## Publications

142 results found

Zhou Y, Muggeridge AH, Berg CF,
et al., 2019, Effect of Layering on Incremental Oil Recovery From Tertiary Polymer Flooding, *SPE RESERVOIR EVALUATION & ENGINEERING*, Vol: 22, Pages: 941-951, ISSN: 1094-6470

Hiller T, Ardevol-Murison J, Muggeridge AH,
et al., The impact of wetting heterogeneity distribution on capillary pressure and macroscopic measures of wettability, *SPE Journal*, ISSN: 1930-0220

This work investigates how the different length scales of pore scale wetting heterogeneities affect the shape of capillary pressure-saturation (CPS) curves and the derived USBM and Amott-Harvey wettability indices. These macroscopic wettability indices are used to describe bulk rock wettability as the local contact angle (the standard physical measure of wettability) in a sample which is difficult to access andmay vary within and between pores due to changes in mineralogy and the surface coverage of organic materials. Our study combines laboratory experiments and full-scale fluid dynamics simulations employing the multi-phase Stochastic Rotation Dynamics (SRDmc) model. Four model systems were created using monodisperse glass beads. The surface properties of the beads were modified so that half of the surface area in each system was strongly hydrophilic and half was hydrophobic but each system had a different length scaleof wetting heterogeneity, ranging from a fraction of the bead diameter to two bead diameters. There is excellent agreement between the experimental and simulation results. All systems are classified as intermediate wet based on their Amott-Harvey and USBM indices. Examination of the capillary pressure curves shows that the opening of the stable hysteresis loop decreases monotonically as the length scale of the wetting heterogeneities is increased. Hence, our results suggest that macroscopic wettability indices may be used as indicators of ultimate recovery, but are not suited to discriminate between the different flows that occur earlier on in a mixed wettability displacement process.

Abdul Hamid SA, Adam A, Jackson MD,
et al., 2019, Impact of truncation error and numerical scheme on the simulation of the early time growth of viscous fingering, *International Journal for Numerical Methods in Fluids*, Vol: 89, Pages: 1-15, ISSN: 0271-2091

The truncation error associated with different numerical schemes (first order finite volume, second order finite difference, control volume finite element) and meshes (fixed Cartesian, fixed structured triangular, fixed unstructured triangular and dynamically adapting unstructured triangular) is quantified in terms of apparent longitudinal and transverse diffusivity in tracer displacements and in terms of the early time growth rate of immiscible viscous fingers. The change in apparent numerical longitudinal diffusivity with element size agrees well with the predictions of Taylor series analysis of truncation error but the apparent, numerical transverse diffusivity is much lower than the longitudinal diffusivity in all cases. Truncation error reduces the growth rate of immiscible viscous fingers for wavenumbers greater than 1 in all cases but does not affect the growth rate of higher wavenumber fingers as much as would be seen if capillary pressure were present. The dynamically adapting mesh in the control volume finite element model gave similar levels of truncation error to much more computationally intensive fine resolution fixed meshes, confirming that these approaches have the potential to significantly reduce the computational effort required to model viscous fingering.

Kostorz W, Muggeridge A, Jackson M, et al., 2019, Non-intrusive reduced order modelling for reconstruction of saturation distributions

Copyright 2019, Society of Petroleum Engineers. Two new Non-Intrusive Reduced Order Modelling approaches to estimate time varying, spatial distributions of variables from arbitrary unseen inputs are introduced. One is a generalization of an existing ‘dynamic’ approach which requires multiple surrogate evaluations to model the solutions at different time instances, the other is a ‘steady-state’ approach that evaluates all time instances simultaneously, reducing the local approximation error. The ability of these approaches to estimate the water saturation distributions expected during a gas flood through a 2D, dipping reservoir is investigating for a range of unseen input parameters. The range of these parameters has been chosen so that a range of flow regimes will occur, from a gravity tongue to a viscous dominated Buckley-Leverett displacement. A number of practically relevant model error measures were employed as opposed to the standard L2 (Euclidean) norm. The influence of the number and the structure of training simulations for the model was also investigated, by employing two simple experimental design methods. The results show that POD based NIROM approaches are prone to significant deviations from the true model. The main sources of error are due to the non-smooth variation of system responses in hyperspace and the transient nature of the flows as well as the underlying dimensionality reduction. Since the first two sources are properties of the physical system modelled it may be expected that similar problems are likely to arise independently of the interpolation method and the reduction process used.

