Dispersion Measurements in Highly Heterogeneous Laboratory Scale Porous Media

Transport in Porous Media 54: 107–124, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

107

Dispersion Measurements in HighlyHeterogeneous Laboratory Scale Porous Media

STEVEN P. K. STERNBERGDepartment of Chemical Engineering, University of Minnesota Duluth, Duluth, MN 55812, USA

(Received: 10 December 2001; in final form: 24 June 2003)

Abstract. Laboratory experiments and numerical simulations investigate conservative contaminanttransport in a heterogeneous porous medium. The laboratory experiments were performed in cyl-indrical columns 1 m long and 3.5 cm inside diameter filled with spherical glass beads. Concentrationbreakthrough curves are measured at a scale much finer than the size of the heterogeneity. Numericalsimulations are based on a random walk in a known constant velocity field. The heterogeneity isa distinct, discontinuous change in the local permeability field. Fluid flow is miscible, flowing in asaturated porous medium. Previous work has shown this to be a very poorly understood phenomenon.The measurements reported here help to better understand how dispersion evolves through and pasta heterogeneity.

Key words: experimental measurement, Monte Carlo, numerical simulation, miscible flow, saturatedflow, scale effect, non-Fickian.

1. Introduction

This project examines the miscible displacement of a fluid by a contaminatedfluid as the fluids flow through a saturated heterogeneous porous medium. Themovement through the porous medium causes the contaminant to disperse (spread)into the other fluid. This phenomenon has been well modeled for homogeneousmedia, but modeling in heterogeneous media is still incomplete. An understandingof the effects on mixing caused by heterogeneities can help determine the rateand direction of the contaminant movement in natural porous media. Applicationsinclude oil, water, and gas production, groundwater development and protection,and packed bed catalysis reactions.

The classical theory describing this spreading or dispersion, based on Fick’s firstand second laws, combines mass conservation and the time rate of change for theflux of a contaminant moving through a porous medium. The equations are basedon the assumptions that the mixing is similar to diffusion (a statistically randomprocess), that the contaminant tracer is conservative (it does not react or sorb to themedium), and that the medium properties (porosity, permeability, and dispersivity)are homogeneous on the scale of measurement. Here a homogeneous material isdefined as media with constant system properties that have a single characteristiclength scale, l, such as a pore size or grain size that is smaller than the measurement

108 STEVEN P. K. STERNBERG

size; L. Fick’s laws are generally useful only after the contaminant has traversedapproximately fifty such characteristic length scales (l � L).

Another property often reported for a porous medium is dispersivity. Dispersionis known to vary with velocity, and an empirical relationship has been proposed toaccount for this:

D = αvm (1)

where D is the measured dispersion coefficient, v is the velocity, α is the disper-sivity (also known as the characteristic length when the coefficient m is equal to1) and m is an empirical exponential factor, often with a value in the range from0.8 to 1.2 for homogeneous media. Dispersivity attempts to factor out the velocityrelationship and hence provides a single value describing the effect that a givenmedium has on mixing. While this relationship appears useful for homogeneousmedia, it is not expected to be a useful parameter in heterogeneous media, as theempirical value m changes in each new region.

Dispersion mechanisms or conditions that cause the spreading of a contaminantduring flow through a porous medium may be lumped into five categories – diffu-sion, fluid effects, medium effects, fluid–medium interactions, and boundary/initialconditions. These mixing mechanisms or system conditions (Greenkorn, 1983)have differing effects based on the strength and direction of the fluid velocity.

• Diffusion causes mixing due to random molecular motions, causing the con-taminant to, on average, move from regions of high concentration to regionsof low concentration. It becomes less important at larger fluid velocities asother mechanisms overwhelm diffusion.

• Fluid effects (gravity, density and viscous differences between fluids, satu-ration levels, and turbulence) cause one fluid to move faster or in differentdirections than the bulk flow. The different fluids may vary simply in the con-taminant amount present (causing viscosity and density difference) or may bedifferent chemical species or mixtures of species. An important considerationis how easily the different fluids mix – miscible fluids freely mix, immisciblefluids do not mix.

• Medium effects (tortuosity, auto-correlation, dead end pores and recirculation)cause spreading due to the nature and alignments of the pores. Some pores willbe longer or wider and will have faster average velocities than others. Thiscauses mixing when the fluid in one pore reaches a downstream location at adifferent time than the fluid in a different pore. The pore structure may alsotrap some contaminant in a dead-end or recirculation zone, causing a time lagbefore it returns to the main flow field, which can be seen in plumes with longtails. These mechanisms may be dominant in cases of anomalous dispersion.

