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Aquifer transmissivity of porous media from resistivity data
Journal of Hydrology, 82 (1985) 143--153 Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands
143
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AQUIFER TRANSMISSIVITY OF POROUS MEDIA FROM RESISTIVITY DATA SRI NIWAS and D.C. SINGHAL Department of Earth Sciences, University of Roorkee, Roorkee 24 7 667 (India) School of Hydrology, University of Roorkee, Roorkee 247 667 (India) (Received April 12, 1984; accepted for publication November 30, 1984)
ABSTRACT
Sri Niwas and Singhal, D.C., 1985. Aquifer transmissivity of porous media from resistivity data. J. Hydrol., 82: 143--153. To optimize the information/cost ratio and avoid the indiscriminate and excessive use of drilling and pump testing to calculate aquifer transmissivity an analytical relationship between modified transverse resistance and aquifer transmissivity has been developed for estimating transmissivity from resistivity sounding data. The relation takes into consideration the variation in the quality of groundwater. The relation has been tested successfully for the glacial aquifers of Rhode Island, U.S.A. and alluvial aquifers of three different areas of Uttar Pradesh, India. The practical applicability of the relation lies in the fact that if hydraulic conductivity is known for any reference point of a porous homogeneous aquifer, one can get fairly good idea of the transmissivity of the aquifer at other locations within a basin, from surface geo-electrical measurements.
INTRODUCTION O n e of the c o m m o n m e t h o d s for evaluating aquifer characteristics, i.e. h y d r a u l i c c o n d u c t i v i t y , transmissivity, and storativity, is the use o f p u m p i n g tests. The f o r m u l a e for calculating these p a r a m e t e r s are valid o n l y if various a s s u m p t i o n s a b o u t t h e t y p e o f aquifer, n a t u r e o f fluid flow, rate o f p u m p i n g a n d welt storage h o l d g o o d u n d e r field c o n d i t i o n s . Large-diameter dug wells, w h i c h are quite c o m m o n in India, pose several p r o b l e m s pertaining to the analysis of p u m p i n g test data. These wells show significant disturbing effects o f well storage in test data. Moreover, if the p e r m e a b i l i t y is low, it m a y n o t be possible to carry o u t l o n g - d u r a t i o n tests w h i c h m a y be necessary for a b e t t e r u n d e r s t a n d i n g o f the h y d r a u l i c characteristics of the aquifer. Surface geo-electrical m e a s u r e m e n t s in w h i c h large v o l u m e s o f earth materials are sampled o f f e r an alternative a p p r o a c h for the e s t i m a t i o n o f aquifer characteristics. If we have i n f o r m a t i o n on transmissivity of the aquifer f r o m o t h e r sources the use o f t e s t - p u m p i n g m e t h o d s t o calculate these p a r a m e t e r s can be reduced. In the present paper, an a t t e m p t has been m a d e to calculate aquifer transmissivity f r o m surface geo-electrical m e a s u r e m e n t s . 0022-1694/85/$03.30
© 1985 Elsevier Science Publishers B.V.
144 BACKGROUND For a homogeneous fluid like water, hydraulic conductivity depends both on matrix and fluid properties -- the relevant matrix properties being mainly grain-size distribution, shape of grain (or pores), tortuosity, specific surface, and porosity. Bear and Bachmat (1966, 1967) showed that intrinsic permeability, which is a property of the medium alone (Nutting, 1930), is directly related to porosity, tortuosity and "conductance of the elementary matrix channels". As the hydraulic conductivity is directly related to the intrinsic permeability, the former would also depend on the tortuosity, porosity and "conductance of the matrix channels". Electric current follows the path of least resistance, as does water. Within and around pores, the mode of conduction of electricity is ionic and thus the resistivity of the medium is controlled more by porosity and water conductivity than by the resistivity of the rock matrix. Thus, at the pore level, the electrical path is similar to the hydraulic path and the resistivity should reflect hydraulic conductivity. Various investigators have studied the relationship between electrical and hydrological parameters of aquifers. Jones and Buford (1951) measured the formation factor and intrinsic permeability of some graded sand samples and found that as the grain size increases, the respective formation factors and the intrinsic permeabilities also increase. Croft ( 1 9 7 1 ) d e v e l o p e d a relation between the aquifer intrinsic permeability and formation factor for given porosity ranges. Pfannkuch (1969) proposed a model relating the intrinsic permeability with resistivity for clean sand and gravel saturated with water. In clean sands, saturated with fresh water, surface conduction becomes the main electrical transport mechanism, in addition to ionic conduction through the electrolyte. Surface conduction is a form of ionic transport which takes place at the solid--liquid interface by means of exchange mechanisms. It was demonstrated by Patnode and Wyllie (1950) and Winsauer and McCardell (1953) that with high-resistivity waters, the role of surface conductance becomes effective. Pfannkuch (1969) explained it by the double-layer theory. Recently, some investigators have tried to establish empirical and semiempirical relationships between various aquifer parameters and the parameters obtained by resistivity measurements (Ungemach et al., 1969; Steeples, 1970; Kelly, 1977a, b; Schimshal, 1981; Kosinsky and Kelly, 1981). Kelly (1977a, b) established an empirical relation between aquifer electrical resistivity and aquifer hydraulic conductivity and a semi-empirical relation between the aquifer formation factor and hydraulic conductivity for glacial outwash materials of the Upper Pawcatuck River Basin in southern Rhode Island, U.S.A. Kosinski and Kelly (1981) have attempted to establish a direct equivalence between "normalized transverse resistance" and aquifer transmissivity. Sri Niwas and Singhal (1981) established an analytical
145 relationship between transverse unit resistance and aquifer transmissivity in homogeneous isotropic media and tested the applicability of the relation using published data for glacial outwash materials of Rhode Island. However, the applicability of the formula was restricted in the sense that an assumption was made that the quality of groundwater remained fairly uniform in a basin. Singhal and Sri Niwas (1983), widened the applicability of a relation between modified transverse resistance and aquifer transmissivity by considering variation in water quality and demonstrated its applicability to field data of southern Uttar Pradesh, India. In the present study the above approach has been tested by using data from Rhode Island (Kosinski and Kelly, 1981), and field data from two other alluvial areas of Uttar Pradesh (U.P.), viz. Saharanpur district (western U.P.) and Varanasi district (east U.P.) with encouraging results in all the four areas.