Alem M, Baig T, Muggeridge A, et al., 2019, Predicting the performance of tight gas reservoirs

Copyright 2019, Society of Petroleum Engineers. Engineers need to predict the production characteristics from hydraulically fractured wells in tight gas fields. Decline curve analysis (DCA) has been widely used over many years in conventional oil and gas fields. It is often applied to tight gas, but there is uncertainty regarding the period of production data needed for accurate prediction. In this paper decline curve analysis of simulated production data from models of hydraulically fractured wells is used to to develop improved methods for calibrating decline curve parameters from production data. The well models were constructed using data from the Khazzan field in Oman. The impact of layering, permeability and drainage area on well performance is also investigated. The contribution of each layer to recovery and the mechanisms controlling that contribution is explored. The investigation shows that increasing the amount of production data used to fit a hyperbolic decline curve does not improve predictions of recovery unless that data comes from many years (20 years for a 1mD reservoir) of production. This is because there is a long period of transient flow in tight gas reservoirs that biases the fitting and results in incorrect predictions of late time performance. Better predictions can be made by estimating the time at which boundary dominated flow is first observed (tb), omitting the preceding transient data and fitting the decline curve to a shorter interval of data starting at tb. For single layer cases, tb can be estimated analytically using the permeability, porosity, compressibility and length scale of the drainage volume associated with the well. Alternatively, tb can be determined from the production data allowing improved prediction of performance from 2-layer reservoirs provided that a) there is high crossflow or b) there is no cross-flow and the lower permeability layer either does not experience BDF during the field life time or it is established qui

Tai I, Muggeridge A, 2019, Evaluation of empirical models for viscous fingering in miscible displacement

© 2019 European Association of Geoscientists and Engineers, EAGE. All Rights Reserved. The performance of miscible gas injection projects can be significantly affected by viscous fingering. This is further complicated by the presence of heterogeneities, as depending on the scale of the heterogeneity, there can be a diffusive, advective or channelling effect. To assess the economic feasibility of a miscible gas injection project, reservoir simulations are needed but very fine grids are required for the fingers to be modelled explicitly. This requires a large amount of computational power and time. To get around this issue, many empirical models have been proposed which model the average behaviour of the viscous fingers, allowing predictions of performance, thus reducing grid size and computational time. Many previous studies have investigated the ability of empirical models to represent fingering in line drives but none have considered flow in a quarter five spot pattern. In this study, a two phase, three component higher-order simulator is used to simulate miscible injection in square line drive and quarter five spot models, with and without heterogeneities. The results of the detailed fingering simulations were compared to the Todd & Longstaff and Fayers empirical models. To account for the effect of heterogeneities, the mixing parameter, w, in the Todd & Longstaff was adjusted using Koval's heterogeneity factor, H_k. The growth rate of the fingers, α, and the final fraction of the cross section occupied by the fingers, a+b, were adjusted in the Fayers model to account for heterogeneities and bypassed oil. The empirical models were implemented in a commercial immiscible reservoir simulator, Eclipse-100 using pseudo relative permeabilities. The detailed simulations indicate that the growth rate of the fingers varies non-linearly with mean concentration in radial flows and this is not captured by either of the empirical models. A modification of th

Kampitsis A, Salinas P, Pain C, et al., 2019, Mesh adaptivity and parallel computing for 3D simulation of immiscible viscous fingering

© 2019 European Association of Geoscientists and Engineers, EAGE. All Rights Reserved. We present the recently developed Double Control Volume Finite Element Method (DCVFEM) in combination with dynamic mesh adaptivity in parallel computing to simulate immiscible viscous fingering in two- and three-dimensions. Immiscible viscous fingering may occur during the waterflooding of oil reservoirs, resulting in early breakthrough and poor areal sweep. Similarly to miscible fingering it is triggered by small-scale permeability heterogeneity while it is controlled by the mobility ratio of the fluid and the level of transverse dispersion / capillary pressure. Up to this day, most viscous fingering studies have focussed on the miscible problem since immiscible fingering is significantly more challenging. It requires numerical simulations capable to capture the interaction of the shock front with the capillary pressure, which is a saturation dependent dispersion term. That leads to models with very fine mesh in order to minimise numerical diffusion, resulting in computationally intensive simulations. In this study, we apply the dynamic mesh adaptive DCVFEM in parallel computing to simulate immiscible viscous fingering with capillary pressure. Parallelisation is achieved by using the MPI libraries. Dynamic mesh adaptivity is achieved by mapping of data between meshes. The governing multiphase flow equations are discretised using double control volumes on tetrahedral finite elements. The discontinuous representation for pressure and velocity allows the use of small control volumes, yielding higher resolution of the saturation field. We demonstrate convergence of fingers using our parallel numerical method in 2d and 3d, on fixed and adaptive meshes, quantifying the speed-up due to parallelisation and mesh adaptivity and the achieved accuracy. Dynamic mesh adaptivity allows resolution to be automatically employed where it is required to resolve the fingers with lower resolution