• Fluid–medium interactions (adsorption, chemical reaction, hydrodynamics,and heterogeneity effects) cause spreading by interfering with the ability ofthe contaminant to pass unhindered through the pore space. Hydrodynamicdispersion is caused by friction between the fluid and the pore walls. The

DISPERSION MEASUREMENTS 109

heterogeneity effect, the main focus of this work, changes the amount ofdispersion in a variety of ways and it operates over all the time and spatialscales of interest (Sternberg et al., 1996a, b). Heterogeneities may alter anyof the medium characteristics, and until the system samples all such changesmultiple times, the fluids will behave preasymptotically.

• The boundary and initial conditions can skew results when they are not prop-erly described. Edge effects, such as fluid short-circuiting, are important onlywhen the medium thickness (in direction perpendicular to flow) is small.Entrance and exit effects can cause large changes in fluid velocities and direc-tions over small distances. Some preliminary work (Sternberg, 1998) showsthat entrance effects can account for over 50% of dispersion in poorly plannedlaboratory measurements. These effects can be compensated for by measuringactual concentrations around the inlet and outlet.

Systems that have non-constant or large length scales in comparison to the scaleof measurement (l ≈ L) are called heterogeneous. At heterogeneous scales, andwhere the system parameters change, the Fick’s law assumptions are no longervalid and the classical theory does not predict the amount of mixing. A system canbe homogeneous at one length scale, heterogeneous at another and again homog-eneous at some other scale. This type of system is said to have an evolving orhierarchical heterogeneity (Cushman and Ginn, 1993a, b).

Another modeling concern is the scale at which the system is heterogeneous.If the heterogeneity is much finer than the system size of interest, its effects canbe averaged or homogenized (Panfilov, 2000). If the heterogeneity is at the size ofthe system (l ≈ L), or larger than the system of interest (l >L), it is very difficultto model and it cannot be homogenized or described with an effective or averagedsystem parameter. There is very little information known about how such systemsbehave; yet these systems can be readily encountered in natural porous materialson scales of human interest. It is these large-scale heterogeneities that are examinedin this work.

Many current theories for transport in heterogeneous porous media place strictlimits on the nature of the heterogeneity such that properties must vary smoothly,have only small changes in property distributions, and/or that the heterogeneitylength scale must be much smaller than the system size (l � L). These constrainedtheories may not seem realistic for all natural media, yet they offer a significantimprovement from theory based on Fick’s laws assumptions. There has been muchwork on the theoretical development of describing this mixing phenomenon inheterogeneous systems with the stochastic and non-local theories (Koch and Brady,1987a, b, 1988; Naff, 1990; Cushman and Ginn, 1993a, b; Neuman, 1993) Thesenew theories use information from many scales simultaneously and recast the de-scriptive equations to use multiple types of information at multiple scales, and arecalled scale-dependent or non-local theories (Tompson, 1988; Cushman and Ginn,1994). This work presents experimental results that may be useful for testing suchmodels, but does not examine the results with these new theories.


Modeling at scale transitions with finite, but changing Peclet numbers (Pe =vL/D) shows great sensitivity of mixing to smaller scale dispersive mechanismswhen observed at sizes below 10 characteristic length scales (Fiori, 1998). Buckleyet al., 1994 perform a numerical study that compares the standard Fick’s laws equa-tion with another transport equation that includes pore scale (fracture) information.Results were more realistic when including the multi-scale information.

Experimental evidence shows that mixing is difficult to model at scales similarto a heterogeneity (Sternberg et al., 1996a, b), yet once a plume travels 20–50characteristic length scales with no major changes in property distribution it willagain behave in a consistent manner (Silliman et al., 1987; Silliman, 2001; Sillimanand Zheng, 2001).

This work experimentally explores dispersion at the transition from homogen-eous to heterogeneous and seeks to quantify these observations. The highly heterog-eneous systems are hard to model because the system properties are discontinuous.Another problem in modeling heterogeneous systems is a lack of knowledge ofwhat occurs in heterogeneous systems. Many experimental results in heteroge-neous systems show very complex behaviors that do not follow the well-establishedhomogeneous results. No general theories model dispersion across a discontinuity,and very little data exists (Greenkorn et al., 1964, 1965; Pakula and Greenkorn,1971; Gupta and Greenkorn, 1973, 1974). No theories account for the transitionbetween layers nor discuss how far a plume needs travel after encountering a dis-tinct property change before it again behaves asymptotically, that is, the dispersionvalue again becomes constant with distance.

