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Appendix II Elements of Geophysics: Surface Electrical Methods
APPENDIX I1
ELEMENTS OF GEOPHYSICS : SURFACE ELECTRICAL METHODS Electrical-sounding techniques are widely used to determine the geometry of aquifers, especially alluvial aquifers which offer electrically well contrasted layers. Described in several books (Kunetz, 1966; Todd, 1959), they are very well summarized by Ungemach (Thesis, 1975), whom we largely quote here. The idea of prospecting the natural electrical fields from the surface is rather old (19th century) and was called spontaneous potential or self potential (SP). Later, Schlumberger (1920) injected a continuous current into the soil and obtained data on the nature and the structure of the subsoil from discrepancies observed with the ideal scheme of a homogeneous and isotropic soil. We shall now examine the various techniques, recalling that for a semi-infinite homogeneous medium of resistivity p , the electrical potential due to a point source is:
U ( r ) = pI/2n where I" is the distance t o the source and I the intensity. The anisotropy coefficient X is:
x=vmFl where pt is the transverse (or lateral) resistivity and pz the longitudinal (or normal) resistivity.
A.2.1. Potential m e t h o d (P.M.) At point M (AM = M B = a, Fig. A.2.1), the electric field is approximately constant and, parallel to AB, can be expressed as:
and at the ground surface E , = p I / n a2. The equipotential lines, obtained from potential measurements at various points, are circular for a homogeneous soil. An increase or a decrease of the resistivity will yield a change of the equipotential net. This change in the ideal cases of simple geometric structures (spheres, cylinders, circular plates) can be measured through an explicit analytical formula and type curves can be derived.
290
APPENDIX I1
M
a
a
iz
A
B
Fig. A.2.1 .Potential method.
A possible application of this method in the case of groundwater pollution is the following: an electrode is put into the aquifer and sends the current, the other electrode being put at an infinite distance (i.e. a very large distance); the first electrode represents potential 100 and the other potential 0 (Fig. A.2.2).
1j
B
Fig. A.2.2. An application of the potential method with an electrode at infinity.
The potentials are expressed as fractions of the potential difference U, and can be measured in M N in two ways: (1)either a reference equipotential line is drawn by moving an electrode in the field t o reach that value; or (2) the potential U, is expressed with respect t o a measured value on a standing electrode ( N for instance), electrically stable (i.e. out of reach of the perturbation). This procedure has proved successful in checking the movement of an electrically well contrasted perturbation.
- U,
A.2.2. Resistivity measurements Consider the device shown in Fig. A.2.3.
+m
A
M
A
M
N
P
N
Fig. A.2.3. Resistivity measurements.
B
SURFACE ELECTRICAL METHODS
291
A B is the emission line and MN the measurement line. The apparent resistivity pa, corresponding to an integrated response of the investigated layers for a given length AB, is:
=
KAU/I
where K appears as a geometrical characteristic constant of the device. Also, pa could be expressed as a function of the potential ratios in a five-electrode device, with a moving B , as represented in Fig. A.2.3: pa = K R
R = AUMp/AUpN
K = (AN - AP)/(AP- A M ) Resistivity measurements are used for resistivity profiles and rectangle measurements.
A . 2.3. Resistivity profiles AB and MN are simultaneously moved along the profile, their lengths being kept constant: the investigation depth, which is proportional t o AB, thus remains constant (Fig. A.2.4).
A
A
M
N
N'
B
8'
M'
Fig. A . 2 . 4 . Determination of resistivity profiles.
A . 2.4. Rectangle measurements Apparent-resistivity measurements are performed, for a given A B , on a rectangular net of electrodes MN contained in a rectangle, the dimensions of which with respect to the emission line are such that the electrical field can be assumed constant with the exception of a few heterogeneities (Fig. A.2.5). This type of measurement is well adapted t o the study of a very local zone: it requires the derivation, by computation, of the coefficient K at each measurement point and the introduction of a correction factor to normalize
APPENDIX I1
292
the investigation depth, for a given AB, which is maximal at the center of the rectangle and minimal at its ends. nai3
A013
t I. .. . .
