Miscible Displacement Research Articles

Investigation of concentration‐dependent diffusion on frontal instabilities and mass transfer in homogeneous porous media

Mass transfer plays an important role in influencing the efficiency of miscible displacements in solvent‐based processes in enhanced oil recovery. The mass transfer rate because of the pure molecular diffusion is very slow. However, this process can be greatly enhanced by the appearance of frontal instabilities called viscous fingering mechanisms, which are beneficial for improving the mixing and mass transfer between the injected solvent and oil. Instead of a piston‐like displacement, the interface between solvent and oil is very convoluted with intricate finger‐like patterns of the less viscous solvent intruding into the highly viscous oil. This intrusion significantly increases the surface area of contact of the two fluids and leads to more efficient mass transfer and mixing. Experimental measurements on the diffusion coefficients of two miscible fluids indicate that, instead of a constant diffusion coefficient (CDC), a concentration‐dependent diffusion coefficient (CDDC) is more realistic. In the present study, a CDDC relation in which the diffusion coefficient is exponentially proportional to concentration is adopted. Its effect on the development of frontal instabilities is examined through highly accurate nonlinear numerical simulations. The differences between the CDDC case and the widely assumed CDC case are discussed. Furthermore, the enhancement of frontal instabilities on mass transfer when the CDDC is considered is investigated at various mobility ratios and Peclet numbers. The special characteristics for the CDDC case indicate its important role in miscible displacements. Eventually, the relation of breakthrough time to parameters is correlated to accurately predict the breakthrough time in any CDDC scenario.

Primary cementing of oil and gas wells in turbulent and mixed regimes

We present a detailed derivation of a practical two-dimensional model for turbulent and mixed regimes in narrow annular displacement flows, such as are found during the primary cementing of oil and gas wells. Such mixed cross regimes, including those in which different regimes exist in the same annular cross section, are relatively common in primary cementing. The modelling approach considers scaling based on the disparity of length-scales, which allows a narrow-gap averaging approach to be effective. With respect to the momentum equations, the leading-order equations correspond to a turbulent shear flow in the direction of the modified pressure gradient. This leads to a nonlinear elliptic problem that is the natural extension of the laminar displacement model in Bittleston et al. (J Eng Math 43:229–253, 2002). The mass transport equations that model the miscible displacement are however quite different. To leading-order turbulence effectively mixes the fluids. Changes in concentrations within the annular gap arise due to the combined effects of advection with the mean flow, anisotropic Taylor dispersion (along the streamlines) and turbulent diffusivity. The diffusive and dispersive effects are modelled for fully turbulent and transitional flows following Maleki and Frigaard (J Non-Newt Fluid Mech 235:1–19, 2016). The model derived allows the investigation of different well geometries and inclinations, pumping sequences and fluid rheologies, all of which can have importance. A number of computed examples are presented with the aim of demonstrating the complexity of turbulent displacements.

Nonlinear simulation of miscible displacements with concentration-dependent diffusion coefficient in homogeneous porous media

The nonlinear dynamics of miscible flow displacements in homogeneous porous media are examined for the special case of concentration-dependent diffusion coefficient (CDDC) models. Highly accurate numerical methods are used to simulate the time evolution of fingering with both mobility and diffusion coefficient contrasts of two components in a fully miscible solution system. The simulations revealed that, when compared with a constant diffusion coefficient (CDC), the CDDC models show drastically different viscous fingering structures and mixing of the two components. In particular, for a pure diffusion process with a stable front, using a CDDC results in less mixing, with a non-symmetric concentration gradient. While for the unstable displacements with relatively large mobility contrasts, it is found that using a CDDC tends to trigger fingers at the very beginning of the displacements. The diffusion-dominated period is therefore very short and can nearly be neglected. Moreover, the fingers reach the downstream boundary much earlier with fingering structures about 2–4 times more complex than the CDC case. Quantitative analysis indicates that the exponential CDDC models can enhance the mixing by shortening the diffusive regime but not change the growth rate of mixing in the convective regime. In addition, in contrast with the CDC case, the time required for the appearance of fingers for the CDDC seems to be independent of Peclet number Pe at a relatively large log mobility ratios R. As Pe increases for the CDDC cases, more complex dynamics of the fingers are observed, which can only be found at very high Pe and R values for the CDC case. In addition, a small increase in R is able to significantly enhance the instabilities since it can result in not only larger mobility contrast but also a steeper diffusion gradient.