2. Methods

2.1. LABORATORY EXPERIMENTS

Experimental measurements of porosity, permeability, and dispersion were made inlaboratory constructed porous media. The porous media was constructed of randompackings of spherical glass beads. The beads were contained in cylindrical polycar-bonate columns. All experiments were done in triplicate, with the averages reportedon the figures. The error in the porosity measurements is approximately 3% (onestandard deviation). The error in the permeability measurements is approximately1.5% within a single column, and 20% between similarly packed columns. Thefirst number shows that the permeability can be measured quite precisely in anyone column, while the second (higher) value provides an estimate on the abilityto reproduce a column packing. The error in the dispersion measurements wasapproximately 22% for the coarse (or large diameter) bead medium and 12% forthe fine bead medium. These errors measure the reproducibility of the results basedon the laboratory conditions (precision) in a single column and are not a measureof accuracy based on comparison with known values. Accuracy error cannot becalculated because there are no published values for these properties for thesemedia.


Table I. Porous media properties

Bead type Bead diameter (mm) Porosity Permeability (Darcy) Dispersivity (cm)

Coarse 0.595 0.35 150 0.0175

Fine 0.163 0.35 8.5 0.0035

2.1.1. Measurements

Porosity was measured by weighing a packed column when dry and weighing itagain when saturated with water. This weight difference is converted to the me-dium’s void volume by dividing by the density of the water. Once the medium voidvolume is known, the porosity is obtained by dividing by the total internal volumeof the column (including voids and solids). Values were adjusted to account forentrance and exit voids that were not a part of the porous medium. Results arepresented in Table I.

Permeability was obtained by measuring the difference in height between mano-meter tubes attached to the column inlet and outlet (�h) for several different fluidflow rates (Q). This information, along with fluid properties of density and viscos-ity and medium properties of cross sectional area (A) and length (L), is used in theDarcy equation to obtain a value for the permeability.

k = QLµ

�hAρg(2)

The values are reported in Table I and use the slope of a best-fit line of �h as afunction of fluid flow rate. Flow rates included values above and below the valuesused during the measurements of dispersion.

Dispersion coefficients were calculated from experimental measurements ofconcentration breakthrough curves. A schematic for the dispersion measurementequipment is shown in Figure 1. Contaminant breakthrough curves are created bydisplacing a weak solution of potassium chloride (0.02 M) with a strong solution(0.2 M). Silver/silver chloride electrodes register the changing chloride concentra-tion as a difference in electrochemical potential between the individual electrodesimbedded in the columns and a reference electrode in the inlet line (Skoog andWest, 1982). The chloride sensitive tip was positioned at the center of the column.The change in concentration is detected as a change in potential at each electrode asthe mixing front passes the electrode location. These potentials are recorded onceevery 2 s at each electrode. Electrodes were spaced every 1.25 cm along the lengthof the column.

The potentials are converted to concentrations using a log-linear calibrationcurve based on the potentials generated at the 0.02 and 0.2 M concentrations.


Figure 1. Schematic of dispersion measurement equipment.

Previous work has shown the system to behave linearly in this range of potentials.Dimensionless concentrations are calculated from the potentials using

C

C0=

[Cmin((Cmax/Cmin)

(Pmax−P)/(Pmax−Pmin)) − Cmin

Cmax − Cmin

](3)

where C/C0 is the dimensionless concentration, Pmax is the maximum potentialreading (approximately 50 mV), Pmin is the minimum potential reading (approxi-mately 0 mV), P is the actual potential reading, Cmax is 0.20 M, and Cmin is 0.02 M.Once the concentrations are calculated, the time associated with the 0.3, 0.5, and0.7 concentrations are obtained from the data. These time values (s) are then used tocalculate the overall dispersion coefficient, which quantifies the amount of mixingbetween the entrance and the electrode position (L) for a given system velocity(v). Dispersion values are calculated based on a modified form of the calculationpresented by Bear (1972)

D = 2Lv

[(t0.70 − t0.30)

2

t20.50

](4)

All calculations were done automatically in the data acquisition and analysis soft-ware (LabView 5.1, National Instruments Corporation, Austin, TX, U.S.A.).

