ELSEVIER

Advances in Water Resources, Vol. 19, No. 6, pp. 343-357, 1996Copyright 0 1996 Published by Elsevier Science Ltd

Printed in Great Britain. All rights reservedPII: S0309-1708(96)00012-7 0309-1708/96/$15.00 + 0.00

Macroscopic representation of structural geometryfor simulating water and solute movement in

dual-porosity media

Horst H. GerkeDepartment of Soil Landscape Research, Center for Agricultural Landscape and Land Use Research, Eberswalder StraJe 84.

D-15374 Miincheberg, Germany

Martinus Th. van GenuchtenU.S. Salinity Laboratory, United States Department of Agriculture, Agriculture Research Service, 450 West Big Springs Road,

Riverside, California 92507-4617, USA

(Received 6 October 1995; accepted 16 April 1996)

The structure of macroporous or aggregated soils and fractured rocks is generallyso complex that it is impractical to measure the geometry at the microscale (i.e., thesize and the shape of soil aggregates or rock matrix blocks, and the myriad offissures or fractures), and use such data in geometry-dependent macroscale flowand transport models. This paper analyzes a first-order type dual-porosity modelwhich contains a geometry-dependent coefficient, p, in the mass transfer term tomacroscopically represent the size and shape of soil or rock matrix blocks. As areference, one- and two-dimensional geometry-based diffusion models were used tosimulate mass transport into and out of porous blocks of defined shapes. Estimatesfor ,0 were obtained analytically for four different matrix block geometries. Valuesfor ,0 were also calculated by directly matching analytical solutions of the diffusionmodels for a number of selected matrix block geometries to results obtained withthe first-order model assuming standard boundary conditions. Direct matchingimproved previous results for cylindrical macropore geometries, especially whenrelatively small ratios between the outer soil mantle and the radius of the innercylinder were used. Results of our analysis show that /3 is closely related to the ratioof the effective surface area available for mass transfer, and the soil matrix volumenormalized by the effective characteristic length of the matrix system. Using valuesof /3 obtained by direct matching, an empirical function is derived to estimatemacroscopic geometry coefficients from medium properties which in principle aremeasurable. The method permits independent estimates of p, thus allowing thedual-porosity approach eventually to be applied to media with complex and mixedtypes of structural geometry. Copyright 0 1996 Published by Elsevier Science Ltd

Key words: preferential flow, variably-saturated structured media, dual-porositymodel, mass transfer coefficient, geometry coefficient.

INTRODUCTION

Structured soils and fractured rock formations oftencontain macropores or fractures through which water andsolutes can move at considerably larger velocities than inthe porous matrix. The resulting microscopic or porescale heterogeneities in flow velocities may eventuallymanifest themselves at the macroscopic scale in terms ofpreferential flow involving transient nonequilibrium

conditions in the pressure head or solute concentrationperpendicular to the flow direction. Preferential flowphenomena have been extensively studied8”7*49 because oftheir implications on the hydrologic and contaminanttransport properties of soil/rock systems. Dual- (ordouble-) porosity approaches’15’ have been applied tovarious problems of water flow in fissured reservoirs,solute transport in aggregated soils, and transient flowand transport in variably-saturated structured media (for

343

344 H. H. Gerke. M. Th. van Genuchten

an extensive review, see Gerke and van Genuchten15).Conceptually, a dual-porosity medium is generallyassumed to involve two overlapping but interactingpore domains having different hydraulic and transportproperties. Water and solute movement in dual-porositymodels (e.g., Dykhuizen’j and Jarvis et a1.24) is simulatedseparately for the fracture and matrix pore systems, whilecoupling terms are used to describe the transfer of waterand solutes between the two pore systems.

A commonality of dual-porosity approaches is thatthey require information about the geometry of the soilor rock system being simulated, i.e., the size and shapeof matrix blocks or soil aggregates, including thegeometry of the interface area through which water andsolutes are exchanged between the two pore systems.The interface area may be associated with the surfacestructure of either the fracture pore system (e.g., planarcracks or cylindrical macropores), or of the matrix poresystem itself (e.g., rock matrix blocks or soil aggregates).Naturally structured media often consist of a mixture ofmatrix blocks having different shapes and sizes, therebyleading to potentially very complex interfacial geome-tries as has been demonstrated recently by means ofcomputed tomography.20925’50

Mass transfer in dual-porosity models is frequentlyapproximated using a first-order differential equa-tion.‘0*‘695’ More sophisticated formulations that con-sider a nalytical solutions of flow and transport into orout of porous matrix blocks of known size and shapeare generally limited to relatively simple situations, suchas for well-defined macropore systems or idealizedmatrix block geometries,34 steady-state water fl~w,~~and for single pulses of water or solutes moving alonga fracture.14y52l53 Microstructure-type mode1s22s23135@describing transport at both scales of the porousmedium, i.e.,, macroscopic transport in the fracturepore s yst e m1 and local-scale transport inside the aggre-gates or matrix blocks, mostly assume a single equiva-lent matrix block geometry (e.g., a sphere) for whichlocal-scale solutions of diffusion-type flow and transportmodels are available. Equivalent geometries may beobtained by deriving a volume-weighted average sphereradiu:-,32 using surface-to-volume ratios and distanceprobability functions for matrix diffusion obtainedfrom fractured rocks or aggregated soi1s,29’33 invokingscaling methods involving shape factors,41,44 usingscaling methods based on ratios between the volumeand surface area of matrix blocks,54 or by means of‘block -geometry functions’ associated with planar, cylin-drical, or spherical block shapes,2 among other methods.

