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Tien, P.-L., Siegl, M.D., Undegraff, C.D., Wahi, K.K., and

Guzowski, R.V., 1985. Repository site data report

for unsaturated tuff, Yucca Mountain. Sandia

National Laboratories Report SAND84-2668 (also

NUREG/CR-4110), 384 p.

Wang, J.S.Y., and Narasimhan, T.N., 1987. Hydrologic

modeling of vertical and lateral movement of

partially saturated fluid flow near a fault zone at

Yucca Mountain. Sandia National Laboratories

Report SAND87-7070 (also LBL-23510), 98 p.

Wittwer, C.S., Bodvarsson, G.S., Chornack, M.P, Flint,

A.L., Flint, L.E., Lewis, B.D., Spengler, R.W., and

Rautman, C.A., 1992. Design of a three-dimensional

site-scale model for the unsaturated zone at Yucca

Mountain, Nevada. Proceedings, Third International

High-Level Radioactive Waste Management

Conference, in press (LBL-31799).

Semianalytical Treatment of Fracture/Matrix Flow in a Dual-Porosity

Simulator for Unsaturated Fractured Rock Masses

R. W.Zimmerman and G. S. Bodvarsson

A potential site for an underground radioactive waste

repository is in Nevada at Yucca Mountain, in a region

consisting of highly fractured volcanic tuff. As part of the

process of characterizing the site for the purposes of

determining its suitability for a repository, we have been

developing, in collaboration with the USGS (Denver),

mathematical and computational models for studying the

flow of water in unsaturated, fractured rock masses having

low matrix permeability. In order to study highly transient

infiltration processes that may occur, we need to be able to

treat flows in tuffaceous geological units that consist of two

intermingled networks of porosity: a (relatively) high-

permeability, low-storativity fracture network and low-

permeability, high-storativity matrix blocks. We

accomplish this by modifying an existing numerical

simulator, TOUGH (Pruess, 1987), so as to treat flow

between the fracture network and matrix blocks by an

analytical expression. This modified code can be used to

study transient flow processes that may be expected to

occur at Yucca Mountain.

DUAL-POROSITY MODELS

For processes that occur on a sufficiently slow time

scale, it is often assumed that the fractured rock mass can

be treated as an equivalent porous medium, with an

effective permeability that is a weighted average of the

permeabilities of the fracture network and the matrix blocks

(Peters and Klavetter, 1988). This approach assumes that

the matrix blocks are always in local equilibrium with their

surrounding fractures; it is therefore capable of simulating

processes that occur slowly enough that pressure

equilibrium can be achieved between the fractures and

matrix blocks. For highly transient processes, such as the

infiltration that would occur after a precipitation event, the

6719

66dual-porosity" nature of the medium must be accounted

for. In a dual-porosity medium, the fractures provide most

of the high permeability of the rock mass, whereas most of

the fluid storage takes place in the relatively low-

permeability matrix blocks. The complex behavior of dual-

porosity systems arises from the fact that there will be

different time scales corresponding to diffusion in the

fracture network and in the matrix blocks.

The most commonly used conceptual model of a

dual-porosity system assumes the existence of two

overlapping continua, the fracture continuum and the

matrix continuum. Flow is assumed to take place not only

through the fractures, but also between the fractures and the

matrix (Warren and Root, 1963). The traditional method of

simulating processes in such systems involves explicit

discretization of both the fracture continuum and the matrix

blocks. Simulations using this approach may require an

enormously large number of computational grid blocks for

each matrix block. For example, if we discretize a cubical

matrix block of side L into smaller, cubical grid blocks, of

size, say, L/5, this will lead to 53 = 125 grid blocks in each

matrix block. A more efficient approach is the MINC

method (Pruess and Narasimhan, 1985), in which the

matrix block is discretized into nested elements, thus

treating flow within each matrix block as being

mathematically one dimensional. We have found that

accurate treatment of transient effects with the MINC

method requires that the representative matrix block

associated with each computational cell of the fracture

continuum must itself be broken up into about ten nested

grid blocks. Hence simulation of transient processes will

require roughly ten times the number of computational cells

needed for quasi-steady-state simulations. Since the CPU

time required by most numerical simulators grows at least

133

as fast as the number of computational cells, simulation of

large-s•ale transient processes using traditional dual-

porosity simulators becomes computationally burdensome.

