2D sections of porosity and water saturation from ...

© 2010 European Association of Geoscientists & Engineers 575

* [email protected]

2D sections of porosity and water saturation from integrated resistivity and seismic surveys

R. Mota1* and F.A. Monteiro Santos2

1 Laboratório Nacional de Engenharia Civil (LNEC), Departamento de Geotecnia, Av. do Brasil, 101, 1799-066 Lisbon, Portugal2 Universidade de Lisboa-IDL, Campo Grande, Edifício C8, 1749-016 Lisbon, Portugal

Received December 2009, revision accepted July 2010

ABSTRACTPorosity and degree of saturation – or the water content – are important parameters for hydrogeo-logical, geotechnical and environmental studies. Geophysical methods, especially the resistivity method, are routinely employed to study the spatial variations of these parameters. Resistivity is highly influenced by the presence of water in pore spaces and hence is well suited for studying the presence of fluids on a site and its saturation condition. However, the non-uniqueness of the solution of resistivity models has led to the joint use of more than one geophysical method in order to reach more accurate geophysical models. In this research, we combined resistivity and seismic refraction profiles. The integration of these methods was particularly aimed to obtain 2D sections for estimat-ing the porosity, water saturation and volumetric water content rather than to obtain a better geo-physical solution. Independently inverted resistivity and P-wave refraction sections are the input data to an iterative analysis process using simulated annealing. Empirical equations taken from the literature are used to relate both the seismic velocity and the electrical resistivity with porosity and water saturation. Several parameters, such as the resistivity of water and clay; the velocity of water, clay, air and matrix; clay percentage and Archie’s parameters remain constant throughout the process. The values considered for each parameter are derive from both the literature and laboratory measurements. Resistivity and seismic refraction profiles were performed on a site located within the LNEC campus. At this site, field measurements of void ratio and volumetric water content were performed at different depths. Soil samples were also collected at three different depths in order to perform laboratory measurements of these parameters and to determine soil composition. The laboratory results were compared with the 2D sections of each parameter. The proposed approach was also applied to two other locations with different and well characterized geology. These tests also allowed us to characterize the dependency of the clay content on the resistivity. This research has potential fields of application in environmental studies, in particular, the deter-mination of probable pathways of pollutants; in hydrological investigations, where it can be useful to transport of nutrients studies; and in geotechnical studies, where, for example, it will be able to give a continuous image of the saturation degree of an embankment.

ties and its subsurface structure, i.e., it reduces the non-unique-ness of the solution. This has led to the joint application of different techniques in geophysical prospecting. The joint inversion of different geophysical data usually needs a common parameter. For resistivity and seismic velocity, there are two ways to perform the joint inversion: by the petrophysical approach, in which both the porosity and the water saturation are the connecting link (e.g., Wempe 2000; Berryman et al. 2002), or by the structural approach, in which the pattern of the variation in properties is used as a common constraint (e.g., Haber and Oldenburg 1997), Gallardo and Meju (2003), de Nardis et al.

INTRODUCTIONThe interpretation of geophysical pseudosections involves solv-ing the inverse problem, i.e., the determination of parameters (i.e., subsoil properties, such as resistivity and seismic velocity), which, by an iterative process, results in a calculated set of data that fits the field data within satisfactory limits. In general, it is possible to obtain more than one model. In sites where the geo-logy is complex, data gathered with different geophysical meth-ods allow us to better constrain the estimation of the site proper-

Near Surface Geophysics, 2010, 8, 575-584 doi:10.3997/1873-0604.2010042

R. Mota and F.A. Monteiro Santos576

© 2010 European Association of Geoscientists & Engineers, Near Surface Geophysics, 2010, 8, 575-584