Equation (4) is a useful tool for analyzing s-shaped breakthrough curves. It isused in this work to calculate the amount of mixing (dispersion) in the experimentaland simulated media between the entrance and the point of measurement at dis-tance L from the entrance. Figure 2 shows an example of the breakthrough profilesobtained in this study. It shows the concentration change with time for several po-sitions in the column. The actual experiments measured concentrations over time atmore than 70 positions. These breakthrough curves are s-shaped, especially in the


Figure 2. Example breakthrough curves.

region used to calculate dispersion (30–70%). The spread in concentration valuesat the high-end tail is an artifact of the log-linear relationship between potentialand concentration (Eq. (3)). Note that this equation is not a model equation for thesystem nor the general solution to any of the advection–dispersion equations, yet itdoes provide a standard method to quantify the spread in a miscible mixing front.

2.1.2. Equipment

2.1.2.1. Beads. The media consisted of two distributions of spherical glass beads(95% of beads have a sphericity of 0.9 or better). A description of each bead typeis given in Table I. The beads are a boro-silicate glass shot (CATAPHOTE Inc.,Jackson, MS, U.S.A.). They are specially made to not dissolve in water. A simpledissolution test of storing the beads in distilled water in a sealed plastic containerfor 3 years showed less than 1% loss of bead mass. Water pH did increase from 6.9to 10.2 over the 3-year time, suggesting that some dissolution does occur over longtimes.

2.1.2.2. Columns. Experiments were conducted in cylindrical polycarbonatecolumns of inside diameter 3.2 cm and outside diameter of 4.5 cm. Column lengthwas 1.0 m. The beads were packed into the column in two arrangements (hetero-geneous), plus two packings of a single bead class (homogeneous), yielding fourpacking arrangements. The two heterogeneous packings consist of two layers, thefirst layer filling one-third of the column length and the second filling the remainingcolumn (see Figure 3). This arrangement was chosen over the half fill arrangementbecause preliminary results showed that mixing in the first layer acted exactly likethe homogeneous case, but that mixing in the second layer evolved over longer


Figure 3. The four column packing arrangements.

distances. Only the top end cap is removable. This allows much tighter and moreuniform packing than previous column models and greatly reduces fluid leaking(Sternberg et al., 1996a, b).

Bead layers were separated using a polycarbonate screen. This screen materialdoes not interact with the electrochemical measurements (unlike metal screens)and does not adsorb chloride on the time scale of these experiments. It also hasbeen shown to not affect the porosity, permeability, or dispersion as measured inthis work.

2.1.2.3. Electrodes. Electrodes were made of 5 cm long, 0.1 cm diameter silverwire (Goodfellow Corporation, Berwyn, PA, U.S.A.). Each electrode was tippedwith a 0.5 cm long chloride coat (Skoog and West, 1982). Total surface area of thecoating was 0.17 cm2 ± 20%. The electrodes are referenced to an electrode placedin the inlet solution (0.2 M KCl). This generates a 50 mV difference with electrodesexposed to 0.02 M KCl. The electrodes are sensitive to the chloride concentrationand not conductivity, so distance between electrode and reference does not affectthe signal at the system scales used in this experiment. However, there must exist acontinuous electrolytic path between the two electrodes for a signal to be produced.

2.1.2.4. Measurement Recording. The potentials generated between the silver/silver chloride electrodes and the reference electrode were measured and record-ed using a National Instrument AT-MIO-16E-10 Multifunction I/O board with 2AMUX 64T multiplexer cards to allow simultaneous measurement of up to 128electrodes. The software interface between the I/O board and computer was theNational Instruments LabView 5.1 package, with user written code (available fromthe author).

2.1.2.5. Column Support. The column is supported with plastic coated clampsand is placed inside a Faraday cage to reduce the amount and intensity of


stray electrical noise picked up by the electrodes, wires, and supportequipment.

2.1.2.6. Tracer Solutions. The 0.02 and 0.2 M KCl solutions were made by di-luting 22.4 and 224.0 g of potassium chloride, KCl (Fisher Scientific Chemicals)with 15 l of distilled and deionized water, respectively. Appropriate mixing insuredhomogeneous solutions. Potassium chloride is used, rather than sodium chloride,because the potassium and chloride ion mobilities are almost the same, whichminimizes the liquid junction potential. The liquid junction potential is causedwhen two electrolytic solutions of different concentrations are placed in contactwith each other and there is a difference in the rate of migration (diffusion) ofthe anions and cations, causing one to pull the other along faster than the averageadvective speed. The 0.02 M KCl has a published specific gravity of 1.0014 andviscosity of 0.999 relative to water and the 0.2 M KCl has a density of 1.0078 andviscosity of 0.997 (Weast, 1986).

