spread over the whole cell and the value for concentration is changed. As a result some of these particles instantly start spilling to the next cell, although the actual concentration front may not have reached that cell boundary yet. Like diffusion this dispersive effect is not reversible, as it is caused by approximations and concentration gradients between cells and not by differences in velocity. Upon flow reversal the numerical artifact is still causing dispersion and spreading continues. In our model, numerical dispersion is caused mostly by the explicit method for time and the upwind scheme applied to the convective term, as those terms are only first-order accurate (section 3.4 and Appendix D). Following the derivation by Lantz (1971) to quantify the effect of numerical dispersion, which is based on Taylor expansion, we obtain a similar result when applied to the convection-diffusion equation (17) C C v x v t C C v D D O t x y t x x y x x x L (,, ) T (6) The derivation of equation (6) is shown in more detail in Appendix D and is an adapted form of equation (17) of second-order-accurate differentials. In this equation, (v x Δx v x Δt)/ is the term leading to first-order numerical dispersion in the longitudinal direction. The form of equation (6) reiterates that the numerical dispersion can be seen as a diffusive term, which is not reversible, and is additive to the longitudinal dispersion coefficient D L. The unit of this term, [m /s], is equal to the unit of the dispersion coefficient. If we insert the values shown in Table 1 for into the numerical dispersion term, we obtain (v x Δx v x Δt)/ = This value is at least an order of magnitude larger than the values we chose for the diffusion coefficient. Because the numerical smearing is additive to the longitudinal diffusion and orders of magnitude larger than D T, which D L should approximately be equal to, we chose to set the value for D L equal to zero; longitudinal diffusion is less than numerical smearing in the longitudinal direction. Whether or not the effect of numerical dispersion in the longitudinal direction is negligible depends on the value for D T, as it counteracts longitudinal dispersion. According to Taylor (1954), the limit where longitudinal diffusion is negligible is met if 4L R vr D T 6.9 (7) Equation (7) is an extension of equation (5) based on the same assumptions. Applying equation (7) to the scenarios shown in Figure 10, using the average velocity in a tube defined as, the width of the pulse L measured between the points where the dimensionless cumulative concentration is between 0.1 and 0.9 and the other parameters as shown in Table 1, yields the results shown in Table. Whether or not the criteria of equation (7) are met or not depends on the interpretation of the ratios. Following Taylor, we assume a ratio of 10:1 is an acceptable ratio between the terms. With this ratio, only scenario D (D T = 10-5 ) meets both the criteria such that longitudinal diffusion may be neglected and scenario C (D T = 10-6 ) nearly satisfies the required conditions as the first ratio is only.98:1 while the second ratio is far larger than 10:1. 1

32 Table Results of applying equation (7) to the parameters of Table 1. Scenario A (D T =0) 518 B (D T = 10-7 ) C (D T = 10-6 ) D (D T = 10-5 ) Assuming that the longitudinal diffusion is negligible and has no influence on the results does not mean numerical dispersion can be ignored. The value for numerical dispersion, (v x Δx v x Δt)/, is much larger than the values we chose for the diffusion coefficient. Taylor s derivations assumed the axial and transverse diffusion coefficients are equal. Therefore we cannot yet assume the numerical dispersion in the longitudinal is negligible. We will discuss this in further detail in section Comparison with high-resolution model In section.1 we ve discussed that axial dispersion in certain scenarios may be negligible and we have based our method on this assumption to attempt to create an accurate model describing in-situ mixing. However, in section 4. we have shown that the magnitude of the numerical dispersion in the longitudinal direction is much larger than the value for the transverse dispersion coefficient D T such that the criteria, as shown in equation (7), may not be sufficient. To verify our model is as accurate as we predict, we compare our simulation results with those of a more-accurate model. From the results shown in Table we conclude that none of the scenarios meet the criteria set by Taylor, if we assume a ratio of 10:1 is sufficient. Scenario D (D T = 10-5 ) almost satisfies the criteria but the first ratio is only 4.6:1. However, from Figure 13 we conclude that the ratio is high enough such that numerical dispersion has little effect. Even scenario C with a ratio of 1.9:1 has little longitudinal numerical dispersion. In section 3.4 we explained that the convective term in the convection-diffusion equation (17) is only firstorder accurate and the diffusive terms are second-order accurate. As such the truncation error (numerical dispersion) is dominated by the first-order accurate convective term. Equation (6) shows the added longitudinal diffusion due to numerical smearing. In this section we compare results of our model with a second-order accurate model to show that the simulation results, for scenario C and D, are (almost) identical. The second-order-accurate model is identical to the model that we used, apart from the convective term. Because the simulations of Taylor dispersion depend on injecting a pulse, central-differencing schemes of second-order accuracy for the convective term are often not monotone; they are oscillatory near large concentration gradients and results can be unphysical (LeVeque 199, Sweby 1984, Zijlema and Wesseling 1995). The method we apply to avoid oscillations is a flux-limiting scheme. A flux-limiting scheme combines a stable and monotone method with a higher-order method to avoid oscillations. Near high gradients the monotone method is preferred due to its stability, while for smooth regions the higher-order method produces more-accurate results. We combine the first-order upwind scheme with a second-order central-difference scheme as a function of the ratio of consecutive gradients: i C 1 1 C i i 1 i 1 i r C C (8) where the flux-limiter ϕ(r i+1/) is a non-linear function of the ratio r of consecutive gradients (LeVeque 199). If the flux-limiter is small, the concentration at the boundary C i+1/ is approximately defined by an upwind scheme while a central difference scheme is used when the flux-limiter approaches unity:

