DISTRIBUTION

DEC 28 1983 WM s/f (3lO9.9)VX JUSTUSWMGT r/f UGOODE-&r/f~NMSS r/f PORNSTEIN & r/fREBROWNING SNEUDER

MEMORANDUM FOR: Malcolm R. Knapp, Acting Chief MBELL DWIDMAYERGeotechnical Branch HJMILLER TMCCARTINDivision of Waste Management JTGREEVES TNICHOLSON

JBUNTING JRANDALLFROM: Peter Ornstein LIGGINBOTHAM MJWISE

Geotechnical Branch NEISENBERGDivision of Waste Management

Dan GoodeGeotechnical BranchDivision of Waste Management

SUBJECT: HARMONIC MEAN AND FLOW PARAMETERS FOR FDMGROUNDWATER MODELS

The recent SWIFT II training seminar offered an opportunity forpresentation of some of the model's numerical techniques. One issuebriefly discussed was the use of harmonic mean for "transmissibility"calculations. This memo is an attempt to continue that discussion andsuggest some alternative techniques.

During finite difference solution of porous media flow equations, it isnecessary to compute the interblock (for grid-centered) or interface (forblock-centered) product of cross-sectional area and hydraulicconductivity. If values are specified for each block, the interfacevalue must be computed as some average of the block values. In general,the best technique for computing these averages is that which results inhead differences and fluxes closest to an analytical solution. Theanalytical solution for flow between two blocks depends on the propertiesof the system and how those properties vary between blocks.

One assumption for the variation of properties between blocks is thatproperties are constant over each block and changes occur (abruptly) onlyat the block interface. The effective hydraulic conductivity oreffective block height (assuming other properties constant) for the blockinterface in this system is exactly equal to the harmonic mean ofhydraulic conductivities or heights (see Bear, 1979 p. 81). SWIFT II(Reeves et.al., 1983) uses harmonic mean for hydraulic conductivity andblock height.

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An alternative second assumption is that properties vary linearly betweennodes or block centers. In this case, as shown in Attachment A, thegeometric and arithmetic mean are better estimates of the effectiveinterface value than the harmonic mean. Haverkamp and Vauclin (1979)discuss the estimation of interblock hydraulic conductivity and recommenduse of geometric mean. Attachment B presents an averaging procedurewhich, like the harmonic mean for blocky systems, yields an exacteffective interface value for linear variation between nodes. Althoughthis technique may, in some instances, provide the most accurate results,we know of no code which uses it.

The issue of the choice of averaging technique is a current one and hasnot been definitively resolved. For example, Milly (Prof. Chris Milly,Princeton, personal communication, 1983) reports instability inunsaturated flow models using geometric mean. No doubt, alternativeconceptualizations of property variation between blocks will yielddifferent "best' estimators.

Given the current work in estimating interblock flow properties, and thesuitability of alternative techniques for at least one realisticconceptualization, we feel that it is important to discuss incorporationof some of these alternative techniques into SWIFT, SWIFT II and othercodes used by NRC.

V IPeter OrnsteinGeotechnical BranchDivision of Waste Management

YI IDan GoodeGeotechnical BranchDivision of Waste Management

AttachmentAs stated

OFC: WMG :

NAME :DGoode:dm stein : :

DATE : 12/AJ/83 : 12/zl/83 :

Draft No. 1 (DM) 83/11/28

Draft No. 2 (DM) 83/11/29

References: Bear, Jacob, Hydraulics of Groundwater, McGraw-Hill,

New York, 1979.

Haverkamp, R., and M. Vauclin, A note on estimating

finite difference interblock hydraulic conductivity

values for transient unsaturated flow problems, Water

Resources Reaearch 15(1), 181-187, 1979.

Reeves, Mark, N. D. Johns, and R. M. Cranwell, Theory and

implementation for SWIFT II, Release 3.83, DRAFT

Sand 83- , 1983.