AB/3
-
.I
Fig. A.2.5. Rectangle measurements.
The displacement of the rectangle requires an overlapping of the measurements and a smoothing at the interfaces, which can be complicated by electrode effects. A .2.5. Elec trica1 soundings Electrical sounding is a vertical exploration of the layers, with cumulative effects, by a progressive increase of AB, the distance between the measurement electrodes remaining small with respect to AB (MN < A B / 5 ) . In the Schlumberger quadripole device (Fig. A.2.6) the variations due t o the contact of the electrodes with a heterogeneous soil are controlled by an overlapping of the measurements for a change of AB (Fig. A.2.7).
M
A
N
B
Fig. A.2.6. Electrical-sounding quadripole-injection-measurement.
t '"'
pa
Fig. A.2.7. Electrical-sounding curve.
SURFACE ELECTRICAL METHODS
293
AU and I are measured with a potentiometer by an opposition method and the values of the apparent resistivity p a are put on log-log paper as a function of AB/2. In this representation the multiplication of the values of the resistivities or of the layer widths is given by a translation of the curve parallel to the y-axis or the x-axis. Electrical soundings thus yield a discrete sequence of apparent-resistivity values as a function of the distance between electrodes and sources. The interpretation of the measurements is a typical identification problem, which is to find a vertical distribution of resistivities p ( z ) corresponding to the measured sequence p a ( r ) . We give now a few hints about this interpretation. We assume a semi-infinite medium with a horizontal homogeneous stratification. A layer i influences the measured response through its transverse resistance Rti and its conductance Cli.which are related to the electrical and geometrical characteristics of the layer by the formulas:
where p i and ei,respectively, are the real resistivities (transverse and longitudinal) and the thickness of the layer i. To take into account the occurrence of water, another parameter is introduced, called the formation factor F and equal to the ratio of the real resistivity of the saturated layer (solid matrix and water) to the resistivity of the water alone p w . F is a function of the porosity of the layer. A widely used representation of this function is Archie’s relationship:
where 4 is the effective porosity and rn a coefficient depending upon the degree of consolidation of the aquifer (1< rn < 2). The electrical-sounding curve represents the global effect of the investigated layers and special numerical processes have to be introduced to obtain the spectrum of the various layers; type curves and a semi-automatic treatment are used according to the following principles. In a stationary regime and under the previously mentioned stratification and isotropy ( p t = p l ) conditions, the potential verifies the partial-differential equation: [A.2.11
APPENDIX I1
294
which has the solution:
Ui( r , z )
=
jirn[ A ,(A) ehz+ B,(X)eChz]Jo(Xr)dX
[A.2.2]
where i is the layer number (1< i < N ) , r the distance to the source, z the vertical coordinate, taken positive downwards, X an arbitrary integration constant, J , ( X r ) the zero-order Bessel function of the first kind, A , and B i the constants computed from the boundary conditions. The boundary conditions are: - at the interface between layers, the continuity of the potential ( Ui= C7i+l) and of the normal component of flow aUi/an = aUi+,/an - at the ground surface, the nullity of the normal component of flow everywhere except at the source -in the Nth layer, of infinite thickness, the nullity of the potential. A , and Bi are thus given by a linear system of 2N equations with 2N unknowns. The function A i(X)ehzf Bi(X)e-" is called Stefanesco's function, S[X, p(z)]. The ground-surface potential can be written: [A.2.3] which yields the value of the reduced apparent resistivity, defined as the ratio of the electrical field measured at the ground surface to its theoretical value, proportional to: -
au
r2 -(r,O) ar [A.2.41
The function S gathers all the information about the stratification and allows the derivation for "N layers"-type curves; for instance, four-layers-type curves have been obtained by an expansion of IA.2.41 in a series of functions. These type of curves can be used for identification purposes of real soils, taking into account some limitations of the method: (1)Equivalence principle. If a conductive layer i lies between two resistive layers (or a resistive layer i between two conductive layers), the electricalsounding curve is not modified by the multiplication of e , and p i by a factor K (or alternatively by the simultaneous multiplication of ei and division of pi by a factor K ) . (2) Suppression principle. A resistive layer between a highly conductive layer and a highly resistive layer has almost no influence on the electrical sounding.