Lei Q, Xie Z, Pavlidis D,
et al., 2018, The shape and motion of gas bubbles in a liquid flowing through a thin annulus, *Journal of Fluid Mechanics*, Vol: 285, Pages: 1017-1039, ISSN: 0022-1120

We study the shape and motion of gas bubbles in a liquid flowing through a horizontal or slightly inclined thin annulus. Experimental data show that in the horizontal annulus, bubbles develop a unique ‘tadpole-like’ shape with a semi-circular cap and a highly stretched tail. As the annulus is inclined, the bubble tail tends to vanish, resulting in a significant decrease of bubble length. To model the bubble evolution, the thin annulus is conceptualised as a ‘Hele-Shaw’ cell in a curvilinear space. The three-dimensional flow within the cell is represented by a gap-averaged, two-dimensional model, which achieved a close match to the experimental data. The numerical model is further used to investigate the effects of gap thickness and pipe diameter on the bubble behaviour. The mechanism for the semi-circular cap formation is interpreted based on an analogous irrotational flow field around a circular cylinder, based on which a theoretical solution to the bubble velocity is derived. The bubble motion and cap geometry is mainly controlled by the gravitational component perpendicular to the flow direction. The bubble elongation in the horizontal annulus is caused by the buoyancy that moves the bubble to the top of the annulus. However, as the annulus is inclined, the gravitational component parallel to the flow direction becomes important, causing bubble separation at the tail and reduction in bubble length.

Gago PA, King P, Muggeridge A, 2018, Fractal growth model for estimating breakthrough time and sweep efficiency when waterflooding geologically heterogeneous rocks, *Physical Review Applied*, Vol: 10, ISSN: 2331-7019

We describe a fast method for estimating flow through a porous medium with a heterogeneous permeability distribution. The main application is to contaminant transport in aquifers and recovery of oil by waterflooding, where such geological heterogeneities can result in regions of bypassed contaminants or oil. The extent of this bypassing is normally assessed by a numerical flow simulation that can take many hours of computer time. Ideally the impact of uncertainty in the geological description is then evaluated by the performing of many such simulations using different realizations of the permeability distribution. Obviously, a proper Monte Carlo evaluation may be impossible when the flow simulations are so computationally intensive. Consequently, methods from statistical mechanics, such as percolation theory and random walkers (such as diffusion-limited aggregation), have been proposed; however, these methods are limited to geological heterogeneities where the correlation lengths are smaller than the system size or to continuous permeability distributions. Here we describe a growth model that can be used to estimate the breakthrough time of the water (and hence the sweep efficiency) in most types of geologically heterogeneous rocks. We show how the model gives good estimates of the breakthrough time of water at the production well in a fraction of the time needed to perform a full flow simulation.

Attar A, Muggeridge A, 2018, Low salinity water injection: Impact of physical diffusion, an aquifer and geological heterogeneity on slug size, *JOURNAL OF PETROLEUM SCIENCE AND ENGINEERING*, Vol: 166, Pages: 1055-1070, ISSN: 0920-4105

Abdul Hamid S, Muggeridge AH, 2018, Analytical solution of polymer slug injection with viscous fingering, *Computational Geosciences*, Vol: 22, Pages: 711-723, ISSN: 1420-0597

We present an analytical solution to estimate the minimum polymer slug size needed to ensure that viscous fingering of chase water does not cause its breakdown during secondary oil recovery. Polymer flooding is typically used to improve oil recovery from more viscous oil reservoirs. The polymer is injected as a slug followed by chase water to reduce costs; however, the water is less viscous than the oil. This can result in miscible viscous fingering of the water into the polymer, breaking down the slug and reducing recovery. The solution assumes that the average effect of fingering can be represented by the empirical Todd and Longstaff model. The analytical calculation of minimum slug size is compared against numerical solutions using the Todd and Longstaff model as well as high resolution first contact miscible simulation of the fingering. The ability to rapidly determine the minimum polymer slug size is potentially very useful during enhanced oil recovery (EOR) screening studies.