inversion methodology of data obtained with several geophysical methods, based on the principle that models for each property are defined at the same points in the physical space considered. This condition is intended to overcome the limitations caused by the absence of a mathematical function that connects the measured properties. A team from Lawrence Livermore National Laboratory, after performing a joint application of field data, mathematical modelling and laboratory data of resistivity and seismic velocity (in the high-frequency range), obtained some resistivity-seismic velocity relationships with porosity, water saturation and clay percentage. The team concluded that the resistivity method leads to better final results, although it shows a strong dependency on soil clay content (Berge et al. 2000; Berryman et al. 2000; Aguirre et al. 2001). Kis (2002) used a serial expansion technique to perform a joint inversion of both refraction and VES data. Meju and Gallardo (2003) used a graphical distribution of resistivity versus seismic velocity data from audio frequency magnetotelluric, transient electromagnetic, resistivity and seismic refraction to obtain a relationship between both properties. When comparing this relationship with that obtained by Rudman et al. (1975), which integrated data from 700–1300 m deep sounding wells (consolidated materials), they concluded that the porosity is a connecting factor between resis-tivity and velocity also in the presence of unconsolidated materi-als. Comina et al. (2004) jointly inverted VES and surface waves data using a weighted, damped, least-squares procedure. De Nardis et al. (2005) combined VES and refraction data using a hybrid joint inversion technique. Archie’s law was also employed by Cooper et al. (2008) to develop porosity maps using airborne electromagnetic surveys in order to study salt concentrations. The growing application of geophysical methods to environ-mental studies has led to research on the relationship between measured geophysical properties and hydrogeologic parameters, particularly the porosity and water saturation (e.g., Wempe 2000; Berryman et al. 2002; Comina et al. 2004). This work presents a method that integrates electrical resistiv-ity models with seismic velocity models, using simulated anneal-ing (SA). The aim is to obtain 2D estimates of the porosity, water saturation and volumetric water content. The method is com-pared with field and laboratory data from samples collected at a controlled site and is applied to two other sites with different geologic environments.

RESISTIvITY-SEISmIC vELOCITY INTEGRATION WITH SImULATED ANNEALING (RSANN)The method presented here uses simulated annealing as an opti-mization tool for integration of independently obtained tomo-graphic models of resistivity and velocity, with a view to produc-ing 2D sections of the porosity, water saturation and volumetric water content. Resistivity models were achieved with Res2DInv software (Loke and Barker 1996; Loke 1999), whereas those for velocity were obtained with Rayfract (Schuster and Bosz 1993; Resources 2004).

(2005) and Nath et al. (2000) presented examples of the applica-tion of the structural approach for both resistivity and the seismic refraction methods, while Comina et al. (2004) applied it to a combination of resistivity and the surface wave methods. Boundaries between layers of resistivity and seismic velocity models may not be coincident but in order to perform a joint inversion, one can assume that they are (Hering et al. 1995). The choice of geophysical methods to be integrated depends on the target and on the geological conditions that one expects to have in the field. At the moment, several techniques using different geophysical data and different approaches have been developed (see, for example, Nath et al. 2000 and de Nardis et al. 2005). The joint inversion of geophysical data was first presented by Vozoff and Jupp (1975) to magnetotelluric and resistivity sur-veys. The joint application of resistivity and seismic data was first accomplished by Dobroka et al. (1991), who performed a joint inversion of vertical seismic profiles with vertical electrical soundings (VES). Carrara et al. (1994, 1999) integrated VES data with seismic profiles, using Archie’s law (Archie 1942) and Wyllie’s equation of mean value (Wyllie et al. 1956), to obtain porosity and saturation curves as a function of resistivity and seismic velocity. Haber and Oldenburg (1997) developed a joint

FIGURE 1

Position of data in resistivity (a) and velocity models (b).

FIGURE 2

How to identify whether velocity model points are within the resistivity

model perimeter.

2D sections of porosity and water saturation 577


(1)

where N is the number of grid points in the section and α and β are weights (α + β = 1); ρcal is obtained with Archie’s law (1942) for unsaturated soils with an added term that accounts for the contribution of clay to the conduction of electrical current:

(2)

where a, m and n are Archie’s parameters, φ is porosity, σw and σcl are the conductivity (inverse of resistivity) of water and clay, respectively and Sw is water saturation. So ρcal is given by:

(3)

while vcal is calculated using the relationship developed by Carrara et al. (1994) based on the mean time equation of Wyllie (1956) and considering that a soil or a rock mass is a four com-ponent system – matrix, clay, water and air – in which the solid component is occupied by the matrix and by clay, whereas the pore spaces between grains are partially filled with water:

(4)

where Pcl is the clay percentage in the matrix; v

m is the propaga-

tion velocity of the seismic wave in the matrix; vcl is the propaga-

tion velocity of the seismic wave in clay; vw is the propagation

velocity of the seismic wave in water and va is the propagation

velocity of the seismic wave in air. Resistivity and velocity variations with porosity were analysed for some values of water saturation using equations (3) and (4) and considering 50% of clay in the solid phase. Figure 3 shows that both properties have a low variation with the porosity for a satu-rated medium and a high resistivity dependence with the saturation