SURFACE ELECTRICAL METHODS
295
A spectral representation of an electrical sounding can be obtained by computing the cumulated resistances and conductances defined by numbering the layers downwards as: R , (p) = CI (P) =
P
C
piei
i= 1
(p = 1,...n )
P
C eilpi i= 1
and drawing the curve:
on log-log paper. To determine e i and p i , supplementary data have to be used: reference borings with a geological log or parametric boring allowing the measurement of the real resistivity of a layer. Two cases of interpretation may occur: (1)the field electrical-sounding curve corresponds to an existing type of curve; and (2) there is no total correspondence. A smoothing will then be introduced.
Fig. A.2.8. Electrical-sounding curve and cumulated resistance-conductance
curve.
296
APPENDIX I1
Fig. A.2.8 is an example of electrical-sounding curves (E.S.) and of cumulated resistance-conductance curves (D.Z.). The interpretation of the experimental curves can now be improved by automatic treatments (Kunetz, 1966), but it should be stressed that to derive resistivities from the electrical field is a rather unstable problem. A small variation of p yields a small variation of the electrical field E , but the converse is not true and unless we have perfect electrical measurements (which is quite unreasonable), no unicity can be reached. A procedure has thus been derived which consists in: (1)Computing a kernel function @ ( X , p ) equivalent to S, by solving an integral equation with a second member made up of measurements. This kernel is the Fourier transform of the sequence of the electrical images of the source at the interfaces of elementary layers. (2) Deriving the sequence of resistivities from @, starting at the ground surface. This procedure applied to a single electrical sounding has been extended to correlations between electrical soundings, under assumptions of regularity and continuity in the stratification.
ELEMENTS OF GEOPHYSICS : SURFACE ELECTRICAL METHODS Electrical-sounding techniques are widely used to determine the geometry of aquifers, especially alluvial aquifers which offer electrically well contrasted layers. Described in several books (Kunetz, 1966; Todd, 1959), they are very well summarized by Ungemach (Thesis, 1975), whom we largely quote here. The idea of prospecting the natural electrical fields from the surface is rather old (19th century) and was called spontaneous potential or self potential (SP). Later, Schlumberger (1920) injected a continuous current into the soil and obtained data on the nature and the structure of the subsoil from discrepancies observed with the ideal scheme of a homogeneous and isotropic soil. We shall now examine the various techniques, recalling that for a semi-infinite homogeneous medium of resistivity p , the electrical potential due to a point source is:
U ( r ) = pI/2n where I" is the distance t o the source and I the intensity. The anisotropy coefficient X is:
x=vmFl where pt is the transverse (or lateral) resistivity and pz the longitudinal (or normal) resistivity.
A.2.1. Potential m e t h o d (P.M.) At point M (AM = M B = a, Fig. A.2.1), the electric field is approximately constant and, parallel to AB, can be expressed as:
and at the ground surface E , = p I / n a2. The equipotential lines, obtained from potential measurements at various points, are circular for a homogeneous soil. An increase or a decrease of the resistivity will yield a change of the equipotential net. This change in the ideal cases of simple geometric structures (spheres, cylinders, circular plates) can be measured through an explicit analytical formula and type curves can be derived.
290
APPENDIX I1
M
a
a
iz
A
B
Fig. A.2.1 .Potential method.
A possible application of this method in the case of groundwater pollution is the following: an electrode is put into the aquifer and sends the current, the other electrode being put at an infinite distance (i.e. a very large distance); the first electrode represents potential 100 and the other potential 0 (Fig. A.2.2).
1j
B
Fig. A.2.2. An application of the potential method with an electrode at infinity.