We have taken the approach of incorporating into a

numerifal simulator an analytical expression for absorption

of water into matrix blocks, which then plays the role of a

source/•ink term for the fractures. This approach is in the

spirit of the MINC method, since it assumes that the

fracture•s surrounding each individual matrix block are in

local eq•ilibrium with each other, and so imbibition into the

matrix block commences simultaneously from its entire

outer b6undary. Our analytical expression accounts for

factors such as block geometry, initial saturation, and the

hydrolokical properties of the matrix blocks. The accuracy

of thesd interaction terms has been verified through

numeridal simulations of imbibition into single matrix

blocks of various shapes and sizes (Zimmerman et al.,

1990). Evaluation of the fracture/matrix flow-coupling term

requires an insignificant amount of CPU time. The

computtitional meshes used in our simulations then consist

only of •'fracture elements," whose permeability, porosity,

and other properties are those of ihe fracture system,

averaged over a suitably large representative elementary

volume • (REV). The matrix blocks are not explicitly

discretized or represented in the computational mesh, but

are accounted for through the source/sink coupling term.

The onl• geometrical properties of the matrix blocks that

are required as input to the simulations are their volumes

and thei• surface areas. Flow from the fractures into the

matrix 6locks at any given location is assumed to begin

when the liquid saturation in the fractures at that location

reaches some nominal value that is taken to signal the

arrival of the liquid front. After this time, the volumetric

flux of •ater into the matrix block is calculated from our

analytical expressions as a function of the matrix block

properties, initial saturation, and elapsed time since the start

of imbibition into that particular block.

FRACTURE/MATRIX INTERACTIONS

In • a large-scale flow process occurring in an

unsaturated dual-porosity medium, the saturations and

pressure• will vary with time in both the fractures and the

matrix blocks. However, because of the much larger

permeability of the fractures, diffusion of liquid will

typically occur much more rapidly in tile fractures than in

the matrix blocks. Hence, if a liquid saturation front is

diffusing through the fracture network, we expect that the

time it takes to travel past a single matrix block will be very

small co•mpared with the characteristic time of diffusion

within that block. This allows us to make the

approximation that any given matrix block is surrounded by

fractures that are either "ahead of' or "behind" the

saturation front in the fracture network. We therefore

assume Liat the boundary conditions for the matrix block,

which are provided by the surrounding fractures, are of the

step-function type. That is to say, the saturation in the

fracture is at its initial value Si for t < to and then abruptly

jumps to some higher value, say So, for t > to In a sense,

we have partially uncoupled the problem, which allows us

to solve the diffusion equation in the matrix block under

boundary conditions that are assumed to be "known."

The rate at which liquid water is imbibed into a

matrix block depends not only on the geometry of the block

and its initial saturation, but also on the hydrological

properties of the matrix rock. The properties that are most

relevant to the rate of imbibition are the absolute

permeability k, the porosity * and its relative permeability

and capillary pressure functions. These two functions

describe the relationship between the liquid saturation S,

the capillary potential $, and the relative permeability kr

One set of such functions that have been used in modeling

the hydraulic behavior of the volcanic tuffs at Yucca

Mountain are those that were proposed by Mualem and van

Genuchten, which are (see Zimmerman and Bodvarsson,

1989)

Solo= ST+(Ss-Sh[1+ (al\1/17]-m ,

krOF) = '

{1 - (0'wly"1[1 + (odwl»}2

I i + (dvIY]*

where a is a scaling parameter that has dimensions of

1/pressure, and m and n are dimensionless parameters that

satisfy m =1- 1/n, n>2 (Fuentes et al., 1991). Sris the

residual air saturation, at which the liquid phase becomes

immobile; S, which is usually very close to 1.0, is the

saturation at which the capillary potential goes to zero.