Assessment of CO2 - EOR and Storage Capacity in South Sumatera and West Java Basins

A study of carbon dioxide-enhanced oil recovery (CO2-EOR) and storage capacity in South Sumatera and West Java Basins, Indonesia has been conducted in conjunctions with the project plans of Indonesia National Electric Company to build carbon capture and storage ready coal power plants in Bojonegara of West Java and Muara Enim of South Sumatera. The Bojonegara power plant comprises 2×1000 MWe Ultra Super Critical Units, while Muara Enim consists of 2×300 MWe subcritical units. Those two power plants will produce approximately 11 and 4 million tons of CO2 per year. The geology of South Sumatera and West Java Basins are proven to have kept oil and gas in place safely for long time periods and there are many oil and gas companies production still active in those basins. Those basins are confirmed to be used as storage of CO2, where capacity, injectivity, and confinement will be suitable to keep CO2 in place for a period of time. CO2 production from those power plants in South Sumatera and West Java may be offered to the nearby oil companies and transported to oil fields for CO2-EOR flooding to get additional revenue. A preliminary CO2-EOR screening has been done for the oil fields surrounding the power plants. There are two types of mechanisms for CO2-EOR, which depend on the nature of the oilfield: immiscible EOR and miscible EOR. Immiscible EOR occurs where the injected CO2 does not mix with the oil but instead displaces the oil from the area where the CO2 is injected and increases the pressure of the oil at the production wells so that it can be pumped out. Miscible EOR occurs where the supercritical CO2 mixes miscibility with the residual oil to make it less viscous and facilitating its flow to the production wells. A total of 127 oil fields in South Sumatera has been analyzed to select which of those fields fulfill CO2-EOR injection criteria. EOR reservoir screenings were performed on those oil fields on which detailed information was available using the screening criteria proposed by Taber et al. The criteria include a set of parameters such as API gravity, oil viscosity, current pressure, temperature, oil saturation, remaining oil, formation depth, thickness, porosity, permeability, and rock type which help to determine whether or not a reservoir is suitable for CO2-EOR injection. 96 fields are classified as miscible displacement while the remaining 31 fields are categorized as immiscible. The result shows that the potency of additional recovery from those fields is around 661 million stock tank barrels with CO2 requirement of approximately 243 million tonnes. In addition to EOR, pure CCS to inject the CO2 emission into depleted gas fields in West Java and South Sumatera regions is also considered. The maximum storage capacity of West Java gas fields is approximately 395 million tonnes, while the storage capacity of South Sumatera is around 537 million tonnes. The potential for storing CO2 in the saline aquifer was also calculated theoretically. The depleted gas fields and aquifers are considered enough to store the CO2 emission from the power plants as long as 25 years.