33 r C C i1 i i 1 Ci Ci 1 (9) The chosen flux-limiter is the one described by Van Leer (1974) and defined as r r r (30) 1 r We chose the van Leer flux limiter because it has less dispersion than other flux-limiters such as the minmod method, but is not as extreme as the Superbee flux limiter, which may cause unphysical results (Roe 1986). This flux-limiter model can be readily applied to our convection-diffusion model and yields results of a higher-order accuracy. The left column of Figure 1 shows the same results as shown in the left column of Figure 10 (low-resolution model) while the right column shows the results obtained when using a higher-order simulator. The input for both models is identical and equal to the parameter values described in Table 1. The color bars of each row are scaled such that the color bar in the left column is equal to the one in the right column. It may first be noticed that the results of scenario A and B for the less-accurate model are different compared to the results of the high-resolution model. This is not surprising as the values for the transverse dispersion coefficient are few and the criteria set in equation (7) are not met. The results for scenario C are more promising as they look very similar when comparing the two models. Based on these results we would say that the shape and local concentrations are close to identical. A similar remark can be made for scenario D although the concentration at the center of the pulse is slightly lower for the low-resolution model. 3

34 Figure 1 Simulation results for the same scenarios as shown in Figure 10 using the parameter values of Table 1. From top to bottom the transverse diffusion is set to A: D T = 0, B: D T = 10-7, C: D T = 10-6 and D: D T = 10-5 (axial diffusion is excluded). The left column shows the results of our first-order-accurate simulator. The results on the right are obtained using a higher-order model. Note that the color scale varies among cases. Although Figure 1 shows the shape of the pulse very well, the values for concentration are not as clear. Figure 13 shows the concentration profiles for a longitudinal cross-section of the tube. The cross-section is taken in the middle of the tube as that is where the velocity is largest, leading to the biggest effect due to longitudinal numerical smearing. Again the results show a big difference between the low- and high-resolution models for scenarios A and B. As expected the values for concentration at the center of the pulse are higher for the more accurate model due to less numerical dispersion. The results for scenario D also show a higher concentration at the center of the pulse for the more-accurate model. We assume this is due to a large transverse dispersion coefficient leading to less dispersion, such that the width of the pulse becomes very small again. As such the pulse contains large concentration gradients, which the low-resolution model is unable to capture accurately; numerical dispersion due to large concentration gradients influences the results. We may note that the difference between the two models for scenario D is noticeable, but small when compared to scenarios A and B. The most-accurate result, based on Figure 1 and Figure 13, seems to be that of scenario C. The shape of the pulse and values for concentration seem to very closely match between the two models. The concentration gradients do not seem to be large enough such that the low-resolution model cannot capture the pulse accurately, and transverse dispersion seems to have mitigated numerical dispersion in the longitudinal direction, as was the main idea behind this thesis. 4

35 Figure 13 Concentration profiles taken in the center of the tube for the low-resolution model (red curve) and high-resolution model (dotted blue curve). The scenarios are equal to those shown in Figure 1 Besides comparing a higher-order model with the first-order model, it is interesting to compare the first-order model with analytical results. By comparing results, we obtain a longitudinal diffusion coefficient yielding the same spread of the initial pulse as was the result of the transverse diffusion coefficient in our D simulations. The analytical equation describing the spreading of a pulse over time in the longitudinal direction is given by (Crank 1975) x vt 1 C x, t exp Dt 4Dt L L (31) where the term determines the position of the center of the pulse in the x-direction. Although we do not yet know the value of D L that gives the same effect as the transverse diffusion coefficient in two-dimensional flow, all the other parameters are known and as shown previously in Table 1. As equation (31) is only onedimensional, we need results with approximately no change in concentration in the transverse direction to use the average concentration over the area flow. We attempt to fit equation (31) to the concentration profile at the center of the tube, as shown in Figure 13, for scenario C and D, as only these two scenarios are at, or close to, 5

36 the Taylor limit. Figure 14 shows the results where the best fit was obtained using D L = and D L = for scenarios C and D respectively. Although the shape of the curves for scenario C (left) are similar, the position on the x-axis is not the same. This is due to the fact that scenario C is not actually at the Taylor limit yet. From Figure 11 C3 we observed earlier that the concentration profile in the y-direction is not uniform. While we used the average velocity in equation (31), the velocity at the center of the tube is slightly higher and therefore the positions on the x-axis are different. As scenario D is in the Taylor limit, the variation in the y- direction is approximately zero and the numerical result does show a good fit to the analytical result for both position and shape. Figure 14 Comparison between numerical concentration profiles from the center of the tube (see also Figure 13) using parameters as shown in Table 1 and one-dimensional analytical results obtained using equation (31). The value for the longitudinal diffusion coefficient used for the analytical result is used as a fitting parameter. Taylor (1953) compared experimental results with an analytical formula to deduce the actual physical molecular diffusion coefficient D mol of his experiment. The formula he uses is very similar to equation (31) except for a factor of πr in the denominator to account for the tubular geometry of his experiments: 1 C R exp kt x vt 4kt (3) k in equation (3) is a diffusion coefficient defined as (Taylor 1953) Rv k (33) max 19D mol By fitting the curve described by equation (3) Taylor obtained the value for k of his experiment and consequently calculated the value for D mol. Fitting equation (3) to the results of scenario C and D yields the results shown Figure 15. The value we found to best fit the simulation result of scenario C is approximately k = yielding D mol = For scenario D the best fit is obtained for approximately k = yielding D mol =

