SIMPLE GROUNDWATER LABORATORY MODELS

By Steven S. Crider,1 Associate Member, ASCE and Benjamin L. Sill,2 Member, ASCE

INTRODUCTION

The purpose of this note is to describe a simple, inexpensive way to build hydraulic models useful in the study of groundwater flow in consolidated rocks. The writers compare results from flow and solute dispersion tests done on a uniform rectangular model to results predicted using mathematical models.

Prior laboratory models have used well-sorted sand-packed beds (Cahill 1966), collections of glass beads (Bear 1961), or blocks of epoxy-sand mixes (Kimbler et al. 1975). Unlike loose sand or glass beads, rigid epoxy-sand mixes maintain constant hydraulic properties during prolonged tests (if bi­ological influences are eliminated), but are expensive and tedious to build. As an alternative to these materials, the writers have used cement mortar mixed with very high proportions of sand and very low percentages of water— for short, "weak mortar." They have produced weak mortar models exhib­iting hydraulic conductivities ranging from 1.4 X 1(T8 m/s to 3.7 X 10~4

m/s depending upon mix proportions. For details of the relationship between mix proportions and hydraulic conductivities see Crider (1987) and Islam and Sill (1985).

EXPERIMENTAL PROGRAM

Fig. 1 shows the model that was constructed: 240 cm long, 117 cm wide, and 15 cm thick. It had a total mass of 994 kg. The model's mix proportions were 1 part Portland Type I cement, 9.5 parts masonry sand (uniformity coefficient = 3.4; d50 = 0.79 mm, where d50 is the median grain diameter), with 9.5% water. The mix had the consistency of damp sand. It was placed by hand between forms atop a table, and sealed on the sides and bottom with a masonry foundation coating to prevent leaks. The 84 13-cm diameter wells (see Fig. 1) were drilled the day after the weak mortar was placed. Water flowed from a variable-head reservoir on the upstream side to a fixed-head reservoir on the other.

To find the total porosity, four different-sized samples were taken from the model; the difference in saturated and oven-dried weight of a sample divided by the density of water gave the volume of the voids. The volume of the voids divided by the total sample volume yielded a total porosity of 0.21. The effective porosity, n (ratio of interconnected void volume to total volume = the specific yield for a phreatic aquifer), was found by dividing the volume of water drained by gravity with the sample's total volume. It

'Asst. Prof., Div. of Engrg. Fundamentals, Virginia Polytechnic Inst, and State Univ., Blacksburg, VA 24061-0218.

2Prof., Dept. of Civ. Engrg., Clemson Univ., Clemson SC 29634-0911. Note. Discussion open until November 1, 1989. To extend the closing date one

month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on March 18, 1988. This paper is part of the Journal of Hydraulic Engineering, Vol. 115, No. 6, June, 1989. ©ASCE, ISSN 0733-9429/89/0006-0818/$ 1.00 + $.15 per page. Paper No. 23569.

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FIG. 1. Dimensions of Rectangular Model

averaged 0.10, which is typical of sandstones and other consolidated sedi­mentary rocks (Freeze and Cherry 1979).

Table 1 contains the results of five hydraulic conductivity tests performed. All were steady-state. The first two assumed a linear piezometric surface for both unconfmed and confined cases; the hydraulic conductivity, K, was found from the linearized version of Darcy's Law: K = (QAx)/(AM), where A is

TABLE 1. Experiments: Hydraulic Conductivities

Flow parameter (D

Volume flow rate, Q (m3/s x 106)

Average pore velocity, v (ra/s x 106)

Average hydraulic conductivity, K (m/s x 106)

Maximum Reynolds number, Rf

Linear Approximation

Unconfineda

(2)

0.46

0.46

96.0

0.0037

Confined" (3)

2.3

13.0

190.0

0.010

Radial Flow to Well

Unconfined0

(4)

0.67

240.0

64.0

0.52

Confined" (5)

1.4

5.2

92.0

0.19

Dupuit-Forchheimer unconfined6

(6)

0.67

330.0

94.0

0.26

"K = (QAx)/(AAh); A = 0.15 m2 (average). bK = (QAx)/(AAh); A = 0.176 m2. °K = [Q In (r,/r2)]/['!:(hl - /i2)], where h{ and h2 are the piezometric heads at distances

r, and r2, respectively, from pumping well. dK = [Q In (r2/ri)]/[2TtKh2 - ft,)]. CK = (2QAx)/[W(h2, - /i2+i)L where W = 117 cm, and ht and fc,+, are the piezometric

heads at wells i and / + 1, respectively. fR = vdso/v; n = 0.10; t; = Q/(nA).

