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ESTIMATION OF WATER CONTENT AND POROSITY USING COMBINED RADAR AND GEOELECTRICAL MEASUREMENTS

GRIT DANNOWSKI and UGUR YARAMANCI Technical University of Berlin, Dept. of Applied Geophysics, Ackerstr. 71-76, D-13355 Berlin, Germany. (Received February 2, 1999; revised version accepted July 28, 1999)

ABSTRACT Dannowski, G. and Yaramanci, U., 1999. Estimation of water content and porosity using combined radar and geoelectrical measurements. European Journal of Environmental and Engineering Geophysics, 4: ………… A primary objective in hydrogeological investigations is to obtain information on the water content and porosity of formations. These parameters can be directly related to resistivity. Hence a reliable estimation of these hydrogeological parameters requires that the resistivity (or its inverse, conductivity) should be measured and mapped accurately. By combining different geophysical measurements and information about the geometric structure of the subsurface, the ambiguity of geoelectric inversion for deriving true resistivities can be reduced. Ground penetrating radar (GPR) offers an additional method for inferring the geometric structure and also the spatial distribution of the physical variable associated with water content, namely the dielectric constant. A combination of geoelectrics and GPR allows a more precise description of the subsurface. This is demonstrated by using measurements from a test site in Spandau/Berlin, Germany, where some borehole data exists and the geology is fairly well known. Common mixing formulas were combined in order to derive water content and porosity from the resistivity and dielectric constant. An estimation of hydrogeological parameters based on one measurement alone could not provide the level of confidence and accuracy required for interpretation. By combining GPR and geoelectrics, however, reliable estimates of water content and porosity were obtained which were consistent with the known geological model.

KEY WORDS: water content, porosity, geoelectrical inversion, GPR, mixing formulas

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INTRODUCTION The vadose zone is very important in groundwater prospecting. Some knowledge of its geometry and physical properties is essential. The main objective of this paper is to focus on the most important hydrogeological parameters, namely water content θ and porosity Φ. The resistivity ρ (or its inverse, the conductivity σ) and the dielectric constant εr of rocks and sediments are highly dependent on the moisture content, the salinity and the distribution of the water in the unsaturated zone. Consequently, electrical and electromagnetic methods are highly suitable for assessing these hydrogeological properties. In this paper, we will demonstrate how the water content and porosity can be determined directly from resistivity and GPR surveys. The limitations of these techniques are also discussed. Geoelectric surveys are based on measuring the distribution of electrical currents caused by an applied electric field. The development of the electrical potential field depends on the resistivity distribution in the subsurface. The measurements are often displayed as 2dimensional pseudosections, which provide the spatial distribution of apparent resistivity ρa. The distribution of the true resistivity ρ can be obtained from numerical inversion. To estimate the water content and porosity from mixing formulas, the precise value of ρ is necessary. Because of the equivalence principle relating depth and resistivity, this problem is non-trivial. The demands on the geoelectric inversion can therefore be very high. A reliable true resistivity model can usually only be achieved by constraining the inversion using additional information such as the geometry of the subsurface. The GPR method which is based on the propagation of electromagnetic waves and reflections at structure boundaries can provide this kind of information. With Common-MidPoint (CMP) measurements the propagation velocity of the multi-phase-system between the reflectors is obtained. The velocity is related to the dielectric constant εr which is the actual parameter of interest. The formation parameters can then be calculated using mixing formulas using the individual properties of the components. Usually, it is not possible to obtain the velocity of the saturated zone due to the absence of reflections below the groundwater level. The two methods by themselves yield insufficient or incomplete information. However, when used in combination, the probable range of equivalent models is considerably reduced. Hence an improvement in the determination of the true resistivity is possible which, in turn, leads to a more reliable estimation of the porosity and water saturation.

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MIXING FORMULAS In rocks containing electrolyte, the pore-water conductivity dominates the formation conductivity in most cases. This means that the conductivity of the rock matrix is negligible, except in media with metallic conductivity (e.g., some minerals) and clays. The oldest empirically determined and most widely used relationship between the conductivity of the saturated rock (σo) and the conductivity of the electrolyte (σw) comes from Archie (1942) and was extended later by Sundberg (Schopper, 1982; Mavko et. Al, 1998),

σ0 =

1 σ w + σq0 . F

(1)

Here σq0 represents the conductivity due to the boundary layer at the internal surface. This is negligible in the case of high pore fluid salinity and need only be considered for materials with a high clay content. The geometrical factor F, also called the formation factor or formation resistivity factor, reflects physical pore properties such as pore geometry and connectivity. Therefore, this factor is closely linked with the porosity. The first Archie equation describes the relationship between F and porosity Φ:

F = Φ −m .

