The Growth of Mixing Zone in Heterogeneous Porous Media

2012 ◽  
Author(s):  
M. R. Othman ◽  
R. Badlishah Ahmad ◽  
Z. May
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Short Duration ◽  
Dispersion Coefficient ◽  
Fluid Velocity ◽  
Heterogeneous Porous Media ◽  
Mixing Zone ◽  
Zone Size ◽  
Model Mixing ◽  
Dispersive Mixing

Dengan menggunakan penyelesaian analitikal yang merangkumi fraktal eksponen, pembesaran jarak pencampuran telah dapat ditentukan bagi model satu dimensi. Size zon pencampuran didapati meningkat apabila media berliang menjadi semakin heterogen. Dalam media berliang yang heterogen, saiz zon pencampuran meningkat apabila pemalar penyerakan meningkat terutama sekali pada aliran jangkamasa singkat relatif. Terdapat tiga faktor penting mempengaruhi saiz zon pencampuran penyerakan, ΔxD. Perkara terpenting dalam kajian ini ialah keheterogenan takungan, yang dipersembahkan oleh fraktal eksponen, β. Hasil kajian mendapati bahawa apabila β menjadi kecil (media berliang menjadi semakin heterogen), saiz zon pencampuran meningkat. Satu lagi faktor mempengaruhi ΔxD ialah pemalar penyerakan bersandar masa, Κ(tD). Di dalam takungan heterogen, zon pencampuran meningkat dengan peningkatan nilai pemalar penyerakan pada aliran jangkamasa singkat relatif. Bagi aliran jangkamasa panjang relatif, bagaimanapun, ΔxD terus meningkat walaupun Κ(tD) menjadi tetap. Faktor ketiga ialah purata kelajuan bendalir, ν. Zon pencampuran mempunyai perkaitan songsang dengan kelajuan bendalir dengan cara ΔxD meningkat apabila ν berkurangan. Kata kunci: Kehomogenan; keheterogenan; pekali penyerakan; eksponen fraktal; zon pencampuran; media berliang Utilizing currently available analytical solutions that incorporate fractal exponent, the growth of mixing length of injected solvent was determined for a one-dimensional model. Mixing zone size was found to increase as porous medium becomes increasingly heterogeneous. In a heterogeneous porous media, mixing zone size increases as dispersion coefficient increases particularly at relatively short duration of flow. There are three important factors influencing the size of the dispersive mixing zone, ΔxD. Of particular importance in this study is reservoir heterogeneity, which is represented by a fractal exponent, β. It was discovered that as β becomes smaller (porous medium becomes increasingly heterogeneous), the size of the mixing zone increases. Another factor affecting ΔxD is time dependent dispersion coefficient, Κ(tD). In a heterogeneous reservoir, mixing zone increases with increasing value of dispersion coefficient at relatively short duration of flow. For relatively long period of flow, however? ΔxD continues to increase even though Κ(tD) remains constant. The third factor is average fluid velocity, ν. Mixing zones have inverse relationship with fluid velocity in that ΔxD increases as ν decreases. Key words: Homogeneity; heterogeneity; dispersion coefficient; fractal exponent; mixing zone; dimensionless concentration; porous media

scholarly journals A Stochastic Two-Scale Model for Rarefied Gas Flow in Highly Heterogeneous Porous Media

2020 ◽  
Vol 135 (1) ◽  
pp. 219-242
Author(s):  
Francesc Pérez-Ràfols ◽  
Fredrik Forsberg ◽  
Gunnar Hellström ◽  
Andreas Almqvist
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Gas Flow ◽  
Present Model ◽  
Rarefied Gas ◽  
Slip Flow ◽  
Local Scale ◽  
Heterogeneous Porous Media ◽  
Scale Model ◽  
Rarefied Gas Flow

Abstract This paper presents the development of a model enabling the analysis of rarefied gas flow through highly heterogeneous porous media. To capture the characteristics associated with the global- and the local-scale topology of the permeable phase in a typical porous medium, the heterogeneous multi-scale method, which is a flexible framework for constructing two-scale models, was employed. The rapid spatial variations associated with the local-scale topology are accounted for stochastically, by treating the permeability of different local-scale domains as a random variable. The results obtained with the present model show that an increase in the spatial variability in the heterogeneous topology of the porous medium significantly reduces the relevance of rarefaction effects. This clearly shows the necessity of considering a realistic description of the pore topology and questions the applicability of the results obtained for topologies exhibiting regular pore patterns. Although the present model is developed to study low Knudsen number flows, i.e. the slip-flow regime, the same development procedure could be readily adapted for other regimes as well.

scholarly journals On the validity of effective formulations for transport through heterogeneous porous media