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the cross-sectional area of the model, Q is the volume flow rate, x is the longitudinal dimension, and h is the piezometric head. The third and fourth tests involved a pumping well near the center of the model and used well-known formulas for unconfihed and confined steady flow to a well, respec­tively (Freeze and Cherry 1979). The last test used in the Dupuit-Forchhei-mer approximation for unconfined flow. The differences in K between tests arise from the varying accuracies of the approximations. Q was measured by recording the time it took the outflow to fill a known volume. The Rey­nolds number, R = vd50/v, never exceeded 0.52, where v is the average pore velocity [v = Q/(An)], and v is the kinematic viscosity of water at 18° C.

There were six dispersion experiments: five in unconfined and one in con­fined flow, all with vertical instantaneous line sources. In each test, after flow became steady, a known mass of dye (Rhodamine WT) was injected with a 30-mL glass syringe into the most upstream well. At hourly intervals, samples from the downstream wells were taken and their fluorescences re­corded. Dye concentrations were chosen to maintain a linear relationship between fluorescence and concentration according to the capacity of the fluo-rometer. The data resulting from each test were spatial concentrations versus time; i.e., C(x,y) versus /, where y is the transverse distance from the source well, C is the dye concentration in a well at position (x,y), and t is time.

Adsorption of the dye onto the weak mortar was approximated by a linear equilibrium isotherm (Freeze and Cherry 1979). To quantify adsorption, dye solutions were passed through a permeameter containing weak mortar; in­fluent and effluent concentrations were measured to compute a distribution coefficient, Kd = [(C0 ~ C)Vi]/(MsC), where C0 and C are the influent and effluent dye concentrations, respectively, Vt is the volume of the solution, and Ms is the mass of the weak mortar in the permeameter. With n = 0.10, the retardation factor, RF = 1 + (pb/n)Kd, where pb is the bulk density of the weak mortar, had an average of 7.5.

For a semi-infinite (x & 0), uniform porous slab, neglecting molecular diffusion, including adsorption, with a vertical instantaneous line source of dye, the solution is (Crider 1987)

C(x,y,t) Mx$ 1/2

4tmb &)'-

exp

\RF; « t

*-fe)' - P y 2

4feK (i)

where M = the mass of dye injected; b = the model's thickness; t = time; aL = the longitudinal dispersivity; and 3 = the ratio of longitudinal to trans­verse dispersivity. Since Eq. 1 is for a confined bed of constant thickness, b, it was applied in the unconfined experiments only to arrive at an order of magnitude approximation of aL and p.

Curve-fitting of Eq. 1 to the data, C(x,y) versus t (see Crider 1987), for the six dispersion tests resulted in the values shown in Table 2. Since the most uncertain value was RF, it was used as a calibration factor; even so, it was consistent from test to test and close to the value obtained in the permeameter test, 7.5. Note the large values of (3 (mean = 56). Grane and Gardner (1961), using natural sandstone with n = 0.22, found 3 could be

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TABLE 2. Experiments: Dispersion/Adsorption Tests

Test (1) l

Unconfined 2

Unconfined 3

Unconfined 4

Unconfined 5

Unconfined 6

Unconfined

Average pore velocity, v

(m/s x 105) (2)

4.2

4.2

4.2

3.1

3.1

13.0

Retardation factor, RF

(3)

6.5

5.7

5.4

. 5.3

4.8

11.0

Longitudinal dispersivity,

«i(m) (4)

0.092

0.095

0.11

0.090

0.096

0.41

Ratio of longitudinal to transverse dispersivity, (3

(5)

73

55

86

53

57

11

as high as 50; Anderson (1984) reported values from 10 to 100 in a survey of field tests that included consolidated rocks. And Koltz et al. (1980) noted that in laboratory tests transverse dispersion is "very small" in relation to longitudinal dispersion. The values of aL in Table 2 are consistent with val­ues of aL reported by Anderson (1984) for field tracer tests in a wide variety of mediums.