The potentials are expressed as fractions of the potential difference U, and can be measured in M N in two ways: (1)either a reference equipotential line is drawn by moving an electrode in the field t o reach that value; or (2) the potential U, is expressed with respect t o a measured value on a standing electrode ( N for instance), electrically stable (i.e. out of reach of the perturbation). This procedure has proved successful in checking the movement of an electrically well contrasted perturbation.
- U,
A.2.2. Resistivity measurements Consider the device shown in Fig. A.2.3.
+m
A
M
A
M
N
P
N
Fig. A.2.3. Resistivity measurements.
B
SURFACE ELECTRICAL METHODS
291
A B is the emission line and MN the measurement line. The apparent resistivity pa, corresponding to an integrated response of the investigated layers for a given length AB, is:
=
KAU/I
where K appears as a geometrical characteristic constant of the device. Also, pa could be expressed as a function of the potential ratios in a five-electrode device, with a moving B , as represented in Fig. A.2.3: pa = K R
R = AUMp/AUpN
K = (AN - AP)/(AP- A M ) Resistivity measurements are used for resistivity profiles and rectangle measurements.
A . 2.3. Resistivity profiles AB and MN are simultaneously moved along the profile, their lengths being kept constant: the investigation depth, which is proportional t o AB, thus remains constant (Fig. A.2.4).
A
A
M
N
N'
B
8'
M'
Fig. A . 2 . 4 . Determination of resistivity profiles.
A . 2.4. Rectangle measurements Apparent-resistivity measurements are performed, for a given A B , on a rectangular net of electrodes MN contained in a rectangle, the dimensions of which with respect to the emission line are such that the electrical field can be assumed constant with the exception of a few heterogeneities (Fig. A.2.5). This type of measurement is well adapted t o the study of a very local zone: it requires the derivation, by computation, of the coefficient K at each measurement point and the introduction of a correction factor to normalize
APPENDIX I1
292
the investigation depth, for a given AB, which is maximal at the center of the rectangle and minimal at its ends. nai3
A013
t I. .. . .
AB/3
-
.I
Fig. A.2.5. Rectangle measurements.
The displacement of the rectangle requires an overlapping of the measurements and a smoothing at the interfaces, which can be complicated by electrode effects. A .2.5. Elec trica1 soundings Electrical sounding is a vertical exploration of the layers, with cumulative effects, by a progressive increase of AB, the distance between the measurement electrodes remaining small with respect to AB (MN < A B / 5 ) . In the Schlumberger quadripole device (Fig. A.2.6) the variations due t o the contact of the electrodes with a heterogeneous soil are controlled by an overlapping of the measurements for a change of AB (Fig. A.2.7).
M
A
N
B
Fig. A.2.6. Electrical-sounding quadripole-injection-measurement.
t '"'
pa
Fig. A.2.7. Electrical-sounding curve.
