Using real surface conductivity component to estimate hydraulic conductivity Mohamed Ahmed Khalil*, Fernando Monteiro Santos, Universidade de Lisboa, Centro de Geofísica da Universidade de Lisboa-

IDL, Campo Grande, Ed. C8, 1749-016

Summary

Estimation of hydraulic conductivity from surface resistivity

measurements is one of the most difficult and challenging

hydrogeophysical targets. The promising side of this

relation is the analogy between electric current flow and

water flow, whereas the grand ambiguity is the non

dimensionality between both two quantities. Imaginary

surface conductivity component is used recently to deduce

the hydraulic conductivity via complex resistivity

measurements (Börner et al 1992, 1996, and Slater and

Lesmes, 2002). Since there are similar properties between

imaginary (out-of phase) and real (in-phase) surface

conductivity components, the latter is used in this paper to

estimate an electrical equivalent parameter to the hydraulic

conductivity. Two mathematical parameters were

determined to express the hydraulic conductivity in two

hydrogeological systems. Highly resistive sand formation

saturated with fresh water is simulated by a parallel resistor

model, whereas highly conductive clayey formation

saturated with salt water is simulated by a parallel

conductor model. The reliability of the proposed method is

tested through applying on different resitivity data sets

resembling the sand and clay systems. Application on the

resistivity data either measured inside the well (Resistivity

well logs) or on the ground surface (Vertical Electrical

Sounding) resulted in a numerical value in the same

magnitude of the measured hydraulic conductivity by

pumping test.

Introduction It is theoretically and empirically deduced that in clay free

sediments that are fully saturated with salt water, all of the

electricity are conducted by the fluid through the pore

space, the rock matrix is non-conducting, and no difference

between intrinsic formation factor (iF ) and apparent

formation factor (aF ) (Archie, 1942, Urish, 1981, Shah,

et. al., 2005). This concept (Archie’s law) is flawed when

the sediments become more clayey and saturated or partially

saturated with high resistivity fresh water (Vinegar and

Waxman, 1984, Martys, 1999, Huntley, 1987). This is

referred to the matrix conduction or surface conductance,

which imparts to the rock conductivity over and above that

due to bulk electrolytic conduction within the free pore

space. So, the validity of Archie's law depends on the value

of the Dukhin number, which is the ratio between surface

conductivity at a given frequency and the conductivity of

the pore water. When the Dukhin number is very low with

respect to 1, Archie's law is valid (Bolève, et al, 2007, and

Crespy, et al, 2007).

Theoretical expressions which include consideration of

conductance or resistance in the solid phase (tR ), the fluid

phase (wR ), as well as a grain surface resistance phase

(mR ) are best represented by an expression in the form of a

parallel resistor model.

Patnode and Wyllie (1950) proposed that the surface

conductivity (mρ ) for low salinity water and/or shaley or

clayey formations could be expressed as a parallel resistor

model.

m

w

ia FF ρ

ρ+=

11 (1)

Waxman-Smits (1968) defined the surface conductivity in

the term of Cation Exchange Capacity (CEC) in clayey

sediments.

w

i

v

ia F

QB

FFρ.

11

+= (2)

Where,vQ is the cation exchange capacity (CEC) per unit

pore volume of the rock (meq/ml). It describes the number

of cations available for conduction that are loosely attached

to the negatively charged clay surface sites. vQ is directly

related to porosity and specific surface area (Revil, et al

1998). B , is the equivalent ionic conductance of clay

exchange cations (mho-cm2/meq) as function of pore water

conductivity.

Urish (1981) defined the surface conductivity for high

resistivity fresh water sands in terms of the geometry of the

matrix system.

w

i

ps

ia F

Sk

FFρ.

11

+= (3)

Where, pS is the specific internal pore area (the total

interstitial surface area of the pores per unit volume of the

sample) and sk is the surface conductance.

Vinegar and Waxman (1984) proposed a complex

conductivity form of Waxman-Smits model (1968), based

on measurements of complex conductivity (*σ ) of shaley

sandstone samples as function of pore water conductivity

(wσ ).

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Fn

Qi

F

BQ

F

v

i

v

a

w λσσ +

+=* (4)

Where, the real part of the equation is the component that

represents the electrolytic conduction in fluid

a

w

F

σ and the

surface conductivity component

i

v

F

BQ which are in phase

with the applied electric field. The imaginary conductivity

component

Fn

Qi v

λ , which is the conductivity, results from

displacement currents that are 90 degrees out-of-phase with

the applied field. Vinegar and Waxman assumed that the

displacement currents were caused by the membrane and

the counter-ion polarization mechanisms. These two

mechanisms were proportional to the effective clay content

or specific surface area represented by the parameter )( vQ .