Although conceptually of considerable interest, theapproaches above are still difficult to apply to fieldproblems since the microscopic model parameters arenot easily measured. A related issue, mostly stillunresolved, is the question of how to obtain adequatemicrostructural information required to extend suchmodels from simple or artificial media like glass beads to

field problems.’ Consequently, the first-order masstransfer coefficient is at present still treated mostly asan empirical parameter which must be calibrated byfitting to observed field data, or to solute breakthroughcurves obtained in displacement experiments.45 One ofthe challenges of dual-porosity models is to identifythose local-scale processes that have the most impact atthe macroscale and to incorporate those processes in amacroscopic model in such a way that the model can beapplied to realistic field problems. With respect to astructured dual-porosity medium, this means that thegeometric effects somehow must be represented by alimited few lumped parameters related to the bulkvolume of the medium. The aim of this paper is to deriveexpressions that appropriately describe the structuralgeometry of a dual-porosity medium using parametersthat can be related to measurable properties.

DUAL-POROSITY MODEL AND FIRST-ORDERMASS TRANSFER

The dual-porosity model used in this study to simulateone-dimensional vertical water flow and solute transportin a variably-saturated structured medium was describedin detail by Gerke and van Genuchten.” The modelassumes that the properties of the bulk porous mediumcan be characterized by two sets of local-scale proper-ties: one set associated with the fracture pore system(subscript f), and the other with the matrix pore system(subscript m), as follows

E = It+& + (1 - Wf)& (la)

0 - M’& + (1 - wr)f& (lb)

4 = VIf + (1 - Qqn, (lc)

where E is the porosity (L3Lp3), 6’ is the water content(L3L-3), q is the fluid flux density (LT-‘), and wf is therelative volumetric proportion of the fracture poresystem (0 < wf < 1). Assuming applicability of Darcy’slaw,45 water flow in the fracture and matrix pore regionsare described by a coupled pair of Richard’s equations:

@I

I‘,+_-1 - Wf (2b)

where h is the pressure head (L), C is the specific watercapacity df?/dh (L-t), K is the hydraulic conductivity(LT-I), z is depth taken to be positive downward (L), tis time (T), and Fw is the water transfer term (T-l) takenas

rw = ‘Yw (hf - h,) (3)

in which cr, is a first-order mass transfer coefficient

Simulation of water and solute movement in dual-porosity media 345

(L-‘T-l) for water flow. The following expression for(Y, was derived by Gerke and van Genuchten”’ using ascaling procedure:

Pow = ;;z %v&(~> (4)

where p is a dimensionless geometry-dependent coeffi-cient, a is the characteristic length of the matrixstructure (L) (e.g., the radius of a spherical or solidcylindrical aggregate, or half the fracture spacing in thecase of parallel rectangular voids), T,,, is a dimensionlessscaling coefficient for which an average value of 0.4 wasobtained empirically assuming parallel rectangular slabs(Fig. la), and K, is the effective hydraulic conductivityfunction of the fracture-matrix interface evaluated interms of both h, and hf as follows

K, = @5K.&) + K,(&Jl (5)Analogous to eqn (2), the transport of solutes in a

variably-saturated dual-porosity medium is describedusing two coupled convection-dispersion equations:

(6a)

where D is the dispersion coefficient (L*T-‘), R is adimensionless retardation factor accounting for linearequilibrium so

‘ption, and Fs is the solute mass transfer

term (MLV3T- ) formulated here in a slightly differentway as done previously in Gerke and van Genuchten.”

rs = %(l - Wfh%l(Cf - CnJ + 1 r&f rw L 0r

wm r,<o(7)c

in which Q, is the first-order solute mass transfercoefficient (T-l) of the form

o.J = -$Da (8)

where Da is an effective diffusion coefficient (L*T-‘)which represents the diffusion properties of the fracture-matrix interface as well as other parameters and which,however, has not yet been evaluated analogously toyWKa in eqn (4) for water transfer.

For steady-state water flow with qm = 0 (no flow inthe matrix pore system), the above variably-saturateddual-porosity model reduces to the two-region mobile-immobile solute transport model of van Genuchten andWierenga.43

4IJG ;tA= a:(cf-c,) (9b)

where the water contents, br and r9,, as well as theretardation factors, RF and R& are expressed in terms of

the bulk soil volume, i.e.,

‘19f = WBfi 6, = (1 -w,)& (lOa; b)

The first-order mass transfer coefficient, (Y: (T-l), in (9b)characterizing diffusive exchange of solutes betweenthe mobile and immobile liquid phases is of the generalform

P%D:,a: = --g- (11)where 0: is the effective diffusion coefficient for solutemass transfer into the immobile pore water region or thematrix pore system. We refer to Gerke and vanGenuchten” and van Genuchten4* for additionaldiscussions of the dual-porosity models presented here,including possible ways of how the different parameters,other than p, may be measured or estimated.