To construct a dual-porosity model that is reasonably

general, we need expressions for the rate of imbibition into

finite, irregularly shaped matrix blocks. To develop such

expressions, we have used the following approach. First,

we used the integral method to develop approximate

expressions for one-dimensional imbibition into an

unbounded formation (Zimmerman and Bodvarsson, 1989).

Next, we extended these solutions to regular geometries

such as spherical, cylindrical, or slab-like matrix blocks

(Zimmerman et al., 1990). Finally, we developed scaling

laws that allow us to extend these results to irregularly

shaped blocks, using knowledge of their volumes and

surface areas. The results show that we can model

imbibition into a matrix block of arbitrary shape by the

equation

Q = $1/(Ss- Si) [0.89VE - 0.19r]

134

(la)

(lb)

(2a)

where r is defined by

2nk(Ss-S I•lin-1(AIV'•(t - t,;)T-a(n + 1) «m(S, - s,)]ilm

MODIFIED TOUGH CODEWe have implemented our expression for the

fracture/matrix flow as a modification to the TOUGH code(Pruess, 1987), which is an integral finite-difference codethat can simulate the flow of liquid water, air, and watervapor in porous or fractured media. The TOUGH codecontains provisions for sources/sinks of mass and heat,which are calculated in the subroutine QU. Thesources/sinks are often used to account for fluid that isinjected or withdrawn from a borehole that penetrates oneof the computational cells. We have modified thissubroutine so as to include a new type of sink, whichrepresents liquid water flowing into the matrix blocks. Themagnitude of the instantaneous flux of the sink associatedwith each computational cell is computed using Eqs. (2a,b).The subroutine QU calculates a "generation" term G foreach cell, at each time step, which represents the averageflux for the source/sink over the time interval tn Sts tn+1.by

G•tn -, t.+1) = -\ Q(40 - Q(4)4+1- 4

Leakage from a given fracture element is assumed to beginwhen the saturation (or capillary pressure) in the fracturereaches some nominal value that is taken to mark the arrivalof the saturation front.

EXAMPLE: LEAKY-FRACTURE PROBLEMOne basic problem that has much relevance to

understanding the hydrological behavior of the YuccaMountain tuffaceous rocks is that of water flowing along afracture, with leakage into the adjacent matrix (Travis et al.,1984). Martinez (1987) discussed this problem for the casewhere the fracture is oriented vertically and the flowdownward along the fracture is gravity driven. In one ofMartinez' models, the fracture was assumed to behave as asmooth-walled "slot" of constant aperture; in anothermodel, the fracture was treated as a porous medium with itsown capillary pressure and relative permeability functions.His results showed that the precise details of the hydraulicproperties of the fracture were relatively unimportant andthat the absolute permeability was the only property of thefracture that influenced the rate of advance of the water.This result provides some justification for our assumptionthat the saturation front moves through the fracture networkin a piston-like manner.

As an example of a simulation using our dual-(2b) porosity code, consider the configuration shown in Figure

1, with the fracture oriented horizontally. This modelwould also apply to the early stages of flow along a verticalfracture. Flow into the fracture is driven by the imposedpotential at the y=0 boundary. For the matrix blocks, weuse the hydrological parameters that have been estimated(Rulon et al., 1986) for the Topopah Spring member of thePaintbrush tuff (Miocene) at Yucca Mountain, which are k= 3.9 x 10-18 m2, a = 1.147 x 10-5 Pa-1,0 = 0.14, Ss =0.984, Sr = 0.318, and n = 3.04. For the fracture we usethe same characteristic curves as for the matrix blocks (forillustrative purposes only), but a transmissivity of 8.33 x10-14 m3 per unit depth perpendicular to the flow. Thistransmissivity corresponds (Zimmerman et al., 1991) to aparallel-plate aperture of 100 Kim. The fracture isdiscretized into 14 elements of successive lengths l m, 2 m,4 m, etc. The temperature is taken to be 20°C. The initialsaturation of the matrix blocks is taken to be 0.6765, whichcorresponds to an initial capillary pressure of-1 x 105 pa( 1 bar).