The parameter )(λ represents an effective quadrature

conductance for these surface polarization

mechanisms. )(λ is slightly dependent on salinity.

Börner et al (1992, 1996) and Slater and Lesmes (2002)

showed that the imaginary (quadrature) surface conductivity

resulted in nearly identical numerical values with the

geometric hydraulic conductivity. The similarity between

real surface conductivity component and imaginary could

be summarized after Börner (2006) as: (1) they have a

constant phase angle behavior for a variety of water-wet

rock material at low frequency (lower than 10 kHZ), (2)

they show an almost identical power law dependence on

frequency for aquifer materials like clean and shaley

sandstones as well as unconsolidated sand and silt, (3) they

showed an experimental dependence on salinity.

Accordingly, it is proposed that the real surface

conductivity could be used also to estimate an equivalent

electric parameter to the hydraulic conductivity.

Theory and method

1-Clayey sediments saturated with salt water

Clayey sediments saturated with salt water could be

simulated theoretically by a parallel conductor model, since

all components are good conductors.

mwto σσσσ

1111++= (5)

Where, oσ is the effective conductivity, tσ is the clay

mineral conductance, wσ is the water conductance, and

mσ is the surface conductance.

It can be expressed also in the form of series resistor model

mwto RRRR ++= (6)

As the resistance of clay (Rt) is very small, so it could be

neglected.

mwo RRR += (7)

Applying the relations between electrical resistance and

resistivity will result in

miwo F ρρρ += (8)

Dividing equation (8) bywρ , will result in

w

mia FF

ρ

ρ+= (9)

Using the surface resistivity mρ as the reciprocal of the

surface conductance parameter defined by Waxman-Smits

(1968) for clay minerals

i

v

F

BQ , will result in

w

v

iia

BQ

FFF σ+= (10)

wvi

ia

BQF

FF

ρ

1=

− (11)

Since the surface conductance in clayey formation is a

function of bulk resistivity and water resistivity (Pfannkuch,

1969), so, multiplying both sides of equation (11) by

woρρ

v

o

i

iawo

BQF

FF ρρρ =

− (12)

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The parameter (

v

o

BQ

ρ ) is proposed to reflect the hydraulic

properties of the rock, where, this parameter should

manifest the relation between water resistivity, porosity,

specific surface area with the bulk resistivity.

2-Clean sand saturated with fresh water

Clean sand saturated with fresh water could be simulated by

a parallel resistor model, since all components have a high

resistivity.

mwto RRRR

1111++= (13)

Neglecting the conductivity of sand (

tR

1 ), applying the

relations between electrical resistance and resistivity, will

result in

mioo F ρρρ

111+= (14)

Multiplying bywρ , and using the surface conductivity in

terms of the geometry of the matrix system (

i

ps

F

Sk ) after

Urish, (1981) will result in

w

i

ps

ia F

Sk

FFρ.

11+= (15)

So,

i

wps

ia F

Sk

FF

ρ=−

11 (16)

The formation factor gradient (

a

i

F

F ) is proposed to convert

the difference between apparent and intrinsic formation

factor into its equivalent hydraulic form.

)()11

(i

wps

a

i

iaa

i

F

Sk

F

F

FFF

F ρ=− (17)

Equation (17) could be simplified as

2)( wps

a

ai

w SkF

FFρρ =

− (18)

The parameter (2

wps Sk ρ ) is proposed to be the

equivalent electric parameter of the hydraulic conductivity

of clean sand fresh water aquifers.

Examples 1-Clayey aquifer

a- Resistivity logs

Wadi El-Assuity is located in the flood plain of the Nile

River in Upper Egypt. Apparent formation factor and water

resistivity are estimated from the resistivity logs (Fig.1) and

the linear relation between water resistivity and (1/Fa) is

shown in fig.(2) based on equation (15).

It is well noticed that the mean value of the proposed

parameter

v

o

BQ

ρ in the sandy clay layers are 3.2 and 3.5

sequentially. The hydraulic conductivity measured in the

same well by pumping test for the first aquifer is 2.5 m/day

(Abu Ella, 1995).

B- Surface resistivity measurements

In the same study area (Fig.3), the same procedures were

carried out using the vertical electrical sounding, which

have been measured nearby some wells with known

hydraulic conductivity value via pumping test. The

empirical linear relation is shown in Fig. (4) and the results

are shown in table (1).