APPLIED ASPECTS Sri Niwas and Singhal ( 1 9 8 1 ) p r o p o s e d the following relationships between aquifer transmissivity and transverse unit resistance for homogeneous and isotropic porous aquifers: T =
~
(1)
where c~ = Ko (some constant); T is the aquifer transmissivity (m 2 day -1 ); K is the hydraulic conductivity (m day -1 ); e is the electrical conductivity (mho m -1 ); and R is the transverse unit resistance (Ohm m 2 ). Equation (1) assumes that changes in aquifer resistivity are due to changes in aquifer material and tortuosity of the interconnected pores. However, it is presumed that the gross chemical quality of the groundwater remains relatively uniform. Equation (1) was modified by Singhal and Sri Niwas (1983) taking into consideration a "modified aquifer resistivity" instead of "aquifer resistivity" (Kosinsky and Kelly, 1981). However, the modification factor is always the ratio of the average aquifer water resistivity (Pw) and the aquifer water resistivity (Pw) at a particular location. Thus we can rewrite eqn. (1) as: T :
(go')R'
(2)
where o' (= o • Pw/Pw ) and R ' (= R • ffw/Pw ) are respectively, "modified conductivity" and "modified transverse resistance" of the aquifer. Here the product K a ' remains constant for a groundwater basin and can be calculated if the hydraulic conductivity of the aquifer at a reference point is known. As mentioned by Singhal and Sri Niwas (1983), a natural corollary of eqn. (2) can be written as: K = c~p'
(3)
V.E.S. a no.
36 20 25 62 59 54 60
Site no.
1 2 3 4 5 6 7
1537 1975 1325 580 1000 375 2275
Aquifer resistivity, p (Ohm ft)
3
82 80 75 210 115 100 51.3
Aquifer thickness (ft)
4
988 1075 1047 533 1304 700 3509
Hydraulic conductivity, K (gallon d a y -1 ft -2)
5
364 449 368 164 164 125 182
Aquifer water resistivity, Pw (Ohmft)
6
8
1095.3 1140.9 934.0 917.4 1581.7 778.2 3242.5
0.0009129 0.0008765 0.0010706 0.00109 0.0006322 0.001285 0.0003084
NormalNormalized ized aquifer aquifer resistivity c o n d u c t i v i t y (P'=P'Pw/(mhoft -l) Pw ) t a k i n g Pw = 2 5 9 . 4 (Ohm ft)
7
0.9020675 0.9421909 1.1210097 0.5809985 0.824493 0.8994794 1.0821931
81,728 83,057.5 63,744 175,311 165,525 70,816 151,369
(KO')R' with Ko'----0.91
R' = RPw / Pw ( O h m f t 2) 89,811.3 91,272.0 70,048.5 192,649.8 181,895.5 77,820 166,339.68
Aquifer transmissivity using eqn. (2) T ----
Normalized transverse resistance
Ko'
Product
11
10
9
81,000 86,000 78,500 110,000 150,000 70,000 180,000
Actual field transmissivity (gallon d a y -1 ft -1)
12
aV.E.S. :- vertical electrical s o u n d i n g s . i O h m f t = 0 . 3 0 4 8 O h m m , 1 ft = 0 . 3 0 4 8 m, 1 gallon day -1 ft -1 = 0 . 0 1 2 4 2 m 3 d a y - l m -1 , 1 gallon day -1 ft -2 ~ 0 . 0 4 0 7 4 m 3 d a y -1 m-2. Geo-electrical p a r a m e t e r s o f c o l u m n s 7 - - 1 1 c a l c u l a t e d b y p r e s e n t a u t h o r s .
2
1
Results of electrical s o u n d i n g a n d p u m p i n g tests for t h e Beaver River a q u i f e r a n d C h i p u x e t River a q u i f e r o f t h e P a w c a t u c k River Basin, R h o d e Island ( D a t a p a r t l y t a k e n f r o m K o s i n s k y a n d Kelly, 1 9 8 1 )
TABLE 1
y.
2 V.E.S.
no.
36 20 25 62 59 54 60
1 Site
no.
1 2 3 4 5 6 7
89,811 91,272 70,048 192,650 181,895 77,820 166,340
by Kosinsky and Kelly (1981)
81,283 82,621 63,356 174,530 164,787 70,500 150,446
by using equation T = 0.91R'
81,728 83,057 63,744 175,311 165,525 70,816 151,369
81,000 86,000 78,500 110,000 150,000 70,000 180,000
in field (actual)
6 T of Kosinsky and Kelly taking actual average aquifer water resistivity (259 Ohm ft)
3 4 5 Transmissivities obtained (gallon day -1 ft -1 )
Error analysis of the transmissivity data calculated by using various approaches
TABLE 2
Kosinsky and Kelly and actual
+ 283 -- 3379 -- 15,144 + 64,530 + 14,787 + 500 --29,554
equation T ----0.91 R' and actual
+728 --2943 -- 14,756 + 65,311 + 15,525 +816 --28,631
7 8 Difference between transmissivities (gallon day -I ft -1 )
+ 8811 + 51,272 --8452 + 82,650 + 31,895 + 7820 -- 13,660
day-1 ft-1)
9 T using Kosinsky and Kelly approach taking average of aquifer water resistivity and field values (gallon
148
PAWCATUCK RIVER BASIN
"= m°t '~ mot g 1/.,0t
~ 120
804 i 60
80
I00 MODIFIED
120×103140 TRANSVERSE
160
180
200
RESISTANCE (Ohm ft2 )
Fig. 1. Plot of transmissivity (observed and estimated) versus modified transverse resistance from Pawcatuck Basin, Rhode Island (data from Kosinsky and Kelly, 1981).
