Estimate of Vertical Anisotropy of Hydraulic Conductivity for Northern Louisiana Aquifers from Grain-Size Data Douglas Carlson Louisiana Geological Survey, Louisiana State University, Baton Rouge, Louisiana 70803
[email protected] ABSTRACT One of the most difficult properties of an aquifer to determine is the vertical anisotropy of hydraulic conductivity. Often it is determined by either running a series of pressure tests (packer tests) within the aquifer at multiple positions vertically or by analysis of a groundwater model where there is a significant vertical gradient of potentiometric level as observed by numerous piezometers with different vertical positions within the aquifer. Both of these techniques are expensive and the second is not possible in many settings where the aquifer is not stressed and yielding significant vertical gradient of potentiometric level. In northern Louisiana neither of these sets of data/observations is available. However, there are several thousand grain-size analysis results available. Often for a municipal well grain size samples were collected every 10 to 20 feet yielding many values within the screened interval of the well that is often over 100 feet. This data is analyzed for hydraulic conductivity (K) using eight standard equations that relate grain-size properties with K. With three to over ten values of K within a stack of sand then the effective horizontal hydraulic conductivity (Kh), effective vertical hydraulic conductivity (Kv) and vertical anisotropy are determined. The number of result values depends on technique of analysis. Techniques based on diameter of d10 grains (grains at 10 percent rank) yield fewer result values than those based on diameter of d17 grains. For this reason, number resulting ratios determined for sand layers within each of the major aquifers in northern Louisiana are: Sparta 109 and 182, Carrizo-Wilcox 35 to 64, Cockfield 19 to 38, Mississippi River Alluvial 13 to 30 values, Catahoula 23 to 33 values, and Upland Terrace 17 to 22. Five standard techniques involved d10 grains and three standard techniques involved d17 grains or d20 grains. Typically these values are determined from a series of three to four grain size results across a screen interval, but there are approximately 20 wells with 10 or more grain-size tests, majority of these are Sparta wells. INTRODUCTION Vertical hydraulic conductivity (Kv) is far less frequently determined for an aquifer than horizontal hydraulic conductivity. This is probably a result of the fact that it is one of the most difficult properties of an aquifer to determine. Often it is determined by either running a series of pressure tests (packer tests) or permeameter tests (Carlson, 2000) within the aquifer at multiple positions vertically within the aquifer or by analysis of a groundwater model where there is a significant vertical gradient of potentiometric level as observed by
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numerous piezometers with different vertical positions within the aquifer (Plomb, 1989; Nader, 1990; Milwaukee Metropolitan Sewerage District, 1992; Carlson, 2001; and Dunning et al, 2004). Both of these techniques are expensive and the second is not possible in many settings where the aquifer is not stressed and yielding significant vertical gradient of potentiometric level. For northern Louisiana aquifers it appears that there are no field scale Kv values for the aquifers in this part of Louisiana. Only permeameter tests associated with oil field studies yield permeability values (at lab scale), for a series of cores. The majority of these cores are samples within the Carrizo-Wilcox (Carlson, in review), which lie towards the southern edges of northern Louisiana. However, there are many grain-size values collected during the drilling of water supply wells in northern Louisiana. In many cases grain size samples were collected every 10 to 20 feet yielding several values within the screened interval of the well and across a large portion of the aquifer. For this study these samples are analyzed for hydraulic conductivity (K) by using eight standard equations that relate grain-size properties with K. With three to over ten values of K within a layer of sand, it was possible to calculate an effective Kh, Kv and estimate vertical anisotropy. METHODS Hydraulic conductivity was determined from grain-size data for intervals within and near the screen intervals of generally public supply and industrial wells in northern Louisiana. These intervals have at least three or more values of grain-size determined that include size information for the 10% (d10), 17% (d17), and 20% (d20) rank grain-size diameter data. There are eight different equations as described in Vukovic and Soro, (1992) and Kasenow (2002) which were used to determine hydraulic conductivity from grain-size data. K = Cg/v(BH)v(n)d210