Considering that the spatial parametrizations of the two mod-els are different, due to the different representation schemes of each software type (Fig. 1), the first step is to find the common area covered by both the resistivity and seismic models. Based on the resistivity model perimeter and using principles of topo-logy (Sunday 2004), all grid points that are within the perimeter are automatically identified (Fig. 2). The electrical resistivity tomographic (ERT) model has more grid points than the seismic one. So, the final number and loca-tion of the grid points are determined by the seismic tomograph-ic model. In order to prevent loss of information due to the reduction in the ERT model grid points, a mean resistivity value is determined within a radius centred at each final grid point. The radius will be equal to the resistivity dipole distance if the resis-tivity range is high (for example between 300–9000 Ωm) or half of it if the range is not too high (for example between 10 – 100 Ωm). After this pre-treatment, we have, at each grid point, a pair of resistivity and seismic velocity values obtained from the independent inversion (ρ

obs,v

obs).

The SA iterative procedure for estimating 2D sections of porosity and water saturation starts with uniform sections of these properties, i.e., the same initial value at all grid points of each section. In each iteration, the Metropolis algorithm (Metropolis et al. 1953) is used to minimize the following cost function (Santos et al. 2006):

FIGURE 3

Resistivity (a) and velocity (b)

variation with porosity for differ-

ent levels of water saturation,

obtained with equations (3) and

(4) and assuming 50% clay in the

solid phase.

TABLE 1

Matrix velocity ranges (vm) (after Press 1966; Dobrin 1976; Darracott

1976; Folque 1988; Lavergne 1989)

material Vp (m/s) Vm (m/s)

Loose soils 180–750 465

Clay and wet marl 750–1200 975

Sand and compact soils 1200–2400 1800

Sandstone 2400–3000 2700

Limestone and granite 3000–6000 4500

Gabbro and basalt 6000–7000 6500



Using a ‘visit function’ defined as x = ΨN (where x is the coordinate; Ψ is a random function (Press et al. 1992), with val-ues in the range [0, 1] and N is the number of cells in the section), a random number of cells is selected, in which both the satura-tion and the porosity values are randomly changed based on the following expressions:

– for SW = 25% resistivity is almost five times higher than with SW = 50% – just as reported by Carrara et al. (1999). The characteristic values of velocity for some materials are presented in Table 1. The right column indicates the mean values for each range, which were those considered for vm at each model cell. For vw, 1690 m/s was used and for va, 330 m/s.

FIGURE 4

Results for LNEC campus.

a) Resistivity model (Res2DInv);

b) resistivity model (RSAnn);

c) velocity model (Rayfract);

d) velocity model (RSAnn);

e) water saturation (RSAnn);

f) porosity (RSAnn); g) volumet-

ric water content (RSAnn). Dotted

grey lines on (e), (f) and (g) show

the positions (vertical line) and

levels (horizontal lines) where

soil samples were collected for

laboratory analysis.

TABLE 2

Laboratory tests and in situ results for soil samples (Mota 2006; Mota and Santos 2006)

Grain size Distribution (%) E E φ * φ * w w Sw** Sw

**

Depth (m)

Silt <0.074 mm

Sand 0.074–2 mm

Gravel >2 mm

lab field lab field lab (%) field (%) lab (%) field (%)

0.00 32.2 61.9 5.9 0.459 0.501 31.5 33.4 4.8 5.0 15.9 15.0

0.10 – – – – 0.421 – 29.6 – 4.6 – 15.5

0.20 – – – – 0.432 – 30.2 – 5.0 – 16.6

0.30 – – – – 0.442 – 30.7 – 4.7 – 15.3

0.65 17.1 35.7 47.2 0.275 0.371 21.6 27.1 6.2 7.1 32.9 26.2

0.75 – – – – 0.246 – 19.7 – 5.9 – 29.9

0.85 – – – – 0.245 – 19.7 – 5.8 – 29.5

0.95 – – – – 0.263 – 20.8 – 5.9 – 28.3

1.30 26.8 68.0 5.2 0.380 0.494 27.5 33.1 8.8 7.8 30.1 23.6

1.40 – – – – 0.425 – 29.8 – 7.6 – 25.5

1.50 – – – – 0.476 – 32.2 – 7.8 – 24.2

1.60 – – – – 0.496 – 33.2 – 7.9 – 23.8

* Porosity was calculated from void ratio values, with .** Water saturation was calculated from volumetric water content and porosity values, using w=φ Sw.