Pawcatuck River Basin, R h o d e Island Table 1 summarizes the data of vertical electrical soundings, aquifer water resistivities, field hydraulic conductivities and transmissivities for seven locations of glacial outwash aquifers of Rhode Island (Kosinsky and Kelly, 1981). With the help of data of aquifer water resistivity, an average value of aquifer water resistivity (Pw = 259.4 Ohm ft) is taken for the purpose of calculation of ratio ~w/Pw, which is useful for calculating the modified aquifer resistivity (p' = p-fi~/Pw ) and modified transverse resistance (R' = hp'). It is clear from Table 1 that the product Ka' is fairly constant with a statistical average value of 0.91 in the basin. From this general value of K o ' for the Upper Pawcatuck River Basin, values of transmissivity using equation T-0 . 9 1 R ' have been computed (column 11 in Table 1). These values of computed transmissivities compare fairly well with the actual field transmissivities (column 12 in Table 1). Figure 1 shows a graph of transmissivities derived from different approaches and the modified transverse resistance. The transmissivities arrived at by Kosinsky and Kelly (1981) are also plotted in the same figure. All the calculated values are given in Table 2. The root mean square error ( = ~ / 1 / n ( T ¢ -- To) 2 ) between the actual field transmissivities and those derived by eqn. (2) a m o u n t to 2.8 x 104 gallon day -1 ft -1 (350 m 2 day -1 ) and are found to be approximately equal to the error between actual field transmissivity and the transmissivities arrived at by Kosinsky and Kelly (349 m 2 day -1 ). Here Tc and To are the computed and observed transmissivities, respectively, and n is the number of data points. It is significant that
149 Kosinsky and Kelly used an average aquifer water resistivity of 235 Ohm ft in their calculations of normalized transverse resistance and tried to equate this parameter with the aquifer transmissivity instead of the actual average of 259.4 Ohm ft. By taking a value of 235 instead of 259.4, Kosinsky and Kelly (1981) have given, probably unknowingly, a weight of 2 3 5 / 2 5 9 . 4 = 0.9059367 to each transverse resistance value. This is the reason for an equal RMS error at individual sites in both the approaches. Had they used the actual average aquifer water resistivity, the transmissivities would have had a RMS error of 4 9 0 m 2 day -~ .
Alluvial aquifers of Uttar Pradesh (India) Encouraged by the results obtained from the published data, we tested the relation for alluvial aquifers of the eastern U.P. (Varanasi area), southern U.P. (Banda Area) and the western U.P. (Saharanpur area). The sites selected are well distributed in the area, and pump-test data were analyzed using Jacob's m e t h o d and the Theis recovery m e t h o d (Papadopulos, 1967; Varanasi area) and Boulton and Streltsova (1976; Banda area), and Theis (1935), Hantush and Jacob (1955), Hantush (1956) and Walton (1962; Saharanpur area). Electrical resistivity soundings were taken as close as possible to the tubewell using the Schlumberger electrode configuration. The resistivity data thus obtained were interpreted using a fast automatic method (Sri Niwas et al., 1982) based upon ridge-regression estimation (Marquardt, 1970). In this scheme, first the resistivity transform function is extracted using digital filters (Ghosh, 1971). Resistivity transform functions for a layered earth can be expressed as a function of two parameter vectors, the vector of u n k n o w n parameters (layer thicknesses and resistivities) and the vector of known parameters (some form of half-current electrode separation), respectively. To apply the scheme, the resistivity transform function is quasilinearized by generating a system of linear equations through a Taylor's series expansion about some point (initial model parameters) in the u n k n o w n parameter space, at a predecided point in the known parameter space, and retaining only the first derivative term. The resulting simultaneous equations are solved using ridge-regression estimation to get stable least-square solutions in the form of layer parameters. Of course, the choice of the model and the subsequent selection of one or more initial model parameters within this model are a priori selection and seem to be limitations of the method, but professional experience minimizes this problem. In cases where some of the unknown model parameters are known a priori, they will be deleted from the vector of u n k n o w n parameters. The value of the Marquardt Lambda (}~) was kept initially at 1.00 and was varied at a rate of ~/1.5 or ~/1.2 in each advancing iteration. The relevant data (geo-electrical and pump test) for all the areas considered are given in Table 3 along with computed values of transmissivity using the relation T = ~R '. Figure 2 presents the variation of transmissivity derived from various approaches together with modified transverse resistance. The
18.74 13.80 5.15 11.17 11.17
Banda area 1 2 3 4 5
Saharanpur area 1 21.49 2 12.52 3 21.18 4 12.85 5 23.63 6 14.14
Varanasi area 1 15.0 2 16.8 3 21.3 4 16.8
Pw at 25°C (Ohm m)
Site no.
42.41 85.29 35.62 83.28 33.08 82.78
49.7 64.5 150.0 100.3
12.18 9.92 33.81 19.14 19.14
p' (Ohmm)
975.4 1594.9 773.0 916.10 302.7 1258.25
2485.0 1980.7 4644.0 3131.4
159.56 119.04 879.06 76.56 76.56
R' (Ohm m 2)
21.0 51.0 27.0 50.0 10.5 70.7
55.5 75.2 161.9 109.3
11.63 15.32 39.85 18.25 19.60
K (m d a y - l )
0.50 0.60 0.75 0.60 0.32 0.84
1.1 1.1 1.08 1.09
0.954 1.544 1.180 0.953 1.023
Ka'
481.0 948.0 584.0 547.8 96.0 1065.0
2773.4 2311.2 5015.1 3413.4
152.34 183.80 1036.20 73.00 78.40
(m 2 day -1 )
T o
Relevant results o f electrical s o u n d i n g and p u m p i n g t e s t s for t h r e e alluvial a q u i f e r s in U t t a r P r a d e s h , India
TABLE 3
586.0 958.0 465.0 550.0 182.0 756.0
2557.0 2038.1 4778.6 3222.2
180.3 134.5 993.3 86.5 86.5
T c : 0~Rt ( m 2 d a y -1)
0
151 TABLE 4 R o o t m e a n square error RMS error using
Area
T~R
Banda area Varanasi area S a h a r a n p u r area P a w c a t u c k Basin
~.1200 1000
F300[
32.6 230 146 350
¢' SAHARANPURAREA
,"
/
/
/"
/ "~oo0
,
800
8
.~.,/~/~
/
/'¢q"
~x
600
//
400 o<
76.1 303 390 490
/
/
5
'
(E) VARANASI AREA ....