37 Figure 15 - Comparison between average numerical concentration in the flow direction using parameters as shown in Table 1 and one-dimensional analytical results obtained using equation (3). The value for the longitudinal diffusion coefficient D mol (see equation (33)) is used as a fitting parameter and yields for scenario C: D mol = For scenario D the best result is obtained for D mol = Although we obtain a good fit in the Taylor limit, we have used trial-and-error to obtain the value of the longitudinal diffusion coefficient used in equation (31). Because we don t know this value beforehand, it is difficult to say whether the streamline model is actually accurate or not. However, we were able to show that by comparing the two-dimensional numerical results with the one-dimensional analytical result, it is possible to obtain a value for the longitudinal diffusion coefficient that results in the same concentration profiles. We will use this method in section 4.6 to compare different one-dimensional analytical results which include adsorption with the two-dimensional model. 4.4 Comparison with particle-tracking model So far we have modeled Taylor dispersion in a tube (section 4.1) and compared the results of our first-order accurate model with results obtained from a higher-order accurate model (section 4.3). By comparing our model with the higher-order model, we have established that for certain values for the diffusion coefficient the difference between the results is very small (see Figure 13). However, by just modeling Taylor dispersion in a tube and comparing the same model with different accuracies, it is not possibly to verify our model works properly. Therefore, we compare our model with a model developed by Berentsen et al. (007). As discussed in section.4, Berentsen et al. (007) developed a particle-tracking simulator to model the convection-diffusion equation. By comparing their simulation results with our simulation with equal parameters, we will verify the functioning of our simulator. The results obtained by Berentsen et al. (007) are from a unidirectional velocity field where tracer particles are uniformly distributed over the height and are subject to convective transport and micro scale mixing mechanisms, representing diffusion. The movement of the particles is described by an up-scaled model using cosine Fourier series and is fourth-order accurate. 7

38 Figure 16 Unidirectional velocity field used to compare simulators (Berentsen et al. 007) Figure 16 shows the unidirectional velocity field, representing a layer-cake model, used by Berentsen et al. (007) to run particle-tracking simulations and to compare their results with those obtained from a conventional two-dimensional simulator. Unlike our model so far, the value of the diffusion coefficient used by Berentsen et al. (007) is not a constant; rather it is a function of the velocity and is defined as D y D / F v y (34) T mol T where D mol represents the molecular diffusion, F the tortuosity and α T the transverse dispersivity. In this section we apply the same definition for the diffusion coefficient D T as shown in equation (34). Although Berentsen et al. (007) apply a similar function for the longitudinal diffusion coefficient, we continue to neglect this term; we assume longitudinal diffusion is negligible for the scenarios where we expect our model to be accurate. The values we use are equal to those of Berentsen et al. (007) and shown in Table 3. Table 3 Values for parameters used to compare our simulation results with the results of Berentsen et al. (007). All parameters are in SI units unless otherwise specified. Parameter Value L x 500 m N x 500 L y 5 m N y 5 N stream 100 v x 0.9, 0.1, 1.0, 0.3, 0.8 m/day T Δt D mol 10-6 m /day A: = 0, B: , C: m α T The boundary conditions are the same as in the previous scenarios: a pulse of tracer is injected during one time-step, as described by equation (0) in section 3.4. Figure 17 shows the simulation results of our model (EL - dashed light blue), the fourth-order accurate analytical model by Berentsen et al. (007)(Berentsen - red) and a conventional two-dimensional simulator described by Berentsen et al. (007)(D dark blue). The curves represent the dimensionless cumulative concentrations in the x-direction. From top to bottom the value for the transverse dispersivity α T increases as shown in Table 3. Scenario A represents results in the convective limit, where the effect of diffusion is negligible. For this scenario the results of our simulator resemble the results of the conventional simulator, more so than the results by Berentsen et al. (007). However, the red curve does not entirely overlap the results of the conventional simulator. This is due to differences in numerical dispersion due to differences in accuracy; our model is only first order accurate while the conventional two-dimensional simulator applies a 8

39 fourth-order accurate Runge-Kutta method. Also, while the conventional model uses a Cartesian grid, the streamline model covers the domain with streamlines to construct a new grid as described in section 3.3. This greatly increases grid size in slow moving layers (second and fourth layer), as there is less flow. On the other hand, the grid size of the fast flowing layers is greatly reduced. For scenario B, showing results between the convective- and Taylor limit, observations are similar. Our results are only slightly different from those obtained by the conventional simulator but represent results closer to the conventional simulator than the analytical results presented by Berentsen. The biggest differences occur for x > 350, which is where we expect numerical dispersion to be the largest as the position is further downstream; the pulses within the fast-moving layers have spread more due to numerical dispersion. 9