(2)

The exponent m increases with compaction, cementation and consolidation and varies between 1.3 and 2.5. Unconsolidated sands have values in the range between 1.3 and 1.5 (e.g. Schopper, 1982; Schön, 1996). If the conductivity of the boundary layer is ignored, equation (2) can be written as the ratio between σw and σ0.

σ0 =Φm σw

⇔

σ0 = Φ m σ w

(3)

For a given conductivity of the saturated rock, m value and pore-water conductivity, it is possible to calculate the porosity from this equation. The insulating effect of air in the unsaturated rock reduces the conductivity. The reduction is expressed by the inclusion of the saturation index (I) into equation (1),

σ=

1 1 σ w + σ q0 . F I

(4)

The saturation index I depends on the degree of saturation Sw. The relationship between the saturation index and saturation degree, also formulated by Archie 1942 (second Archie’s equation), can be defined in the same way as the first Archie equation (e.g., Schopper, 1982).

I = S −wn

(5)

The saturation exponent n varies between 1.4 and 2.2 (Schön, 1996) and has to be estimated based on experience. In practice, a value of n = 2 is often adopted, although this is not always justified. 3

In contrast to the empirical mixing formulas for the conductivity by Archie and Sundberg as in equation (1), the mixing formulas for the dielectric constant are based on physical considerations. The starting point of derivation is the dielectric displacement in a homogeneous medium of ε1 with a spherical inclusion of ε2 in an electromagnetic field. This situation can be described by Maxwell’s law. Different transformations of Maxwell’s law give the law of Hanai and Bruggeman (Dukhin, 1971) for different boundary conditions. 1

ε − ε1 æ ε 2 ö 3 =Φ ç ε 2 − ε1 è ε

(6)

In this equation ε represents the dielectric constant of the heterogeneous mixture, ε1 the dielectric constant of the matrix and ε2 the dielectric constant of the inclusion in the pore space Φ. The dielectric constant of the heterogeneous mixture ε can be derived from the velocity of propagation of electromagnetic waves. Equation (7) relates the measured velocity v to the dielectric constant ε, using the velocity of light c, whereas the magnetic permeability µ = 1 is constant.

ε=

c2

(7)

v2

The condition for the derivation of the equation (6) is the independence of one inclusion from another. This means, that the inclusions do not polarise each other. Therefore (6) is valid only for porosities less than 0.4 and ε1/ε2 < 1. For Φ > 0.4 the equation is valid if ε2/ε1 < 10. Another effect neglected in (6) is the deviation from an ideal dielectric medium, so that the motion of free electrons theoretically has to be considered. Using a complex dielectric constant ε* the conductivity is then taken into account. In that case and for different grain shapes, the law of Hanai, Bruggeman, Shen and Sen (BHSS-law), (see Shen et al, 1985 and Sen et al,1981), is as follows: d

ε* − ε1* æç ε 2 * ö = Φ. ε 2* − ε1* çè ε *

(8)

This equation includes the exponent d, the so-called depolarisation factor, the shape of the inclusion (for example d=1/3 represents a spherical inclusion). The depolarisation factor d can even be a complex parameter and the salinity of the inclusion thereby affects the imaginary part of d, which is commonly negative (Shen et al, 1985). However, at low salinities, as in this work, d is a real value. For 3 phases, equation (8) can be extended by successive use for 2 phases (i.e., by calculating the mixture of air and water and then calculating the mixture of rock grain and the water-air-mixture).

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Using the general relationship between the complex conductivity and complex *

*

dielectric constant : σ = i ω ε 0 ε r = σ + i ω ε 0 ε r (Nabighian, 1989), the BHSS-mixing formula can also be written in terms of conductivity: d

σ* − σ1* æç σ 2 * ö÷ = Φ. * * ç * ÷ σ σ 2 − σ1 è

(9)

Here too, the subscript 1 characterises the matrix and subscript 2 the inclusion in the pore space Φ. At low frequencies the complex conductivity σ* converges to the real part σ, because d

for ω→ 0

σ − σ1 æ σ 2 ö =Φ . ç σ 2 − σ1 è σ

(10)

Equation (10) cannot be solved directly for σ. However, re-writing (10) for σ yields (Bussian, 1983): 1