Abdul Hamid SA, Muggeridge A, 2018, Viscous fingering in reservoirs with long aspect ratios, SPE Improved Oil Recovery Conference

© 2018, Society of Petroleum Engineers. This paper investigates the impact of aspect ratio on the growth rate of viscous fingers using high resolution numerical simulation in reservoirs with aspect ratios of up to 30:1. The behaviour of fingers in porous media with such high aspect ratios has been overlooked previously in many previous simulation studies due to limited computational power. Viscous fingering is likely to adversely affect the sweep obtained from any miscible gas injection project. It can also occur during polymer flooding when using chase water following the injection of a polymer slug. It depends upon the viscosity ratio, physical diffusion and dispersion, the geometry of the system and the permeability heterogeneity. It occurs because the interface between a lower viscosity displacing fluid and a higher viscosity displaced fluid is intrinsically unstable. This means that any small perturbation to the interface will cause fingers to grow. It is therefore almost impossible to predict the exact fingering pattern in any given displacement although many previous researchers have shown that it is possible predict average behaviour (such as gas breakthrough time and oil recovery) provided a very refined grid is used such that physical diffusion dominates over numerical diffusion. It is impossible to use such fine grids in field scale simulations. Instead engineers will tend to use standard empirical models such as the Todd and Longstaff or Koval models, calibrated to detailed simulations, to estimate field scale performance. At late times in high aspect ratio systems, we find that one finger dominates the displacement and that this finger grows with the square root of time, rather than linearly. We also observe that this single finger tends to split, during which time the solvent oil interface length grows linearly with time before one finger again dominates and grows with the square root of time. This cycle can repeat several times. We also find that

Muggeridge AH, Smalley PCC, Dalland M, et al., Screening for EOR and Estimating Potential Incremental Oil Recovery on the Norwegian Continental Shelf, SPE Improved Oil Recovery Conference

Muggeridge AH, Alshawaf M, Krevor S, Experimental Investigation of Cross-flow in Stratified Reservoirs During Polymer Flooding, SPE IMproved Oil Recovery Conference

Santo A, Muggeridge A, 2018, An investigation into the benefits of combined polymer-low salinity waterflooding

Copyright 2018, Society of Petroleum Engineers Polymer flooding and low salinity waterflooding are two different but potentially complementary Enhanced Oil Recovery (EOR) techniques. Polymer flooding improves fractional flow and sweep efficiency by improving the mobility ratio for the displacement. Low salinity waterflooding improves pore scale displacement efficiency by changing the wettability of the reservoir rocks toward more water-wet. Reduced salinity water is often used in polymer injection to reduce hydrolysis however the water salinity in this case is typically higher than that needed to obtain a true low salinity effect. This paper describes the outcomes of a systematic study into the potential benefits of combined polymer-low salinity waterflooding versus polymer-high salinity waterflooding, polymer-reduced salinity waterflooding and conventional waterflooding. Numerical simulation, validated against analytical solutions, was used to evaluate the relative performance of these processes. The impacts of layering and reservoir heterogeneity were investigated using two-dimensional (2D) and three-dimensional (3D) reservoir models. Sensitivity studies of injected water salinity and the start time of injection were carried out in each of these models. Outcomes were compared against the recoveries and water cuts predicted using a one-dimensional (1D) analytic solution for the EOR processes to evaluate the impact of sweep versus displacement efficiency on incremental oil recovery and water cut. Combined polymer-low salinity waterflooding shows an improvement in recovery and reduction in water cut compared with the other EOR processes in all cases. We show this is partly due to improving the fractional flow (increasing shock front saturation) but is also due to both the leading and trailing shock fronts in polymer-low salinity waterflooding being more stable than in the other EOR processes, reducing the possibility of viscous finger growth and thus increasing perfo

Al-Dhuwaihi A, King P, Muggeridge A, 2018, Upscaling for polymer flooding

Copyright 2018, Society of Petroleum Engineers. Polymer flooding is a proven EOR/IOR process for viscous and light oil reservoirs alike. However, it results in the formation of two shocks front that require simulation models with fine grid blocks to represent field scale fluid movement. Therefore, upscaling is required to transfer such fluid behavior to coarser models. However, most upscaling methods are designed for waterflood only, while upscaling techniques for polymer flood are rarely discussed in the literature. In this paper, A new upscaling methodology specifically designed for polymer flooding is presented to address such impracticality. The methodology allows the average flow behavior to be captured, including the effects of small scale heterogeneity whilst compensating for the impact of increased numerical diffusion present in coarse grid models. The method is based on the pore volume weighted method for relative permeability pseudoization first derived by Emanuel and Cook (1974) for waterflooding but extends its implementation to model polymer specific parameters such as adsorption isotherm and viscosity-concentration function. The method is demonstrated on a series of simple reservoir models for a range of different aggregation ratios, showing overall improvement in the prediction of oil recovery, water cut, produced polymer concentration with time, and pressure response in the coarse grid models. This is demonstrated by comparing the predictions from the coarse grid, upscaled models with those from fine grid simulations and coarse grid simulations of the same model reservoir without the new upscaling methodology.