E
z
T~CZR
(m 2 day -1 )
BANDA AREA
T
>
'
(m 2 day -1 )
,ooo[
200 2.,,/ / --
,
200
1000 / q
z.oo
6oo
TRANSVERSE
,
~
/ 5
,
L
,
,
8oo 1o,oo lOOO 2000 2ooo 400o 5ooo 6~oo
', T.R) RESISTANCE/
MOOIFIED
T.R.(©hr'q
nq 2 .,
Fig. 2. Plot o f transmissivity (observed and e s t i m a t e d ) versus m o d i f i e d transverse resistance for alluvial aquifers in the districts Banda, Varanasi and S a h a r a n p u r , India.
various approaches are pump tests, from the relation T = R' (as suggested by Kosinski and Kelly, 1981), and from the relation T = oaR' (as suggested by present authors). The root mean square error calculated for each area between the observed transmissivity and those calculated using T = R' and T = 0~R' are summarized in Table 4. Table 4 clearly demonstrates the utility of the relation T = a . R ' . The groundwater quality seems uniform for Varanasi area, therefore, the parameters used were aquifer resistivity and transverse resistance, rather than modified aquifer resistivity and modified transverse resistance.
CONCLUSIONS It is concluded that the aquifer transmissivity can be estimated more accurately using the linear relation T - - ~ R ' between transmissivity and
152
modified transverse resistance than using T = R'. The analytical relation is based on the fact that K o ' = o~ is relatively constant for homogeneous porous formations. This was investigated for different areas successfully as described in the present article. The method is quite useful in that if the hydraulic conductivity of the aquifer at a reference point is known, it can be estimated at other locations with the help of surface geo-electrical measurements.
ACKNOWLEDGEMENTS
The authors are thankful to Sri J.J. Mathew and Sri S.K. Dimri for providing data help in computations and to Dr. B.B.S. Singhal for useful discussions. Financial support by the University Grants Commission for collecting field data is thankfully acknowledged. Organisations such as the Groundwater Investigation Organization and Action for Food Production helped at various stages of data collection.
REFERENCES Bear, J. and Bachmat, Y., 1966. Hydrodynamic dispersion in nonuniform flow through porous media, taking into account density and viscosity differences. Hydrol. Lab. Tech., Haifa, Israel, I.A.S.H., P.N. 4166. Bear, J. and Bachmat, Y., 1967. A generalized theory on hydrodynamic dispersion in porous media. I.A.S.H. Symp. Artificial Recharge and Management of Aquifers, Israel, P.N. 72: 7--16. Boulton, N.S. and Streltsova, T.D., 1976. The drawdown near an abstraction well of large diameter under nonsteady conditions in an unconfined aquifer. J. Hydrol., 30: 29--46. Croft, M.G., 1971. A method of calculating permeability from electric logs. In: Geological Survey Research. U.S. Geol. Surv., Prof. Pap. 750-B, pp. 265--269. Ghosh, D.P., 1971. The application of linear filter theory to the direct interpretations of geoelectrical resistivity sounding measurements. Geophys. Prospect., 19: 192--217. Hantush, M.S., 1956. Analysis of data from pumping tests in leaky aquifers. Trans. Am. Geophys. Union, 37: 702--714. Hantush, M.S. and Jacob, C.E., 1955. Nonsteady radial flow in an infinite leaky aquifer. Trans. Am. Geophys. Union, 36: 95--100. Jones, P.H. and Buford, T.B., 1951. Electric logging applied to ground water exploration. Geophysics, 16: 115--139. Kelly, W.E., 1977a. Electrical resistivity for estimating permeability. J. Geotech. Eng. Div., 103: 1165--1168. Kelly, W.E., 1977b. Geoelectrical sounding for estimating aquifer hydraulic conductivity. Groundwater, 15 : 420--425. Kosinsky, W.K. and Kelly, W.E., 1981. Geoeletrical soundings for predicting aquifer properties. Ground Water, 19 : 163--171. Marquardt, D.W., 1970. Generalized inverses, ridge regression, biased linear estimation and non-linear estimation. Technometrics, 12: 591--612. Nutting, P.G., 1930. Physical analysis of oil sands. Am. Assoc. Pet. Geol. Bull., 14: 1337--1349. Papadopulos, I.S., 1967. Drawdown distribution around a large diameter well. Proc. Syrup. Groundwater Hydrology, San Fransisco. Am. Water Resour. Assoc., pp. 156-167.