æ σ1 ö 1−d 1 ç 1− σ2 σ = σ 2 Φ 1−d ç ç σ1 ç 1− σ è

(11)

If the conductivity of matrix (σ1) is small, then equation (11) goes over to the Archie law in (3) with a negligible layer bounded conductivity and m=1/(1-d) or d=(m-1)/m (Bussian, 1983). For a 2-phase-system, with matrix and pore fluid, the water is the inclusion with index 2, which is described in the law of Archie (3) with index w. In the case of direct current or low frequency conductivity the BHSS-mixing formula represents Archie’s law. A further mixing-law is also widely used to estimate the dielectric constant of a multiphase-system and is based on the principle of time average relationship; the addition of weighted travel time. This relation is called CRIM (Complex Refractive Index Method) (Graeves et. al., 1996, and Wilson, et. al.,1996).

ε = ( 1 − Φ ) ε m + S w Φ ε w + ( 1 − S w ) Φ εa

(12)

The subscripts m, w and a represent matrix, water and air respectively. Sw is the degree of saturation.

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TEST SITE AND MEASUREMENTS The test site was selected where the available subsurface information suggested the layering was not complicated and the water table was nearly horizontal. These conditions simplify the verification of the interpretation results. The selected area in Berlin-Spandau (Germany) comprises a glacial spillway. The main material is a medium grained sand (Assmann, 1957). The result of the core drilling is shown in Fig. 1 as a stratigraphic section. The position and orientation of the survey lines are shown in Fig. 2.

Fig.. 1: Stratigraphic section at the test site based on core drilling.

Fig. 2: Overview and location of all measurements conducted at the test site. All radar and geoelectrical measurements were taken on a grid in an east-west (x) and north-south direction (y). The radar profiles (trace spacing 0.1 m) were separated by about 3 m from each other. The CMP-measurements had a profile spacing of approximately 6 m. All radar data were measured at a frequency of 200 MHz, using RAMAC/GPR equipment from MALÅ GeoScience. 6

The geoelectric sections were 10 m apart. The basic electrode spacing of the Wenner-array was 1.5 m. The Campus Geopulse equipment was used for the measurements. In addition, a seismic refraction profile was measured at x = 37 m, in order to obtain geometric information from another independent geophysical method. The additional geoelectrical Schlumberger sounding (mid-point at 37;37) served also to verify the groundwater depth. The core drilling (at 37;24) yielded the exact depth of the groundwater table and the stratigraphic sequence in the area of investigation at one point. The conductivity of the groundwater was measured in another borehole. Two samples (P1, P2) from the core drilling were investigated in the laboratory to determine the porosity. Both samples consisted of a medium grained sand, with porosity of 0.42 ± 0.01. INTERPRETATION Two representative radar sections are shown in Fig. 3. The event at 60 ns represents the reflection from the water table. The velocity analysis of CMP data from 28 different points yields a velocity of 0.156 ± 0.001 m/ns between the surface and the reflection from the groundwater. With this velocity the distance-travel time-section can be converted to a distance-depth section. The depth of the reflectors was picked from all profile measurements to show the variation of groundwater level, (Fig. 4). The groundwater level has a slope of 0.20 m over 75 m (corresponding to 0.3 %) to the east, which cannot be explained by variations in the topography or lateral velocity changes. The location of the test site is within the area of influence (cone of depression) of a pumping well. The distance to the pumping well (2.7 km) and the direction of the slope both support this explanation. However, the gradient could also be due to the natural drainage into a nearby lake and river. Here the drainage is also expected to be in the same direction. The velocity of propagation, ascertained from the CMPs, yields a dielectric constant of 3.70 ± 0.06 and by using equation (8), a water content of 0.029 ± 0.002 was obtained. A value of 0.42 was adopted for the porosity. This value is based on laboratory analysis of the sand, and is in agreement with values known previously for this sand. In using equation (8) the dielectric constant of the matrix is set to 4.5 (quartz), to 1.0 for air and to 80.8 for the water. With m = 1.5 in Archie's law for a slightly cemented sand, a depolarisation factor d = 1/3 is used. The error is calculated from the error of the dielectric constant. With the CRIM formula (12) and the same dielectric constants, a water content of 0.034 ± 0.003 is calculated. Both results agree within the error range.

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Fig. 3: Example of radar profile and CMP using 200 MHz antennas at the test site.