Obidi O, Muggeridge AH, Vesovic V, 2017, Analytical solution for compositional profile driven by gravitational segregation and diffusion, *Physical Review E*, Vol: 95, ISSN: 2470-0045

An improved analytical solution is presented, based on irreversible thermodynamics, that describes the equilibrium distribution of the components of a non-ideal fluid mixture in an 1D, hydrostatic and isothermal system. In such a system, the vertical compositional profile of the fluid at equilibrium will be determined by the interaction of gravitational and chemical potentials. The new analytical solution estimates this profile from the overall composition of the fluid. It is thus more general than the existing solution which requires a knowledge of the fluid composition at a given depth and assumes that the vertical compositional profile of this fluid is already at equilibrium. The solution is demonstrated by comparison against results obtained from previously published molecular dynamics simulations of segregation in a binary mixture and against numerical simulations of a real hydrocarbon reservoir system.

Maes J, Muggeridge AH, Jackson MD,
et al., 2017, Scaling analysis of the In-Situ Upgrading of heavy oil and oil shale, *FUEL*, Vol: 195, Pages: 299-313, ISSN: 0016-2361

The In-Situ Upgrading (ISU) of heavy oil and oil shale is investigated. We develop a mathematical model for the process and identify the full set of dimensionless numbers describing the model. We demonstrate that for a model with nf fluid components (gas and oil), ns solid components and k chemical reactions, the model was represented by 9+k×(3+nf+ns-2)+8nf+2ns dimensionless numbers. We calculated a range of values for each dimensionless numbers from a literature study. Then, we perform a sensitivity analysis using Design of Experiments (DOE) and Response Surface Methodology (RSM) to identify the primary parameters controlling the production time and energy efficiency of the process. The Damköhler numbers, quantifying the ratio of chemical reaction rate to heat conduction rate for each reaction, are found to be the most important parameters of the study. They depend mostly on the activation energy of the reactions and of the heaters temperature. The reduced reaction enthalpies are also important parameters and should be evaluated accurately. We show that for the two test cases considered in this paper, the Damköhler numbers needed to be at least 10 for the process to be efficient. We demonstrate the existence of an optimal heater temperature for the process and obtain a correlation that can be used to estimate it using the minimum of the Damköhler numbers of all reactions.

Alshawaf MH, Krevor S, Muggeridge A, 2017, Analysis of viscous crossflow in polymer flooding, EAGE IOR Symposium 2017

Polymer flooding improves oil recovery by improving flood front conformance compared with waterflooding as well as, in some cases, extracting more oil from lower permeability zones in the reservoir by viscous cross-flow. However viscous cross-flow of water from the low permeability zone may also adversely affect the polymer flood by causing the polymer slug to be diluted and possibly to lose its integrity. The extent to which viscous cross-flow improves or reduces recovery depends upon the permeability contrast between the low and high permeability zones, the viscosity ratios of the fluids (oil, water and polymer solution) and the geometry of the layers. This paper uses inspectional analysis to derive the minimum set of 6 dimensionless numbers that can be used to characterise a polymer flood in a two layered model. A series of finely gridded numerical simulations are then performed to determine the contribution of viscous crossflow to oil recovery from secondary and tertiary polymer flooding in this system. We show that viscous cross-flow will only make a positive impact on oil recovery from secondary polymer flooding when the viscosity ratio values of oil to polymer solution is less than 1 and permeability ratio between the layers is less than 50. Furthermore, we show that there is an inverse relationship between the permeability ratio between layers and the amount of degradation the polymer slug experiences due to viscous crossflow in the high permeability layer. As the permeability contrast between layers increases, the slug degradation decreases. Also, the results show that the desired positive impact from viscous crossflow is higher in secondary polymer foods when compared to tertiary polymer floods. Finally, the results can be used to make initial estimates of the contribution of both viscous cross-flow and mobility control in polymer flooding applications without the need to perform extensive and time consuming numerical simulations.