Hazen Formula (Harzen, 1892):
(1)
where K is hydraulic conductivity is in m/day (1 m/day = 3.281 ft/day), C is a conversion factor to convert hydraulic conductivity from cm/s to m/day, g is acceleration of gravity is 9.81 m/s2 (32 ft/ s2), v is kinematic viscosity is 8 x 10-7 m2/s (8 x 10-6 ft2/s) at water temperature of 30oC (86oF), BH is 6 x 10-4, and v(n) 2.4, were it is assumed that porosity (n) is 0.4), and d10 is diameter of grains in the 10th percentile expressed in millimeters. Beyer Formula (Beyer, 1966):
K = CFg/v(BH)log(500/C)d210
(2)
where C = d60/d10, K is hydraulic conductivity (m/day), CF is a conversion factor to convert hydraulic conductivity from cm/s to m/day, g is acceleration of gravity (9.81 m/s2), v is kinematic viscosity at water temperature of 30oC, BH is 6 x 10-4, d60 is diameter of grains in the 60th percentile expressed in millimeters., and d10 is diameter of grains in the 10th percentile expressed in millimeters. Sauerbrei Formula (Vukovic and Soro, 1992): K = 3.49C(n3/(1-n)2)_d217 (3)
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where K is hydraulic conductivity (m/day), C is a conversion factor to convert hydraulic conductivity from cm/s to m/day, n is porosity which is assumed to be 0.4, _ temperature correction, which is 1.313 for a temperature of 30oC, and d17 is diameter of grains in the 17th percentile expressed in millimeters. K = 5400(n3/(1-n)2)d210
Kozeny Formula (Kozeny, 1953):
(4)
where K is hydraulic conductivity (m/day), n is porosity which is assumed to be 0.4, and d10 is diameter of grains in the 10th percentile expressed in millimeters. United States Bureau of Reclamation (USBR) Formula (Vukovic and Soro, 1992): K = C0.36d2.320 (5) where K is hydraulic conductivity (m/day), C is a conversion factor to convert hydraulic conductivity from cm/s to m/day, and d20 is diameter of grains in the 20th percentile expressed in millimeters. Pavchich’s Formula (Pravedny, 1966):
K = C_d217
(6)
where K is hydraulic conductivity (m/day), C is a conversion factor to convert hydraulic conductivity from cm/s to m/day, _ temperature correction which is 1.313 for a temperature of 30oC, and d17 is diameter of grains in the 17th percentile expressed in millimeters. K = 4960n3.287 d210
Slichter Formula (Slichter 1898):
(7)
where K is hydraulic conductivity (m/day), n is porosity which is assumed to be 0.4, and d10 is diameter of grains in the 10th percentile expressed in millimeters. Terzaghi Formula (Terzaghi, 1925): K = CRg/v((n-0.13)2/(1-n)1/3)d210
(8)
where K is hydraulic conductivity (m/day), C is a conversion factor to convert hydraulic conductivity from cm/s to m/day, g is acceleration of gravity, R is an empirical coefficient dependent on nature of grain surface, which is assumed to be the median value (8.4 x 10-3) within the range listed by Kasenow (2002), v is kinematic viscosity at water temperature of 30oC, n is porosity which is assumed to be 0.4, and d10 is diameter of grains in the 10th percentile expressed in millimeters. For each of the equations an estimate of K was determined. After a series of K values were determined then the effective Kh and Kv were determined from the sequence of layers each with their own value of K using the below equations: and
Kh = ∑miKi/∑mi
(9)
Kv= ∑mi/∑(mi/Ki)
(10)
Where mi is thickness of i th interval included within grain-size sample, Ki is the hydraulic conductivity of i th interval included within a series of grain-size samples (Domenico and
34
Schwartz, 1990) as determine by any one of the eight equations noted previously. Lastly, with effective Kh and Kv determined the value of vertical anisotropy was determined by dividing Kh by Kv. RESULTS The results of this study were determine by analyzing three or more grain-size results within the screen interval of wells or zones of interest for development as water source for usually public supply and industrial wells in northern Louisiana. A significant number of anisotropy results were determined for six aquifers in northern Louisiana: Carrizo-Wilcox , Catahoula, Cockfield, Mississippi River Alluvial, Sparta, and Upland Terrace. Most of the anisotropies were result of analysis of between 3 and 6 observations (Fig. 1). This is the case for between 62.5% of the observations for the Mississippi River Alluvial Aquifer and 93.9% of the observations for the Catahoula Aquifer.