Attachment A

Estimates of interblock flow parameters with linear variation betweennodes

Consider steady-state one-dimensional flow in a layer with changingthickness (see Figure).

Vertical flow components are ignored. For finite difference methods,layer thickness, B. is specified at the node points. Computation of fluxbetween node "1" at x7-xl and node "2"1 at x--xl + Ax requires theestimation of effective layer thickness between 1 and 2. The goal ofestimating effective layer thickness, B', is not that the actual layerthickness at the interface between I and 2 be computed accurately (i.e.,B' = B. + a,&x/2), rather it is that the computed head values and fluxesare accurate. To evaluate this accuracy, we can compute the change inhead from 1 to 2 for a given flux using several averaging techniques, andcompare these results to an analytical solution.

Analytical Solution

The governing equation for steady-state one-dimensional aquifer flow canbe written:

in which q [L2/T] is volumetric discharge per unit width (Ay). This is a"hydraulic approach" equation in that vertical flux is ignored and(1) reflects integration over the thickness of the layer.

Integrating (1) once we obtain

q = constant

and from Darcy's law we recognize

q =-KB dhiA )

TR -

where K [L/T] is hydraulic conductivity; B[L] is layerh[L] is piezometric head.

thickness; and

Integrating (2) assuming K is constant*, we solve for h:

A dk -It S.c4.K (3)

For a linear varying thickness

(Li)

Substituting (4) into (3):2%

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*The following analysis applieswith linearly varying hydraulic

equally to a constant thickness layerconductivity.

A-2

This solution is compared to corresponding solutions from a finitedifference technique using averaged interface layer thickness.

Finite Difference Computation

The finite difference representation of flux between 1 and 2 is:

'C - -1<11 A X (7i)in which K' and B' are the interface values of hydraulic conductivity andlayer thickness. Thus the head difference between 1 and 2 is:

~% : K '& Y (zq)

The accuracy of various techniques for computing B' can be investigatedby comparing (8), with constant K' = K, to the analytical solution (6).

Harmonic Average

The harmonic average layer thickiness is:

a6k II4

('0Ifok=O, layer thickness is constant, and (9) reduces to B' = B1.Substituting (9) into (8):

0 4, -. r t J + -.w

The ratio of (10) to (6) can be a relative measure of error.1 is an exact solution with no error. For harmonic average,is:

(10)A ratio of

this ratio

L&~ K I ':)k !��I-

61Ij �'� I +

'-04&YC II

This ratio is a functionthickness change between

of i I alone, which is the ratio ofnodes to thickness at node 1.

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The arithmetic average layer thickness is:A-3

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A-H

Comparison

The relative accuracy of the harmonic, arithmetic and geometric means inestimating effective interface value are illustrated in Figures 2 and 3.Figure 2 is a semilog plot of relative error versus small relativechanges in layer thickness. Relative error is the absolute differencebetween unity and the ratio of finite difference head change toanalytical head change between nodes. The harmonic mean is the leastaccurate and the geometric is most accurate. For a 50% increase inthickness between nodes (Cow/ -_ O-r) , use of harmonic mean resultsin an error of about 3%, while arithmetic mean yields about 1.5% errorand geometric mean yields about 0.7% error.

Figure 3 illustrate accuracy of the three techniques when hydraulicconductivity changes by orders of magnitude between nodes. The ratio offinite difference head change to analytical head change between nodes isplotted versus the relative change in hydraulic conductivity on a log-logscale. The finite difference head change using harmonic and geometricmeans is larger than the analytical solution while the head change usingarithmetic mean is smaller than analytical. For a three order ofmagnitude increase in hydraulic conductivity the head difference ratiosfor the harmonic, arithmetic, and geometric mean techniques are about 70,C3, and H.T, respectively.

If the system flow parameters are conceptualized as varying linearlybetween nodes, the harmonic mean is the least accurate technique forestimating effective interface values.

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