First CO2-Enhanced-Oil-Recovery Demonstration Project in Saudi Arabia

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 181729, “Design and Implementation of the First CO2-Enhanced-Oil-Recovery Demonstration Project in Saudi Arabia,” by Sunil Kokal, SPE, Modiu Sanni, SPE, and Almohannad Alhashboul, SPE, Saudi Aramco, prepared for the 2016 SPE Annual Technical Conference and Exhibition, Dubai, 26–28 September. The paper has not been peer reviewed. An operator has designed a demonstration project for carbon dioxide (CO2) enhanced oil recovery (EOR) and has implemented it in one of its fields. The main objectives of the demonstration project are estimation of sequestered CO2, determination of incremental oil recovery, and evaluating the risks and uncertainties involved, including migration of CO2 within the reservoir and operational concerns. It is estimated that approximately 40% of the injected CO2 will be sequestered permanently in the reservoir. Project Design Conceptual Road Map and Screening Studies. Given the relatively light nature of crude oils and generally high reservoir pressures in Saudi Arabia, CO2 injection is a viable recovery method, especially in flooded reservoirs. An initial screening highlighted several good candidates for CO2 injection. A mature, waterflooded part of a large oil field with a carbonate reservoir was selected as a candidate for CO2 injection. Further studies were conducted for the candidate reservoir that included laboratory, feasibility, and detailed reservoir-simulation studies. This reservoir has been flooded for decades in a peripheral water-injection mode, and considerable reservoir and production data were available. Laboratory Studies. Two sets of experimental studies must be conducted for any given CO2-EOR prospect: fluid/fluid and fluid/rock interactions. The important laboratory experiments include the minimum miscibility pressure of the crude with CO2, swelling and fluid properties of CO2/oil mixtures, asphaltene precipitation onset and bulk asphaltene deposition, and oil-recovery potential by use of coreflooding studies. It must be emphasized that these experiments need to be conducted at reservoir conditions with live reservoir fluids and supercritical CO2; otherwise, the data have limited value at best. Simulation Studies. The reservoir selected for CO2 injection is a Jurassic carbonate reservoir, and the area selected is in a downflank, flooded part of the field. The selected area has been on peripheral water injection for more than 50 years and has been well-waterflooded because of its proximity to the peripheral injectors. Approximately 40 MMscf/D of relatively pure CO2 was available from a gas plant approximately 85 km from the pilot site. The slimtube data show that the minimum miscibility pressure is lower than the reservoir pressure, indicating that the CO2 will develop a miscible displacement in the reservoir at current reservoir pressures. The main objectives of the simulation study were as follows: Carry out screening and mechanistic studies and find areas suitable for a CO2-injection pilot. Assess the amount of CO2 sequestered over the period of the pilot testing. Assess incremental oil recoveries associated with different modes of CO2 injection. Optimize the pilot design within the reservoir and operational constraints.

Transport of Cadmium and Phosphate in Soils

This study quantified and modeled cadmium (Cd) transport in the presence to phosphate (P) in three different soils, Mahan (acidic, kaolinitic), Webster (neutral, mixed mineralogy), and Windsor (acidic, sandy). Two sets of miscible displacement experiments were carried out: (1) sequential pulses of 200 mg L−1 Cd as Cd(NO3)2 then 200 mg L−1 P as KH2PO4, both in 0.01 M KNO3 background; and (2) two consecutive pulses of mixed 200 mg L−1 Cd and P in background solution. Isotherms for Cd and P (1-d equilibration) were also determined. Breakthrough curves (BTCs) of Cd applied without P were generally consistent with the (Freundlich) isotherms and soil properties, Cd retention decreasing: Webster, 72% > Mahan, 61% > Windsor, 37%. Retention of Cd increased when applied with P in Webster and Windsor but decreased in Mahan. Phosphate BTCs indicated high mobility (≥92% recovery) and little effect of Cd, thus limited Cd-P precipitation. Residual distribution of Cd with depth also showed no clear evidence for precipitation. Cadmium BTCs could not be described solely by either linear or nonlinear equilibrium isotherms. Rather, the data were best described by nonlinear, reversible kinetic sorption with an irreversible phase. However, P sorption during transport was well described without irreversible sorption, thus indirect evidence of limited Cd-P precipitation. Although transport of Cd applied without P was well described (r2 = 0.95, Webster and Windsor), description of BTCs was more qualitative when applied with P, accurately locating effluent peak but approximating the sorption and desorption sides.

