Groundwater Quality: Remediation and Protection (Proceedings of the Prague Conference, May 1995). IAHS Publ. no. 225,1995. 157

Facies dependent scale behaviour of hydraulic conductivity and longitudinal dispersivity

DIRK SCHULZE-MAKUCH & DOUGLAS S. CHERKAUER Department of Geosciences, University of Wisconsin-Milwaukee, PO Box 413, Milwaukee, Wisconsin 53201, USA

Abstract Tests on several scales were conducted and analysed in a dolomite aquifer of southeastern Wisconsin to examine the variation of hydraulic conductivity and longitudinal dispersivity with scale of measurement. Scale behaviour of hydraulic conductivity was shown to be dependent on the hydrofacies. In vuggy facies dominated by porous flow hydraulic conductivity increased relative to scale linearly with an upper bound on a log-log plot. In mudstone facies dominated by fracture-flow hydraulic conductivity also increased with scale. However, its log-log slope was steeper, and neither an upper nor a lower bound was apparent. A similar increase of longitudinal dispersivity with scale for these carbonate facies is probable but not certain, due to insufficient data. Scale dependency in coarse, unconsolidated clastic sediments is similar to that for hydraulic conductivity in porous-flow facies of the carbonate aquifer (log-log linear with an upper bound). Data are insufficient at this time to separate the role of individual facies.

INTRODUCTION

There are various reports of hydraulic conductivity and dispersivity increases with scale of measurement in the literature (Wheatcraft & Tyler, 1988; Kinzelbach, 1990; Neuman, 1990; Clauser, 1992). Most of the papers summarize different types of rocks and aquifers or treat scale behaviour statistically (Gelhar et al., 1985; Neuman, 1994). Only a few authors compiled data from the same aquifer or geologic formation and analysed scale behaviour in a specific rock type (Herzog & Morse, 1984; Rovey & Cherkauer, 1995). This paper will go a step further and introduce a facies approach to the scale behaviour of hydraulic conductivity and longitudinal dispersivity. The study area is a carbonate aquifer of southeast Wisconsin; an aquifer where porous, double-porosity and fractured hydrofacies are present.

Three formations of this aquifer will be introduced: the Thiensville Formation, the Racine Formation and the Mayville Formation. The Thiensville Formation consists of four different hydrofacies: a soil facies, a solutioned facies with large vugs, a mudstone facies and a mixed facies, which is mainly made up by recrystallized dolomite, algal laminates and packstones with thin layers of the previously mentioned facies. The four different facies can be distinguished by their hydraulic conductivities on the scale of pressure-packer tests (Fig. 1). The Thiensville Formation as a unit shows the porous-flow, Theis-type responses in pumping tests, and has a unimodal hydraulic conductivity distribution (as indicated by specific capacity data).

158 Dirk Schidze-Makuch & Douglas S. Cherkauer

1.00E-04 T

K (m/sec)

1.00E-06

1.00E-07

OM m M OS

A S

ixed Facies udstone Facies

olut ioned Facies

oil Facies

5 6

Scale (m)

10

Fig. 1 Differentiation of the hydrofacies of the Thiensville Formation at the scale of pressure-packer tests. Some increase of hydraulic conductivity with scale can be recognized. Scale was calculated by assuming cylindrical fiow to the packed-off interval.

The Racine Formation consists of a fracture-flow dominated mudstone facies and a vuggy, porous facies called the Romeo beds. The hydraulic conductivity distribution of the Racine Formation as a unit is bimodal (as indicated by pressure-packer test results; Rovey, 1990).

The final facies examined is the vuggy facies of the May ville Formation. Fluid flow through the vuggy Mayville facies is primarily transmitted by the porous medium, secondarily by fractures.

DIMENSIONS OF SCALE OF MEASUREMENT

While the scale measure for dispersive behaviour is commonly taken as a distance, such as the distance to the advective front or a radius of influence, the scale measure of

1.00E-04

K (m/sec) I.00E-05 -

1.00E-06

Discharge Volume (m3)

Fig. 2 Scale dependency during pumping test. Hydraulic conductivity of the geological unit increases with discharge volume. Hydraulic conductivities were calculated for successive five-point sets of observation data utilizing the Theis matching curve procedure.

Fades dependent scale behaviour of hydraulic conductivity 159

Table 1 Determination of dimension of scale of measurement for various types of hydraulic tests.