GEOMETRY REPRESENTATION OF THE MASSTRANSFER COEFFICIENT

Analytical evaluation

In order to relate the mass transfer coefficient, ai, to thediffusion properties of soil aggregates or rock matrixblocks, the first-order formulation given by eqns (9)and (11) may be compared with more comprehensivegeometry-based microstructure-type models for diffusioninto porous blocks having spherical, cylindrical, orrectangular slab-type geometries. A variety of techniqueshave been used to derive expressions for CX: in eqn (1 l),including Laplace transform comparisons,31~39~44 momentanalysis,30l40 and the use of algebraic expansions.4 TheLaplace transform method may also be used to deriveexpressions for the first-order water transfer coefficient,ow, in e n (4) by linearizing the horizontal flowequation. 18 Such analyses lead to first-order mass transfercoefficients in eqns (4), (8), and (11) which are of thegeneral form

cu,* PQ,3(Y,~---_-

6, a*’ (12)with P = D& in the two-region mobile-immobile typetransport model (9), D, in the dual-porosity solutetransport model (6), and -yWKa in the dual-porositywater flow model (2). Thus, we hypothesize that valuesof @ can be derived that are identical for these threemodels, and hence a function only of the geometry.

Mass transfer coefficients obtained analytically usingLaplace transform comparisons differ only in the valueof the geometry-dependent coefficient, p. Most of thestudies mentioned above derived values for ,0 of 3 forrectangular slabs, 8 for solid cylinders, and 15 forspheres, although other studies obtained slightly differ-ent values.“y3’ For a hollow cylindrical dual-porositymedium involving macropores surrounded by cylindri-cal soil or rock matrix mantles, van Genuchten and

346 H. H. Gerke. M. Th. van Genuchten

Dalton4 derived the following expression for 0:

2(Co - u2(13)

where co = (a + b)/b, in which b is the radius of thecylindrical macropore and a is the thickness of the soilmatrix mantle (Fig. lb).

For hollow cylindrical geometries the flux across thefracture-matrix interface (the surface of the innercylinder) is very similar to the soil water flux into aroot located in the center of a cylindrical soil segment.

/

/

The latter process has been widely studied.28 Analogousto the water transfer term, rw, root water uptake isgenerally included as a sink term in the Richard’sequation (2), and often formulated as an effective first-order approach. Such an approach was actually used byMoldrup et aL2’ who derived a soil resistance term byintegrating the flow equation for a cylindrical coordi-nate system between the inner and outer radius of thesoil mantle and assuming dO/dt to be constant (>O).This approach leads to another equivalent expressionfor the geometry-dependent coefficient, p. Using thenotation of this paper, this alternative expression is

b a

/

(b)

a b

(4

Fig. 1. Schematic illustration of porous blocks having (a) rectangular slab-type, (b) hollow cylindrical, (c) solid cylindrical, and(d) spherical geometries. Widths of 2a and 26 are associated with the matrix and fracture pore systems, respectively.

Simulation of water and solute movement in dual-porosity media 347

given by

P =[lnK0)

d-1

--c _ 1 0.5 ] ; b > 1 (14)

Analytically derived mass transfer coefficientsare limited to basically four geometries for whichsolutions of the diffusion equations are available.Additional limitations are due to simplifying assump-tions made during mathematical derivation of thecoefficients, e.g., neglecting higher-order terms in theseries expansions.

Direct matching

Alternatively, estimates for the mass transfer coefficient,crf, may also be obtained by directly matching solutionsof the first-order approach, i.e., of eqn (9b), with calcu-lated average solute concentrations of soil aggregates orrock matrix blocks of known geometry. For this purposeone could use analytical solutions of the equationsgoverning diffusion into spherical, rectangular (slab-type), or solid cylindrical aggregates, or those pertainingto diffusion from hollow cylindrical macropores into thesurrounding soil matrix. The direct matching techniquewas previously used by van Genuchten41 to comparethe diffusion properties of a soil aggregate with givengeometry and size with those having different shapes andsizes in a nonflowing system. His method is employedhere to obtain geometry-dependent coefficients forvarious matrix block sizes for use in the first-ordertransfer terms. The direct matching technique is demon-strated below for the case of a hollow cylindricalmacropore geometry.

Radial diffusion from the macropore through thesurface of the inner cylinder (Fig. lb) into thesurrounding mantle (assuming an infinite longitudinallength) is described by

bLr<(a+b) (15)

where c,(r, t) is the local solute concentration inside thecylindrical soil mantle, r is the local radial coordinate,and the subscript m indicates that Rk and 0:represent matrix pore system properties which areconsistent with the transport model in eqn (9) at themacroscopic scale. We assume here that the soluteconcentration of the matrix pore system, c,, defines theaverage solute concentration, c,, of the liquid phase inthe hollow cylindrical (subscript h) porous matrix asfollows

2

s

atbE,(f) = (a+b)2 -62 b rca(r’t)dr (16)

in which a is the characteristic length (the radial width of

the soil mantle surrounding the macropore). Solvingeqn (15) for a uniform initial solute concentration,ca(r, 0), of zero, a constant concentration of unity at theinner, ca(b, t) = 1, and a zero concentration gradient (noflow) at the outer boundary (at r = a + b), and sub-stituting that solution into eqn (16) leads to9

c,(Th) = ’ - [; _ 1 n=,““-2

x exp(-k&-b) (17)where Jo, J1, Y,, and Y1 are Bessel functions and p,, arepositive roots of

Jo(&) YI (&lCo) - J1 MO) Yo(&) = 0 (18)and where C~ is expressed in terms of dimensionless time,Th, as follows

J&f‘% = R$a;

and where the subscript h on T and a is used to indicate ahollow cylindrical macropore.