Fig. 4: Depth of groundwater level as derived from radar measurements The geoelectric pseudosections have to be inverted in order to determine the distribution of true resistivity in the subsurface. The inversion has to be optimised, because the resistivities required for the calculation of formation parameters have to be very accurate.

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Fig. 5: Example of geoelectric measurement results using smooth inversion The smooth inversion of geoelectrical data using RES2DINV (1995) (see also Loke and Barker, 1996) for a selected section is shown in Fig. 5. The model shows a near-continuous decrease of resistivity in the vertical direction. This suggests that a 1D model should be suitable for the area. However, it is impossible to draw a boundary between high resistivity and low resistivity regions, i.e. between the unsaturated and the saturated zone corresponding 9

to the groundwater level. Low resistivities are present near the surface, but below this horizon there is an extremely high resistivity zone followed by another low resistivity zone. The smooth inversion provides a good qualitative resistivity image of the subsurface. However, it is difficult to interpret the model in terms of layer geometry or depths, as the variation in intrinsic resistivities is gradational rather than discrete. Therefore, the estimation of hydrogeological parameters with depth becomes problematic. In order to map the expected sharp resistivity contrast a block inversion scheme (RESXIP2DI, 1996, and Inman, 1975) is used. In this scheme supplementary additional available information can be incorporated.

Fig. 6: Geometric modelling of the data in Fig. 5 using block inversion.

resistivity [Ωm]

depth range[m]

400 3600 22000 1980

0-0.2 0.2-0.6 0.6-2.0 2.0-4.25

water content with Φ = 0.42 0.19 ± 0.04 0.06 ± 0.02 0.026 ± 0.006 0.08 ± 0.02

water content with Φ = 0.35 0.18 ± 0.04 0.06 ± 0.02 0.024 ± 0.005 0.08 ± 0.02

Table 1: Water content from different resistivities and porosities Φ in the unsaturated zone. The starting model requires input of the number of bodies, their geometry and coordinates and intrinsic resistivities. The disadvantage of this kind of inversion is the dependence of the results on the starting model. If no information about the subsurface is available, the starting model has to be guessed somewhat arbitrarily and the inverted model may not accurately characterise the subsurface, (Dannowski, 1998). The combination of qualitative information from the smooth inversion and geometric information from the radar measurements provided a good starting model. The inverted model with block inversion (Fig. 6) characterises the subsurface more precisely. These results can be verified against borehole and other measurements.

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A porosity of 0.35 ± 0.06 was calculated from the resistivity of the saturated zone (ρ = 105 ± 22 Ωm), using equation (3). The other used parameters were: resistivity of water ρw = 22.1 Ωm (measured in the above mentioned additional borehole) and a cementation exponent m = 1.5. With this porosity it is possible to calculate the water content θ for the unsaturated zone (see Table 1). Another way to calculate the water content is the use of the porosity ascertained in the laboratory. In Table 1 the two possibilities are compared. It shows quite explicitly, that the different porosities of 42 % or 35 % have no decisive influence on the calculated water content in the unsaturated zone. The actual error source can be attributed to the varying values of resistivity between the profiles. The seismic refraction measurement was evaluated with the intercept-time method. Fig. 7 shows the corresponding travel-time curve and the model of the subsurface. The ground water table is mapped as a good refractor that corresponds to the known depth. In addition, the layer of surficial organic soil, observed in the borehole (see Fig. 1), is also resolved. No attempt was made to derive any water content and porosity from these very low velocities, as this would be unreliable regardless of the mixing rule for seismic velocities that might be used. The water content ascertained from the radar and geoelectric methods are displayed as water-content depth curves in Fig. 8. Fig. 8c shows the expected water content distribution for the soil and sand layers indicated in the borehole (Fig. 1).

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Fig. 7: Seismic travel-time curve and model.

Fig. 8: Distribution of water content with depth as derived from: (a) radar measurements, (b) block inversion of the geoelectric pseudosections, (c) expectation from borehole log.

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The water content from the radar measurements is indicative of dry sand. However, the radar method depends on contrasts in dielectric and resistivity impedances for the development of reflections. Here these contrasts are not available in the unsaturated zone so radar cannot yield a more exact differentiation of water content in this zone. The geoelectric sections can yield a more precise image of the subsurface in the unsaturated zone. The expectation, shown in Fig. 8c, was only partly realised. The first 0.2 m is typified by a comparatively high water content (0.19 ± 0.04). According to the borehole record (Fig. 1) this layer corresponds to organic soil. The next layer (to 0.6 m) with a water content of 0.06 ± 0.02 agrees with the expected capillary zone. Also the following layer with a low water content is indicative of a well drained sand. The thickness cannot be verified with other geophysical methods. This layer goes down to a depth of 3.7 m. There is a capillary zone with slightly increased water content, however, which would be associated with a gradual change in properties. The fourth layer identified by the geoelectric method (Table 1), in the depth interval 2 m to 4.25 m, is probably due to seeping water, held by capillary force, causing water accumulation to disrupt the water-phases above the capillary zone (Busch et. al., 1974). Together with the following capillary zone this layer is characterised by a water content of 0.08 ± 0.02.