40 A B C Figure 17 Comparison between cumulative concentrations in the x-direction for different simulators: a conventional two-dimensional simulator of fourth order accuracy (solid blue), a fourth-order upscaled analytical model developed by Berentsen et al. (007) (solid yellow) and our first-order accurate streamline model (dashed red). The underlying velocity field is shown in Figure 16. Three different scenarios are shown for dispersion (see equation (34) for the definition of the transverse diffusion coefficient); A: D mol = 10-6, α T = 10-6, B: D mol = 10-6, α T = and C: D mol = 10-6, α T =

41 Berentsen et al. (007) conclude that scenario C is in the Taylor limit, although a ratio of 10:1 is not yet obtained when applying the parameters to equation (7). If we assume R equals approximately half the height of the domain the first ratio is approximately 5.:1 and the second ratio approximately 4.5:1. However, we are now inusing a rectilinear geometry rather than a cylindrical geometry. Even though the ratios are not high enough yet and we have neglected longitudinal diffusion, the figure shows that all three models have the same approximate result. The S-shaped curve that is obtained is characteristic for low at the Taylor limit (Berentsen et al. 007). As the results of our simulator are qualitatively similar to the conventional two-dimensional simulator, and quantitatively similar at the Taylor limit similar to both the up-scaled fourth-order model by Berentsen et al. (007) and the conventional simulator, we conclude our model functions properly and as hypothesized. Another particle-tracking simulator is described by John et al. (010). Although John s model is three dimensional, the results they show describing the variance of the x-position of the particles with respect to the mean particle position, are expected to be similar to the results we obtain using the same parameters. The mean velocity of particles in the z-direction equals zero and should not affect the x-direction variance. The dimensions of John s model are m with a cell size of 1 m. The base-case, the case we compare our results with, uses an uncorrelated permeability field with a mean of 1700 md and a variance (natural logarithm of permeability) of 1D and no anisotropy (K z = K x) (John et al. 010). Ten thousand particles are used for transport simulations, injected uniformly over one boundary. The pressure boundary conditions are set such that the mean velocity equals 0.7 m/d. To model the normalized relative variance of the particles, we use: Nx, n 1 x i i C i1 n C x x (35) where is the mean position of the particles at timestep n and C i is the cumulative concentration in the y- direction at position x i. To compare our results with those obtained by John et al. (010), we use equation (34) to define the diffusion coefficient and apply our first-order accurate model. The parameter values are equal to the values used by John for two scenarios: Scenario A describes the particle position with no local mixing (D mol = 0 m /d, α t = 0 m) and for scenario B local mixing is included (D mol = 10-4 m /d, α T = m). Scenario B can be considered to be an intermediate scenario; the Taylor limit is not yet reached. We also add a scenario C which is in the Taylor limit (D mol = 10-4 m /d, α T = 0.01 m). The results are shown in Figure

42 Figure 18 Variance of the particle distribution in the x-direction with respect to the mean particle position. Flow is reversed after 300 days. Input parameters are equal to those described by John et al. (010) for scenario A with no diffusion (D mol = 0 m /d, α t = 0 m) and B with diffusion (D mol = 10-4 m /d, α T = m). Scenario C is added to show the variance at the Taylor limit (D mol = 10-4 m /d, α T = 0.01 m). We observe that the results we obtain are different from the results as described by John et al. (010). Not only is the quantitative value different, qualitatively the curves do not show the same behavior (John et al. 010). The variance is a result of the spreading of particles within the pulse. As observed earlier, increasing the transverse diffusion coefficient yields a smaller pulse width (Figure 10). Lake (1986) describes spreading of the mixing zone L, the distance between the points where the cumulative dimensionless concentration is 0.1 and 0.9, to be proportional to time in the convective limit and proportional to the square root of time at the Taylor limit. Figure 19 shows the growth of the mixing zone as function of time for the same parameter values as described for Figure 18. The behavior shown in Figure 18 is as described by Lake (1986) except for very small times, where the central limit theorem has not yet kicked in. Figure 19 Growth of the mixing zone L, the distance between the points where the cumulative dimensionless concentration is 0.1 and 0.9, over time. The scenarios are equal to the scenarios of Figure 18. The mixing zone grows proportional to time for the convective scenario and proportional to the square root of time at the Taylor limit. As the variance is a quadratic relation with regards to the spreading zone (see equation (35)), we expect the variance to grow quadratically for the convective scenario and linearly at the Taylor limit. We observe this 3