153 Patnode, H.W. and Wyllie, M.R.J., 1950. The presence of conductive solids in reservoir rocks as a factor in electric log interpretation. J. Pet. Technol., 189: 47--52. Pfannkuch, H.O., 1969. On the correlation of electrical conductivity properties of porous systems with viscous flow transport coefficient. Proc. I.A.H.R. Int. Symp. Fundamentals of Transport Phenomenon in Porous Media, Haifa, pp. 45--54. Schimschal, U., 1981. The relationship of geophysical measurements to hydraulic conductivity at the Brantley Dam Site, New Mexico. Geoexploration, 19: 115--126. Singhal, D.C. and Sri Niwas, 1983. Estimation of aquifer transmissivity from surface geoelectrical measurements. Proc. Symp. Methods and Instrumentation of Investigating Groundwater Systems, The Netherlands, pp. 405--414. Sri Niwas and Singhal, D.C., 1981. Estimation of aquifer transmissivity from Dar-Zarrouk parameters in porous media. J. Hydrol., 50: 393--399. Sri Niwas, Pawan Kumar and Wason, H.R., 1982. Fast automatic solution of the inverse resistivity problem. Proc. Ind. Acad. Sci. (Earth Planet. Sci.), 91: 29--41. Steeples, D.W., 1970. Resistivity method in prospecting for groundwater. M.S. Thesis, Kansas State Univ., Manhattan, Kans. Theis, C.V., 1935. The relation between the lowering of the piezometric surface and the rate of duration of discharge of a well using groundwater storage. Trans. Am. Geophys. Union, 16: 519--524. Ungemach, P., Mostaghimi, F. and Duprat, A., 1969. Essais de d~termination du coefficient demanagesinement en nappe. Libre Application ~ la nappe alluvial du Rhin. Bull. Int. Assoc. Sci. Hydrol., 14: 169--190. Walton, W.C., 1962. Selected analytical methods for well and aquifer evaluation. Ill. State Water Surv. Bull., 49. Winsauer, W.O. and McCardell, W.M., 1953. Ionic double layer conductivity in reservoir rock. Pet. Trans. AIME, 198: 129--134. Wyllie, M.R.J. and Spangler, M.B., 1952. Application of electrical resistivity measurements to the problem of fluid flow in porous media. Bull. Am. Assoc. Pet. Geol., 36: 359-403.
143
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AQUIFER TRANSMISSIVITY OF POROUS MEDIA FROM RESISTIVITY DATA SRI NIWAS and D.C. SINGHAL Department of Earth Sciences, University of Roorkee, Roorkee 24 7 667 (India) School of Hydrology, University of Roorkee, Roorkee 247 667 (India) (Received April 12, 1984; accepted for publication November 30, 1984)
ABSTRACT
Sri Niwas and Singhal, D.C., 1985. Aquifer transmissivity of porous media from resistivity data. J. Hydrol., 82: 143--153. To optimize the information/cost ratio and avoid the indiscriminate and excessive use of drilling and pump testing to calculate aquifer transmissivity an analytical relationship between modified transverse resistance and aquifer transmissivity has been developed for estimating transmissivity from resistivity sounding data. The relation takes into consideration the variation in the quality of groundwater. The relation has been tested successfully for the glacial aquifers of Rhode Island, U.S.A. and alluvial aquifers of three different areas of Uttar Pradesh, India. The practical applicability of the relation lies in the fact that if hydraulic conductivity is known for any reference point of a porous homogeneous aquifer, one can get fairly good idea of the transmissivity of the aquifer at other locations within a basin, from surface geo-electrical measurements.
INTRODUCTION O n e of the c o m m o n m e t h o d s for evaluating aquifer characteristics, i.e. h y d r a u l i c c o n d u c t i v i t y , transmissivity, and storativity, is the use o f p u m p i n g tests. The f o r m u l a e for calculating these p a r a m e t e r s are valid o n l y if various a s s u m p t i o n s a b o u t t h e t y p e o f aquifer, n a t u r e o f fluid flow, rate o f p u m p i n g a n d welt storage h o l d g o o d u n d e r field c o n d i t i o n s . Large-diameter dug wells, w h i c h are quite c o m m o n in India, pose several p r o b l e m s pertaining to the analysis of p u m p i n g test data. These wells show significant disturbing effects o f well storage in test data. Moreover, if the p e r m e a b i l i t y is low, it m a y n o t be possible to carry o u t l o n g - d u r a t i o n tests w h i c h m a y be necessary for a b e t t e r u n d e r s t a n d i n g o f the h y d r a u l i c characteristics of the aquifer. Surface geo-electrical m e a s u r e m e n t s in w h i c h large v o l u m e s o f earth materials are sampled o f f e r an alternative a p p r o a c h for the e s t i m a t i o n o f aquifer characteristics. If we have i n f o r m a t i o n on transmissivity of the aquifer f r o m o t h e r sources the use o f t e s t - p u m p i n g m e t h o d s t o calculate these p a r a m e t e r s can be reduced. In the present paper, an a t t e m p t has been m a d e to calculate aquifer transmissivity f r o m surface geo-electrical m e a s u r e m e n t s . 0022-1694/85/$03.30
© 1985 Elsevier Science Publishers B.V.
144 BACKGROUND For a homogeneous fluid like water, hydraulic conductivity depends both on matrix and fluid properties -- the relevant matrix properties being mainly grain-size distribution, shape of grain (or pores), tortuosity, specific surface, and porosity. Bear and Bachmat (1966, 1967) showed that intrinsic permeability, which is a property of the medium alone (Nutting, 1930), is directly related to porosity, tortuosity and "conductance of the elementary matrix channels". As the hydraulic conductivity is directly related to the intrinsic permeability, the former would also depend on the tortuosity, porosity and "conductance of the matrix channels". Electric current follows the path of least resistance, as does water. Within and around pores, the mode of conduction of electricity is ionic and thus the resistivity of the medium is controlled more by porosity and water conductivity than by the resistivity of the rock matrix. Thus, at the pore level, the electrical path is similar to the hydraulic path and the resistivity should reflect hydraulic conductivity. Various investigators have studied the relationship between electrical and hydrological parameters of aquifers. Jones and Buford (1951) measured the formation factor and intrinsic permeability of some graded sand samples and found that as the grain size increases, the respective formation factors and the intrinsic permeabilities also increase. Croft ( 1 9 7 1 ) d e v e l o p e d a relation between the aquifer intrinsic permeability and formation factor for given porosity ranges. Pfannkuch (1969) proposed a model relating the intrinsic permeability with resistivity for clean sand and gravel saturated with water. In clean sands, saturated with fresh water, surface conduction becomes the main electrical transport mechanism, in addition to ionic conduction through the electrolyte. Surface conduction is a form of ionic transport which takes place at the solid--liquid interface by means of exchange mechanisms. It was demonstrated by Patnode and Wyllie (1950) and Winsauer and McCardell (1953) that with high-resistivity waters, the role of surface conductance becomes effective. Pfannkuch (1969) explained it by the double-layer theory. Recently, some investigators have tried to establish empirical and semiempirical relationships between various aquifer parameters and the parameters obtained by resistivity measurements (Ungemach et al., 1969; Steeples, 1970; Kelly, 1977a, b; Schimshal, 1981; Kosinsky and Kelly, 1981). Kelly (1977a, b) established an empirical relation between aquifer electrical resistivity and aquifer hydraulic conductivity and a semi-empirical relation between the aquifer formation factor and hydraulic conductivity for glacial outwash materials of the Upper Pawcatuck River Basin in southern Rhode Island, U.S.A. Kosinski and Kelly (1981) have attempted to establish a direct equivalence between "normalized transverse resistance" and aquifer transmissivity. Sri Niwas and Singhal (1981) established an analytical
145 relationship between transverse unit resistance and aquifer transmissivity in homogeneous isotropic media and tested the applicability of the relation using published data for glacial outwash materials of Rhode Island. However, the applicability of the formula was restricted in the sense that an assumption was made that the quality of groundwater remained fairly uniform in a basin. Singhal and Sri Niwas (1983), widened the applicability of a relation between modified transverse resistance and aquifer transmissivity by considering variation in water quality and demonstrated its applicability to field data of southern Uttar Pradesh, India. In the present study the above approach has been tested by using data from Rhode Island (Kosinski and Kelly, 1981), and field data from two other alluvial areas of Uttar Pradesh (U.P.), viz. Saharanpur district (western U.P.) and Varanasi district (east U.P.) with encouraging results in all the four areas.