Test Measure of scale for hydraulic conductivity Measure of scale for longitudinal dispersivity

Permeameter tests

Piezometer tests

Packer tests

Single and multiple well pumping tests

Volume of rock sample

Volume of water introduced or removed divided by fades porosity3

Volume of introduced water divided by fades porosity*

Average pumping rate multi- plied by half of the elapsed time of the testb divided by faciès porosity3

Flow-through distance

Well construction tests Volume removed divided by fades (specific capacity data) porosity"

Tracer tests Volume removed from the pumped well (until appearance of advective front) divided by fades porositya and pumping well

Distance between observation well, where tracer was injected,

Breakthrough of Lake Distance from Lake Michigan shore to Distance from Lake Michigan Michigan water into the piezometer (where breakthrough was shore to piezometer, where carbonate aquifer faciès measured) squared and multiplied by thickness breakthrough was measured

Computer model Modelled flow distance squared multiplied by faciès thickness

a The porosity of a porous fades was obtained by the Principle of Archimedes, the porosity of a fractured faciès by the following equation, which was modified from Snow (1968):

nf=2 Nh bh + Nv bv

where nf is the fracture porosity, Nh and Nv are the number of fractures per distance in horizontal and vertical direction, and bh and bv are the mean apertures of the fractures in horizontal and vertical direction, respectively.

b Pumping rate was multiplied by half of the elapsed time to obtain an average value of hydraulic conductivity relative to scale, since hydraulic conductivity was often found to increase during the pumping tests (Schulze-Makuch & Cherkauer, 1995b).

hydraulic conductivity behaviour is better expressed as a volume of effected geological unit. For example, consider comparing piezometer tests and permeameter tests. The radius of influence of a piezometer test (the dimension parallel to flow in the test) is only several centimetres into the geological unit, but the procedure is testing 1 to 2 m of material transverse to flow (along the screen). In contrast, the flow distance through a lab sample in permeameter tests is also several cm, but the transverse radius is only centimetres. Using either a radius of influence or distance parallel to flow approach would project both tests basically on the same scale even though they are not. The effected volume of geological unit approach separates these two tests by orders of magnitude. The effected volume of the geological unit can generally be calculated simply by dividing the volume of water used in the test by the unit's porosity (Table 1; Schulze-Makuch & Cherkauer, 1995a). An averaging procedure had only to be applied to pumping tests (Schulze-Makuch & Cherkauer, 1995b), since most of them show an increase of hydraulic conductivity with discharge volume during the pumping test (Fig. 2).

160 Dirk Schulze-Makuch & Douglas S. Cherkauer

TESTS USED FOR DIFFERENT SCALES OF MEASUREMENT

The test with the smallest dimension is a falling head permeameter test for low conductivity carbonate units ( < 5 X 10"8 m s4) and a constant head permeameter test for high conductivity carbonate units (>5 x 10"8 m s"1). The tested samples had a length between 3 and 4 cm and are 3.3 cm in diameter. Lab columns have been designed to measure both hydraulic conductivity and longitudinal dispersivity. The hydraulic conductivity is measured first using the flow rate under a specified gradient. Then a KC1 tracer is applied to measure the breakthrough curve via electrical resistance to obtain the longitudinal dispersivity.

Piezometer tests were conducted by adding or removing a slug of water into piezometers screened in a specific geological unit. The head drop or rise was measured and the hydraulic conductivity was obtained (Hvorslev, 1951).

Packer tests were conducted in sets (MMSD, 1984). Successive pressures of 0.07, 0.14, 0.21, 0.34 and 0.69 MPa were applied, and a geometric mean of the hydraulic conductivity was calculated for each set.

Single and multiple well pumping tests were analysed depending on the fluid medium. The Theis and Cooper-Jacob methods were used to obtain hydraulic conductivities for porous media, while the linear slopes on a log-log plot were used for fractured media. Double-porosity responses were segmented into fractured and porous responses and analysed as above.

During construction of each water supply well in Wisconsin a specific capacity test is done. These were analysed using an iterative estimation technique to determine hydraulic conductivity (Bradbury & Rothschild, 1985).

One multiple well tracer test conducted in the Mayville Formation was used to determine a longitudinal dispersivity from the measured breakthrough curve (MMSD, 1983). Due to both high pumpage and the presence of a tunnel in the dolomite aquifer acting as a drain Lake Michigan water is induced into the dolomite aquifer of southeast Wisconsin. Since Lake Michigan water has a different chemical composition than ambient groundwater, the breakthrough of Lake Michigan water can be observed by analysing the chemical composition of water in nearby piezometers and longitudinal dispersivities and mean hydraulic conductivities can be obtained. Finally, one computer model (Mueller, 1992) was used to obtain hydraulic conductivity values for the porous-flow facies on the largest scale.

HYDRAULIC CONDUCTIVITY BEHAVIOUR

The above described tests were used to analyse scale behaviour of hydraulic conductivity over several orders of magnitude. Enough data were collected for the mixed facies of the Thiensville Formation to establish a scale behaviour (Fig. 3). Even though the mixed facies is quite heterogeneous, hydraulic conductivity seems to increase linearly on a log-log plot. At a higher scale the increase of hydraulic conductivity with scale levels off (upper cutoff), and the average hydraulic conductivity is constant with scale. This levelling off is a homogenation effect. The effect of the inherent heterogeneities persists with scale to a certain point, where the geological unit becomes so homogeneous, that no further increase of hydraulic conductivity with scale occurs.