2.1.2.7. Experimental Design. Four individual column packings were investi-gated, see Figure 3. Each column used one (homogeneous) or two (heterogeneous)of the two distinct bead distributions. Heterogeneous packings consisted of two dis-tinct layers of beads, as discussed in Section 2.1.2.2. Each packing was measuredfor porosity and permeability, results in Table I. Dispersion measurements weretriplicates for each column at each of three velocities. Multiple velocities allowvelocity effects to be examined, and allow a dispersivity value to be calculated.This gives a total of 36 experimental runs. Triplicate results were averaged, and theaverage values are shown in the dispersion graphs.

Common experimental concerns and errors were addressed in this study. Poorelectrode response: Electrical noise such as spikes or large fluctuations can happenat any electrode at any time. Such recordings yield poor results when it occurswhile the mixing front is passing a given electrode. Such results are discardedwhen (error/N) was more than 2 standard deviations from the average electrodestandard deviation for all runs. Poor packing and bed settling: This tends to causevery large increases in dispersion values near the entrance or exit and it occurs overperiods of weeks. Whenever detected the column was repacked and any previousresult is discarded. Electrode deterioration: Occasionally an electrode would losesensitivity to the tracer. It would yield very large dispersion values and unrealisticadvective times. These results would be discarded if they were observed for severalruns at the same position. Leaks can occur at fluid entrance and exit points in thecolumn or the electrode ports. Such error is unacceptable and requires a resealingof the column. Fluid short-circuiting may happen if the column diameter is smallcompared to the bead size. The standard rule is that the column diameter must beat least 20 times larger than the packed bead or there will be an increase in porosityalong the column wall. This creates a short circuit that the fluid will preferentiallyfollow. The column diameter in this study is 32.0 mm, which is 50 times larger than


the coarse bead and 200 times larger than the fine bead. Layer transition: Fine meshpolycarbonate screens are used to separate the layers of beads and at the entranceand exit of the column. Preliminary work showed these screens to have no effecton the porosity, permeability, or dispersion measurements.

2.2. NUMERICAL SIMULATION

Experimental results were simulated using a Monte Carlo particle tracking tech-nique. This method of describing the dispersion process in porous media has beenused since the 1950s (de Josselin de Jong, 1958; Scheidegger, 1958) and is based onthe analogy between diffusion theory and stochastic processes. It has been shownthat the results are analogous to the analytical solution of the advection–dispersionequation with the appropriate initial and boundary conditions. This implies that theresults of either method (numerical or analytical) will provide equal results, thoughthe random walk solution will be specific to the particular case and the analyticalsolution will be more general.

In this work, the system is discretized into layers, each with a dispersion param-eter based on the permeability of the modeled media (Sternberg et al., 1996a). Timeis a discrete variable and position is a continuous variable. Velocity is constantthroughout the simulations in both time and space. This realistically models theexperiments, since each layer has a constant, and equal, porosity. This somewhatcounter intuitive fact is the result of all layers being constructed of a narrow dia-meter range of spherical glass beads. Dispersion is treated as a random variable,and it modeled using a Gaussian random variable whose magnitude is proportionalto the permeability of the bead layers. This type of random walk simulation isclassified as an Eulerian–Lagrangian system.

The system was trained to provide accurate values of dispersion for the ho-mogeneous layers for all the velocities used in the laboratory experiments. Thesystem is capable of modeling dispersion for a variety of velocities based on asingle value (the permeability) describing each medium, and one other number,used to fit the simulations to the observed data. This single fit parameter is thesame for all simulations and includes all the velocities and bead sizes.

The formula for moving a particle is:

xnew = xold + v(xold)�t + [M(xold)�t]0.5 RN (5)

where xnew denotes the new time particle position (at t+�t) and xold is the old timeposition (t), v is the velocity at the old time position, M is the dispersion multiplier(a constant that depends on the layer permeability), �t is the time step, and RNis a Gaussian random number of zero mean and unit variance. Uniform randomnumbers in [0, 1] were obtained using the random number generator described inPress et al. (1992) (subroutine RAN2) and then adjusted to provide a Gaussianrandom number. The one-dimensional media is discretized into 1.25 cm layers,


each described with the medium permeability that corresponds to the media usedin the laboratory experiments (see Figure 3).