2016 ◽  
Vol 20 (4) ◽  
pp. 1319-1330 ◽  
Author(s):  
Jean-Raynald de Dreuzy ◽  
Jesus Carrera
Keyword(s):  
Porous Media ◽  
Initial Conditions ◽  
Heterogeneous Porous Media ◽  
Local Concentration ◽  
Correlation Pattern ◽  
Effective Transport ◽  
Modeling Transport ◽  
Fine Structures ◽  
Dispersive Mixing ◽  
Permeability Field

Abstract. Geological heterogeneity enhances spreading of solutes and causes transport to be anomalous (i.e., non-Fickian), with much less mixing than suggested by dispersion. This implies that modeling transport requires adopting either stochastic approaches that model heterogeneity explicitly or effective transport formulations that acknowledge the effects of heterogeneity. A number of such formulations have been developed and tested as upscaled representations of enhanced spreading. However, their ability to represent mixing has not been formally tested, which is required for proper reproduction of chemical reactions and which motivates our work. We propose that, for an effective transport formulation to be considered a valid representation of transport through heterogeneous porous media (HPM), it should honor mean advection, mixing and spreading. It should also be flexible enough to be applicable to real problems. We test the capacity of the multi-rate mass transfer (MRMT) model to reproduce mixing observed in HPM, as represented by the classical multi-Gaussian log-permeability field with a Gaussian correlation pattern. Non-dispersive mixing comes from heterogeneity structures in the concentration fields that are not captured by macrodispersion. These fine structures limit mixing initially, but eventually enhance it. Numerical results show that, relative to HPM, MRMT models display a much stronger memory of initial conditions on mixing than on dispersion because of the sensitivity of the mixing state to the actual values of concentration. Because MRMT does not restitute the local concentration structures, it induces smaller non-dispersive mixing than HPM. However long-lived trapping in the immobile zones may sustain the deviation from dispersive mixing over much longer times. While spreading can be well captured by MRMT models, in general non-dispersive mixing cannot.

Two Types of Nonlinear Pressure-Drop Versus Flow-Rate Relation Observed for Saturated Porous Media

1997 ◽  
Vol 119 (3) ◽  
pp. 700-706 ◽  
Author(s):  
J. L. Lage ◽  
B. V. Antohe ◽  
D. A. Nield
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Hydrostatic Pressure ◽  
Pressure Gradient ◽  
Fluid Velocity ◽  
Rate Of Change ◽  
Transition To Turbulence ◽  
Abnormal Behavior ◽  
Hydrostatic Pressure Gradient ◽  
Transition Speed

Previous reports of experiments performed with water (Fund et at., 1987 and Kececioglu and Jiang, 1994) indicated that beyond the Forchheimer regime the rate of change of the hydrostatic pressure gradient along a porous medium suddenly decreases. This abnormal behavior has been termed “transition to turbulence in a porous medium.” We investigate the relationship between the hydrostatic pressure gradient of a fluid (air) through a porous medium and the average seepage fluid velocity. Our experimental results, reported here, indicate an increase in the hydrostatic pressure rate beyond a certain transition speed, not a decrease. Physical arguments based on a consideration of internal versus external incompressible viscous flow are used to justify this distinct behavior, a consequence of the competition between a form dominated transition and a viscous dominated transition. We establish a criterion for the viscous dominated transition from consideration of the results of three porous media with distinct hydraulic characteristics. A theoretical analysis based on the semivariance model validation principle indicates that the pressure gradient versus fluid speed relation indeed departs from the quadratic Forchheimer-extended Darcy flow model, and can be correlated by a cubic function of fluid speed for the velocity range of our experiments.

scholarly journals Dispersion Coefficients for Gases Flowing in Consolidated Porous Media

1967 ◽  
Vol 7 (01) ◽  
pp. 43-53 ◽  
Author(s):  
Max W. Legatski ◽  
Donald L. Katz
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Electrical Resistivity ◽  
Natural Gas ◽  
Dispersion Coefficient ◽  
Longitudinal Dispersion ◽  
Rock Properties ◽  
Dispersion Coefficients ◽  
On Line ◽  
Mixing Problem