43 behavior in Figure 18. It is unclear why the variance results we obtain are different from those described by John et al. (010). Differences could occur due to the difference in dimensions of the models (three-dimensional versus two-dimensional) or the resolution of the model. We may also note the difference in variance after the flow reversal test for the convective scenario. At time t = 600 the variance we observe has not reduced back to the initial variance of zero. This is due to numerical dispersion as described in section. and Comparison with Virtual Asset In the previous sections we have applied our model to Taylor dispersion in a tube (section 4.1) and a layer cake model (section 4.4). In both cases we obtained results as we had expected. However, in both cases a Cartesian grid also lines up with the flow direction, which is unidirectional. Although numerical dispersion occurs even in these cases, the worst case scenario would be when the flow direction is diagonal to the grid as discussed in section.. To show the potential of a streamline simulator we will apply the model to a more realistic, although still two-dimensional, permeability field which was developed by Overbeek et al. (004). This permeability field is called the virtual asset in their study. We compare our results based on streamline simulations with results obtained from a conventional simulator which applies a Cartesian grid and uses the same first-order upwind scheme to model convection and second-order central-difference scheme to model diffusion. The permeability field is shown in Figure 0, left. The high permeability channels (red) are representative for a meandering or braided river system in a low-permeable matrix. Due to this permeability heterogeneity, flow directions are no longer unidirectional. Rather, flow patterns resemble the permeability heterogeneity. Figure 0, right, shows the resulting streamlines when we apply our simulator to the permeability field, using 50 streamlines. Due to the permeability heterogeneity, which varies from approximately 00mD within the matrix to approximately 10D within the highly permeable channels, about much of the flow goes through the highpermeability zones. As a result, most of the streamlines go through the high-permeability zones while the streamline density of the slow areas is poor. Figure 0 The left figure shows the permeability field of the virtual asset (Overbeek et al. 004). The permeability ranges from approximately 0.0D in the matrix up to 10D in the high-permeability zones. The right figure shows the resulting streamlines when the number of streamlines is set equal to 50. As in the previous sections, we apply three different scenarios for the transverse diffusion coefficient D T. The values for the parameters that we use in this section are shown in Table 4 below. 33

44 Table 4 - Values for parameters used for simulation of flow through the virtual asset shown in Figure 0 for both the streamline model and conventional model. All parameters are in SI units unless otherwise specified. Parameter Value L x 110 m N x 110 L y 110 m N y 110 N stream 50 T 50,000 s Δt 0.1 s D T A: = 0, B: 10 -, C: 10-1 m /s To better represent the results when applying the streamline model to the virtual asset, we change the boundary conditions. Rather than injecting a pulse, we continuously inject tracer particles at the inlet with unit concentration C( x0, y, t) 1 (36) When we simulate the convection-diffusion equation at the convective limit, where diffusion is negligible, exact results would consist of concentrations equal to unity and zero only; only through numerical dispersion or diffusion do we see smearing of the shock front. This scenario is represented as scenario A in Figure 1. The left column shows results obtained with our streamline model while the right column shows results obtained from a conventional simulator. Even though we use only 50 streamlines, the results of the streamline model seem to be less smeared than the results of the conventional simulator. The numerical dispersion of the streamline model is greatly reduced compared to the conventional model. As expected the conventional simulator shows a lot of smearing when the flow direction is diagonal to the Cartesian grid. On the other hand, the streamline model loses accuracy in the slow flow areas. Detail of small permeability variations within the slowly flowing zones is lost because of the low streamline density. The second and third scenario, scenario s B and C, shows results that include some diffusion but has not reached the Taylor limit yet. The results of the streamline- and conventional model are now very similar and it is hard to say which model is more accurate. A small difference may be noted for scenario B as the spread in the high-permeability zones is less. As we have seen in section 4.1, the variance in particle distribution and effect of numerical dispersion is reduced when the transverse diffusion coefficient is increased. Scenario D does approach the Taylor limit and the differences between the two models are even less. Although the difference between the two models for scenario D is hardly observable, this is not a realistic representation of true in-situ mixing within a reservoir. It is very unlikely for a particle to travel from the bottom of a reservoir to the top of the reservoir in a time-frame where changes due to convection do not occur. Even on the time scale of a particle traveling from a producer to injector, a particle may not travel from the bottom of a reservoir to the top as a result of molecular diffusion. More likely is a scenario similar to the ones described by A and B. These are scenarios where the streamline model shows more accurate results than the conventional simulator and the streamline model is preferred. 34

45 A B C D Figure 1 Comparison between results obtained from our streamline simulator (left) and a conventional simulator of similar accuracy (right) when applied to the permeability field shown in Figure 0. Three different scenarios are shown; A: D T = 0, B: D T = 10-4, C: D T = 10 -, D: D T =

46 Although the streamline simulator shows more-accurate results than the conventional simulator for the convective scenario, there still is significant numerical smearing along every streamline. As we assume numerical smearing to be mitigated when diffusion is included, especially towards the Taylor limit, numerical dispersion is expected when transverse diffusion is excluded. To better show the difference between the two models, Figure shows an enlargement of scenario A of the area where numerical dispersion seems largest (40 < x < 85, 60 < y < 95). Although there is still significant numerical dispersion along a streamline, the result of the streamline model is significantly better than the conventional model. Figure Enlargement of the results shown in Figure 1 scenario for 40 < x < 85, 60 < y < 95. Numerical dispersion is reduced significantly by using a streamline simulator. To further improve the streamline model, the accuracy may be increased. By using the second-order model described earlier in section 4.3, we obtain the results shown in Figure 3. The numerical dispersion is now greatly reduced compared to both results shown in Figure. Figure 3 Results using the second-order accurate model described in section 4.3 applied to the permeability field shown in Figure 1. (LEFT) Full-scale results. (RIGHT) An enlargement of 40 < x < 85, 60 < y < 95. The parameters used are equal to those shown in Table 4. Although we have not actually quantified the numerical dispersion of the different models, Figures 1, and 3 show the potential of using streamline based models to reduce numerical dispersion. Especially in the convective limit the difference between the accuracy of the models is easily distinguishable and quite significant. 36