Adam A, Pavlidis D, Percival JR, et al., 2017, Dynamic mesh adaptivity for immiscible viscous fingering, Pages: 788-802

© Copyright 2017, Society of Petroleum Engineers The unstable displacement of one fluid by another in a porous medium occurs frequently in various branches of enhanced oil recovery. It is now well known that when the invading fluid is of lower viscosity than the resident fluid, the displacement front is subject to a Saffman-Taylor instability and is unstable to transverse perturbations. These instabilities can grow, leading to fingering of the invading fluid. Numerical simulation of viscous fingering is challenging. The physics is controlled by a complex interplay of viscous and diffusive forces and it is necessary to ensure physical diffusion dominates numerical diffusion to obtain converged solutions. This typically requires the use of high mesh resolution and high order numerical methods. This is computationally expensive, particularly in 3D. We use IC-FERST, a novel control volume finite element (CVFE) code that uses dynamic mesh adaptivity on unstructured meshes to simulate 2D and 3D viscous fingering with higher accuracy and lower computational cost than conventional methods. We provide evidence that these unstructured mesh simulations in fact yield better results that are less influenced by grid orientation error than their structured counterparts. We also include the effect of capillary pressure and show three examples that are very challenging to simulate using more conventional approaches.

Mostaghimi P, Kamali F, Jackson MD,
et al., 2016, Adaptive mesh optimization for simulation of immiscible viscous fingering, *SPE Journal*, Vol: 21, Pages: 2250-2259, ISSN: 1930-0220

Viscous fingering can be a major concern when waterflooding heavy oil reservoirs. Most commercial reservoir simulatorsemploy low-order finite volume/difference methods on structured grids to resolve this phenomenon. However, this approachsuffers from a significant numerical dispersion error due to insufficient mesh resolution which smears out some importantfeatures of the flow. We simulate immiscible incompressible two-phase displacements and propose the use of unstructuredcontrol volume finite element (CVFE) methods for capturing viscous fingering in porous media. Our approach usesanisotropic mesh adaptation where the mesh resolution is optimized based on the evolving features of flow. The adaptivealgorithm uses a metric tensor field based on solution interpolation error estimates to locally control the size and shape ofelements in the metric. The mesh optimization generates an unstructured finer mesh in areas of the domain where flowproperties change more quickly and a coarser mesh in other regions where properties do not vary so rapidly. We analyze thecomputational cost of mesh adaptivity on unstructured mesh and compare its results with those obtained by a commercialreservoir simulator based on the finite volume methods.

Xiao D, fang F, pain C, et al., Non-intrusive Reduced Order Modelling of Waterflooding in Geologically Heterogeneous Reservoirs, ECMOR XV - 15th European Conference on the Mathematics of Oil Recovery

Xiao D, Lin Z, Fang F,
et al., 2016, Non-intrusive reduced order modeling for multiphase porous media flows using smolyak sparse grids, *International Journal for Numerical Methods in Fluids*, Vol: 83, Pages: 205-219, ISSN: 0271-2091

In this article, we describe a non-intrusive reduction method for porous media multiphase flows using Smolyak sparse grids. This is the first attempt at applying such an non-intrusive reduced-order modelling (NIROM) based on Smolyak sparse grids to porous media multiphase flows. The advantage of this NIROM for porous media multiphase flows resides in that its non-intrusiveness, which means it does not require modifications to the source code of full model. Another novelty is that it uses Smolyak sparse grids to construct a set of hypersurfaces representing the reduced-porous media multiphase problem. This NIROM is implemented under the framework of an unstructured mesh control volume finite element multiphase model. Numerical examples show that the NIROM accuracy relative to the high-fidelity model is maintained, whilst the computational cost is reduced by several orders of magnitude.

Adam A, Pavlidis D, Percival J,
et al., 2016, Higher-order conservative interpolation between control-volume meshes: Application to advection and multiphase flow problems with dynamic mesh adaptivity, *Journal of Computational Physics*, Vol: 321, Pages: 512-531, ISSN: 1090-2716

A general, higher-order, conservative and bounded interpolation for the dynamic and adaptive meshing of control-volume fields dual to continuous and discontinuous finite element representations is presented. Existing techniques such as node-wise interpolation are not conservative and do not readily generalise to discontinuous fields, whilst conservative methods such as Grandy interpolation are often too diffusive. The new method uses control-volume Galerkin projection to interpolate between control-volume fields. Bounded solutions are ensured by using a post-interpolation diffusive correction. Example applications of the method to interface capturing during advection and also to the modelling of multiphase porous media flow are presented to demonstrate the generality and robustness of the approach.

