Miscible Flow Research Articles

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.

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.

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.

Mixing and spreading of multiphase fluids in heterogeneous bimodal porous media

We investigate the impact of heterogeneous bimodal porous media on fluid mixing and spreading in viscously-unstable flows. Previous studies have mostly studied miscible mixing and spreading behavior in unimodal media. We characterize the temporal evolution of these processes in bimodal media and under both miscible and partially miscible conditions. We model advection-diffusion transport of a finite volume of $$\hbox {CO}_2$$ diluting within a rectilinear background flow of a multicomponent compressible hydrocarbon fluid. Accurate numerical simulations are performed to capture the details of hydrodynamic instabilities as well as the heterogeneity channelling. Thermodynamic phase behavior and Fickian diffusion are represented based on rigorous equation of state. We generate by means of a Markov Chain approach the permeability fields that represent facies architecture by volume fractions of each facies unit. Our results show that bimodal media significantly alter the flow pattern and spreading dynamics, especially at lower proportions of the high-permeability facies. However, we find the miscible mixing to be less sensitive to bimodal media, as a result of a delicate balance between fingering and permeability channeling. On other hand, spreading is usually lower in partially miscible flows, but this can be overridden by channeling due to bimodal media. The new hydrothermodynamic mechanisms that predominantly drive most of the mixing in partially miscible systems are still in effect even in bimodal media regardless of the facies architecture. Therefore, partially miscible mixing is negligibly impacted by bimodal heterogeneity. Our qualitative and quantitative results elucidate the key flow processes resulting from bimodal structures, and also rationalize the distinct mixing and spreading behaviors that emerge differently from the interplay between hydrothermodynamic mechanisms and flow channeling.

Highly pressurized partially miscible liquid-liquid flow in a micro-T-junction. I. Experimental observations.

This is the first part of a two-part study on a partially miscible liquid-liquid flow (liquid carbon dioxide and deionized water) which is highly pressurized and confined in a microfluidic T-junction. Our main focuses are to understand the flow regimes as a result of varying flow conditions and investigate the characteristics of drop flow distinct from coflow, with a capillary number, Ca_{c}, that is calculated based on the continuous liquid, ranging from 10^{-3} to 10^{-2} (10^{-4} for coflow). Here in part I, we present our experimental observation of drop formation cycle by tracking drop length, spacing, frequency, and after-generation speed using high-speed video and image analysis. The drop flow is chronologically composed of a stagnating and filling stage, an elongating and squeezing stage, and a truncating stage. The common "necking" time during the elongating and squeezing stage (with Ca_{c}∼10^{-3}) for the truncation of the dispersed liquid stream is extended, and the truncation point is subsequently shifted downstream from the T-junction corner. This temporal postponement effect modifies the scaling function reported in the literature for droplet formation with two immiscible fluids. Our experimental measurements also demonstrate the drop speed immediately following their generations can be approximated by the mean velocity from averaging the total flow rate over the channel cross section. Further justifications of the quantitative analysis by considering the mass transfer at the interface of the two partially miscible fluids are provided in part II.

Highly pressurized partially miscible liquid-liquid flow in a micro-T-junction. II. Theoretical justifications and modeling.

This is the second part of a two-part study on a partially miscible liquid-liquid flow (carbon dioxide and deionized water) that is highly pressurized and confined in a microfluidic T-junction. In the first part of this study, we reported experimental observations of the development of flow regimes under various flow conditions and the quantitative characteristics of the drop flow including the drop length, after-generation drop speed, and periodic spacing development between an emerging drop and the newly produced one. Here in part II we provide theoretical justifications to our quantitative studies on the drop flow by considering (1) CO_{2} hydration at the interface with water, (2) the diffusion-controlled dissolution of CO_{2} molecules in water, and (3) the diffusion distance of the dissolved CO_{2} molecules. Our analyses show that (1) the CO_{2} hydration at the interface is overall negligible, (2) a saturation scenario of the dissolved CO_{2} molecules in the vicinity of the interface will not be reached within the contact time between the two fluids, and (3) molecular diffusion does play a role in transferring the dissolved molecules, but the diffusion distance is very limited compared with the channel geometry. In addition, mathematical models for the drop length and the drop spacing are developed based on the observations in part I, and their predictions are compared to our experimental results.

Hydrothermodynamic mixing of fluids across phases in porous media

AbstractWe investigate the coupled dynamics of fluid mixing and viscously unstable flow under both miscible (single‐phase) and partially miscible (two‐phase) conditions, and in both homogeneous and heterogeneous porous media. Higher‐order finite element methods and fine grids are used to resolve the small‐scale onset of fingering and tip splitting. An equation of state determines the thermodynamic phase behavior and Fickian diffusion. We compute global quantitative measures of the spreading and mixing of a diluting slug to elucidate key differences between miscible and partially miscible systems. Hydrodynamic instabilities are the main driver for mixing in miscible flow. In partially miscible flow, however, we find that relative permeabilities spread the two‐phase zone. Within this mixing zone dissolution and evaporation drive mixing thermodynamically while reducing mobility contrasts and thus fingering instabilities. The different mixing dynamics in systems involving multiple phases with mutual solubilities have important implications in hydrogeology and energy applications, such as geological carbon sequestration and gas transport in hydrocarbon reservoirs.