Analogously, solutions for the average concentra-tions, i;,, as a function of dimensionless time, T, areavailable for spherical, solid cylindrical, and rectangulargeometries.9”’ Plots of the resulting curves of &(T)versus r for different aggregate geometries exhibitrelatively similar, but not identical, sigmoidal shapes.41In the original paper, van Genuchten41 introduced a‘shape factor’, f, for conversion of one geometry intoan equivalent other geometry by matching the F,(T)

curves of two geometries through modification of thecharacteristic length a in the dimensionless time (e.g.,eqn (19)) pertaining to one of those geometries. Theshape factor was evaluated at an average value of T, =0.5, thus ignoring any observed dependency off on &.

A similar approach can be used for estimating shapefactors in the first-order transfer term.4’ Integratingeqn (9b) for c,(O) = 0 and cf( t) = 1 gives

I?~ E C,(To) = 1 - eXp(-To) (20)

where

cY*t7-O = $--& (21)

is the dimensionless time in the first-order model(subscript 0). The shape factor fh,o for converting ahollow cylindrical into the equivalent first-order ratemodel was defined by van Genuchten4’ as

fh$ = ($ “2/4=o.1 (22)

in which the dimensionless times Th and 7. were bothevaluated at C~ = c,,, = 0.5. Substituting eqns (19) and

348 H. H. Gerke. M. Th. van Genuchten

(21) into eqn (22) shows that the shape factor,f, is relatedto the geometry coefficient, /3, in the first-order transferterm of the dual-porosity model by means of p = l/fz.

By using the direct matching method, we obtainedvalues for 0 of 3.5 for rectangular slabs, 11 for solidcylinders, 22.7 for spheres (see also van Genuchten4’),and the following empirical regression function(2 = 0.996) for a hollow cylindrical geometry:

’ = 10.19lnil6Q]‘;1 < <s < 100 (23)

Figure 2 compares the geometry coefficients forhollow cylindrical matrix geometries as obtained bymeans of analytical derivation (eqns (13) and (14)) withthose obtained using eqn (23) which is based on thedirect matching approach. Equation (13) closely agreeswith the direct matching results in the limit of relativelythick cylindrical soil mantles (large b), whereas eqn (14)approximates the direct matching results more closely inthe limit when <s -t 1. The results in Fig. 2 are aconsequence of simplifying assumptions invoked in thederivations of eqns (13) and (14), i.e., by having systemswith relatively few macropores per unit area (& > lo),or relatively dense root systems (<a < 5). The effect ofusing different values of fl in eqn (4) on soil water flow

simulations will be shown later by directly comparingthe dual-porosity model with equivalent two-dimen-sional models for different matrix geometries. We notethat the scaling procedure described by Gerke and vanGenuchten16 was employed to evaluate the hydraulicconductivity of the fracture-matrix interface, K,(h), ineqn (4) using the analytically-derived geometry coeffi-cient for parallel rectangular slabs u = 3).

COMPARISONS USING EQUIVALENTTWO-DIMENSIONAL MODELS

To study the geometry coefficients in the mass transferterm (4) as evaluated by different methods, considertransient unsaturated water flow in both the fractureand matrix pore systems. For convenience, and tofacilitate a direct comparison, vertical movement in thematrix pore system has been neglected. Results obtainedwith the dual-porosity model using the first-ordertransfer term will be compared with those generatedusing the SWMS_2D finite element code.” The flowregion was designed to imitate rectangular, hollowcylindrical, and solid cylindrical geometries of thematrix blocks (Fig. 1). Hydraulic properties of relatively

j ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,(,, I ,,,,,,,” “,““,

0 10 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100

Outer-to-Inner-Radius Ratio, c0Fig. 2. The geometry coefficient, @, as a function of the ratio, CO, between the outer and inner radius, assuming hollow cylindricalgeometry. Plotted are values and curves obtained by (a) analytical derivation using eqn (13), (b) direct matching using eqn (23), and

(c) analytical derivation using eqn (14).

Simulation of water and solute movement in dual-porosity media 349

Table 1. Model parameters and initial conditions used in the twodimensional comparisons

(Y(1/cm)

n 4 4 KS 4 (2) 4 (2)(cm/day) (cm)

Matrix 0.005 1.5 0.11 0.5. 1.05 - 1000 0.277Fracture 0.100 2.0 0.0 0.5 2000 -1000 0.005

coarse- and fine-textured soil materials (Table 1) wereassigned to the fracture and matrix pore systems,respectively. Water flow for the parallel rectangularfracture (slab) geometry was simulated by using a two-dimensional cross-section (x-z plane) where x is thehorizontal and z is the vertical spatial direction. Forcylindrical geometries we assumed radial symmetry, andsimulated flow in a quasi three-dimensional region (r-zplane) using the flow equation for a cylindricalcoordinate system. In order to compare the results ofthe two-dimensional calculations with those obtainedwith the dual-porosity model (2), nodal water contentsat given simulation times were averaged over thehorizontal or radial distance and transformed intoequivalent pressure head values of the fracture andmatrix pore systems using the inverse of the waterretention function (e.g., van Genuchten et a1.47):

h = i (s;‘/” _ 1)“”

where Q and n are empirical parameters, m = 1 - l/n,s, = (0 - &>l(& - 0,) is effective saturation, and 0, and8, are the residual and saturated water contents,respectively. The procedure used for comparing model-ing results is described in detail by Gerke and vanGenuchten.16

For the idealized geometries (Fig. 1) used here, thecharacteristic length, b (e.g., the half-width in the caseof a rectangular slab, or the radius in the case of acylinder or sphere) of the fracture pore system is relatedto the relative proportion of the fracture system, wf, andthe characteristic length, a, of the matrix system by