Abstract The best currently available description of the longitudinal mixing properties of a porous medium is an equation of the formEquation 1 which relates the effective longitudinal dispersion coefficient Dl to the molecular diffusion coefficient D0, the electrical resistivity factor F, the porosity f and a Peclet number. If the parameters dps and m are determined for a porous medium of known porosity and electrical resistivity factor, then a dispersion coefficient may be estimated for a given flow rate and a given gas pair. A new method, featuring on-line gas analysis by thermal conductivity and on-line data reduction by analog computation, was developed and used to determine these mixing parameters for eight naturally occurring sandstones and two dolomite samples. The exponent m of the above equation was found to vary between 1.0 and 1.5. The characteristic length dp s in the above equation was found to vary between 0.25 and 1.9 cm, with an average value of 0.4 cm for sandstones. Measurements were made on two cores in which paraffin wax had been deposited by evaporation from a pentane solution. They indicated that the presence of an immobile phase such as connate water could increase the dispersion coefficients significantly. INTRODUCTION While the petroleum and chemical industries have studied the mixing of miscible liquids flowing in consolidated porous media and of miscible gases flowing in unconsolidated porous media, relatively little data have been presented to describe the mixing of gases flowing through consolidated porous media. Such data are of particular interest to the gas storage industry. For instance, the U.S. Bureau of Mines is storing large quantities of a rich helium-nitrogen gas in contact with a natural gas in a dolomite reservoir. Since the rich gas occupies only 15 percent of the total reservoir volume, it is essential that the extent of rich gas-natural gas mixing be predicted and understood as a function of rock properties, pressure and rate of movement. This investigation was concerned only with the determination of longitudinal dispersion coefficients. It is understood that a transverse dispersion coefficient, which characterizes mixing perpendicular to the direction of flow, may be an order of magnitude less than the coefficient characterizing mixing in the direction of bulk flow.5,19 It should also be recognized that the use of any dispersion coefficient is in itself a simplification. It is necessary to assume that mixing in a porous medium may be characterized by the equationEquation 2 for flow in a single direction. A number of authors1 have pursued the mixing problem, not in terms of the so-called "dispersion model" described by Eq. 1, but in terms of a "mixing cell model". This model supposes that a porous medium is constructed of a large number of small mixing chambers and that the concentration of the diffusing component within each mixing chamber is uniform. Fick's law (Eq. 1) assumes that there is no gross by-passing of one fluid by another, and that there are not stagnant pockets of gas in the system under consideration as discussed by Coats and Smith.8 These assumptions are not always valid for flow through porous media and it is important to recognize the limitations upon Eq. 1.

scholarly journals Experimental investigation of dispersion phenomenon in a fractured porous medium

2015 ◽  
Vol 4 (1) ◽  
pp. 209
Author(s):  
Ali Sanati ◽  
Mohammad Yousefi Khoshdaregi
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Dispersion Coefficient ◽  
Flow Direction ◽  
Fractured Reservoirs ◽  
Miscible Displacement ◽  
Naturally Fractured Reservoirs ◽  
Volume Calculation ◽  
Fractured Porous Medium ◽  
Core Samples

Dispersion of fluids flowing through porous media is an important phenomenon in miscible displacement. Dispersion causes instability of miscible displacement flooding; therefore, to obtain and maintain the miscibility zone, the porous medium dispersivity should be considered in displacing fluid volume calculation. Many works have been carried out to investigate the dispersion phenomenon in porous media in terms of theory, laboratory experiments and modeling. What is still necessary is to study the effects of presence of fracture in a porous medium on dispersion coefficient or dispersivity. In this work dispersion phenomenon in a fractured porous medium has been investigated through a series of miscible displacement tests on homogeneous sandstone core samples. Tests were repeated on the same core samples with induced fracture in the flow direction. The effects of fracture on miscible displacement flooding have been studied by comparison of the results of dispersion tests in the absence and presence of fracture. In the presence of fracture, breakthrough time reduced and the tail of effluent S-shaped curve smeared. Moreover, the slope of S-shaped curve at 1 pore volume of injected fluid was lower than homogeneous case which means dispersion coefficient increased. The results presented in this work provide an insight to the understanding of dispersion phenomenon for modeling of miscible displacement process through naturally fractured reservoirs.

scholarly journals Effects of Velocity and Permeability on Tracer Dispersion in Porous Media

2021 ◽  
Vol 11 (10) ◽  
pp. 4411
Author(s):  
Yulong Yang ◽  
Tongjing Liu ◽  
Yanyue Li ◽  
Yuqi Li ◽  
Zhenjiang You ◽  
...  
Keyword(s):  
Porous Media ◽  
Flow Velocity ◽  
Dispersion Coefficient ◽  
Fluid Velocity ◽  
Tracer Tests ◽  
Volumetric Flow Rate ◽  
Flow In Porous Media ◽  
Low Permeability ◽  
Regression Equations ◽  
Medium Permeability