SURFACE ELECTRICAL METHODS
293
AU and I are measured with a potentiometer by an opposition method and the values of the apparent resistivity p a are put on log-log paper as a function of AB/2. In this representation the multiplication of the values of the resistivities or of the layer widths is given by a translation of the curve parallel to the y-axis or the x-axis. Electrical soundings thus yield a discrete sequence of apparent-resistivity values as a function of the distance between electrodes and sources. The interpretation of the measurements is a typical identification problem, which is to find a vertical distribution of resistivities p ( z ) corresponding to the measured sequence p a ( r ) . We give now a few hints about this interpretation. We assume a semi-infinite medium with a horizontal homogeneous stratification. A layer i influences the measured response through its transverse resistance Rti and its conductance Cli.which are related to the electrical and geometrical characteristics of the layer by the formulas:
where p i and ei,respectively, are the real resistivities (transverse and longitudinal) and the thickness of the layer i. To take into account the occurrence of water, another parameter is introduced, called the formation factor F and equal to the ratio of the real resistivity of the saturated layer (solid matrix and water) to the resistivity of the water alone p w . F is a function of the porosity of the layer. A widely used representation of this function is Archie’s relationship:
where 4 is the effective porosity and rn a coefficient depending upon the degree of consolidation of the aquifer (1< rn < 2). The electrical-sounding curve represents the global effect of the investigated layers and special numerical processes have to be introduced to obtain the spectrum of the various layers; type curves and a semi-automatic treatment are used according to the following principles. In a stationary regime and under the previously mentioned stratification and isotropy ( p t = p l ) conditions, the potential verifies the partial-differential equation: [A.2.11
APPENDIX I1
294
which has the solution:
Ui( r , z )
=
jirn[ A ,(A) ehz+ B,(X)eChz]Jo(Xr)dX
[A.2.2]
where i is the layer number (1< i < N ) , r the distance to the source, z the vertical coordinate, taken positive downwards, X an arbitrary integration constant, J , ( X r ) the zero-order Bessel function of the first kind, A , and B i the constants computed from the boundary conditions. The boundary conditions are: - at the interface between layers, the continuity of the potential ( Ui= C7i+l) and of the normal component of flow aUi/an = aUi+,/an - at the ground surface, the nullity of the normal component of flow everywhere except at the source -in the Nth layer, of infinite thickness, the nullity of the potential. A , and Bi are thus given by a linear system of 2N equations with 2N unknowns. The function A i(X)ehzf Bi(X)e-" is called Stefanesco's function, S[X, p(z)]. The ground-surface potential can be written: [A.2.3] which yields the value of the reduced apparent resistivity, defined as the ratio of the electrical field measured at the ground surface to its theoretical value, proportional to: -
au
r2 -(r,O) ar [A.2.41
The function S gathers all the information about the stratification and allows the derivation for "N layers"-type curves; for instance, four-layers-type curves have been obtained by an expansion of IA.2.41 in a series of functions. These type of curves can be used for identification purposes of real soils, taking into account some limitations of the method: (1)Equivalence principle. If a conductive layer i lies between two resistive layers (or a resistive layer i between two conductive layers), the electricalsounding curve is not modified by the multiplication of e , and p i by a factor K (or alternatively by the simultaneous multiplication of ei and division of pi by a factor K ) . (2) Suppression principle. A resistive layer between a highly conductive layer and a highly resistive layer has almost no influence on the electrical sounding.
SURFACE ELECTRICAL METHODS
295
A spectral representation of an electrical sounding can be obtained by computing the cumulated resistances and conductances defined by numbering the layers downwards as: R , (p) = CI (P) =
P
C
piei
i= 1
(p = 1,...n )
P
C eilpi i= 1
and drawing the curve:
on log-log paper. To determine e i and p i , supplementary data have to be used: reference borings with a geological log or parametric boring allowing the measurement of the real resistivity of a layer. Two cases of interpretation may occur: (1)the field electrical-sounding curve corresponds to an existing type of curve; and (2) there is no total correspondence. A smoothing will then be introduced.
Fig. A.2.8. Electrical-sounding curve and cumulated resistance-conductance
curve.
296
APPENDIX I1
Fig. A.2.8 is an example of electrical-sounding curves (E.S.) and of cumulated resistance-conductance curves (D.Z.). The interpretation of the experimental curves can now be improved by automatic treatments (Kunetz, 1966), but it should be stressed that to derive resistivities from the electrical field is a rather unstable problem. A small variation of p yields a small variation of the electrical field E , but the converse is not true and unless we have perfect electrical measurements (which is quite unreasonable), no unicity can be reached. A procedure has thus been derived which consists in: (1)Computing a kernel function @ ( X , p ) equivalent to S, by solving an integral equation with a second member made up of measurements. This kernel is the Fourier transform of the sequence of the electrical images of the source at the interfaces of elementary layers. (2) Deriving the sequence of resistivities from @, starting at the ground surface. This procedure applied to a single electrical sounding has been extended to correlations between electrical soundings, under assumptions of regularity and continuity in the stratification.