Attar A, Muggeridge AH, 2016, Evaluation of mixing in low salinity waterflooding

Copyright 2016, Society of Petroleum Engineers. The impact of heterogeneity induced mixing between the injected low salinity water and the high salinity connate water is investigated in secondary low salinity waterflooding (LSW). Although LSW has proved to be a promising EOR process in laboratory experiments and field trials, its efficiency can be reduced due to mixing with in situ connate water. This mixing is caused by molecular diffusion and dispersion and some workers suggest it can be exacerbated by heterogeneity. We investigated the impact of 1) conformance caused by layering in the sandstone and 2) mixing caused by water-saturated shale layers adjacent to the reservoir sand. The study used a commercial reservoir simulator in which low salinity flooding is modelled by varying relative permeability as a function of salinity. All simulations used very fine grids so that physical (rather than numerical) diffusion and dispersion dominated the displacement. In single phase flow, it was found that physical dispersion can only be modelled for when the grid Peclet number is less than 60, for higher values numerical dispersion dominates over physical values. This threshold Peclet number varies with the number of grid blocks, e.g.The optimum number of grid blocks was found to be 10 for (Pe <12) compared with 1000 grid blocks (Pe = 60). In two phase flow, the transverse dispersion number (NTD) originally proposed by Lake and Hirasaki (1981) is shown to be a very robust way to measure the impact of mixing on the performance of LSW. Generally, for any (NTD > 1), diffusion dominates the flow and thus the efficiency of LSW is reduced. In layered high net to gross reservoirs the impact of molecular diffusion increases as the thickness of the higher permeability layer decreases. In lower net to gross reservoirs containing thin, shaly zones filled with higher salinity connate water molecular diffusion reduces the effectiveness of LSW as the thickness of the sand layers be

Ajibola J, Adam A, Muggeridge A, 2016, Gravity driven fingering and mixing during CO<inf>2</inf> sequestration

© 2016 Society of Petroleum Engineers. All rights reserved. Injection of carbon dioxide into deep saline aquifers is one way to reduce greenhouse gas emissions. Carbon dioxide, usually a super critical fluid at aquifer pressure and temperature conditions, is lighter than the resident brine and so forms a gas cap above the water. However, over time it dissolves in the water, creating a density inversion which induces gravitational instability. Understanding whether the dominant mixing mechanism is convective mixing rather than pure diffusion is important as this controls the timescale over which the carbon dioxide-saturated brine mixes with the unsaturated brine. This paper presents numerical simulations, using a finite difference reservoir simulator, to evaluate the predictions of analytical solutions for stability analysis and growth rate of the fingers of different wavenumbers at different Rayleigh numbers (Ra). The effects of density difference, permeability anisotropy and diffusion (both longitudinal and transverse) on fingering behaviour were investigated through the dimensionless Rayleigh number. The density difference and the vertical permeability were found to mainly control the degree of instability. At Rayleigh numbers greater than 800, fingers are present and the degree of fingering increases with Rayleigh number. Growth rate analysis showed that growth rate is directly proportional to Rayleigh number and time. The critical time (at which flow becomes unstable) varies inversely with the Rayleigh number whilst the corresponding critical wavenumber number varies linearly with the Rayleigh number. These results are consistent with previously reported linear stability analyses providing a validation of the simulator. Numerical simulation results were also validated against experiments. These validations both show that the simulator is robust and can thus be used to investigate more complex situations (heterogeneity) that cannot be analysed mathematically

Gago PA, King PR, Muggeridge AH, 2016, Fast estimation of effective permeability and sweep efficiency of waterflooding in geologically heterogeneous reservoirs

Geological heterogeneity can adversely affect the macroscopic sweep efficiency when waterflooding oil reservoirs, however the exact distribution of permeability and porosity is generally not known. Engineers try to estimate the range of impacts heterogeneity might have on waterflood efficiency by creating multiple geological models and then simulating a waterflood through each of those realizations. Unfortunately each simulation can be computationally intensive meaning that it is generally not possible to obtain a statistically valid estimate of the expected sweep and the associated standard deviation. In this paper we show how the volume of unswept oil can be estimated rapidly (without flow simulations) from a geometrical characterization of the spatial permeability distribution. A "constriction" factor is defined which quantifies the effective cross-section area of the zones perpendicular to the principal flow direction. This is combined with a 'net-To-gross ratio' (which quantifies the fractional reservoir volume occupied by the zones that contribute to flow) to estimate effective permeability and the expected recovery factor for that realization. The method is tested using a range of realistic geological models, including SPE10 model 2 and its predictions are shown to agree well with values obtained using a well established commercial flow simulator.