(grain size distribution, volumetric water content (w) and void ratio (e)). A Surface Moisture-Density Gauge was used to obtain in situ values of both the volumetric water content and the void ratio at four different levels, 0.0 m, 0.1 m, 0.2 m and 0.3 m, at each ground-surface level sampled (Table 2). At the three sites, P-wave refraction profiles were recorded with a 24 channel Bison 9000 seismograph. The energy source was an 11 kg hammer, except for the Guadiana river site where explosives were the energy source. Geophone spacing was designed to match, at each site, the resistivity survey. Data were recorded from two overlapping receiver spreads, each with 24 geophones, with the purpose of constructing a 41 geophone gather with the number of receivers corresponding to the number of resistivity electrodes. The resistivity surveys were performed with an ABEM Lund Imaging System. At the LNEC campus, a Wenner array was used, because it was known that there was a top-layer stratifica-tion for which this array is more suitable. At the other sites, as lateral resistivity variations were expected, the dipole-dipole array was chosen because it is more sensitive to these geological conditions. For the SA procedure, several combinations of ρ w , ρcl, vcl, Pcl, α and β as well as different values of the Archie’s parameters, a and m, were used. Table 3 shows the final combination of para-meters used for each site. With these tests, it was observed that those clay-related parameters have more influence on the variations in the final results. This is attributed to the fact that the clay is one of the governing factors for resistivity, as has already been concluded by the team from Lawrence Livermore National Laboratory (Berge et al. 2000; Berryman et al. 2000; Aguirre et al. 2001). Furthermore, a distinction is made with respect to the velocities for solid phase constituents (matrix and clay). The variations in Archie’s parameters as well as the weight parameters α and β are also highly relevant. The latter cannot differ too much from the 50–50% relation. It was observed that in the presence of an anomaly in low resistivity, the RSAnn sections are more sensitive to variations in weights. Figure 4 shows the models obtained from LNEC field data inversion with Rayfract and Res2DInv, as well as the models for water saturation, porosity and volumetric water content from RSAnn. Due to the very shallow investigation both the resistivity and the velocity ranges are very low. However, observation of the velocity and resistivity models seems to indicate that the low

(5)

(6)

where dSw and dφ are water saturation and porosity ranges (these are initial input parameters) and ζ is a division factor that reduc-es the increase/decrease in the variation of these properties, as the cost function diminishes and assumes the following values:

ues:

The random function, Ψ, assumes, also randomly, both a positive and a negative sign to obtain either an increase or a decrease in Sw and φ. The selected random function was ran3 from numerical recipes (Press et al. 1992). After calculating φ and Sw, it is pos-sible to obtain the volumetric water content, which is the product of φ and Sw.

FIELD AND LABORATORY TESTSThree different sites with known geology were selected to test the proposed methodology – the left bank side of Sabor River (north of Portugal), where the geology is granite with some weathered outcrops and loose top soil (residual granite soil with some clay) (Mota 2004); the left bank side of Guadiana River (south of Portugal), where the geology is more complex since it has a granite/schist contact with clean and uniform grain-size sand covering granite and a clay-rich soil covering schist (Mota 2001); and, the third one, the LNEC campus, in Lisbon, where the near-surface geology is mainly sand and gravel. At the Sabor River site, the survey was performed in September (dry soil con-ditions); at the Guadiana River site the survey was performed in May, after some rainfall and the survey at LNEC was performed in February, with dry soil conditions (following a dry winter). The field test performed at the LNEC campus, included pro-files, in situ measurements of some geotechnical parameters and laboratory tests on soil samples (Mota 2006; Mota and Santos 2006). After profile execution, the test site was excavated to col-lect soil samples by the sand bottle test at 0.0 m, 0.65 m and 1.3 m depths, at coordinate 9.0 m (Fig. 4), for laboratory tests

TABLE 3

RSAnn model parameters for each location

Location Iter. no. Final energy α (m/s) β (ohm.m) Vcl (m/s) ρ

cl (ohm.m) ρ

w (ohm.m) P

cl (%) a, m, n

LNEC 82 33.98 0.4 0.6 2000 65 70 25 0.88; 1.2; 2

Sabor 74 21.29 0.4 0.6 4000 3500 100 20 0.95; 1.1; 2

Guadiana 72 15.35 0.4 0.6 2500 1000 15 20 0.88; 1.1; 2



FIGURE 5

Arithmetic differences between

resistivity models obtained with

Res2DInv and RSAnn (a) and

between velocity models from

Rayfract and RSAnn (b) for

LNEC campus.