Fades dependent scale behaviour of hydraulic conductivity 161

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Effected Volume of Geological Unit, Volume/porosity(m3)

Pig. 3 Scale dependency of the mixed facies of the Thiensville Formation. Hydraulic conductivity increases relative to scale but displays an upper bound. Number of observations for each type of measurement is given in parenthesis.

The Racine Formation was divided into two hydrofacies: the fractured mudstone facies and the vuggy Romeo facies. The mudstone facies appears as one line on a log-log plot without upper or lower bound (Fig. 4). The increase of hydraulic conductivity with scale of the vuggy Romeo facies starts somewhere near the scale of the slug tests. The

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Effected Volume of Geological Unit, Volume/porosity (m3)

Fig. 4 Scale dependency for facies of the Racine Formation. The Romeo facies shows a linear increase of hydraulic conductivity with upper and lower bounds, while the mudstone facies shows an unbounded linear increase with a steeper slope. Points plotted are geometric means of multiple tests. Numbers of observations upon which geometric mean is based: from left to right; Romeo facies N = (3,4,18,2,51,2,1,1); mudstone facies N = (10,7,1,64). WCR = well construction reports (specific capacity data).

162 Dirk Schulze-Makuch & Douglas S. Cherkauer

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Fig. 5 Scale dependency of the vuggy Mayville faciès indicating a hydraulic conductivity increase relative to scale with an upper bound. Packer test results are given as geometric mean with standard deviation. Other data points are single observations.

slope of the line on the log-log plot is less than that of the mudstone facies (Fig. 4). At the scale of the well construction tests the homogenation effect occurs, and hydraulic conductivity remains constant with scale thereafter.

The vuggy facies of the Mayville Formation shows an increase of hydraulic conductivity with scale (Fig. 5) very similar to that for the mixed facies of the Thiensville Formation. Although the absolute hydraulic conductivities of the Mayville facies are somewhat lower, the linear increase of hydraulic conductivity with an upper bound on a log-log plot is also present.

DISPERSIVITY BEHAVIOUR

Scale behaviour of dispersivity is more difficult to observe, since most aquifer tests are not designed to measure dispersivity. Due to this lack of data, different facies were summarized to reach some preliminary assessment of dispersivity-scale relationships. Dispersivities measured in situ in the Wisconsin carbonate aquifer were obtained from one multiple well tracer test in the vuggy Mayville facies and from one breakthrough of Lake Michigan water in the Romeo facies. Lab test dispersivities were obtained from the Romeo, vuggy Mayville and the mixed facies (Thiensville Formation). Due to the shortage of test results, data from other carbonate aquifers were added for comparison (Arya, 1986; Lallemand-Barrès & Peaudecerf, 1978; Gelhar et al., 1985). Figure 6 shows that lab dispersivities are much smaller than dispersivities measured in situ. Data are still needed on the scale of a metre. The relationship appears to be similar to that of the hydraulic conductivity-scale plots (line on log-log plot with an upper bound).

Because of the limited data it is not possible to examine the role for carbonate facies in the relation of dispersivity to scale. However, enough measurements are available for clastic material (from data collections provided by Arya, 1986 and Gelhar et al., 1985)

Faciès dependent scale behaviour of hydraulic conductivity 163

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Scale (m) Fig. 6 Scale dependency of longitudinal dispersivity in carbonate aquifers. Lab tests produce lower dispersivities than field tests. Insufficient data are available to separate individual carbonate fades at this time. The data from other carbonate aquifers were selected from Gelhar et al. (1985), Lallemand-Barrés & Peaudecerf (1978), and from Arya (1986).

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Fig. 7 Scale dependency of longitudinal dispersivity in coarse unconsolidated sediments indicating a linear increase of dispersivity relative to scale with an upper bound. No facies dependency is apparent. Data were selected from Gelhar et al. (1985) and from Arya (1986).

to examine facies effects. Coarse, unconsolidated sediments can be separated into sand aquifers, gravel aquifers, and sand and gravel aquifers (Fig. 7). There is an obvious log-log linear increase of dispersivity up to a scale of measurement of about 100 m, limited by an upper bound beyond that. The gravel aquifers seem to occur on the upper limit of the variation, but otherwise there is no obvious separation of data on the basis of facies.