Two thousand particles are used in each simulation. Each of these simulationsis replicated four times, and averages of these four replications are reported asresults. Each replication uses a different random number seed. Initially, the numberof particles and number of replications was varied until results were consistent tothe second significant figure (Botz et al., 1993). No greater precision was desiredsince the experiments that are being modeled have a 10–25% precision error. Thismay be noticed in the results, where the numerical simulations are not completelysmooth.

Statistics (first and second moment) are generated at the end of each 1.25 cmlayer. Homogeneous media are simulated by using a single, constant value of M inevery layer. Heterogeneous media are simulated with two values of M, correspond-ing to the experimental work. The output distribution of each layer is used as theinput to the next layer. No averaging or mixing rules are applied between layers.Dispersion is calculated from these statistics using:

D = Lv

2

[var(t)

ave(t)2

](6)

where L is the total length between the entrance and the calculation point, v isthe average one-dimensional velocity (obtained by dividing the length by the timerequired to reach that length), var(t) is the variance of the time required by theparticles to travel the distance L, ave(t) is the average time required by the particleto travel the distance L. This calculation is based on the method recommended byBear (1972) for experimental calculations for dispersion, and it is similar to thecalculation used for the laboratory experiments described in this paper (Eq. (4)).

Dispersion profiles are simulated for each of the experimental media at all ofthe experimental positions and velocities. No averaging or mixing rules are used atthe permeability transitions. The dispersion multiplier is always calculated basedon the simple local values; no attempt is made to average, homogenize, or smoothaway the transitions. The output from one layer is used as the input for the nextlayer.

3. Results and Discussion

3.1. MEDIUM PROPERTIES

The properties for the two bead sizes examined in this study are presented inTable I. It includes the mean bead diameter, homogeneous column porosity, per-meability, and dispersivity (as calculated by Eq. (1)). These measurements quantifythe description of the experimental media. The results are well within values report-ed for similar media.


3.2. EXPERIMENTAL SCALES

There are a number of characteristic length scales in these experiments. The timescale for the numerical simulation discretization is 1 s. The time scale of the ex-periments is 2 s. These values represent the time elapsed between measurements.The spatial scales include the size of the beads (0.1–0.5 mm) the size of the chloridesensitive electrode (0.5 cm), the distance between measurements (1.25 cm), the sizeof the individual layers or each medium (30 cm) and finally the total overall lengthof the medium (90 cm).

The time scale values were chosen for convenience (simulation) or as a lim-itation of the measurement system. No trends, other than the concentration frontbreakthrough curves, were noticeable concerning the time scales used. While someof the heterogeneous theories allow for temporal memory effects, no such interest-ing behaviors were readily observable in the collected data.

The spatial scales were chosen to allow observation of the evolution of the mix-ing front as it passes a transition in the system permeability. The bead sizes werechosen to be much smaller than the column diameter to prevent wall channeling.The two bead distributions have distinctly different measured properties from eachother to create the heterogeneities. The electrodes are large enough so that they canmeasure concentration over several pores (between 3 and 10 on average). This isenough to be representative of the local system around the electrode, yet still giveprecise measurement of the mixing front as it passes each measurement location.The distance between the electrodes is much smaller than the size of the porousmedium layers, allowing direct observation of the phenomenon of interest – theevolution of the mixing front. The layer length scale (30 cm for the first layer,60 cm for the second layer) was chosen based on previous work (Sternberg andGreenkorn, 1994). The final length scale, the entire system, was chosen so that thetime required to run an experiment would be in the 1–8 h range.

3.3. DISPERSION MEASUREMENTS

Homogeneous results are shown in Figures 4 and 5. These plots show the dis-persion coefficient measured at each electrode in the fine and coarse media. Eachfigure includes experimental results (points) with the one standard deviation out-lines (dashes) from the three measurements, and a line representing the averagevalue. These graphs give the reader a sense of how experimental error is distrib-uted in these experiments. Only one velocity is shown since the other velocitiesappear similar. It should be noted that the coarse beads, in all cases, yield largererror in their measurement. This is probably due to difficulty in creating a trulyhomogeneous packing of the beads. This is in contrast to the fine beads, whichpack more uniformly, yet also tend to settle more with time. The experimentalvalues and standard errors for all runs are presented in Table II.

Dispersion profiles for the two heterogeneous columns are presented inFigures 6–9. The coarse layer first models are shown for two velocities in


Figure 4. Dispersion profile in fine homogeneous media (velocity = 0.027 cm/s).