Temporal localized nonlinear model reduction with a priori error estimate

This work presents a model reduction framework using a temporal localized basis approach to efficiently reduce the simulation time for nonlinear dynamical systems with rapid changes over a short time period, and proposes a corresponding a priori error bound. This framework employs the proper orthogonal decomposition (POD) to construct localized basis sets from different temporal subdomains, which can be used in the Galerkin projection to accurately capture the important local dynamics of the system. The discrete empirical interpolation method (DEIM) with the corresponding temporal localized basis sets is then applied to efficiently compute the projected nonlinear terms. A heuristic procedure for subdividing snapshots over the temporal domain is proposed. This procedure first partitions the set of snapshots where there are possible significant changes in system dynamics, and then uses the notion of distance between subspaces to later remove unnecessary partitioning. An a priori error bound is derived to confirm the convergence of this framework and to explain how the propagated errors from the localized reduced systems affect the overall accuracy. Numerical experiments demonstrate the accuracy improvement of the temporal localized framework through a parametrized nonlinear miscible flow simulation. The results show the applicability of the proposed approach to various parameter values that are not necessary used for generating the POD and DEIM localized basis sets.

Analytical and variational numerical methods for unstable miscible displacement flows in porous media

The miscible displacement of one fluid by another in a porous medium has received considerable attention in subsurface, environmental and petroleum engineering applications. When a fluid of higher mobility displaces another of lower mobility, unstable patterns – referred to as viscous fingering – may arise. Their physical and mathematical study has been the object of numerous investigations over the past century. The objective of this paper is to present a review of these contributions with particular emphasis on variational methods. These algorithms are tailored to real field applications thanks to their advanced features: handling of general complex geometries, robustness in the presence of rough tensor coefficients, low sensitivity to mesh orientation in advection dominated scenarios, and provable convergence with fully unstructured grids.This paper is dedicated to the memory of Dr. Jim Douglas Jr., for his seminal contributions to miscible displacement and variational numerical methods.

Numerical validation of an upscaled sharp–diffuse interface model for stratified miscible flows

A new model based on a mixed sharp and diffuse interface approach has been derived recently for miscible stratified displacements in porous media. It is an upscaled two-dimensional model tracking the interfaces between the fluid layers. The aim of the paper is to compare numerically the results provided by this upscaled model with the ones obtained using the classical three-dimensional model for miscible and compressible displacements in slightly deformable porous media. We construct for each model a numerical scheme based on a Euler semi-implicit method for the time discretization coupled with a mixed finite element approach for the space discretization. For the numerical simulations we focus on the example of the pollution of an aquifer. Computations are provided for various settings, especially when the flow is not quasi-horizontal.

The frequency domain approach to analyse field-scale miscible flow transport experiments in the soils

A new approach to the estimate of the parameters u (advective velocity) and λ (dispersivity) characterizing solute transport in soils is presented. The pair (u, λ) is estimated by matching in the frequency domain (FD) the theoretical expression of moments pertaining to the breakthrough curve (BTC) against to the one evaluated by means of the experimental data. In particular, we demonstrate that to reduce the impact of the random measurement-errors upon such an estimate, it is worth retaining in the Fourier's expansion of the moments only the harmonics associated to the smaller frequencies. This is due to the fact that the Fourier transform moves most of the measurement-errors affecting moments in the high-frequency range. As a consequence, by adopting a relatively small number of harmonics to compute the Fourier transform of the experimental moments, one may filter out most of the noise. It is also shown that the number of harmonics to retain (cut-off) depends upon the soil's water content as well as the magnitude of the characteristic length l E of the error relative to the dispersivity λ. The proposed methodology has been applied to a recently conducted plot-scale transport experiment. For comparison purposes, we have also estimated the pair (u, λ) by the classical method of moments (MM). Both the methods lead to the same value of the advective velocity u. This is explained by recalling that u depends upon the first-order moment, a quantity that is scarcely influenced by the measurement-errors. Instead, the estimate of the dispersivity λ (which is related to the second-order moment) is largely different (with the value achieved by the MM larger than the one obtained by the FD approach). Such a difference is addressed to the fact that in the MM the distortion-effect due to the measurement-errors amplifies with the increasing order of the moments, a phenomenon which is completely avoided in the FD approach by adopting the above mentioned cut-off.