percentage of anisotropies determined
60 50 Catahoula Cockfield Mississippi River Sparta Upland Terrace
40 30 20
Carrizo-Wilcox
10 0 3
4
5
6
7
8
9
10
11
12 over 12
number of observations Figure 1. The percentage of anisotropy results by major aquifer in northern Louisiana by number of grain-size observations that were analyzed to determine the vertical anisotropy. The grain-size values often increase with depth for given sand, which in turn yields increasing values of hydraulic conductivity downwards within sand (Fig. 2). This increase of grain-size and resulting hydraulic conductivity is typical as indicated by the fact that bottom samples usually have a larger value of hydraulic conductivity than the top samples for all six of the aquifers of this study: Mississippi River Alluvial (80%), Catahoula (75%), Carrizo-Wilcox (70%), Cockfield (62%), Upland Terrace (59%) and Sparta (58%). In addition when bottom samples have a higher hydraulic conductivity than top samples the difference is usually greater than when the reverse is the case (Fig. 3). Approximately 80%
35
of the K bottom divided by K top ratios range between 0.5 and 4, with about two thirds over 1, which is reasonable given fluvial sands within the Mississippi River Alluvium and the Upland Terrace typically become coarser downwards (Fisk, 1938; Wang, 1952; Visher, 1965; and Carlson, 2006). The other aquifers are generally also considered to have formed under fluvial and/or deltaic conditions Catahoula (Maher, 1940), Carrizo-Wilcox (Anderson, 1960), Cockfield (Fisk, 1938; Anderson, 1960), and Sparta (Wang, 1952). 15.5 to 18.6 (51 to 61)
depth below surface meter (feet)
18.6 to 21.6 (61 to 71) 21.6 to 24.7 (71 to 81) (25 to 28 (82 to 92) 28 to 31.1 (92 to 102) 31.1 to 34.1 (102 to 112) 34.1 to 37.5 (112 to 123) 37.5 to 40.5 (123 to 133) 40.5 ro 43.6 (133 to 143) 43.6 to 46.6 (143 to 153) 46.6 to 50 (153 to 164) 50 to 53 (164 to 174) 53 to 56.1 (174 to 184)
1 10 100 1000 (3.28) (32.8) (328) (3280) hydraulic conductivity m/day (ft/day)
Figure 2. Above is an example of how hydraulic conductivity as function of depth below the surface for the Mississippi River Alluvial Aquifer in Madison Parish. Initially eight different techniques were used to determine the anisotropy of hydraulic conductivity from grain-size data. These results were compared with each other using a ttest. For all aquifers seven techniques: Hazen, Beyer, Sauerbrei, Kozeny, Pavchich’s Slichter and Terzaghi, yield a t-test results indicating that results were the same, that is the confidence of difference is less than 95% which is the standard for defining a statistical comparison indicating that two data sets are significantly different (Kirk, 1990). Only the USBR technique yielded t-test results that indicate that its anisotropy results are significantly different from the other seven tests (Tables 1 and 2). With this in mind further analysis was limited to the seven techniques that yield anisotropy values that are similar to each other. The results in the following figures were determined from only samples that include at least 7 techniques used for the determination of vertical anisotropy of hydraulic conductivity.
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percentage of observations
40 35 30 25 20 15 10 5 0 under 0.125
0.125 to 0.25 to 0.25 0.5
0.5 to 1
1 to 2
2 to 4
4 to 8
8 to 16
over 16
bottom K/top K ranges
Figure 3. Above is display of the ratio of hydraulic conductivity for bottom sample divided by hydraulic conductivity for top sample by Pavchich technique for the six aquifers considered in this study. As a result of the fact that hydraulic conductivities generally increase downwards for the sand aquifers of northern Louisiana the typical vertical anisotropy is over one and generally between 1 and 2 (Figs. 4 to 6). Generally the anisotropy values are under 1.25 (Figs. 4 to 6). This is probably a result of the procedure that only determines anisotropy from a stack of layers (K values) and ignores any anisotropy within these individual layers. This probably yields a low value of vertical anisotropy because it ignores small scale features such as stratification (bedding planes) (Domenico and Schwartz, 1990; Anderson and Woesnner, 1992), lamina (Anderson and Woesner, 1992), imbrication (Bouwer, 1978) and layering on a small scale of a fewer millimeters (Bouwer, 1978), which can yield a vertical anisotropy at the scale of core samples and will be missed by grain-size samples for intervals typically 10 to 20 feet. Table 1. Average t-test values for comparison of the eight technique results for vertical anisotropy, read by matching result in each column with a row. Hazen Beyer Sauerbre Kozeny USBR Pavchich Slichter Terzaghi