Figure 5. Dispersion profile in coarse homogeneous media (velocity = 0.027 cm/s).

Table II. Homogeneous media dispersion measurements

Velocity Fine media Coarse media(cm/s)

Dispersion Standard Error (%) Dispersion Standard Error (%)

(×104 cm/s) deviation (×104 cm/s) deviation

0.0036 2.04 0.16 7.8 2.94 0.43 14.6

0.011 6.82 0.86 12.6 12.4 2.99 24.1

0.016 12.29 1.74 14.1

0.027 20.87 3.28 15.7 54.38 12.12 22.3


Figure 6. Dispersion profile in coarse/fine heterogeneous media (velocity = 0.013 cm/s).

Figure 7. Dispersion profile in coarse/fine heterogeneous media (velocity = 0.032 cm/s).

Figures 6 and 7. The fine layer first models are shown in Figures 8 and 9. In eachfigure, the circles represent the average value of three experimentally measureddispersion coefficients. The dashed line represents the local value of dispersion (asif the measurement was made in just the media around that electrode position andnot including the remainder of the media). The solid line represents the numericalsimulation of dispersion.

In each of these figures, the experimental results in the first layer of the columnsare well modeled by the local values and the numerical simulations. This is as itshould be, since the columns are, up to the transition, homogeneous media. Afterthe transition (and perhaps before it for the fine/coarse media) the simulation doesnot match the experiments. The experimental measurements quickly approach thelocal value; whereas the numerical simulations require over 25 length scales beforeapproaching the local value, see Figures 10 and 11. Note that here the length scaleis the length of the first layer (30 cm). This length corresponds to the standardFickian assumption of distance required before asymptotic behavior is reached


Figure 8. Dispersion profile in fine/coarse heterogeneous media (velocity = 0.013 cm/s).

Figure 9. Dispersion profile in fine/coarse heterogeneous media (velocity = 0.027 cm/s).

(20–50 length scales). Since the simulations are based on the Fickian dispersionmodel, this type of result is expected.

What was unexpected was the speed with which the laboratory-measured val-ues approached the local value – within two or less length scales. This suggeststhat some type of additional mechanism is operating at the transition. The actualmixing is not just the simple average of the layers, as the simulation depicts. Thismechanism is described in Section 1, though it is not quantified. It is not yet knownif this effect is scale dependent, or if it occurs at a specific distance, unrelated tolength of the layers. These results begin to demonstrate what actually occurs ata transition. It is not the behavior expected from the solution of the advection–dispersion equation in layered or stratified porous media. It is also interesting tonote that, for certain arrangements of media, that the dispersion coefficient candecrease with increasing travel time and distance. It will be interesting to expandthese observations to a greater variety of heterogeneities and to include two- andthree-dimensional media as well, as is planned in future work.


Figure 10. Heterogeneous dispersion profile simulation in coarse/fine system (24 lengthscales).

Figure 11. Heterogeneous dispersion profile simulation in fine/coarse system (24 lengthscales).

4. Conclusion

These results begin to show the effects that heterogeneity has on dispersion. Theexperimental observations show that the approach to asymptotic mixing after apermeability transition happens very quickly, on the order of two length scales.This is in contrast to the usual assumption of 20–50 length scales usually assumedwith mixing in a porous medium, which is observed in the numerical simulationsbased on the advection–dispersion equation. It is obvious that the Fickian model ismost pessimistic about the length needed for asymptotic behavior to begin.


References

Bear, L.: 1972, Dynamics of Fluids in Porous Media, Dover, New York.Botz, M. M., Sternberg, S. P. K. and Greenkorn, R. A.: 1993, A multifractal analysis of dispersion

during miscible flow in porous media, Int. Ser. Numer. Math. 114, 15–23.Buckley, R. L., Loyalka, S. K. and Williams, M. M. R.: 1994, Numerical studies of radionuclide

migration in heterogeneous porous media, Nucl. Sci. Eng. 118, 133–144.Cushman, J. H. and Ginn, T. R.: 1993a, Nonlocal dispersion in media with continuously evolving

scales of heterogeneity, Transport in Porous Media 13, 123–138.Cushman, J. H. and Ginn, T. R.: 1993b, On dispersion in fractal porous media, Water Resour. Res.

29(10), 3513–3515.de Josselin de Jong, G.: 1958, Longitudinal and transverse diffusion in granular deposits, Trans. Am.