Bt = al&f

for rectangular slabs (subscript l), by

for hollow cylinders (subscript h), and by

bcir = a,;,[(1 - wf)-li2 - l] (27)

for solid cylinders (subscript c); for rectangular prisms(subscript r) we assumed a square base area of (2a)2.Flow systems containing a spherical geometry wereapproximated by assuming that the spherical matrixblocks or soil aggregates were surrounded by thinspherical mantles imitating the fracture pore system.In case the spheres are completely surrounded by thefractures, the equivalent characteristic length of thefracture system, b:, is

b: = a,[(1 - wf)-lj3 - l] (28)

Assuming that the spherical matrix blocks are in contactwith each other (Fig. Id), the volume, V,,, of the top andbottom sphere-caps (subscript SC) of the fracture mantle

V, = : bi (3a, + 2b,) (29)

has to be distributed in the fracture pore systemresulting in a value of b, which is slightly larger thanbl as follows

b3 _ 3as s

‘a5 - wf 62 _ 3a2 ’ - wf

Wf ss -bb,+az = 0

Wf(30)

Equation (30) can either be solved exactly, using aclosed-form trigonometric solution for cubic equations,or solved approximately in an iterative manner begin-ning with 6: obtained from eqn (28) as the initialestimate. For example, assuming a = 1 cm and wr = 0.05,one obtains bf = 0.017245 cm and b, M 0.0174cm. Thevalues of b for the other geometries (Table 2) were used todesign equivalent two-dimensional flow regions andassociated numerical finite element discretizations.

To illustrate physical nonequilibrium during transientflow in an unsaturated dual-porosity-type structuredporous medium, the simulation example used pre-viously’5~‘6 was selected. We assumed the applicationof water at a constant rate of 50 cm/day to a 40-cm-deepdual-porosity medium having an initial uniform pressurehead of h,i = h,,i = -1000 cm. Water was allowed toinfiltrate exclusively into the fracture pore system(wr = 0.05), such that a local infiltration rate ofqf,o = l000 cm/day was imposed at the upper and afree-drainage condition at the lower boundary of the

Table 2. Model parameters in first-order transfer term and two-dimensional representation

P P a xv KWanalytic matching (cm) (A) (cm/day)

Rectangular slab 3 3.5 1 0.02563 0.4 0.01 0.1Solid cylinder 8 11.0 1 0.02598 0.4 0.01 0.1Hollow cylinder 2.42 1.5 1 0.28800 0.4 0.01 0.1

Simulation of water and solute movement in dual-porosity media 351

fracture pore system. Parameters of the soil waterretention function ((Y, n, O,, O,), the hydraulic conduc-tivity at saturation, K,, the initial pressure head, hi, andwater content, ei, and the first-order mass transfercoefficients can be found in Tables 1 and 2, respectively.

Figure 3 shows calculated two-dimensional pressurehead distributions at t = 0.04 days for (a) a parallelrectangular slab, (b) a hollow cylindrical geometry, and(c) a solid cylindrical geometry assuming relatively low(K,,a = 0.01 cm/day) water transfer rates between thefracture and matrix pore systems. Note that thehorizontal axis is enlarged 20 times. The simulationexamples in Fig. 3 illustrate significant preferential flowand pressure head nonequilibrium between the fractureand matrix pore domains for all three geometries. Afterabout 1 hour (t = 0.04 days) of constant infiltration,water moved downward to a depth of 23cm in therectangular pore system (Fig. 3a), to z = 35 cm in thehollow cylindrical pore system (Fig. 3b), and toz = 15cm in the fracture pore system surrounding thesolid cylindrical soil matrix (Fig. 3c). Since all modelparameters, including the relative volume, wf, of thefracture pore system and the initial and boundaryconditions were identical, differences in calculatedpressure distributions were caused only by the geometry-dependent rates of water transfer from the fracture into

0

10

(a)

the matrix pore system. Two-dimensional model simula-tions were also carried out for the case of relatively highwater transfer rates assuming K,+ = 0.1 cm/day (resultsnot shown here).

Horizontally averaged two-dimensional simulationresults are compared with results obtained from thedual-porosity model for relatively small water transferrates (I& = 0.0 1 cm/day) assuming matrix structuresinvolving rectangular slabs (Fig. 4), hollow cylinders(Fig. 5), and solid cylinders (Fig. 6). As was discussedpreviously,‘6 the first-order term generally underesti-mates the transfer rate at the infiltration front due tofailure of the linear first-order term to capture the highlynonlinear transient flow process near the front (see alsoZimmerman et al.“). However, except in areas close tothe infiltration front, the proposed linear term yieldedrelatively good approximations for the transfer of water(parts b of Figs 4-7), so that the calculated pressurehead (parts a) and water content profiles (parts c) stillclosely matched the reference results. The matchgenerally improved when allowing higher water transferrates by using KS,, = 0.1 cm/day (cfX, Figs 5 and 7 for ahollow cylindrical geometry and p = 1.5). The results inFigs 4-6 reveal similar dependencies of the transfer rateon p among the different geometries, i.e., the larger thegeometry-dependent mass transfer rates along the

Pressure Head, h (cm) Water Transfer Rate, I’, (l/day) Volumetric Water Content, 0

Fig. 5. Pressure head, h (a), water transfer rate, Pw (b), and total volumetric water content, 0 (c), versus depth, z, at t = 0.04 days ascalculated with the one-dimensional dual-porosity (Dual) and two-dimensional (2-D) models, assuming a hollow cylindricalgeometry and relatively small water transfer rates (K,+ = 0.01 cm/day); results are for geometry coefficients derived analytically

(p = 2.4) and by direct matching (p = 1.5).