2- Clean sand aquifer

a- Resistivity log

Well No. W-2 is located in the western bank of the Nile

River, about 12 km north from Aswan city. It penetrates the

the Nubian sandstone aquifer in this area (Fig.5). A

pumping test, carried out in the well W-2 resulted in a

hydraulic conductivity of about 10.6 m/day. The estimated 2

wps Sk ρ mean value from the linear relation in Fig.(6) is

10.4.

B- Surface Resistivity measurments

The study area is Keritis basin in Chania, Crete – Greece

The proposed parameter 2

wps Sk ρ is calculated for each

sample (Fig. 7). It resulted in a very close numerical value

to the estimated geometrical hydraulic conductivity by

Soupios et al. (2007), and the highest water resistivity

samples (samples 1 and 12) showed very close numerical

values to the measured hydraulic conductivity (58.7 m/day)

in the area (table.2).

Conclusion The salt water clayey aquifer is simulated electrically in the

present study as a series resistor model, whereas fresh water

sand aquifer is simulated as a parallel resistor model. From

both models electrical equivalent parameters to hydraulic

conductivity are estimated. They are

v

o

BQ

ρ for clayey

aquifers and 2

wps Sk ρ for sand aquifers. The two proposed

parameters are tested in different resistivity data either

measured inside the well (Resistivity well logs) or on the

ground surface (Vertical Electrical Sounding) resulted in a

numerical value in the same magnitude of the measured

hydraulic conductivity by pumping test.

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Fig. (1) Resistivity logs of Well No. 1 Fig. (2) Empirical linear relationships between (Rw) and (1/Fa) for the

at Wadi El-Assiuty different formations of well No.1

Fig. (4) Water resistivity versus (1/Fa) of VES

and wells in Wadi El-Assuity

Fig. (3) Location map of VESes and wells in

Wadi El-Assuity

Fig.(5) Resistiviy logs of well -2 in Aswan area.

Table (1) Measured and estimated hydraulic cductivity Fig.(6) Water resistivity versus 1/Fa

in wadi El-Assiuty. in well -2 at Aswan area.

Table (2) Measured and estimated hydraulic

cductivity Keritis basin in Chania, Crete –

Greece Fig. (7) Water resistivity versus 1/Fa in

Keritis basin in Chania, Crete – Greece .

Depth

in m

ete

r

0100

200

300

0

60

Resistivity (Ohm.m)

SH N (R-16")

LO N (R-64")

Clay and sandy clay

Clay

92m

120m

160m

214m

Clayey sand

Clay and sandy clay

Clayey sand

4 6 8 10 12 14

0

0.4

0.8

1.2

1.6

Y = 0.1530623796 * X - 0.3946461241 R-squared = 0.807945

Rw

1/Fa

Fi= 2.5BQv = 0.38

From 160-214 m depth

Kest.=18-42 m/day (mean=30)

4 6 8 10 12 14

0.6

0.8

1

1.2

1.4

1.6

Y = 0.143195374 * X - 0.06647259027 R-squared = 0.782649

From 120-160m depth

Fi= 15BQv = 2.1

K est. = 2.8-4.3m/day (mean=3.5)

Rw

1/Fa

4 6 8 10 12

0.6

0.8

1

1.2

1.4

1.6

Y = 0.1484619312 * X + 0.07532054646 R-squared = 0.683141

Rw

1/Fa

From 40-92m depth

Fi = 13.2

BQv = 1.97

Kest.=2.4-4 m/day (mean=3.2)

4 6 8 10 12 14

0

0.4

0.8

1.2

1.6

Y = 0.1646809529 * X - 0.4703357158

R-squared = 0.96392

Fi=2.1BQv= 0.35

Kest. = 22-45 m/day (mean= 32)

Rw

1/Fa

From 92-120m depth

Sandy Clay

Sandy Clay

Clayey Sand

Clayey Sand

0 2 4 6 8

0

0.4

0.8

1.2 1/Fa = 0.1658294442 * Rw - 0.08780850546

R-squared = 0.74

1/Fa

Water Resistivity (Ohm.m)

2 4 6 8 10

0

0.2

0.4

0.6

0.8

Y = 0.1080159798 * X - 0.3575450369 R-squared = 0.523666

Fi= 2.4

Ks Sp=0.3Ks Sp Rw2=4.4-27 (mean=10.4)

Rw

1/Fa

5 10 15 20 25 30

0.08

0.12

0.16

0.2

0.24

0.28

1

5

6

7

812

Rw (ohm.m)

1/Fa

Y = 0.005411076225 * X + 0.09122379713 R-squared = 0.815507

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EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2010 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

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