Buoyancy effects on micro-annulus formation: Density stable displacement of Newtonian–Bingham fluids

Buoyant miscible displacement flow of a Bingham fluid by a Newtonian fluid along a vertical plane-channel is studied, in the high Péclet number regime. The displacing fluid is denser than the displaced fluid and the flow direction is density-stable (upwards). The flow is effectively governed by 4 dimensionless parameters: the Newtonian Bingham number (BN), the viscosity ratio (m), the Reynolds number (Re), and modified Froude number (Fr). This is a simple model for micro-annulus formation in the primary cementing of oil and gas wells. We show that the residual layer thickness is largely determined by two parameters: (BN/m, χ*/m), where χ*=2ReFr2. Residual wall layers may be either static or mobile, and mobile layers either evolve to become static or are washed from the channel at long times. We show that the different behaviours of the residual layers are linked to different characteristic behaviours of the displacement front, and we show how the latter behaviours can be predicted using a lubrication/thin-film approximation.

Field-scale simulation of CO2 enhanced oil recovery and storage through SWAG injection using laboratory estimated relative permeabilities

Numerical simulations are widely used to investigate the performance of simultaneous water and gas (SWAG) injection in an oil field. However, they usually use two-phase water and gas relative permeability functions (krg) and one of the known correlations (e.g., Stone, Baker) to calculate oil relative permeability in a three-phase displacement. Moreover, the same krg is used for all SWAG injections, irrespective of miscibility conditions or fraction of gas injected (FGI).Recent experiments have shown that gas relative permeability functions in three-phase SWAG are significantly different from the conventionally used two-phase functions. This paper investigates the impact of using experimentally estimated three-phase gas relative permeability functions on co-optimization study of CO2 storage and enhanced oil recovery (EOR). A three-dimensional, layered reservoir model initially saturated with oil and connate water is used to examine different injection schemes for co-optimizing oil recovery and CO2 storage efficiency. The oil phase consists of a mixture of 0.65 hexane and 0.35 decane by molar fraction. Numerical simulations are run at 70 °C and at three pressures (1300, 1700, and 2100 psi) to represent immiscible, near-miscible, and miscible displacements, respectively. SWAG displacements are run at an FGI of 0.5 for immiscible, near-miscible and miscible conditions. Then the effect of FGI dependent relative permeability on co-optimization of CO2 storage and EOR is investigated in the near-miscible condition for FGI values 0.25, 0.5, 0.75 and 1.0.Experimentally estimated three-phase gas relative permeability results in lower CO2 mobility. This leads to higher CO2 saturation and lower water saturation at the end of simulation. The significant increase in CO2 storage efficiency and hence the co-optimization function are more prominent in near-miscible and miscible displacements than in immiscible. Although, for the homogeneous model, ultimate oil recovery is the same when two-phase or experimentally estimated three-phase relative permeabilities are used, ultimate recovery value is significantly affected in the heterogeneous reservoir model.

Dynamics of a Highly Viscous Circular Blob in Homogeneous Porous Media

Viscous fingering is ubiquitous in miscible displacements in porous media, in particular, oil recovery, contaminant transport in aquifers, chromatography separation, and geological CO2 sequestration. The viscosity contrasts between heavy oil and water is several orders of magnitude larger than typical viscosity contrasts considered in the majority of the literature. We use the finite element method (FEM)-based COMSOL Multiphysics simulator to simulate miscible displacements in homogeneous porous media with very large viscosity contrasts. Our numerical model is suitable for a wide range of viscosity contrasts covering chromatographic separation as well as heavy oil recovery. We have successfully captured some interesting and previously unexplored dynamics of miscible blobs with very large viscosity contrasts in homogeneous porous media. We study the effect of viscosity contrast on the spreading and the degree of mixing of the blob. Spreading (variance of transversely averaged concentration) follows the power law t 3 . 34 for the blobs with viscosity ∼ O ( 10 2 ) and higher, while degree of mixing is found to vary non-monotonically with log-mobility ratio. Moreover, in the limit of very large viscosity contrast, the circular blob behaves like an erodible solid body and the degree of mixing approaches the viscosity-matched case.