Hazen XXXXXX 0.24 0.51 0.07 3.62 0.51 0.17 0.13
Beyer 0.24 XXXXXX 0.41 0.19 3.39 0.42 0.08 0.11
Sauerbre 0.51 0.41 XXXXXX 0.49 4.11 0.00 0.35 0.34
Kozeny 0.07 0.19 0.49 XXXXXX 3.67 0.49 0.02 0.08
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USBR 3.62 3.39 4.11 3.67 XXXXXX 4.11 3.53 3.45
Pavchich 0.51 0.42 0.00 0.49 4.11 XXXXXX 0.38 0.38
Slichter 0.17 0.08 0.35 0.02 3.53 0.38 XXXXXX 0.04
Terzaghi 0.13 0.11 0.34 0.08 3.45 0.38 0.04 XXXXXX
Table2. Confidence of difference for t-test comparisons between results for the eight different techniques used to determined vertical anisotropy of hydraulic conductivity determined from grain-size data. Hazen Hazen Beyer Sauerbre Kozeny USBR Pavchich Slichter Terzaghi
Beyer 6 no
XXXX
XXXX
6 no 6 no 6 no 2n4y 6 no 6 no 6 no
6 no 6 no 2n4y 6 no 6 no 6 no
Sauerbre 6 no 6 no
XXXX 6 no 1n5y 6 no 6 no 6 no
Kozeny 6 no 6 no 6 no
XXXX 2n4y 6 no 6 no 6 no
USBR 2n4y 2n4y 1n5y 2n4y
XXXX 1n5y 2n4y 2n4y
Pavchich 6 no 6 no 6 no 6 no 1n5y
XXXX 6 no 6 no
Slichter 6 no 6 no 6 no 6 no 2n4y 6 no
XXXX
Terzaghi 6 no 6 no 6 no 6 no 2n4y 6 no 6 no
6 no
XXXX
percentage of observations
Note n is no and y is yes.
1 1. .75 75 to 2 2 to 2 2. .5 5 to 3 3 to 3 3. .5 5 to 4 4 to 6 6 to 8 ov er 8
1. 5
to
1. 5
to
25
1.
1
to
1.
25
90 80 70 60 50 40 30 20 10 0
Kh/Kv anisotropy ranges Mississippi River
Upland Terrace
Figure 4. Vertical anisotropy results for Pleistocene age or younger units of northern Louisiana, which include Upland Terrace and Mississippi River Alluvial Aquifers. There are 10 and 17 results determined for Mississippi River Alluvial and Upland Terrace Aquifers respectively. What appears to be the case for the six aquifers considered is that, in general, anisotropy values are larger for the aquifers which have higher geometric mean of hydraulic conductivity than lower geometric mean of hydraulic conductivity. The portion of anisotropy results greater than 1.25 is 40% for Upland Terrace and 20% for Mississippi River Alluvium. By comparison for tighter, lower conductivity units such as Cockfield and Carrizo-Wilcox the portion of anisotropy results greater than 1.25 is 10% and 5.7% respectively.
38
7 75 5 to 2 2 to 2 2. .5 5 to 3 3 to 3. 3. 5 5 to 4 4 to 6 6 to 8 ov er 8 1.
1.
5 1.
to
1.
5
to
25 25
1.
1
to
1.
percentage of observations
100 90 80 70 60 50 40 30 20 10 0
Kh/Kv anisotropy ranges Catahoula
Cockfield
Figure 5. Vertical anisotropy results for Catahoula and Cockfield Aquifers within northern Louisiana. There are 23 and 19 results determined for Catahoula and Cockfield Alluvial Aquifers respectively. SUMMARY Grain-size analysis will yield vertical anisotropies such that Kh is typically 1 to 2 times that of Kv. However, in general, vertical anisotropies are slightly larger for aquifers with typically higher hydraulic conductivity such as Upland Terrace and Mississippi River Alluvial than for aquifers with typically lower hydraulic conductivity such as Cockfield or Carrizo-Wilcox. Lastly, these small anisotropies are probably a result of two factors: (1) missing smaller scale features such as very thin beds of lower K material that is just mixed into a sample 10 to 20 feet long and (2) missing the impact of bedding planes, lamina, and grain orientation within a sediment for sediments that contain oval grains. ACKNOWLEDGEMENTS I thank Zahir “Bo” Bolourchi and Brad Hanson for their assistance and the access to the vast data set of well completion reports of the Water Resources Division of the Department of Transportation and Development.
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percentage of observations
100 90 80 70 60 50 40 30 20 10 0
75
0.
to
1 1
25
to
1.
25 1.
75
5
to
1.
5 1.
to
1.
75
1.
to
5
2 2
to
2.
.5
2
to
3 3
to
5 3.