CONCLUSIONS

The uniform, rectangular weak mortar model described here performed according to simple analytical models. Solute dispersion was similar to that found in field and laboratory tests using consolidated (and even some un­consolidated) materials; the model seemed to behave most like sandstone. Hydraulic conductivity remained constant during tests that usually lasted 5 days; the model was in use for 3 months. Wells were easily drilled anytime and anywhere they were needed. Cost of materials and labor (10 hr of work for one person) was less than $500.

Consequently, weak mortar models provide the desired features of rigid­ity, simplicity, availability, and low cost. They overcome a problem earlier groundwater models have had with channeling resulting from consolidation and repacking of the medium. Hence, weak mortar could conceivably be used to build a three-dimensional scaled model of an actual site. (In fact, an initial three-dimensional, layered, scaled model has only recently been constructed and tested; these results will be reported in the near future.) Furthermore, since the hydraulic properties of weak mortar can be deter­mined more cheaply than natural materials, weak mortar models can be used to verify mathematical and numerical models.

ACKNOWLEDGMENT

The results presented here were supported in part by a grant from the U.S. Department of Energy's Savannah River Laboratory.

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APPENDIX I. REFERENCES

Anderson, M. P. (1984). "Movement of contaminants in groundwater: Groundwater transport—Advection and dispersion." Groundwater Contamination, National Academy Press, Washington, D.C., 37-45.

Bear, J. (1961). "Some experiments in dispersion." J. Geophysics Res., 66(8), 2455-2467.

Cahill, J. M. (1966). "Preliminary evaluation of three tracers used in hydraulic ex­periments on sand models." U.S. Geol. Survey Professional Paper, No. 550-B, 213-217.

Crider, S. S. (1987). "Groundwater solute transport modeling using a three-dimen­sional scaled model," thesis presented to Clemson University, at Clemson, S.C., in partial fulfillment of the requirements for the degree of Doctor of Philosophy.

Freeze, R. A., and Cherry, J. A. (1979). Groundwater, Prentice-Hall, Inc., Engle-wood Cliffs, N.J.

Grane, F. E., and Gardner, G. H. F. (1961). "Measurements of transverse dispersion in granular media." J. Chem. andEngrg. Data, 6(2), 283-287.

Islam, M., and Sill, B. L. (1985). "Feasibility of physical modeling of groundwater pollutant transport." Final Kept. forEA-4229, Project 2280-2, Electric Power Res. Inst.

Kimbler, O. K., Kazmann, R. G., and Whitehead, W. R. (1975). "Cyclical storage of fresh water in saline aquifers." Bulletin 10, Louisiana Water Resour. Res. Inst., Louisiana State Univ., Baton Rouge, La.

Klotz, D., et al. (1980). "Dispersivity and velocity relationship from laboratory and field experiments." J. Hydrology, 45, 169-184.

APPENDIX II. NOTATION

The following symbols are used in this paper:

A b C

Co d50

h K

Kd

M Ms

n Q R t

V

v, X

y « i

P ?b V

= = = = = = = = = = = = = = = = = = = = = =

cross-sectional area of model; model thickness; dye concentration in solution; initial dye concentration; median grain diameter; piezometric head; hydraulic conductivity; distribution coefficient; mass of injected dye; mass of weak mortar in permeameter; effective porosity; volume flow rate; Reynolds number based upon d50; variable for time; average pore velocity volume of dye solution; dimension parallel to flow; dimension transverse to flow; longitudinal dispersivity; ratio of longitudinal to transverse dispersivity; mass density of the weak mortar; and kinematic viscosity of water at 18° C.

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