Stability of viscosity stratified flows down an incline: Role of miscibility and wall slip

The effects of wall velocity slip on the linear stability of a gravity-driven miscible two-fluid flow down an incline are examined. The fluids have the matched density but different viscosity. A smooth viscosity stratification is achieved due to the presence of a thin mixed layer between the fluids. The results show that the presence of slip exhibits a promise for stabilizing the miscible flow system by raising the critical Reynolds number at the onset and decreasing the bandwidth of unstable wave numbers beyond the threshold of the dominant instability. This is different from its role in the case of a single fluid down a slippery substrate where slip destabilizes the flow system at the onset. Though the stability properties are analogous to the same flow system down a rigid substrate, slip is shown to delay the surface mode instability for any viscosity contrast. It has a damping/promoting effect on the overlap modes (which exist due to the overlap of critical layer of dominant disturbance with the mixed layer) when the mixed layer is away/close from/to the slippery inclined wall. The trend of slip effect is influenced by the location of the mixed layer, the location of more viscous fluid, and the mass diffusivity of the two fluids. The stabilizing characteristics of slip can be favourably used to suppress the non-linear breakdown which may happen due to the coexistence of the unstable modes in a flow over a substrate with no slip. The results of the present study suggest that it is desirable to design a slippery surface with appropriate slip sensitivity in order to meet a particular need for a specific application.

Linear stability analysis and direct numerical simulation of two-layer channel flow

We study the stability of two-fluid flow through a plane channel at Reynolds numbers of 100–1000 in the linear and nonlinear regimes. The two fluids have the same density but different viscosities. The fluids, when miscible, are separated from each other by a mixed layer of small but finite thickness, across which the viscosity changes from that of one fluid to that of the other. When immiscible, the interface is sharp. Our study spans a range of Schmidt numbers, viscosity ratios and locations and thicknesses of the mixed layer. A region of instability distinct from that of the Tollmien–Schlichting mode is obtained at moderate Reynolds numbers. We show that the overlap of the layer of viscosity-stratification with the critical layer of the dominant disturbance provides a mechanism for this instability. At very low values of diffusivity, the miscible flow behaves exactly like the immiscible one in terms of stability characteristics. High levels of miscibility make the flow more stable. At intermediate levels of diffusivity however, in both linear and nonlinear regimes, miscible flow can be more unstable than the corresponding immiscible flow without surface tension. This difference is greater when the thickness of the mixed layer is decreased, since the thinner the layer of viscosity stratification, the more unstable the miscible flow. In direct numerical simulations, disturbance growth occurs at much earlier times in the miscible flow, and also the miscible flow breaks spanwise symmetry more readily to go into three-dimensionality. The following observations hold for both miscible and immiscible flows without surface tension. The stability of the flow is moderately sensitive to the location of the interface between the two fluids. The response is non-monotonic, with the least stable location of the layer being mid-way between the wall and the centreline. As expected, flow at higher Reynolds numbers is more unstable.

Utilização de Modelos Autoregressivos na Quantificação de Incertezas em Problemas de Transporte Linear

The goal of this work is to study the uncertainty quantification in inverseproblems of miscible flows in heterogeneous porous media in Bayesian framework and propose a permeability update based on observed measurements of spatially sparse tracer concentration at certain times. The Successive Sum of Independent Gaussian Fields method is used to parametrization of the uncertainty in heterogeneous porous media. A two-stageMarkov chain Monte Carlo (MCMC) method is used to solving the inverse problem. For the construction of Markov chains is used the Metropolis-Hastings algorithm based on a random walk sampling process employs an auto-regressive instrumental distribution. The main idea of MCMC is to generate a Markov chain with an equilibrium distribution equal to the posterior distribution of interest. Numerical results are presented for a set sampled realizations of the permeability fields.

3 次元多孔質内混和性二相流におけるヴィスカスフィンガリングの挙動

Carbon dioxide flooding, which is one of enhanced oil recovery schemes, improves oil recovery due to the miscibility between supercritical carbon dioxide and oil. When a less viscous fluid displaces a more viscous one, however, viscous fingering reduces oil recovery. In this study, the there-dimensional development of viscous fingering in miscible two-phase flows in porous media is visualized using an X-ray CT scanner. Interaction among the fingers such as shielding, merging, and tip splitting is observed. Three-dimensionally involved structure of the fingers results in low sweep efficiency compared with two-dimensional viscous fingering.