Estimating Hofmeister energy in ion-clay mineral interactions from the Gouy-Chapman theory

Hofmeister effects are of scientific importance in cation-clay mineral interactions. In this study, we present a kinetics-based approach to estimate the Hofmeister energy of such interactions from the Gouy-Chapman theory. Montmorillonite was employed as the experimental material; the Hofmeister energies of Cs+ and Na+ were estimated. A mathematical relationship between the Hofmeister energy (as well as the classical Coulomb energy and the total energy) of a cation and its equilibrium adsorption amount was first established based on the modified Gouy-Chapman theory; the miscible displacement technique was then employed to estimate the amount of the cation adsorbed at equilibrium. The Hofmeister energy was calculated by introducing the equilibrium adsorption amount into the established mathematical relationship. The validity of the suggested approach was further corroborated. Our study indicated that the observed Hofmeister energy in cation-clay mineral interactions cannot be explained by cationic size, hydration, or dispersion forces. The observed Hofmeister energy bears two important characteristics: it increased with an increase of electric field strength, and it was much larger for the softer Cs+ cation than for the harder Na+ cation. These characteristics implied that Hofmeister effects in cation-clay mineral interactions are closely related to the changes of cationic energy in the strong electric field near the clay mineral surface. Because of the strong Hofmeister energy from the Gouy-Chapman theory, the apparent charge of Cs+ increased from the normal/typical value of +1 to an effective charge of +2.354; in contrast, the apparent charge of Na+ merely increased from the normal/typical +1 to an effective charge of +1.112 in this clay mineral system.

A Numerical Study on Miscible Viscoelastic Fingering Instability

In this study, the viscoelastic fingering instability of miscible displacement is investigated using computational fluid dynamics (CFD). The Maxwell's model has been used as the constitutive equation of viscoelastic fluid and Hartley transformation (as a spectral method) and fourth-order Adams-Bashforth technique are used to solve the governing equations. The development of the finger structures is discussed using concentration contours and diagrams of transversely average concentration, mixing length and sweep efficiency. It is found that the flow is more stabilized when relaxation time of displacing fluid is increased.

Overview of Carbon Dioxide Injection and Water-Alternating-Gas Sensitivity

This article, written by Special Publications Editor Adam Wilson, contains highlights of paper SPE 179569, “Overview of CO2 Injection and WAG Sensitivity in SACROC,” by Reza Barati Ghahfarokhi, SPE, The University of Kansas, and Steve Pennell, Michael Matson, SPE, and Mark Linroth, SPE, Kinder Morgan, prepared for the 2016 SPE Improved Oil Recovery Conference, Tulsa, 11–13 April. The paper has not been peer reviewed. This paper presents an overview of the SACROC Unit’s activity focusing on different carbon dioxide (CO2) injection and water-alternating-gas (WAG) projects that have made the SACROC unit one of the most successful CO2 injection projects in the world. The main objectives of this were was to review CO2 injection and injection-rate losses with respect to the CO2/WAG miscible displacement process in the SACROC Unit and recommend an injection strategy for WAG-sensitive patterns. Introduction The Kelly-Snyder field is the largest of a chain of fields along the Pennsylvanian Horseshoe atoll in the Midland Basin. Within this field, the Scurry Area Canyon Reef Operators Committee (SACROC) Unit covers approximately 56,000 acres with 2,800 million STB of original oil in place (OOIP). Limestone is the dominant mineral within the Canyon Reef formation, and less than 3% of the formation exists as thin sections of shale (1–10 ft in thickness) that are important stratigraphic markers. The formation is divided into four major zones: the Cisco, the Green Zone (GZ), the Upper Middle Canyon, and the Lower Middle Canyon (LMC). Of these, the GZ shows the highest matrix permeability, significant nonmatrix-flow features, and high-conductivity channels. Moreover, the transition zone (TZ) below the oil/water contact has recently been developed in parts of the SACROC Unit. Review Primary Production. Completed in November 1948, the Standard No. 1 Jessie Brown 2 was drilled to 6,700 ft and produced 530 B/D from the Canyon Reef Formation. Located 9 miles northwest of Snyder, Texas, this well was the discovery well of the North Snyder field. The Texas Railroad Commission eventually merged this field with the neighboring Kelly field upon recognizing that both fields produce from the same reservoir. The reservoir thickness varies from 10 ft on the flanks to 900 ft on the crest of the reef. A map of the unit’s 3D structure and thickness is shown in Fig. 1. By late 1950, 1,600 production wells had been drilled on the Kelly- Snyder field on irregular 40-acre spacing.