352 H. H. Gerke, M. Th. van Genuchten

sequence: ‘hollow cylinders < rectangular slabs < solidcylinders’, the closer the match with the referenceresults. Thus, the closest correspondence was obtainedfor the solid cylindrical geometry (Fig. 6) assumingp=g

In t ontrast to the above, the analysis becomes morecomplicated when, for a particular geometry, differentmethods for evaluating the geometry coefficient, p, arecomp ared. Because of the approximate nature of thefirst-order approach, and the nonlinear flow regimesinvolved, the comparisons will differ for each particularset o f hydraulic parameters and at each particularsimul ation time. We again present results at t = 0.04 foran infiltration event, and selected a situation with fairlytypical results. In case of a hollow cylindrical geometry(Figs 5 and 7), the geometry coefficient obtained bydirect matching, i.e., p = 1.5 calculated using (23),substantially improved the correspondence with the two-dimen sional reference results, as compared to theanaly tically derived value (p = 2.42) using eqn (13).These results confirm that eqn (13) is limited to dual-porosity systems with relatively large values of Q., Onthe other hand, results obtained with different values offl (as estimated analytically or using direct matching)were quite similar for rectangular slabs (Fig. 4), whereasfor solid cylinders the results using the analytically derived

value turned out to be superior as compared to thedirect matching method (Fig. 6). The results in Figs 4-6only in part reflect the different sensitivities of the masstransfer rate on 0 for different matrix geometries. Theresults also support the assumption that a singleconstant value of 0.4 for the scaling coefficient, T,,,, ineqn (4) in combination with an arithmetically averagedK, (eqn (5)) can be reasonably well applied to all threegeometries.

EXTENSION TO MEDIA WITH COMPLEXSTRUCTURAL GEOMETRIES

To be able to extend the approach to porous media withmore complex structural geometries, we assume thatsomehow a macroscopic geometry-dependent coefficientmay be defined which effectively captures the overalldiffusion properties of the structured system. So far, /3 ineqn (4) was quantitatively related to the shape of a well-defined matrix structure. Alternatively, /3 could berelated also to the ratio between the surface area ofthe interface between the fracture and matrix poresystem and the volume of the matrix blocks. Both ofthese quantities are, at least in principle, measurable.The surface-to-volume ratio has been used in various

(b)

-100 -10 -1 0 5 10 15

Pressure Head, h (cm) Water Transfer Rate, Tw, (l/day)

(c)wJltlI)IlIIJIIIl~lIII’

0.30 0.35 0.40 0.45 0.50Volumetric Water Content, 8

Fig. 6. Pressure head, h (a), water transfer rate, I’, (b), and total volumetric water content, fI (c), versus depth, z, at t = 0.04 days ascalculated with the one-dimensional dual-porosity (Dual) and two-dimensional (2-D) models, assuming a solid cylindrical geometryand relatively small water transfer rates (& = 0.01 cm/day); results are for geometry coefficients derived analytically (J = 8) and by

direct matching @ = 11).

Simulation of water and solute movement in dual-porosity media 353

approaches to obtain geometric information of soil orrock structures?9z3’” mostly from microscopic points ofview.

The ratio between the surface area, A, (L2), of thematrix-fracture interface (e.g., the soil aggregate surface)and the volume, Vm (L3), in the case of a matrix blockformed by parallel rectangular slabs (plates), is given by

A 1m,l_v, -z (31)

which assumes that the contributions of the sides (topand bottom ends) can be ignored. For solid cylindersand rectangular prisms we obtain similarly

(32)

while for spheres the surface-area-to-volume ratio is

A 3m,sv,=Q, (33)

For hollow cylinders, assuming that water and solutemass transfer is restricted to the surface area of the innercylinder, one obtains

Am h 2 2A=vm bh(&f - 1) = ah(cO f 1) ’

co > 1 (34)

where ah = bh(CO - 1) is the characteristic length of thematrix structure.

For application to media with arbitrary geometriesand various characteristic diffusion lengths, we intro-duce a dimensionless surface-area-to-volume ratio, c(-),normalized per unit length of the matrix structure forrectangular and solid cylindrical geometry

*c=$a*

m(35)

and for hollow cylindrical structures

<=&Ty co > 1

where Ah is the effective surface area and a* is theeffective length of the matrix pore system, both beinglumped macroscopic scale parameters which integrateover all geometries of a defined volume of the structuredmedium.

By assuming lumped parameters, we may plot thegeometry coefficient, p, versus the normalized surface-area-to-matrix-volume ratio, r;, obtained from thedirectly-matched shape factors for various sizes andshapes for conversion into the equivalent first-order ratemodel using eqn (22). Results are shown in Fig. 8 where,for comparison, the analytically-derived coefficients for

15-

(a) (b)20-sm. 1 . . ,..“‘I . . ,““‘. . 1 I ” ’ ‘1””-1000 -100 -10 -10 5 10

Pressure Head, h (cm) Water Transfer Rate, I’, (l/day) Volumetric Water Content, 8

Fig. 7. Pressure head, h (a), water transfer rate, lTW (b), and total volumetric water content, 8 (c), versus depth, z, at t = 0.04 days ascalculated with the one-dimensional dual-porosity (Dual) and two-dimensional (2-D) models, assuming a hollow cylindricalgeometry and relatively large water transfer rates (KS+ = 0.1 cm/day); results are for geometry coefficients derived analytically

(@ = 2.4) and by direct matching (p = 1.5).