A coupled lattice Boltzmann model for fluid flow and diffusion in a porous medium

The flow and diffusion of miscible fluid in a porous medium with a high Plcet number (Pe) and large viscosity ratio widely exist in industrial processes, such as oil recovery, geological sequestration of carbon dioxide, and chemical engineering process. When these problems are studied by numerical methods, the key point is to accurately describe the flow dynamics and diffusion process in a porous medium at the same time. As an alternative to conventional numerical methods, the lattice Boltzmann method based on kinetic theory is well suited to pore-scale simulations of miscible fluid flows and molecular diffusion. However, most of the existing lattice Boltzmann models have many difficulties (e.g. robustness and numerical stability) in simulating such systems at high Pe and large viscosity ratio. In this paper, in order to overcome the above difficulties, we propose a coupled lattice Boltzmann model based on the multiple-relaxation-time model and the lattice kinetic scheme for the fluid flow and diffusion, respectively. It can be shown that the incompressible Navier-Stokes equations and the convection-diffusion equation can be derived from the presented coupled model through the Chapman-Enskog procedure. The proposed model is validated by simulating a concentration gradient driven flow in a porous channel. Numerical results demonstrate that the model is of second-order accuracy in space. We further simulate a flow through two types of artificial porous media. The robustness of the presented model is investigated by measuring the permeability and diffusivity under different relaxation times. It is found that the model is insensitive to relaxation parameters. In addition, the miscible viscous displacement in two parallel plates is simulated to test the numerical stability of the model. It is observed that the results accord well with those reported in previous work, and the model is very stable at high Pe and large viscosity ratio in comparison with the standard lattice Bhatnagar-Gross-Krook model. Overall, the coupled lattice Boltzmann model can serve as an effective tool for directly simulating the fluid flow and diffusion at high Pe and large viscosity ratio in the pores of a porous medium.

Numerical study of compositional compressible degenerate two-phase flow in saturated–unsaturated heterogeneous porous media

We study the convergence of a combined finite volume–nonconforming finite element scheme on general meshes for a partially miscible two-phase flow model in anisotropic porous media. This model includes capillary effects and exchange between the phases. The diffusion term, which can be anisotropic and heterogeneous, is discretized by piecewise linear nonconforming triangular finite elements. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh. The relative permeability of each phase is decentered according to the sign of the velocity at the dual interface. The convergence of the scheme is proved thanks to an estimate on the two pressures which allows to show estimates on the discrete time and compactness results in the case of degenerate relative permeabilities. A key point in the scheme is to use particular averaging formula for the dissolution function arising in the diffusion term. We show also a simulation of hydrogen production in nuclear waste management. Numerical results are obtained by in-house numerical code.

Uniform temporal convergence of numerical schemes for miscible flow through porous media

The Hybrid Mimetic Mixed (HMM) family of schemes contains the Hybrid Finite Volume, Mimetic Finite Difference and Mixed Finite Volume methods. This note proves that HMM schemes, when applied to a model of the miscible displacement of one incompressible fluid by another through a porous medium, produce approximations of the concentration variable that converge uniformly towards the exact concentration.

Radial flows in heterogeneous porous media with a linear injection scheme

Abstract A robust diffuse-interface formulation is applied to simulate miscible and immiscible radial flows in heterogeneous porous media, in which the permeability is characterized by a log Gaussian distribution. The stabilizing effects of linear injection scheme, determined by the fingering interfacial length, are investigated to verify its applicability in various conditions of miscibility and permeability heterogeneity. For fully miscible conditions, the linear injection scheme shows destabilizing effects both in homogeneous and heterogeneous media. On the other hand, even the immiscible fingering instability in a homogenous medium can be suppressed effectively by applying the linear injection scheme, the stabilizing effect on such a linear injection scheme is insignificant in a heterogeneous medium.

Multiple-relaxation-time lattice Boltzmann model for incompressible miscible flow with large viscosity ratio and high Péclet number.

A lattice Boltzmann model with a multiple-relaxation-time (MRT) collision operator is proposed for incompressible miscible flow with a large viscosity ratio as well as a high Péclet number in this paper. The equilibria in the present model are motivated by the lattice kinetic scheme previously developed by Inamuro et al. [Philos. Trans. R. Soc. London, Ser. A 360, 477 (2002)]. The fluid viscosity and diffusion coefficient depend on both the corresponding relaxation times and additional adjustable parameters in this model. As a result, the corresponding relaxation times can be adjusted in proper ranges to enhance the performance of the model. Numerical validations of the Poiseuille flow and a diffusion-reaction problem demonstrate that the proposed model has second-order accuracy in space. Thereafter, the model is used to simulate flow through a porous medium, and the results show that the proposed model has the advantage to obtain a viscosity-independent permeability, which makes it a robust method for simulating flow in porous media. Finally, a set of simulations are conducted on the viscous miscible displacement between two parallel plates. The results reveal that the present model can be used to simulate, to a high level of accuracy, flows with large viscosity ratios and/or high Péclet numbers. Moreover, the present model is shown to provide superior stability in the limit of high kinematic viscosity. In summary, the numerical results indicate that the present lattice Boltzmann model is an ideal numerical tool for simulating flow with a large viscosity ratio and/or a high Péclet number.