During micro-scale tracer flow in porous media, the permeability and fluid velocity significantly affect the fluid dispersion properties of the media. However, the relationships between the dispersion coefficient, permeability, and fluid velocity in core samples are still not clearly understood. Two sets of experiments were designed to study the effects of tracer fluid flow velocity and porous medium permeability on the dispersion phenomenon in a core environment, using natural and sand-filled cores, respectively. From experimental data-fitting by a mathematical model, the relationship between the dispersion coefficient, flow velocity, and permeability was identified, allowing the analysis of the underlying mechanism behind this phenomenon. The results show that a higher volumetric flow rate and lower permeability cause a delay in the tracer breakthrough time and an increase in the dispersion coefficient. The core experimental results show that the dispersion coefficient is negatively correlated with the permeability and positively correlated with the superficial velocity. The corresponding regression equations indicate linear relations between the dispersion coefficient, core permeability, and fluid velocity, resulting from the micron scale of grain diameters in cores. The combination of high velocity and low permeability yields a large dispersion coefficient. The effects of latitudinal dispersion in porous media cannot be ignored in low-permeability cores or formations. These findings can help to improve the understanding of tracer flow in porous media, the design of injection parameters, and the interpretation of tracer concentration distribution in inter-well tracer tests.

scholarly journals On the validity of effective formulations for transport through heterogeneous porous media

2015 ◽  
Vol 12 (11) ◽  
pp. 12281-12310 ◽  
Author(s):  
J.-R. de Dreuzy ◽  
J. Carrera
Keyword(s):  
Porous Media ◽  
Initial Conditions ◽  
Heterogeneous Porous Media ◽  
Local Concentration ◽  
Correlation Pattern ◽  
Effective Transport ◽  
Modeling Transport ◽  
Fine Structures ◽  
Dispersive Mixing ◽  
Permeability Field

Abstract. Geological heterogeneity enhances spreading of solutes, and causes transport to be anomalous (i.e., non-Fickian), with much less mixing than suggested by dispersion. This implies that modeling transport requires adopting either stochastic approaches that model heterogeneity explicitly or effective transport formulations that acknowledge the effects of heterogeneity. A number of such formulations have been developed and tested as upscaled representations of enhanced spreading. However, their ability to represent mixing has not been formally tested, which is required for proper reproduction of chemical reactions and which motivates our work. We propose that, for an effective transport formulation to be considered a valid representation of transport through Heterogeneous Porous Media (HPM), it should honor mean advection, mixing and spreading. It should also be flexible enough to be applicable to real problems. We test the capacity of the Multi-Rate Mass Transfer (MRMT) to reproduce mixing observed in HPM, as represented by the classical multi-Gaussian log-permeability field with a Gaussian correlation pattern. Non-dispersive mixing comes from heterogeneity structures in the concentration fields that are not captured by macrodispersion. These fine structures limit mixing initially, but eventually enhance it. Numerical results show that, relative to HPM, MRMT models display a much stronger memory of initial conditions on mixing than on dispersion because of the sensitivity of the mixing state to the actual values of concentration. Because MRMT does not restitute the local concentration structures, it induces smaller non-dispersive mixing than HPM. However long-lived trapping in the immobile zones may sustain the deviation from dispersive mixing over much longer times. While spreading can be well captured by MRMT models, non-dispersive mixing cannot.

On oscillating flows in randomly heterogeneous porous media

2010 ◽  
Vol 368 (1910) ◽  
pp. 197-216 ◽  
Author(s):  
M. G. Trefry ◽  
D. McLaughlin ◽  
G. Metcalfe ◽  
D. Lester ◽  
A. Ord ◽  
...  
Keyword(s):  
Porous Medium ◽  
Porous Media ◽  
Frequency Domain ◽  
Stochastic Perturbation ◽  
Chemical Properties ◽  
Momentum Equation ◽  
Heterogeneous Porous Media ◽  
Spatial Density ◽  
Perturbation Approach ◽  
Spectral Densities

The emergence of structure in reactive geofluid systems is of current interest. In geofluid systems, the fluids are supported by a porous medium whose physical and chemical properties may vary in space and time, sometimes sharply, and which may also evolve in reaction with the local fluids. Geofluids may also experience pressure and temperature conditions within the porous medium that drive their momentum relations beyond the normal Darcy regime. Furthermore, natural geofluid systems may experience forcings that are periodic in nature, or at least episodic. The combination of transient forcing, near-critical fluid dynamics and heterogeneous porous media yields a rich array of emergent geofluid phenomena that are only now beginning to be understood. One of the barriers to forward analysis in these geofluid systems is the problem of data scarcity. It is most often the case that fluid properties are reasonably well known, but that data on porous medium properties are measured with much less precision and spatial density. It is common to seek to perform an estimation of the porous medium properties by an inverse approach, that is, by expressing porous medium properties in terms of observed fluid characteristics. In this paper, we move toward such an inversion for the case of a generalized geofluid momentum equation in the context of time-periodic boundary conditions. We show that the generalized momentum equation results in frequency-domain responses that are governed by a second-order equation which is amenable to numerical solution. A stochastic perturbation approach demonstrates that frequency-domain responses of the fluids migrating in heterogeneous domains have spatial spectral densities that can be expressed in terms of the spectral densities of porous media properties.

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