354 H. H. Gerke, M. Th. van Genuchten

plane sheets, solid cylinders, and spheres have beenincluded. Figure 8 shows that the difference in the valueof p between structures that have the same reducedsurface-to-volume ratio, are much smaller than differ-ences in 0 with respect to the method of evaluation(e.g., ,0 = 22.7 for a sphere and 20.7 for a cube). Thisfact permits one to describe the geometry coefficientover a wide range of c-values by the fitted empiricalexpression

which is simpler but also somewhat less accurate thaneqn (37).

The proposed functions in eqns (37) or (38) relate thegeometry-dependent coefficient, p, of the macroscopicscale mass transfer terms in eqn (12) to macroscaleproperties of an arbitrarily structured medium. Thefunctional interpretation of structure used here focusesonly on the size and shape of the fracture-matrixinterface area through which diffusive mass transferactually takes place. An important lumped parameter inthis context is the effective surface area, A;, whichintegrates all ‘internal’ structural surfaces over thevolume of the porous medium; its value also containscontributions from imperfectly developed (sub-) struc-tures such as small fissures in spherical or cylindricalmatrix blocks. Values of the normalized surface-area-to-volume ratio smaller than unity represent structuredsoils or fractured rocks in which the ‘internal’ surfacearea for diffusive mass transfer is primarily concave(e.g., hollow cylinders). For porous media having valuesof 5 > 1, the fracture-matrix interface areas are pre-dominantly convex (e.g., spheres). The effective char-acteristic length parameter, a*, may be interpreted to bethe smallest average distance for diffusion between the

[0191n{16(~-1)}]-z 0.0198<<<1 (37)

3.6~“’ lLc12Ul + u2c + u3c2 2<s110

in which u, = 11.4275, u2 = -7.4438, and u3 = 3.5473and where 2 = 1. For 1 < < < 2, the appropriategeometry coefficients may be obtained by using theinterp olation function suggested in eqn (37). Alterna-tively. the dependency may be described also by

[0491n{16(f-1))1-2 0.0198<<< 1 (38)

0.65 -2[ 10.09 + <I<r<lO

_ _ , . ( , ,. . . ,. 1 .’ ._’ _‘_ _‘_ : ‘. : ‘. _’ ‘...,.. 7 Equation 37

1

Normalized Surface-to-Volume Ratio, s

Fig. 8. Plot of the geometry coefficient 0 as a function of the normalized effective surface-to-volume ratio, c = a*Ak/ If,,,, as given byeqn (37), as well as p-values obtained by analytical evaluation and direct matching.

Simulation of water and solute movement in dual-porosity media 355

fracture-matrix interface and the center of the matrixblocks perpendicular to the macroscopic scale flowdirection. The results in Fig. 8 suggest that the effect of pon flow and transport calculations will be most apparentwhen c is close to 1.0. The effect of ,LI declines for c < 1where /3 approaches a relatively constant value of 0.5(i.e., the asymptotic behavior of fl for c < 0.1 isdemonstrated in Fig. 2) and total mass transfer ratesdecrease because of a reduced effective surface area fortransfer, Ah. For c > 1, the effect of p may also declineif, in the case of relatively large mass transfer rates, thesituation of a homogeneous (unstructured) medium isapproached which will not exhibit any preferential flow.We emphasize that the coefficient /I as estimated witheqn (37) or eqn (38) accounts for the shape dependencyof the mass transfer process, and indirectly still is afunction of the characteristic length, a*. The macroscaleparameters a*, b*, and wr, however, are no longer relatedto each other in well-defined ways as for idealizedgeometries using eqns (31)-(34); they have to bemeasured independently.

POSSIBLE PARAMETERIZATION OFSTRUCTURAL GEOMETRY

Most approaches involving microstructure-type mobile-immobile models assume prior knowledge of the type ofgeometry, or rely on detailed micromorphologicalmeasurements.29V32V33,54 In addition, an equivalent dia-meter concept has been proposed for tubular macroporesin saturated media.12 However, we believe that realisticsimulations of preferential flow require the use ofmacroscopic scale flow and transport parameters forboth pore systems.

The macroscopic approach presented in this paperpermits structural effects to be included without assum-ing any type of geometry a priori, and hence follows amore functional view as suggested previously by Boumaet al7 The proposed macroscopic scale representation ofthe structural geometry leading to diffusive masstransfer relies on one of the most basic properties of astructured medium, i.e., the fracture-matrix interfacearea. The extent, size and shape of the interface, as wellas its physical, chemical, and biological properties (e.g.,those of fracture coatings) are major factors determiningtransport processes in aggregated soils or fracturedrocks. Of relevance here are, as discussed earlier,t6 notonly the interfacial permeability, K,, but also theeffective diffusion coefficient, D,, the chemical composi-tion and sorption sites, biological activities alongfractures, as well as other properties occurring at ornear such interfaces.

Unfortunately, the effective fracture-matrix interfacearea is not easily determined experimentally. Undercertain conditions the area and shape of the interfacemay actually depend on the water content and flow rate,

while also partial wetting of aggregate surfaces orswelling and shrinking of the matrix may occur.Recently, methods have been suggested to calculatethe surface areas of structured soils using dyes that stainmacropore area&‘* or by using an image analyzer.6Measurements of air permeability have also beensuggested36 to determine the effects of pore geometryon transport processes in macroporous soils. Birgerssonet al.’ used two different tracers at the same time toestimate the flow-wetted surface of fractured rocks intowhich the dyes can diffuse and sorb. In another study,McKay et a1.26 used nonreactive solute and colloid-sizedbacteriophage tracers to reveal large differences in trans-port rates between the two tracers because of exclusion ofthe large tracers from diffusing into the matrix.