47 4.6 Sorption of particles In this final section of this chapter we will apply the model to a scenario a particle-tracking model cannot easily simulate accurately: flow including Langmuir adsorption of particles. Particle-tracking models were briefly discussed in section.4 where we concluded that particle-tracking models cannot accurately model in-situ concentrations; they can only infer mixing after flow-reversal or from particle variance. The example we use includes adsorption of particles by the solid-phase of a porous medium as a function of flowing particle concentrations. The derivations of the formulae we use have been accurately described by Lake et al. (00). As there are now two particle concentrations to model, one within the flowing phase and the other adsorbed by the solid phase, a new mass balance is needed. In dimensionless form and including the adsorbed concentration, the mass-balance equation (1) is modified to C C C C t t x y D D D D DT 0 D D D D (37) where again the longitudinal diffusion is neglected. The interaction between the solid and flowing concentration is represented by the non-linear Langmuir isotherm and is a function of the local in-situ concentration, defined as (Lake et al. 00) C eq Cmax (38) 1 bc bc eq In this equation is the maximum concentration of particles that are adsorbed by the solid phase, C eq a concentration that is in equilibrium with the solid phase and b a parameter that represents how strongly a component is adsorbed by, or attracted to, the solid phase. When b is large, only a small equilibrium concentration is needed for the solid phase to reach the maximum adsorbed concentration. We assume the flowing phase instantly reaches a local equilibrium with the adsorbed phase such that C eq = C. This assumption simplifies equation (38) and allows us to replace the solid-phase concentration time derivative of equation (37) with a term that is dependent on the flowing concentration only (Lake et al. 00) t C t t C C C bc max C D D 1 bc D (39) Substituting equation (39) into equation (37) for the adsorbed concentration derivative yields bc max C C C 1 0 v DT 1 bc td xd yd (40) We will compare the results of our model using equation (40) with an analytical one-dimensional result. To obtain the analytical result we will use the method of characteristics together with an equation called the traveling-wave solution describing the concentrations around the shock front. As time increases, we assume the shock front reaches an equilibrium with the dispersive and adsorption properties and will travel downstream without change of profile (Lake et al. 00). The traveling wave can then be defined as function of a new position zd xd utd (41) where u is a specific constant velocity the coordinate of the traveling wave, z D, flows downstream with as a function of the adsorption parameters. The definition of this velocity can be obtained using a material balance 37

48 and definitions for concentrations upstream and downstream of the shock. The derivation is shown by Lake et al. (00) and yields u 1 1 (4) bc max By substituting equation (41) into equation (37) and applying the definition of differential change we obtain a mass-balance for the traveling in the new coordinate system z D (Lake et al. 00) 1 b C C C C 1 C 1 u u 0 t t z z N z D D D D Pe D (43) Because we assumed the shape of the traveling wave does not change after a sufficient amount of time has passed, such that equilibrium has been reached, the time derivatives of equation (43) are equal to zero. After substitution of equation (4) into equation (43) and applying an integration step, the traveling wave solution is obtained which is defined as (Lake et al. 00) C NPeC maxb exp z 1 b D C 1bC max 1 (44) The traveling wave equation (44) now describes the concentration as a function of the adsorption parameters, diffusion parameter (N Pe is the inverse of the diffusion coefficient) and dimensionless distance z D. Equation (44) is a one-dimensional equation to describe the concentration profile of the traveling wave centered around z D = 0. The solution to equation (44) can be obtained numerically and, by use of equation (41), the traveling wave coordinates system z D may be transformed to the original coordinate system x D. As was shown in section 4.3, it is possible to obtain an effective longitudinal diffusion coefficient by comparing a concentration profile from the two-dimensional results with a one-dimensional analytical result. Although we used equation (31) for a pulse, we will now use a boundary condition for continuous injection on one side of the domain. The analytical equation describing the concentration profile is then given by equation (3). Table 5 Parameters used to model adsorption. All parameters are dimensionless Parameter Value x D 1 Δx D 10-3 y D 1 Δy D 0.0 N stream 50 Δt D 0.75 Δt D T D L 0 b 1 C max 1 To be able to compare our model including adsorption with the analytical equation, we follow three steps. First we will run a two dimensional model to simulate convection-diffusion in a tube in the Taylor limit without adsorption. The parameters used are shown in Table 5 above. Secondly, we fit the analytical equation (3) to the 38