APPLIED ASPECTS Sri Niwas and Singhal ( 1 9 8 1 ) p r o p o s e d the following relationships between aquifer transmissivity and transverse unit resistance for homogeneous and isotropic porous aquifers: T =
~
(1)
where c~ = Ko (some constant); T is the aquifer transmissivity (m 2 day -1 ); K is the hydraulic conductivity (m day -1 ); e is the electrical conductivity (mho m -1 ); and R is the transverse unit resistance (Ohm m 2 ). Equation (1) assumes that changes in aquifer resistivity are due to changes in aquifer material and tortuosity of the interconnected pores. However, it is presumed that the gross chemical quality of the groundwater remains relatively uniform. Equation (1) was modified by Singhal and Sri Niwas (1983) taking into consideration a "modified aquifer resistivity" instead of "aquifer resistivity" (Kosinsky and Kelly, 1981). However, the modification factor is always the ratio of the average aquifer water resistivity (Pw) and the aquifer water resistivity (Pw) at a particular location. Thus we can rewrite eqn. (1) as: T :
(go')R'
(2)
where o' (= o • Pw/Pw ) and R ' (= R • ffw/Pw ) are respectively, "modified conductivity" and "modified transverse resistance" of the aquifer. Here the product K a ' remains constant for a groundwater basin and can be calculated if the hydraulic conductivity of the aquifer at a reference point is known. As mentioned by Singhal and Sri Niwas (1983), a natural corollary of eqn. (2) can be written as: K = c~p'
(3)
V.E.S. a no.
36 20 25 62 59 54 60
Site no.
1 2 3 4 5 6 7
1537 1975 1325 580 1000 375 2275
Aquifer resistivity, p (Ohm ft)
3
82 80 75 210 115 100 51.3
Aquifer thickness (ft)
4
988 1075 1047 533 1304 700 3509
Hydraulic conductivity, K (gallon d a y -1 ft -2)
5
364 449 368 164 164 125 182
Aquifer water resistivity, Pw (Ohmft)
6
8
1095.3 1140.9 934.0 917.4 1581.7 778.2 3242.5
0.0009129 0.0008765 0.0010706 0.00109 0.0006322 0.001285 0.0003084
NormalNormalized ized aquifer aquifer resistivity c o n d u c t i v i t y (P'=P'Pw/(mhoft -l) Pw ) t a k i n g Pw = 2 5 9 . 4 (Ohm ft)
7
0.9020675 0.9421909 1.1210097 0.5809985 0.824493 0.8994794 1.0821931
81,728 83,057.5 63,744 175,311 165,525 70,816 151,369
(KO')R' with Ko'----0.91
R' = RPw / Pw ( O h m f t 2) 89,811.3 91,272.0 70,048.5 192,649.8 181,895.5 77,820 166,339.68
Aquifer transmissivity using eqn. (2) T ----
Normalized transverse resistance
Ko'
Product
11
10
9
81,000 86,000 78,500 110,000 150,000 70,000 180,000
Actual field transmissivity (gallon d a y -1 ft -1)
12
aV.E.S. :- vertical electrical s o u n d i n g s . i O h m f t = 0 . 3 0 4 8 O h m m , 1 ft = 0 . 3 0 4 8 m, 1 gallon day -1 ft -1 = 0 . 0 1 2 4 2 m 3 d a y - l m -1 , 1 gallon day -1 ft -2 ~ 0 . 0 4 0 7 4 m 3 d a y -1 m-2. Geo-electrical p a r a m e t e r s o f c o l u m n s 7 - - 1 1 c a l c u l a t e d b y p r e s e n t a u t h o r s .
2
1
Results of electrical s o u n d i n g a n d p u m p i n g tests for t h e Beaver River a q u i f e r a n d C h i p u x e t River a q u i f e r o f t h e P a w c a t u c k River Basin, R h o d e Island ( D a t a p a r t l y t a k e n f r o m K o s i n s k y a n d Kelly, 1 9 8 1 )
TABLE 1
y.
2 V.E.S.
no.
36 20 25 62 59 54 60
1 Site
no.