We have solved this problem using both the modifiedversion of TOUGH and also using TOUGH without thesource/sink expressions but with ten grid blocks extendinginto the matrix adjacent to each fracture element. Whenusing the modified code, we input a very small value ofA/V

(3) for the matrix blocks, in order to simulate the unboundedregion adjacent to the fracture. When solving the problemwith explicit discretization of the fracture and matrixregions, the matrix elements must be extended sufficientlyfar into the formation so as to effectively simulate a semi-infinite region. The saturation profile in the fracture afteran elapsed time of 1 x 103 sec is shown in Figure 2 for bothmethods of calculation. The agreement is excellent, and thedecrease in CPU time obtained using the semi-analyticalmethod was 88%. Similar savings in computational timewere found for the problem of vertical infiltration into apervasively fractured formation (Zimmerman andBodvarsson, 1992). Future work will include using the

• Matrix

9- 0 ' FractureMatrix

Figure 1. Schematic diagram of flow along a horizontal fracture,with transverse leakage into the matrix. [XBL 894-7532]

1 ry

135

10 100DISTANCE FROM THE INLET (m)

Figure 2. Saturation profile in the fracture, during flow along asingle Ihorizontal fracture. Parameters used in the simulations arelisted ib the text. [XBL 924-694]

modified TOUGH code to study the effect of block-sizediswib•tions, as well as to determine the ranges of validityof the quivalent porous medium approximation.

REFERENCESFuente•, C., Haverkamp, R., Parlange, J.-Y., Brutsaert, W.,

Zayani, K., and Vachaud, G., 1991. Constraints on•arameters in three soil-water capillary retentione•uations. Transp. Porous Media, v. 6, p. 445-449.

Martin•, M.J., 1987. Capillary-driven flow in a fracture16cated in a porous medium. Sandia NationalIjaboratories Report SAND84-1697.

Peters, • R.R., and Klavetter, E.A., 1988. A continuumrr•odel for water movement in an unsaturatedfrBctured rock mass. Water Resour. Res., v. 24, p.416-430.

Pruess, K., 1987. TOUGH user's guide. LawrenceBerkeley Laboratory Report LBL-20700.

Pruess, K., and Narasimhan, T.N., 1985. A practicalmethod for modeling fluid and heat flow in fracturedporous media. Soc. Pet. Eng. J., v. 25, p. 14-26(LBL-21274).

Rulon, J., Bodvarsson, G.S., and Montazer, P., 1986.Preliminary numerical simulations of groundwaterflow in the unsaturated zone, Yucca Mountain,Nevada. Lawrence Berkeley Laboratory ReportLBL-20553.

Travis, B.J., Hodson, S.W., Nuttall, H.E., Cook, T.L., andRundberg, R.S., 1984. Preliminary estimates ofwater flow and radionucIide transport in YuccaMountain. Los Alamos National Laboratory ReportLA-UR-84-40.

Warren, J.E., and Root, P.J., 1963. The behavior ofnaturally fractured reservoirs. Soc. Petrol. Eng. J., v.3, p. 245-255.

Zimmerman, R.W., and Bodvarsson, G.S., 1989. Anapproximate solution for one-dimensional absorptionin unsaturated porous media. Water Resour. Res., v.25, p. 1422-1428 (LBL-25629).

Zimmerman, R.W., and Bodvarsson, G.S., 1992. Semi-analytical treatment of fracture/matrix flow in a dual-porosity simulator for unsaturated fractured rockmasses. In Proceedings, 3rd Int. High-LevelRadioactive Waste Management Conference, LasVegas, Nevada, April 12-16,1992, p. 272-278 (LBL-31975).

Zimmerman, R.W., Bodvarsson, G.S., and Kwicklis, E.M.,1990. Absorption of water into porous blocks ofvarious shapes and sizes. Yfater Resour. Res., v. 26,p. 2797-2806 (LBL-27511).

Zimmerman, R.W., Kumar, S., and Bodvarsson, G.S.,1991. Lubrication theory analysis of the permeabilityof rough-walled fractures. Int. J. Rock Mech. Min.Sci. & Geomech. Abstr., v. 28, p. 325-331 (LBL-30534).

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