49 concentration profile obtained of the streamline at the center of the tube to obtain the Péclet number that will result in the same spread of the front (see also section 4.3). As the two-dimensional simulation is run in the Taylor limit, the concentration in the transverse direction is approximately uniform such that all streamlines show approximately identical concentration profiles and all will be equal to the analytical result. The result of these first two steps, running the two-dimensional simulator and fitting the analytical solution, is shown in Figure 4. The Péclet number yielding the best fit equals 30. Figure 4 (LEFT) Comparison between numerical results for dispersion in a tube in the Taylor limit (blue) with the analytical result from equation (3) (red). The Péclet number is used to fit the shape of the front and yields N Pe = (RIGHT) Close-up of the front. Finally, we run a two-dimensional simulation including adsorption as described by equation (40) with the parameters as shown in Table 5 and the Péclet number obtained from the first two steps (see Figure 4). The results are then compared with the results of the one-dimensional analytical equation (44), where we use equation (41) to transform the domain of the analytical equation back to the x D coordinate system. The results including adsorption are shown in Figure 5. Figure 5 (LEFT) Comparison between the numerical results (blue) including adsorption as described by equation (40), with the analytical result from equation (44). The parameters used are as shown in Table 5 with the Péclet number equal to the Péclet number obtained from the fit shown in Figure 4 (N Pe = 1000). (RIGHT) A close-up of the front. Although there are very small differences between the numerical- and analytical results shown in Figure 5, the position and the shape of the front are almost identical. The purpose of this example was to show that our streamline model is able to accurately model flow that is dependent on in-situ concentrations (see equation (40)), a scenario particle-tracking methods cannot easily model (see section.4). As the simulation results 39

50 shown in Figure 5 are very similar, we conclude the streamline model is indeed able to capture flow dependent on local concentrations. 40

51 5 Conclusion 5.1 Conclusions Accurately modeling in-situ mixing is important for various EOR processes. Recent studies have focused on particle-tracking methods as they are free of numerical dispersion. However, particle-tracking methods are unable to measure in-situ concentrations and can only infer mixing through flow-reversal tests. We have focused on developing a first-order accurate two-dimensional streamline-based model in MATLAB to numerically solve the convection-diffusion equation. As a streamline model lines up with local flow directions, the main numerical dispersion occurs in the longitudinal direction. However, Taylor (1953) described that diffusion in the longitudinal direction, which has the same effect as longitudinal numerical dispersion, is negligible under certain conditions. We have focused on modeling Taylor dispersion, compared the results with a higher-order streamline model and analytical results and have attempted to show the potential of streamline based simulators with regards to numerical dispersion. A summary of the main findings obtained from this thesis is given below. We have studied the model by simulating Taylor dispersion in a tube and established that the firstorder accurate model shows a lot of numerical dispersion in the convective limit. However, by comparing the results to a higher-order model and to a particle-tracking simulator, we have shown that the effect of numerical dispersion diminishes towards the Taylor limit; the larger the diffusion coefficient, the smaller the effect of longitudinal dispersion The model is compared to a conventional simulator of the same accuracy for a more realistic twodimensional heterogeneous permeability field, called the Virtual Asset (Overbeek et al. 004). Although we have established that numerical dispersion in the streamline model is still significant in the convective limit, by aligning the grid to the flow direction numerical dispersion for this scenario is still significantly less than for a conventional simulator. Finally, the streamline model is used to simulate adsorption of particles by the solid phase. Flow then depends on local concentrations, a scenario particle-tracking models cannot easily simulate. The results in the Taylor limit were compared with an analytical solution and were found to match very closely, showing that the two-dimensional simulator can accurately simulate flow depending on local concentrations. 5. Recommendations Recommendations for future studies are: We were unable to reproduce the results of the change of the variance in the x-direction of particles over time as shown by John et al. (010). Although our results match the prediction of Lake (1986), the possible cause of the difference is still unclear but may include: the difference in dimensions and the resolution of the model. This should be studied further to better understand the underlying principles. The two-dimensional model can readily be applied in three dimensions. However, the effect of complex geometries on streamlines and transverse diffusion should be evaluated as streamtube geometries may not easily be determined. 41

52 4

53 6 Nomenclature Symbol Subscript A velocity gradient 1, cell face indices b adsorption parameter D dimensionless C concentration e exit D dispersion coefficient entry entry (injector) J flux eq equilibrium L length of core / mixing zone exit exit (producer) N number of streamlines / grid cells i,j cell indices n normal vector L longitudinal N Pe Péclet number max maximum P pressure mol molecular P pressure n timestep q flow rate p particle r radial coordinate or ratio r radial R radius of a tube stream streamline s distance along a streamline / T,τ transverse streamline seed S surface x,y directions t time T number of time steps v velocity V volume x,y coordinates Γ interface λ mobility ρ density ς variance τ time-of-flight φ porosity ϕ flux limiter ψ stream function 43