Geophys. Union 39, 67–74.Fiori, A.: 1998, On the influence of pore-scale dispersion in non-ergodic transport in heterogeneous

formations, Transport in Porous Media 30, 57–73.Greenkorn, R. A.: 1983, Flow Phenomena in Porous Media, Fundamentals and Applications in

Petroleum, Water, and Food Production, Marcel Dekker, New York.Greenkorn, R. A., Haring, R. E., Johns, H. O. and Shallenberger, L. K.: 1964, Flow in heterogeneous

Hele–Shaw models, Petroleum Trans. AIME 231, SPEJ 124.Greenkorn, R. A., Johnson, C. R. and Haring, R. E.: 1965, Miscible displacement in a controlled

natural system, Pet. Trans. AIME 234, 1229.Gupta, S. P. and Greenkorn, R. A.: 1973, Dispersion during flow in porous media with bilinear

adsorption, Water Resour. Res. 9(5), 1357.Gupta, S. P. and Greenkorn, R. A.: 1974, Determination of dispersion and nonlinear adsorption

parameters for flow in porous media, Water Resour. Res. 10(4), 839.Koch, D. L. and Brady, J. F.: 1987a, A non-local description of advection–diffusion with application

to dispersion in porous media, J. Fluid Mech. 180, 387–403.Koch, D. L. and Brady, J. F.: 1987b, Non-local dispersion in porous media; nonmechanical effects,

Chem. Eng. Sci. 42(6), 1377–1392.Koch, D. L. and Brady, J. F.: 1988, Anomalous diffusion in heterogeneous porous media, Phys.

Fluids 31(5), 965–973.Naff, R. L.: 1990, On the nature of the dispersive flux in saturated heterogeneous porous media,

Water Resour. Res. 26(5), 1013–1026.Neuman, S. P.: 1993, Eulerian–Lagrangian theory of transport in space–time nonstationary veloc-

ity fields: exact non-local formalism by conditional moments and weak approximation, WaterResour. Res. 29, 633–645.

Pakula, R. J. and Greenkorn, R. A.: 1971, An experimental investigation of a porous media modelwith nonuniform pores, AIChE J. 17, 1265–1267.

Panfilov, M.: 2000, Macroscale Models of Flow Through Highly Heterogeneous Porous Media,Kluwer Academic Publishers, Boston, MA.

Press, W. H., Vetterling, W. T., Teukolsky, S. A. and Flannery, B. P.: 1992, Numerical RecipesinFORTRAN, The Art of Scientific Computing, 2nd edn, Cambridge University Press, New York.

Scheidegger, A. E.: 1958, The random-walk model with autocorrelation of flow through porousmedia, Can. J. Phys. 36, 649–659.

Silliman, S. E.: 2001, Laboratory study of chemical transport to wells within heterogeneous porousmedia, Water Resour. Res. 37(7), 1883–1892.

Silliman, S. E. and Zheng, L.: 2001, Comparison of observations from a laboratory model withstochastic theory: initial analysis of hydraulic and tracer experiments, Transport in Porous Media42(1–2), 85–107.

Silliman, S. E., Konikow, L. F. and Voss, C. I.: 1987, Laboratory investigation of longitudinaldispersion in anisotropic porous media, Water Resour. Res. 23(11), 2145–2151.


Skoog, D. A. and West, D. M.: 1982, Fundamentals of Analytical Chemistry, Saunders, New York.Sternberg, S. P. K.: 1998, Unpublished experimental data. Laboratory notes available upon request.Sternberg, S. P. K. and Greenkorn, R. A.: 1994, An experimental investigation of dispersion in

layered heterogeneous porous media, Transport in Porous Media 15, 15–30.Sternberg, S. P. K., Cushman, J. H. and Greenkorn, R. A.: 1996a, Random walks in scale dependent

porous media, AIChE J. 42(4), 921–926.Sternberg, S. P. K., Cushman, J. H. and Greenkorn, R. A.: 1996b, Laboratory observation of non-local

dispersion, Transport in Porous Media 23, 135–151.Tompson, A. F. B.: 1988, On a new functional form for the dispersive flux in porous media, Water

Resour. Res. 24(11) 1939–1947.Weast, R. C.: 1986, CRC Handbook of Chemistry and Physics, 67th edn, CRC Press, Boca Raton,

FL.

Documents

Dispersion Measurements in Highly Heterogeneous Laboratory Scale Porous Media