1 2 3 4 5 6 7
89,811 91,272 70,048 192,650 181,895 77,820 166,340
by Kosinsky and Kelly (1981)
81,283 82,621 63,356 174,530 164,787 70,500 150,446
by using equation T = 0.91R'
81,728 83,057 63,744 175,311 165,525 70,816 151,369
81,000 86,000 78,500 110,000 150,000 70,000 180,000
in field (actual)
6 T of Kosinsky and Kelly taking actual average aquifer water resistivity (259 Ohm ft)
3 4 5 Transmissivities obtained (gallon day -1 ft -1 )
Error analysis of the transmissivity data calculated by using various approaches
TABLE 2
Kosinsky and Kelly and actual
+ 283 -- 3379 -- 15,144 + 64,530 + 14,787 + 500 --29,554
equation T ----0.91 R' and actual
+728 --2943 -- 14,756 + 65,311 + 15,525 +816 --28,631
7 8 Difference between transmissivities (gallon day -I ft -1 )
+ 8811 + 51,272 --8452 + 82,650 + 31,895 + 7820 -- 13,660
day-1 ft-1)
9 T using Kosinsky and Kelly approach taking average of aquifer water resistivity and field values (gallon
148
PAWCATUCK RIVER BASIN
"= m°t '~ mot g 1/.,0t
~ 120
804 i 60
80
I00 MODIFIED
120×103140 TRANSVERSE
160
180
200
RESISTANCE (Ohm ft2 )
Fig. 1. Plot of transmissivity (observed and estimated) versus modified transverse resistance from Pawcatuck Basin, Rhode Island (data from Kosinsky and Kelly, 1981).
Pawcatuck River Basin, R h o d e Island Table 1 summarizes the data of vertical electrical soundings, aquifer water resistivities, field hydraulic conductivities and transmissivities for seven locations of glacial outwash aquifers of Rhode Island (Kosinsky and Kelly, 1981). With the help of data of aquifer water resistivity, an average value of aquifer water resistivity (Pw = 259.4 Ohm ft) is taken for the purpose of calculation of ratio ~w/Pw, which is useful for calculating the modified aquifer resistivity (p' = p-fi~/Pw ) and modified transverse resistance (R' = hp'). It is clear from Table 1 that the product Ka' is fairly constant with a statistical average value of 0.91 in the basin. From this general value of K o ' for the Upper Pawcatuck River Basin, values of transmissivity using equation T-0 . 9 1 R ' have been computed (column 11 in Table 1). These values of computed transmissivities compare fairly well with the actual field transmissivities (column 12 in Table 1). Figure 1 shows a graph of transmissivities derived from different approaches and the modified transverse resistance. The transmissivities arrived at by Kosinsky and Kelly (1981) are also plotted in the same figure. All the calculated values are given in Table 2. The root mean square error ( = ~ / 1 / n ( T ¢ -- To) 2 ) between the actual field transmissivities and those derived by eqn. (2) a m o u n t to 2.8 x 104 gallon day -1 ft -1 (350 m 2 day -1 ) and are found to be approximately equal to the error between actual field transmissivity and the transmissivities arrived at by Kosinsky and Kelly (349 m 2 day -1 ). Here Tc and To are the computed and observed transmissivities, respectively, and n is the number of data points. It is significant that
149 Kosinsky and Kelly used an average aquifer water resistivity of 235 Ohm ft in their calculations of normalized transverse resistance and tried to equate this parameter with the aquifer transmissivity instead of the actual average of 259.4 Ohm ft. By taking a value of 235 instead of 259.4, Kosinsky and Kelly (1981) have given, probably unknowingly, a weight of 2 3 5 / 2 5 9 . 4 = 0.9059367 to each transverse resistance value. This is the reason for an equal RMS error at individual sites in both the approaches. Had they used the actual average aquifer water resistivity, the transmissivities would have had a RMS error of 4 9 0 m 2 day -~ .
Alluvial aquifers of Uttar Pradesh (India) Encouraged by the results obtained from the published data, we tested the relation for alluvial aquifers of the eastern U.P. (Varanasi area), southern U.P. (Banda Area) and the western U.P. (Saharanpur area). The sites selected are well distributed in the area, and pump-test data were analyzed using Jacob's m e t h o d and the Theis recovery m e t h o d (Papadopulos, 1967; Varanasi area) and Boulton and Streltsova (1976; Banda area), and Theis (1935), Hantush and Jacob (1955), Hantush (1956) and Walton (1962; Saharanpur area). Electrical resistivity soundings were taken as close as possible to the tubewell using the Schlumberger electrode configuration. The resistivity data thus obtained were interpreted using a fast automatic method (Sri Niwas et al., 1982) based upon ridge-regression estimation (Marquardt, 1970). In this scheme, first the resistivity transform function is extracted using digital filters (Ghosh, 1971). Resistivity transform functions for a layered earth can be expressed as a function of two parameter vectors, the vector of u n k n o w n parameters (layer thicknesses and resistivities) and the vector of known parameters (some form of half-current electrode separation), respectively. To apply the scheme, the resistivity transform function is quasilinearized by generating a system of linear equations through a Taylor's series expansion about some point (initial model parameters) in the u n k n o w n parameter space, at a predecided point in the known parameter space, and retaining only the first derivative term. The resulting simultaneous equations are solved using ridge-regression estimation to get stable least-square solutions in the form of layer parameters. Of course, the choice of the model and the subsequent selection of one or more initial model parameters within this model are a priori selection and seem to be limitations of the method, but professional experience minimizes this problem. In cases where some of the unknown model parameters are known a priori, they will be deleted from the vector of u n k n o w n parameters. The value of the Marquardt Lambda (}~) was kept initially at 1.00 and was varied at a rate of ~/1.5 or ~/1.2 in each advancing iteration. The relevant data (geo-electrical and pump test) for all the areas considered are given in Table 3 along with computed values of transmissivity using the relation T = ~R '. Figure 2 presents the variation of transmissivity derived from various approaches together with modified transverse resistance. The
18.74 13.80 5.15 11.17 11.17
Banda area 1 2 3 4 5
Saharanpur area 1 21.49 2 12.52 3 21.18 4 12.85 5 23.63 6 14.14
Varanasi area 1 15.0 2 16.8 3 21.3 4 16.8
Pw at 25°C (Ohm m)
Site no.