SUMMARY AND IMPLICATIONS

For the carbonate system examined, a universal exponential (log-log linear) increase of hydraulic conductivity has been observed. The rate of increase appears to be different

164 Dirk Schulze-Makuch & Douglas S. Cherkauer

for different facies. In addition, most of the individual faciès exhibit an upper bound on the relationship. It is an hypothesis that the rate of the hydraulic conductivity increase and the position of the upper bound are uniquely related to the distribution of heterogeneities within a geologic material and that, in turn, this is controlled by the environment of deposition. Further examination at the facies level should ultimately provide sufficient data that the scale relationship of hydraulic conductivity can be predicted for a material a priori. It is anticipated that a similar result will be possible for dispersivity, but to date there are insufficient measurements of this property to undertake such an analysis.

REFERENCES

Arya, A. (1986) Dispersion and reservoir heterogeneity. PhD dissertation, University of Texas, Austin. Bradbury, K.R. & Rothschild, E.R. (1985) A computerized technique for estimating the hydraulic conductivity of aquifers

from specific capacity data. Groundwater 23(2), 192-198. Clauser, C. (1992) Permeability of crystalline rocks. EOS Trans. AGU 73(21), 233, 237-238. Gelhar, L.W., Mantoglou, A., Welty, C. & Rehfeldt, K.R. (1985) A review of field- scale physical solute transport

processes in saturated and unsaturated media. Electric Power Research Institute, EA-4190 Report, project 2485-5, Norris, Tennessee.

Herzog, B.L. & Morse, W.J. (1984) A comparison of laboratory and field determined values of hydraulic conductivity at a waste disposal site. In: Proc. Seventh Annual Madison Waste Conference. Dept. of Engineering Professional Development, University of Wisconsin-Madison, Madison, Wisconsin, 30-52.

Hvorslev, M.J. (1951) Time lag and soil permeability in groundwater observations. US Army Corps of Engineers Waterway Experimental Station Bull. 36, Vicksburg, Missouri.

Kinzelbach, W. (1990) Modelling and applications of transport phenomena in porous media. Lecture Series 1990-01 of the Karman Institute for Fluid Dynamics, GesamthochschuleKassel, Germany.

Lallemand-Barrés, A. & Peaudecerf, P. (1978) Recherchedes relations entre la valeurde la dispersivitémacroscopiqued'un milieu aquifère, ses autres caractéristiques et les conditions de mesure (Research about the relation of the value for macroscopic dispersivity of an aquifer environment, to its characteristics and measuring approach). Bulletin de Recherches Géologiques et Minières, 2e Série, Section III, no. 4, Orléans, France.

Milwaukee Metropolitan Sewage District (1983) Infiltration/ exfiltration analysis. Milwaukee Water Pollution Abatement Program, Milwaukee, Wisconsin.

Milwaukee Metropolitan Sewage District (1984) Pressure test data. Project ID I30A11, file code D5126, T.O. no. 54, Milwaukee, Wisconsin.

Mueller, S.D. (1992) Three-dimensional digital simulation of the ground-water contribution to Lake Michigan from the Silurian aquifer of southeastern Wisconsin. UnpublishedMS Thesis, University of Wisconsin-Milwaukee,Milwaukee, Wisconsin.

Neuman, S.P. (1990) Universal scaling of hydraulic conductivities and dispersivities in geologic media. Wat. Resour. Res. 26(8), 1749-1758.

Neuman, S.P. (1994) Generalized scaling of permeabilities: validation and effect of support scale. Geophys. Res. Letters 21(5), 349-352.

Rovey, C.W.II (1990) Stratigraphy and sedimentology of Silurian and Devonian carbonates, eastern Wisconsin, with implications for ground-water discharge into Lake Michigan. Unpublished PhD Dissertation, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin.

Rovey, C.W.II & Cherkauer, D.S. (1995) Scale dependency of hydraulic conductivity measurements in a carbonate aquifer. Groundwater (in press).

Schulze-Makuch,D. & Cherkauer, D.S. (1995a) Relationofhydraulicconductivity and dispersivityto scale of measurement in a carbonate aquifer. In: Models for Assessing and Monitoring Groundater Quality (ed. by B. J. Wagner, T. H. Illangasekare& K. H. Jensen) (Proc. Boulder Symp., July 1995). IAHS Publ. no. 227 (in press).

Schulze-Makuch, D. and Cherkauer, D.S. (1995b) Scale behaviour of hydraulic conductivity during a pumping test. In: Effects of Scale on Interpretation and Management of Sediment and Water Quality (ed. by W. Osterkamp) (Proc. Boulder Symp., July 1995). IAHS Publ. no. 226 (in press).

Snow, D.T. (1968) Rock fracture spacings, openings and porosities./. SoilMech. Found. Div., Proc. Am. Soc. Civil Engrs 94,73-91.

Wheatcraft, S.W. & Tyler, S.W. (1988) An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry. Wat. Resour. Res. 24(4), 566-578.