54 44

55 Appendix A Velocity field model In order to generate a velocity field we solve the mass-conservation law for flow of a single phase through a porous medium (Bird et al. 007): P q t (45) where is the fluid density, is the porosity, λ the mobility and q a source term. We assume incompressibility of the carrier fluid and porous medium and no source term (q = 0) which reduces equation (45) to P 0 (46) As equation (46) is elliptic it is easily solvable with the correct boundary conditions. Applying a finite volume method to discretize equation (46) in two dimensions, for any given cell not at a boundary, yields i1, j i1, j i, j1 i, j1 n Ax n A n Ay n A x y ( Pi, j Pi 1, j ) ( Pi, j Pi 1, j ) ( Pi, j Pi, j1 ) ( Pi, j Pi, j1 ) qi, j x x y y (47) Equation (47) is applied to a Cartesian grid where the mobilities are defined at the grid-block interfaces, applying harmonic averaging. The boundary conditions we apply are P( x 0, y 1... N ) P P( x N, y 1... N ) P q( x 1... N, y 0) 0 q( x 1... N, y N ) 0 y entry y y exit x x After solving the system of equations for pressure we calculate the velocity v at all interior grid interfaces using y v v i1 i1 j1 j1 Pi 1 Pi x P P j1 y j (48) Appendix B Streamline seeds How well streamlines cover a heterogeneous reservoir and the accuracy of the results is very dependent on defining the initial coordinates ψ(x 0,y 0) of every streamline, i.e. the streamline seeds. In our model we inject tracer with unit concentration over the entire inlet boundary. The y-direction coordinates then need to be spaced such that they are in accordance with streamline definitions: the higher the velocity, the smaller the distance between streamlines such that the flow through every streamtube is equal. The distance between consequent streamlines has to reflect the inverse of the local flux. As we are injecting over one boundary and we set the gridlines, including the boundary, in the y-direction to be strictly vertical (see Figure 6) everywhere, only the inverse flux in the x-direction has to be considered. The flux through every grid interface j in the y-direction at x = 0 can be defined as q v y (49) j x, j 45

56 By dividing the total flux at the inlet by the number of streamlines, N stream, we determine the flow of every streamline s q s q j (50) N stream Using equation (49) and (50) we can easily determine the position of every streamline seed. Appendix C Convection-diffusion model In order to numerically model the convection-diffusion equation (17) it has to be discretized. We apply a finite volume method to do so as it is by definition mass-conservative. The convection-diffusion equation (17) can be written in vector notation as C t vc D C 0 (51) Integrating equation (51) with respect to a control volume V gives t Applying Gauss theorem to equation (5) yields CdV v CdV ( D C) dv 0 (5) V V V t dc CdV v C nds D ds dn V S S (53) where S is the surface of the control volume and the unit normal vector to this surface pointing outwards. Equation (53) is suitable to be used over any control volume and is easily applied to a Cartesian grid. When doing so we have to keep in mind the transformation of the grid from an (x,y)-domain to an (x,ψ)-domain (see section 3.4). We apply equation (53) in the (x,ψ)-domain but use the geometric information stored on the streamtubes (see chapter 3 and Figure 6 below); the fluxes obtained from equation (53) have to be appropriately scaled with respect to the original geometry when applying the convection-diffusion equation to the transformed grid. To calculate diffusive fluxes on the (x,ψ)-domain we adjust the values for the transverse dispersion coefficient to reflect the variation in streamline density or distances between streamlines. As the diffusive flux over a boundary is scaled directly by the value of the transverse dispersion coefficient, and the flux scales linearly with distance, it is possible to redefine the dispersion coefficient to include the changing streamline density; wherever streamline densities are high, the dispersion coefficient is large and where streamline density is low, the dispersion coefficient is small. Thus the dispersion coefficient that was constant on the (x,y)-domain now varies in space. This can be defined as: D,, i j y DT y i, j i, j1 (54) D τ,i,j in equation (54) represents the scaled transverse dispersion coefficient to be used in the convectiondiffusion equation on the (x,ψ)-domain. 46

57 Figure 6 Example grid cell (i,j) on the corner-point grid ((x,y)-domain), from where the geometry information is stored. The black dots represent cell nodes while the dotted lines represent cell interfaces. C is the concentration of a grid cell, x and y the coordinates of a cell node, the velocity at a boundary, V the volume of a cell, an example cell face and i and j the cell indices of cell (i,j). We apply the definition of a streamline where convection occurs in the longitudinal direction only; cross-flow occurs only through diffusion. For the convective part of the equation we apply a first-order upwind scheme in space to determine the concentration at the cell interface. Higher-order methods are often oscillatory without applying additional constraints (LeVeque 199). Also, we attempt to show that truncation errors in the longitudinal direction have negligible effect on simulation results. As flow is in the positive -direction for our example of Taylor dispersion, an upwind scheme is appropriate. Discretizing equation (53) over all cell interfaces Γ, using an explicit method in time, yields in a simplified notation C C V J J J J t n1 n i, j i, j n n n n i, j 1, 1, 1, 1,, 1, 1 i j i j i j i j i j i j i, j 1 i, j 1 where J is the flux over a cell interface. We define the volume V of cell i,j, obtained from the original domain, as Vi, j (55) x y (56) The fluxes in equation (55) are separated over each individual cell face and may consist of a diffusion and convective part. In the longitudinal direction, the flux J over boundary i 1 for a cell (i,j) on an irregular grid, j (see Figure 6), can be defined as n n 1 1 1,,, i j i j i j We have now used the scaled value for the transverse dispersion coefficient to include the distance consecutive streamlines, as defined in equation (54). Substituting equation (57) and (58) into equation (55) for all cell interfaces, and rewriting for, yields 47 J C v n (57) where we apply Pythagoras theorem to define the distance between two grid nodes. In the transverse direction there is only a diffusion term to consider. Because of how we structure the grid, the gridlines are strictly vertical and the flux over an interface in the transverse direction, such as 1, can be defined as n n n i, j 1, i, j1 i, j1 i, j i, j J D C C (58)