The second macroscopic property, the effectivecharacteristic length of the matrix structure, a*, maybe determined using diffusion experiments. As a reason-able initial estimate for a*, one-half of the volume-surface mean diameter d (as defined by Gregg andSing”) may be used:

&,A 1 c(d)a* =2=2C(n&) (39)

where ni are the number of particles (cubes) of edgelength li of a specific volume. Often, the values of thelengths li in eqn (39) have to be estimated, for example,by using the mean projected diameter,” which is thediameter of a circle having the same area as theprojected image of the particle when viewed in adirection normal to the plane of greatest stability. Onecould apply a similar approach to aggregate mixtures soas to obtain a weighted average of the geometry andlength coefficient. Mass transfer rates could also beobtained by multiple calculations using observed shapeand size distributions of the matrix pore system, andapplying a volumetric weighing procedure.

The above modeling concept offers also the possibilityto combine mass transfer with preferential flow insituations involving partial wetting of matrix surfaces ofcracked clays or fractured rocks, for instance, byassuming

A:, = A,$, OIS<l (40)

where s(t) is a dimensionless wetting parameter whichmay be a function of time or certain soil properties.2’The effects of swelling and shrinking in soils in thisapproach could be described directly by means of theparameter wf.

CONCLUSIONS

Preferential flow of water in a dual-porosity system ofvarious matrix geometries was simulated using a first-order type water transfer term. Direct comparisons with

356 H. H. Gerke, M.. Th. van Genuchten

reference two-dimensional model simulations demon-strate (1) inherent limitations in the approach because ofsimplification of highly nonlinear transfer processes intoan approximate first-order rate model, and (2) sensitivityof the mass transfer coefficient in the dual-porositymodel to the exact geometry of the matrix pore system(e.g., rectangular slabs, and hollow or solid cylinders).The comparisons revealed similar tendencies for allthree geometries when using an appropriate value of thecoefficient, & describing the specific geometry of thematrix pore system. By scaling the diffusion propertiesand relating the geometry coefficient to the ratiobetween effective surface area at the fracture-matrixinterface and matrix volume, the approach may beextended to media with arbitrary and complex internalgeometries.

4. Bolt, G. H., Movement of solutes in soil: Principles ofadsorption/exchange chromatography. In Soil Chemistry,B. Physico-Chemical Models, Dev. Soil Sci. 5B, ed. G. H.Bolt, 1979, pp. 295-348.

5. Booltink, H. W. G. & Bouma, J., Physical and morpho-logical characterization of bypass flow in a well-structuredclay soil. Soil Sci. Soc. Am. J., 55 (1991) 1249-1254.

6. Booltink, H. W. G., Hatano, R. & Bouma, J., Measure-ment and simulation of bypass flow in a structured claysoil: A physico-morphological approach., J. Hydrol., 148(1993) 149-168.

7. Bouma, J., Belmans, C. F. M. & Dekker, L. W., Waterinfiltration and redistribution in a silt loam subsoil withvertical worm channels. Soil Sci. Soc. Am. J., 46 (1992)917-921.

The methodology in this paper gives qualitatively agood approximation of more realistic two-dimensionalrepresentations of the mass transfer process. Animportant advantage of the scaling approach is thatthe method can be readily incorporated in multi-dimensional numerical models (e.g., Vogel et ~1.~‘).Also, the model may be used as a tool to represent orbetter understand preferential flow and transportphenomena in a mechanistic (process-based) fashionwith a reasonable numerical effort. The approachenhances the utility of dual-porosity models in thatotherwise empirical fitting parameters now can beexpressed in terms of measurable soil/rock properties.Also, the approach is shown to be not limited to porousmedia containing a single type of aggregate shape andsize (geometry) or uniformly-sized aggregates, nor is ana priori qualitative description of the geometry of theporous matrix required. The approach is also notlimited to experimentally determined aggregate sizedistributions, which often are difficult to obtain forspecific soils. Nevertheless, the study of interfacialmacroscale properties will require new or improvedexperimental methods for determining effective surfaceareas, i.e., those of soil aggregates or fracture walls.The approach may help to further increase theapplicability of the dual-porosity approach to analyzetransport in media with complex and widely-differingstructural geometries.

9.

10.

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Crank, J., The Mathematics of Diffusion. Oxford UuiversityPress, London, 1956, 347 pp.Dunn, G. H. & Phillips, R. E., Equivalent diameter ofsimulated macropore systems during saturated flow. SoilSci. Soc. Am. J., 55 (1991) 1244-1248.Dykhuizen, R. C., Transport of solutes through unsatu-rated fractured media. Water Res., 21 (1987) 1531-1539.Dykhuizen, R. C., A new coupling term for double-porosity models. Water Resour. Res., 26 (1990) 351-356.Gerke, II. H. & van Genuchten, M. T., A dual-porositymodel for simulating the preferential movement of waterand solutes in structured porous media. Water Resour.Res., 29 (1993), 305-319.Gerke, H. H. & van Genuchten, M. T., Evaluation ofa first-order water transfer term for variably-saturateddual-porosity models. Water Resour. Res., 29 (1993) 1225-1238.

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