42.41 85.29 35.62 83.28 33.08 82.78
49.7 64.5 150.0 100.3
12.18 9.92 33.81 19.14 19.14
p' (Ohmm)
975.4 1594.9 773.0 916.10 302.7 1258.25
2485.0 1980.7 4644.0 3131.4
159.56 119.04 879.06 76.56 76.56
R' (Ohm m 2)
21.0 51.0 27.0 50.0 10.5 70.7
55.5 75.2 161.9 109.3
11.63 15.32 39.85 18.25 19.60
K (m d a y - l )
0.50 0.60 0.75 0.60 0.32 0.84
1.1 1.1 1.08 1.09
0.954 1.544 1.180 0.953 1.023
Ka'
481.0 948.0 584.0 547.8 96.0 1065.0
2773.4 2311.2 5015.1 3413.4
152.34 183.80 1036.20 73.00 78.40
(m 2 day -1 )
T o
Relevant results o f electrical s o u n d i n g and p u m p i n g t e s t s for t h r e e alluvial a q u i f e r s in U t t a r P r a d e s h , India
TABLE 3
586.0 958.0 465.0 550.0 182.0 756.0
2557.0 2038.1 4778.6 3222.2
180.3 134.5 993.3 86.5 86.5
T c : 0~Rt ( m 2 d a y -1)
0
151 TABLE 4 R o o t m e a n square error RMS error using
Area
T~R
Banda area Varanasi area S a h a r a n p u r area P a w c a t u c k Basin
~.1200 1000
F300[
32.6 230 146 350
¢' SAHARANPURAREA
,"
/
/
/"
/ "~oo0
,
800
8
.~.,/~/~
/
/'¢q"
~x
600
//
400 o<
76.1 303 390 490
/
/
5
'
(E) VARANASI AREA ....
E
z
T~CZR
(m 2 day -1 )
BANDA AREA
T
>
'
(m 2 day -1 )
,ooo[
200 2.,,/ / --
,
200
1000 / q
z.oo
6oo
TRANSVERSE
,
~
/ 5
,
L
,
,
8oo 1o,oo lOOO 2000 2ooo 400o 5ooo 6~oo
', T.R) RESISTANCE/
MOOIFIED
T.R.(©hr'q
nq 2 .,
Fig. 2. Plot o f transmissivity (observed and e s t i m a t e d ) versus m o d i f i e d transverse resistance for alluvial aquifers in the districts Banda, Varanasi and S a h a r a n p u r , India.
various approaches are pump tests, from the relation T = R' (as suggested by Kosinski and Kelly, 1981), and from the relation T = oaR' (as suggested by present authors). The root mean square error calculated for each area between the observed transmissivity and those calculated using T = R' and T = 0~R' are summarized in Table 4. Table 4 clearly demonstrates the utility of the relation T = a . R ' . The groundwater quality seems uniform for Varanasi area, therefore, the parameters used were aquifer resistivity and transverse resistance, rather than modified aquifer resistivity and modified transverse resistance.
CONCLUSIONS It is concluded that the aquifer transmissivity can be estimated more accurately using the linear relation T - - ~ R ' between transmissivity and
152
modified transverse resistance than using T = R'. The analytical relation is based on the fact that K o ' = o~ is relatively constant for homogeneous porous formations. This was investigated for different areas successfully as described in the present article. The method is quite useful in that if the hydraulic conductivity of the aquifer at a reference point is known, it can be estimated at other locations with the help of surface geo-electrical measurements.
ACKNOWLEDGEMENTS
The authors are thankful to Sri J.J. Mathew and Sri S.K. Dimri for providing data help in computations and to Dr. B.B.S. Singhal for useful discussions. Financial support by the University Grants Commission for collecting field data is thankfully acknowledged. Organisations such as the Groundwater Investigation Organization and Action for Food Production helped at various stages of data collection.
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153 Patnode, H.W. and Wyllie, M.R.J., 1950. The presence of conductive solids in reservoir rocks as a factor in electric log interpretation. J. Pet. Technol., 189: 47--52. Pfannkuch, H.O., 1969. On the correlation of electrical conductivity properties of porous systems with viscous flow transport coefficient. Proc. I.A.H.R. Int. Symp. Fundamentals of Transport Phenomenon in Porous Media, Haifa, pp. 45--54. Schimschal, U., 1981. The relationship of geophysical measurements to hydraulic conductivity at the Brantley Dam Site, New Mexico. Geoexploration, 19: 115--126. Singhal, D.C. and Sri Niwas, 1983. Estimation of aquifer transmissivity from surface geoelectrical measurements. Proc. Symp. Methods and Instrumentation of Investigating Groundwater Systems, The Netherlands, pp. 405--414. Sri Niwas and Singhal, D.C., 1981. Estimation of aquifer transmissivity from Dar-Zarrouk parameters in porous media. J. Hydrol., 50: 393--399. Sri Niwas, Pawan Kumar and Wason, H.R., 1982. Fast automatic solution of the inverse resistivity problem. Proc. Ind. Acad. Sci. (Earth Planet. Sci.), 91: 29--41. Steeples, D.W., 1970. Resistivity method in prospecting for groundwater. M.S. Thesis, Kansas State Univ., Manhattan, Kans. Theis, C.V., 1935. The relation between the lowering of the piezometric surface and the rate of duration of discharge of a well using groundwater storage. Trans. Am. Geophys. Union, 16: 519--524. Ungemach, P., Mostaghimi, F. and Duprat, A., 1969. Essais de d~termination du coefficient demanagesinement en nappe. Libre Application ~ la nappe alluvial du Rhin. Bull. Int. Assoc. Sci. Hydrol., 14: 169--190. Walton, W.C., 1962. Selected analytical methods for well and aquifer evaluation. Ill. State Water Surv. Bull., 49. Winsauer, W.O. and McCardell, W.M., 1953. Ionic double layer conductivity in reservoir rock. Pet. Trans. AIME, 198: 129--134. Wyllie, M.R.J. and Spangler, M.B., 1952. Application of electrical resistivity measurements to the problem of fluid flow in porous media. Bull. Am. Assoc. Pet. Geol., 36: 359-403.