CONCLUSION The radar method by itself cannot resolve the structure of the subsurface sufficiently, because reflectors above the groundwater level are often absent. For this reason the variable water content in the unsaturated zone can not be defined closely. Without additional information concerning the porosity (e.g. from investigations on samples in the laboratory), a hydrogeological interpretation of the radar measurements (i.e. velocity) cannot succeed. If the porosity is available, the errors in the estimation of water content from the velocities are small. The error will be much larger if using a porosity value derived from the literature or experience. If only direct current geoelectric data are available, the error in the estimation of water content will be very large. Unconstrained inversion may give resistivity values which are in error by 100 %. Therefore, this method by itself cannot specify the structure and the resistivity distribution precisely enough for hydrogeological determinations. The necessity of exact determination of the true resistivity for estimating hydrogeological parameters requires more accurate inversion models. This can be accomplished by determining the true geometry of the structure by other means. The radar method can provide this information. Seismic surveys may also provide the same or similar information but they are time-consuming and expensive. Similarly, boreholes can provide invaluable ground-truth but are costly. These alternatives may be necessary in cases where the radar measurements do not yield the required information, e.g. in the presence of saline water, a large capillary zone or conductive clay layers.

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The radar data used here adequately restricted the variety of models. The porosity could be defined only with a relative error of 17 %, but this error had no significant influence (maximum deviation of 1 % ) on the calculated water content of the unsaturated zone. At this location also the determination of the vertical stratigraphic sequence at one point (e.g. the borehole) would be sufficient, because the subsurface is mainly horizontally layered. A good hydrogeological interpretation was possible by constraining the geoelectric inversion using geometric information from the GPR survey. The limited resolution of structures at larger depths, particularly below groundwater level, can be overcome using direct current resistivity surveys. It has been demonstrated in this paper, that the determination of water content and porosity using classical techniques such as geoelectrics and radar, will always be accompanied by some ambiguity since these hydrogeological properties are derived indirectly from geophysical properties. This limitation may be off-set to some extent by the introduction of new techniques such as surface nuclear magnetic resonance (SNMR), which allows the direct determination of water content (Schirov et al.1991, Legchenko and Shushakov, 1998, Yaramanci et al. 1998, 1999). Even then the use of multiple methods and datasets to constrain the inferred model is recommended. ACKNOWLEDGEMENTS The authors would like to thank Gerhard Lange and the BGR (Federal Institute for Geosciences and Resources) who made it possible to conduct the measurements at the test site, and also for helpful discussion of the results. Thanks are also due to the two reviewers for their constructive comments and to the Editor for correcting the use of English. REFERENCES Archie, G.E.,1942. Electrical resistivity as an aid in core analysis interpretation. Transaction of the American Institute of Mining Engineers, 146: 54-52 Assmann, P., 1957. Der geologische Aufbau der Gegend von Berlin. Senat für Bau- und Wohnungswesen, Berlin. Busch, K.F. und Luckner, L., 1974. Geohydraulik für Studium und Praxis. Ferdinand Enke Verlag Stuttgart. Bussian, A.E., 1983. Electrical conductance in a porous medium. Geophysics, 48: 1258-1268. Dannowski, G., 1998. Abschätzung hydrologisch relevanter gesteinsphysikalischer Merkmale durch kombinierte Radar- und Geoelektrikerkundung. MSc. thesis, TU Berlin. Davis, J.L. and Annan, A.P., 1989. Ground-Penetrating-Radar for high-resolution mapping of soil and rock stratigraphy. Geophysical Prospecting, 37: 531-551. Dukhin, S.S., 1971. Dielectric properties of disperse systems. Surface and Colloid Science, 3: 85-165. Graeves, R.J., Lesmes, D.P., Lee, J.M. and Toksöz, M.N., 1996. Velocity variations and water content estimated from multi-offset ground-penetrating radar. Geophysics, 61: 683-695. 14

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