Convergence Analysis for Mixed Finite Element Method of Positive Semi-definite Problems

A mixed element-characteristic finite element method is put forward to approximate three-dimensional incompressible miscible positive semi-definite displacement problems in porous media. The mathematical model is formulated by a nonlinear partial differential system. The flow equation is approximated by a mixed element scheme, and the pressure and Darcy velocity are computed at the same time. The concentration equation is treated by the method of characteristic finite element, where the convection term is discretized along the characteristics and the diffusion term is computed by the scheme of finite element. The method of characteristics can confirm strong computation stability at the sharp fronts and avoid numerical dispersion and nonphysical oscillation. Furthermore, a large step is adopted while small time truncation error and high order accuracy are obtained. It is an important feature in numerical simulation of seepage mechanics that the mixed volume element can compute the pressure and Darcy velocity simultaneously and the accuracy of Darcy velocity is improved one order. Using the form of variation, energy method, $L^2$ projection and the technique of priori estimates of differential equations, we show convergence analysis for positive semi-definite problems. Then a powerfu tool is given to solve international famous problems.

A numerical study of two-phase miscible flow through porous media with a Lagrangian model

The multiphase flow mechanism in miscible displacement through porous media is an important topic in various applications, such as petroleum engineering, low Reynolds number suspension flows, dusty gas dynamics, and fluidized beds. To simulate such flows, volume averaging spatial operators are considered to incorporate pressure drag and skin friction experienced by a porous medium. In this work, a streamline-based Lagrangian methodology is extended for an efficient numerical approach to handle dispersion and diffusion of solvent saturation during a miscible flow. Overall pressure drag on the diffusion and dispersion of solvent saturation is investigated. Numerical results show excellent agreement with the results obtained from asymptotic analysis. The present numerical simulations indicate that the nonlinear effects due to skin friction and pressure drag cannot be accurately captured by Darcy’s method if the contribution of the skin friction dominates over that of the pressure drag. Moreover, mass conservation law is investigated, which is an important feature for enhanced oil recovery, and the results help to guide a good agreement with theory. This investigation examines how the flow regime may be optimized for enhanced oil recovery methods.

The effect of numerical integration in mixed finite element approximation in the simulation of miscible displacement

We consider the effect of numerical integration in finite element procedures applied to a nonlinear system of two coupled partial differential equations describing the miscible displacement of one incompressible fluid by another in a porous meduim. We consider the use of the numerical quadrature scheme for approximating the pressure and velocity by a mixed method using Raviart - Thomas space of index  and the concentration by a standard Galerkin method. We also give some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numerical integration. Optimal order estimates are derived when the imposed external flows are smoothly distributed.

Research on microscopic oil displacement mechanism of CO2 EOR in extra-high water cut reservoirs

In China, CO2 EOR technology is mostly applied in small scale field test after water flooding. Microscopic mechanism of CO2 EOR at extra-high water cut stage is not clear to us. Based on the characteristics of residual oil after water flooding, three types of ideal core models were designed including dead-end model, shaped island model and cluster model. The microscopic experiments were carried out in high temperature high pressure micro-visualization experiment apparatus of independent research and development. The microscopic mathematic model of CO2-water-oil interaction was established. The process of CO2 dissolution and extraction during CO2 injection at extra-high water cut stage (water cut>98%) were monitored in the experiments for the first time. The microscopic mechanism of CO2 EOR on trapped oil droplets and the influence of water barrier on miscible displacement were discussed. The results from experiments and mathematic calculation conclude that water barrier should postpone and complicate miscible process between CO2 and trapped oil. No matter how thick the water film, CO2 can diffuse through the water film into the trapped oil. The higher injection pressure is, the faster CO2 diffuses into oil, the more easily miscibility is achieved. CO2 miscible displacement contributes to oil recovery largely after water flooding. CO2 EOR might benefit from the increasing of injection pressure and the full contact time between CO2 and fluid (water and trapped oil) underground in extra-high water cut reservoir.