Adam AG, Pavlidis D, Percival JR, et al., 2016, Simulation of immiscible viscous fingering using adaptive unstructured meshes and controlvolume galerkin interpolation

Displacement of one fluid by another in porous media occurs in various settings including hydrocarbon recovery, CO2 storage and water purification. When the invading fluid is of lower viscosity than the resident fluid, the displacement front is subject to a Saffman-Taylor instability and is unstable to transverse perturbations. These instabilities can grow, leading to fingering of the invading fluid. Numerical simulation of viscous fingering is challenging. The physics is controlled by a complex interplay of viscous and diffusive forces and it is necessary to ensure physical diffusion dominates numerical diffusion to obtain converged solutions. This typically requires the use of high mesh resolution and high order numerical methods. This is computationally expensive. We demonstrate here the use of a novel control volume - finite element (CVFE) method along with dynamic unstructured mesh adaptivity to simulate viscous fingering with higher accuracy and lower computational cost than conventional methods. Our CVFE method employs a discontinuous representation for both pressure and velocity, allowing the use of smaller control volumes (CVs). This yields higher resolution of the saturation field which is represented CV-wise. Moreover, dynamic mesh adaptivity allows high mesh resolution to be employed where it is required to resolve the fingers and lower resolution elsewhere. We use our results to re-examine the existing criteria that have been proposed to govern the onset of instability. Mesh adaptivity requires the mapping of data from one mesh to another. Conventional methods such as collocation interpolation do not readily generalise to discontinuous fields and are non-conservative. We further contribute a general framework for interpolation of CV fields by Galerkin projection. The method is conservative, higher order and yields improved results, particularly with higher order or discontinuous elements where existing approaches are often excessively diffusive.

Djabbarov S, Jones ADW, Krevor S, et al., 2016, Experimental and numerical studies of first contact miscible injection in a quarter five spot pattern

Copyright 2016, Society of Petroleum Engineers. We quantify the impact of mobility, simple heterogeneities and grid orientation error on the performance of first contact miscible gas flooding in a quarter five spot configuration by comparing the outputs from experimental and numerical models. The aim is to quantify the errors that may arise during simulation and to identify a workflow for minimizing these when conducting field scale fingering studies. A commercial reservoir simulator was validated by comparing its predictions with the results obtained from physical experiments. An uncorrelated, random permeability distribution was used to trigger fingering in the simulations. The physical experiments were carried out using a Hele-Shaw cell (40x40cm) designed and constructed for this study. The impact of a square low permeability inclusion (20x20cm) on flow was investigated by varying its permeability, location and orientation. For lower mobility ratios (M=2 to M=10) the commercial numerical simulator was able to reproduce the experimental observations within the uncertainty range of the permeability distribution used to trigger the fingers, provided a nine-point scheme was used for the pressure solution. At higher mobility ratios (M=20 to M=100) the grid orientation effect meant that the simulator overestimated the areal sweep even when a nine-point scheme was used. The introduction of a square, low permeability inclusion near the injection well reduced the discrepancy between experimental and numerical results, bringing it back within uncertainty limits in some of the cases. This was mainly because the real flow was then forced to move parallel to the edges of the Hele-Shaw cell and thus parallel to the simulation grid. Breakthrough times were well predicted by the numerical simulator at all mobility ratios.

Hamid SAA, Muggeridge AH, 2016, Characterizing the impact of heterogeneity on miscible gas injection

The performance of miscible gas injection processes is typically adversely affected by geological heterogeneity and viscous fingering. It is well known that very high grid resolutions are needed to model viscous fingering explicitly so most field scale simulations will use an empirical model such as the Todd and Longstaff (1972) model to describe the average effects of viscous fingering on recovery. In homogeneous reservoirs it is usual to assume to use a Todd and Longstaff mixing parameter with a value of 2/3 in these simulations. It is, however, unclear how to model the influence of heterogeneities on the viscous fingering. Previous work by Fayers et al. (1992) identified that there are two flow regimes associated with heterogeneity - diffusive when the permeability distribution is uncorrelated and random tending to advective as the permeability distribution has a larger standard deviation and a greater correlation length. In the former case the effect of heterogeneity can be modelled using a dispersivity whilst in the latter case it can be modelled by an effective viscosity ratio of heterogeneity factor (Koval, 1963). In this work we identify a third heterogeneity controlled flow regime -That of channelized flow. In this regime the effect of heterogeneity can best be modelled using a bypassed oil volume. We propose the use of a phase diagram for identifying which of these three flow regimes is dominant in a given heterogeneous reservoir and hence which method of upscaling flow is best suited to that reservoir. This phase diagram uses only static measures of the permeability distribution. We further provide a simple method for estimating the effective mobility ratio for use when the heterogeneity can be modelled by a Koval heterogeneity factor. The results are benchmarked against high resolution, finite-difference, first-contact miscibility simulation using SPE 10 Model 2 and synthetic permeability fields.

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