.5
to
4
3
4
to
6 6
to
8
8 er v o
Kh/Kv anisotropy ranges Sparta
Carrizo-Wilcox
Figure 6. Vertical anisotropy results for Sparta and Carrizo-Wilcox Aquifers within northern Louisiana. There are 35 and 110 results determined for Carrizo-Wilcox and Sparta Aquifers respectively. REFERENCES Anderson, H.V., 1960, Geology of Sabine Parish: Department of Conservation Louisiana Geological Survey, Geological Bulletin, no. 34, 164p. Beyer, W., 1966, Hydrogeologische Untersuchungen bei der Ablagerung von Wasserschadstoffen: Zeitschrift fuer Angewandte Geologie, v 12 no. 11, p 599-606. Bouwer, H., 1978, Groundwater Hydrology: McGraw Hill Inc., New York, New York, 480p. Carlson, D.A., in review, Louisiana Aquifer Hydraulic Properties, Louisiana Geological Survey, report of investigations, unnumbered. Carlson, D.A., 2006, Systematic Variability of Hydraulic Conductivity within the Mississippi River Alluvial Aquifer in Northeastern Louisiana: Gulf Coast Association of Geological Societies, Transactions, v 56, p.121-136. Carlson, D.A., 2001, Dependence of Conductivities and Anisotropies on Geologic Properties within the Near-Surface Aquifer in Milwaukee, Wisconsin: Doctoral Dissertation, Milwaukee, Wisconsin, University of Wisconsin-Milwaukee, 768p. Carlson, D.A., 2000, A Hydrogeophysical Examination of the Paleozoic Rock of Southeastern Wisconsin-Northeastern Illinois, unpublished report for U.S. Geological Survey Wisconsin office, 67p.
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Domenico, P.A., and F.W. Schwartz, 1990, Physical and Chemical Hydrogeology: John Wiley and Sons, New York, New York, 824p. Dunning, C.P., D.T. Feinstein, R.J. Hunt, and J.T. Krohelski, 2004, Simulation of GroundWater Flow, Surface-Water Flow, and a Deep Sewer Tunnel System in Menomonee Valley, Milwaukee, Wisconsin: U.S. Geological Survey Scientific Investigations Report 2004-5031, 40p. Fisk, H.N, 1938, Geology of Grant and LaSalle Parishes: Department of Conservation Louisiana Geological Survey, Geological Bulletin, no. 10, 246p. Hazen, A., 1892, Some physical properties of sands and gravels: Report Massachusetts State Board of Health. Kasenow, M., 2002, Determination of Hydraulic Conductivity from Grain Size Analysis: Water Resources Publications, LLC, Highlands Ranch, Colorado, 97p. Kirk, R.E., 1990, Statistics an Introduction, 3 rd edition: Holt, Rinehart and Winston, Inc., Forth Worth, Texas, 711p. Kozeny, J., 1953, Hydraulik: Springer, Wien, Germany, 588p. Milwaukee Metropolitan Sewerage District, 1992, North Shore Interceptor Phase 1A Lining Report no. 2, contract documents, unnumbered. Maher, J.C., Ground-Water Resources of Rapides Parish, Louisiana: Department of Conservation Louisiana Geological Survey, Geological Bulletin, no. 17, 96p. Nader, D.C., 1990, Three-Dimensional Digital Simulation of the Ground Water-Lake Michigan Hydraulic Connection: Master’s Thesis, Milwaukee, Wisconsin, University of Wisconsin-Milwaukee, 159p. Pravedny, G.H., 1966, Design and selection of grain-size composition of filter beds for the transition zones of large dams: Energiia, Moscow Plomb, D.J., 1989, A 3-D Finite Element Model to Predict Drawdown Caused by Infiltration into a 32 ft. Diameter Tunnel in Solving Ground Water Problems with Models: Associate of Groundwater Scientist and Engineers, National Water Well Association, 4th international conference on the use of models to analyze and find working solutions to groundwater problems, Indianapolis, Indiana, February 7-9, 1989, proceedings, p 955-978. Slichter, C.S., 1898, Theoretical investigations of the motion of ground waters: United States Geological Survey, 19 th Annual Report, p 295-384. Terzaghi, K., 1925, Principles of soil mechanics: Engineering News-Record, v. 95, p 832.
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Visher, G.S., 1965, Use of vertical profile in environmental reconstruction: American Association of Petroleum Geologists, Bulletin, v. 49, p 41-61. Vukovic, M., and Soro, A., 1992, Determination of hydraulic conductivity of porous media from grain-size distribution: Water Resources Publications, LLC Highlands Ranch, Colorado. Wang, K.K., 1952, Geology of Ouachita Parish: Department of Conservation Louisiana Geological Survey, Geological Bulletin, no. 28,1296p.
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