FIGURE 6

Results for Sabor River left bank

(Mota 2006).







ric water content (RSAnn).



at each site and by analysing the 2D sections of arithmetic differ-ences between original models of the resistivity and velocity and those obtained with RSAnn, it is possible to perform a qualitative evaluation. At the Sabor River site, the central region of higher water saturation, porosity and volumetric water content corre-sponds to a region of thicker soil deposits, whereas at depths where granite is not so weathered the values of these properties are low. At the Guadiana River site, the low porosity region is correlated with the granite rock mass and the higher volumetric water content at higher coordinate distances is representative of the clay-rich soil covering the schist. This clay soil was wet at the time of the survey. The sections of arithmetic differences evidence those zones of each site where the misfit of RSAnn sections from the original models is higher. These zones correspond to places of anomalies in original models of resistivity or velocity when compared with the surrounding environment. This shows that the proposed methodology has some limitations when there is a high contrast in resistivity, in velocity or in both.

CONCLUSIONSThis work addresses a new method to integrate electrical resistiv-ity models with seismic velocity models so as to obtain estima-tions of the porosity, water saturation and volumetric water con-tent, the corresponding results being presented as 2D sections. The presented method consists of an integrated use of 2D electrical resistivity tomographic models and P-wave velocity models, in an energy cost function, for a Simulated Annealing procedure, with a view to obtaining 2D sections of the porosity, water saturation and volumetric water content. It is essential to

velocity zone somehow corresponds to the higher resistivity top zone, which is a sign of low water content. Figure 5 presents 2D sections of the arithmetic differences between velocity models obtained with Rayfract and RSAnn and between resistivity models obtained with Res2DInv and RSAnn. Dashed lines were used in water saturation, porosity and volumetric water content models to mark the profile position (9.0 m) and the depths (0.65 m and 1.3 m) at which the soil sam-ples were collected, in order to compare laboratory and field values with those of the models. Good correlations were observed for water saturation and volumetric water content while a poor correlation was noticed for porosity. The latter seems to be the result of a poor match between Rayfract and RSAnn velocity models. The results obtained for the Sabor and Guadiana sites are presented in Figs 6 and 8, respectively and the corresponding 2D sections of the arithmetic differences between velocity models obtained with Rayfract and RSAnn and between resistivity models obtained with Res2DInv and RSAnn are presented in Figs 7 and 9. The variation in the geophysical properties at these sites is high due to the presence of granite (high resistivity and high velocity values), as well as soil, sand and clay (low values of resistivity and seismic velocity). No soil samples were collected at these sites during the field work. Hence, the porosity and water saturation ranges resulted from estimations based on tables taken from the literature and no laboratory values for the porosity, water saturation and volumetric water content were available in order to quantitatively evaluate the results obtained with the RSAnn procedure. Nonetheless, by knowing the geology present

FIGURE 7





Rayfract and RSAnn (b) for the

Sabor River site.



collect soil samples, so as to obtain an estimate of the local range of variation in the main parameters of porosity and water satura-tion and of the percentage of clay content. Otherwise, achieving the proposed goal will be very difficult and time consuming, as was the case of the Sabor and Gaudiana River sites where soil

samples were not collected during field work. The main drawback associated with such soil sampling is that we only have access to the top soil, while the geophysical investigation depth can reach several tens of metres. In the future, we plan to collect soil at several depths from a borehole made in the vicinity of the site

FIGURE 8

Results for Guadiana River left

bank (Mota 2006).







ric water content (RSAnn).

FIGURE 9





Rayfract and RSAnn (b) for the

Guadiana River site.



where the method will be used to calibrate the initial parameters. It was observed that the resistivity has a high influence on the final results, as this property is partially governed by the clay content of the soil present at the site, of which the percentage is very difficult to estimate. As a result of the method presented, it is possible to evaluate depth and lateral variations in the porosity, water saturation and volumetric water content. Potential fields of application of this method are: environmental studies (e.g., mapping of preferential pathways for leachate), geotechnical engineering (e.g., producing geomechanical zonation maps of rock masses and volumetric water content maps of embankment works), hydrogeology (e.g., studying water pathways) or agriculture (e.g., mapping of preferential path-ways for plant nutrient transport and for water supply to plants).

ACkNOWLEDGEmENTSThe authors would like to acknowledge LNEC for fieldwork sup-port. Improvements to the original version of this work are a result of constructive comments by two referees, an anonymous one and Prof. Dr Mahmut G. Drahor and by guest editors Nigel Cassidy and Jamie Pringle, to whom we wish to express our sincere appreciation.

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