Minimum miscibility pressure of CO2 and crude oil during CO2 injection in the reservoir

Recently, carbon dioxide (CO2) flooding into depleted reservoirs regardless of miscible or immiscible displacement is widely investigated not only to improve oil recovery but also to reduce the greenhouse effect of this gas produced by numbers of industries in the globe. In the light of this fact, in the first stage of this investigation, the minimum miscibility pressure (MMP) of CO2 and light crude oil (API°=35) with low asphaltene content was determined at temperatures of 30, 50 and 80°C using vanishing interfacial tension (VIT) method. The obtained results demonstrated that the MMP of the studied system is almost linear function of temperature with slope of 0.15MPa/K. The interesting point is that the results revealed a reduction in slope as the composition of light components of crude oil increases.

Variational inequality approach to enforcing the non-negative constraint for advection–diffusion equations

Predictive simulations are crucial for the success of many subsurface applications, and it is highly desirable to obtain accurate non-negative solutions for transport equations in these numerical simulations. To this end, optimization-based methodologies based on quadratic programming (QP) have been shown to be a viable approach to ensuring discrete maximum principles and the non-negative constraint for anisotropic diffusion equations. In this paper, we propose a computational framework based on the variational inequality (VI) which can also be used to enforce important mathematical properties (e.g., maximum principles) and physical constraints (e.g., the non-negative constraint). We demonstrate that this framework is not only applicable to diffusion equations but also to non-symmetric advection–diffusion equations. An attractive feature of the proposed framework is that it works with any weak formulation for the advection–diffusion equations, including single-field formulations, which are computationally attractive. A particular emphasis is placed on the parallel and algorithmic performance of the VI approach across large-scale and heterogeneous problems. It is also shown that QP and VI are equivalent under certain conditions. State-of-the-art QP and VI solvers available from the PETSc library are used on a variety of steady-state 2D and 3D benchmarks, and a comparative study on the scalability between the QP and VI solvers is presented. We then extend the proposed framework to transient problems by simulating the miscible displacement of fluids in a heterogeneous porous medium and illustrate the importance of enforcing maximum principles for these types of coupled problems. Our numerical experiments indicate that VIs are indeed a viable approach for enforcing the maximum principles and the non-negative constraint in a large-scale computing environment. Also provided are Firedrake project files as well as a discussion on the computer implementation to help facilitate readers in understanding the proposed framework.

Analysis of miscible displacement through porous media with vanishing molecular diffusion and singular wells

This article proves the existence of solutions to a model of incompressible miscible displacement through a porous medium, with zero molecular diffusion and modelling wells by spatial measures. We obtain the solution by passing to the limit on problems indexed by vanishing molecular diffusion coefficients. The proof employs cutoff functions to excise the supports of the measures and the discontinuities in the permeability tensor, thus enabling compensated compactness arguments used by Y. Amirat and A. Ziani for the analysis of the problem with L2 wells (Amirat and Ziani, 2004 [1]). We give a novel treatment of the diffusion–dispersion term, which requires delicate use of the Aubin–Simon lemma to ensure the strong convergence of the pressure gradient, owing to the troublesome lower-order terms introduced by the localisation procedure.

Saffman-Taylor Instability and the Inner Splitting Mechanism.

For viscously unstable, miscible Hele-Shaw displacements, we investigate the origin of the streamwise vorticity shown to be responsible for the inner splitting mechanism by Oliveira and Meiburg [J. Fluid Mech. 687 431 (2011)JFLSA70022-112010.1017/jfm.2011.367]. Towards this end, we compare 3D Navier-Stokes simulation results for neutrally buoyant, viscously unstable displacements and gravitationally unstable, constant viscosity ones. Only the former exhibit the generation of streamwise vorticity. The simulation results show that it is caused by the lateral displacement of the more viscous fluid by the less viscous one, with the variable viscosity terms playing a dominant role.