Application of the DEXP Method to the Streaming Potential Data

Abstract

We interpret the self-potential data related to groundwater flow by the depth from extreme points (DEXP) method; a multiscale method in which the data are upward-continued and scaled by a scaling law depend on the structural index. The depth to the water table is estimated from extreme points of the DEXP image without a priori estimate of the hydraulic coupling coefficient. The method is tested with a synthetic model of the water table and applied to a real self-potential dataset near a pumping well. The obtained results agree well with the known information.

Full text

Near Surface Geoscience
Turin, Italy, 6-10 September 2015
We 21P1 15
Application of the DEXP Method to the Streaming
Potential Data
M.A. Abbas* (Univ. of Naples Federico II & South Valley Univ.) & M. Fedi
(University of Naples Federico II)
SUMMARY
We interpret the self-potential data related to groundwater flow by the depth from extreme points (DEXP)
method; a multiscale method in which the data are upward-continued and scaled by a scaling law depend
on the structural index. The depth to the water table is estimated from extreme points of the DEXP image
without a priori estimate of the hydraulic coupling coefficient. The method is tested with a synthetic model
of the water table and applied to a real self-potential dataset near a pumping well. The obtained results
agree well with the known information.

Near Surface Geoscience
Turin, Italy, 6-10 September 2015
Introduction
The self-potential anomalies have two main mechanisms; the oxidation-reduction associated with ore
deposits and streaming potential (SP) associated with ground water flow through the electrokinetic
coupling (Revil et al., 2004). There are two approaches for interpreting the origin of the SP anomalies
related to ground water flow. The first approach supposes that SP signals could be mainly related to
the thickness of the unsaturated (vadose) zone in which water penetrate vertically until the water table
(e.g., Zablocki, 1978; Jackson &Kauahikaua, 1987). Another approach presumes that the main SP
contribution is located along the water table and that the variations of the hydraulic head are directly
responsible for the electrical potential anomalies measured at the ground surface (Fournier, 1989;
Revil et al., 2004).
In this work, we consider the application of a multiscale method; the Depth from EXtreme Points
(DEXP) method (Fedi, 2007; Fedi and Abbas, 2013), for the interpretation of the streaming potential
data related to the flow of groundwater.
Theory
Considering that the SP sources are along the water table (Figure 1), the electric potential U(r) (in
mV) measured at the point P(r) located on the Earth’s surface is related to the hydraulic head h at
point M(rq) at the water table (in m) by a Fredholm equation of the first kind (Fournier, 1989; Revil et
al., 2004),




dSh
c
rU
q
qq
q
 

2
)(
2
)(
rr
r.nrr
r

(1)
where n is the unit normal vector over the water table


, dS is a surface element of the water table, c
is an apparent streaming potential coupling coefficient (in mV m-1) given by s
CCc 

, where C and
Cs are the electric potential coupling coefficients of the vadose and saturated zone respectively.

is the
ratio between the electrical conductivity of the saturated and unsaturated zones (Revil et al., 2004).
For a 2D profile, assuming that the variation in y direction can be neglected, the electric potential is
obtained by (Revil et al., 2004):






L
q
qq
qdh
c
rU
02
)(
2
)(


rr
r.nrr
r (2)
where

represents the curvilinear coordinate along the water table line.
Figure 1 Sketch of the geometry of the water table for a pumping well (modified after Rizzo et al.,
2004).
If we assume a flat measurement surface, so that n is constant, equation (1) will be equivalent to the
potential field due to a sheet or a simple layer (Baranov, 1975). Hence, as we mentioned in the
previous chapter, the streaming potential has the same mathematical form as the magnetic potential.
In fact, any deviation from the horizontal level of the water table can be interpreted as a one-point
source (point-dipole, line of dipoles, or others). Consequently, we can apply the DEXP method to the
streaming potential anomaly. The DEXP method is a multiscale method in which the potential or its
derivative is continued and scaled with a scaling-law dependent on the geometry of the source. The

Near Surface Geoscience
Turin, Italy, 6-10 September 2015
scaling exponent is not fixed but can be determined directly from the data (Fedi, 2007; Fedi and
Abbas, 2013). An important issue is that we do not need any a priori estimate of the hydraulic
coupling coefficient; one more reason to use the DEXP method is that it is fast and stable.
Let us assume Cartesian coordinates where x and z are the horizontal and vertical directions,
respectively, and the depth is positive downward. We can discretize equation (2) in terms of the
depths of the dipoles along the water table zq assuming a vertical electric polarization as:



 Q
qqq
q
qzxx
z
zz
cd
rU
122
0.
2
)(

(3)
where d is the horizontal step. We can note from equation (3) that the two above-mentioned
approaches for the origin of SP anomalies are in fact equivalent. We have already said that any
deviation from the horizontal level of the water table can be interpreted as a point source (point-
dipole, line of dipoles, or others) so that the DEXP transformation (Fedi, 2007; and Fedi and Abbas,
2013) can be applied. The DEXP transformation p

is given by Fedi (2007) as:
 
zUzz p
p
N
p
2/
 (4)
where Up is the p-th order derivative of U and z is the scale or altitude. The depth to the water table
can be estimated from the extreme points of the DEXP image p

. In case of non-vertical polarization,
the DEXP transformation can be applied to the analytic signal modulus of the SP anomaly. This is
equivalent to the dipole occurrence probability (DOP) of Revil et al. (2001) but, once more, in an
easier way: we do not need normalization. We also do not use a fixed scaling exponent (Fedi and
Pilkington,. 2012), which makes the DEXP method more suitable to be used in different situations.
The application of the DEXP transformation method requires upward continuation and vertical
differentiation of order p for the computation of Up and p

. Recalling that the upward continuation is
not allowed for SP anomalies, being the air infinitely resistive. However, the upward continuation of
SP data, from a measurement level zm to the level zm+a, is equivalent to the SP data at zm of the same
source located not at the depth z0 but at the deeper depth z0+a. In this way, the concept of vertical
derivative of SP data may be introduced for SP anomalies as well (Fedi and Abbas, 2013).
Synthetic Example
The SP anomaly of the water table model shown in Figure 2b is calculated using equation (3). The SP
anomaly (Figure 2a) was upward continued from 0 to 60 m and its 2nd order vertical derivative is
calculated. Figure 3a shows the continued derivative with the ridges. The scaling function was
calculated along two ridges at horizontal distances 500 m and 100 m (Figure 3b, c). Both scaling
functions yield a scaling exponent equal 0.6. The scaling functions calculated along the two other
ridges in the central part of Figure 3a did not give reasonable values of homogeneity degree due to the
interference effects connected to the complexity of the source in this part. We used the estimated
scaling exponent to calculate the DEXP image shown in Figure 4a. The extreme points of the DEXP
image give a good estimate of the depth of the water table. Also, we used a geometrical method to
estimate the depth to the water table at the intersection of the ridges of the 4th order derivative of the
SP anomaly (Figure 4b), this order warranting a nice degree of resolution to the problem. The results
agree will with the true model of the water table.
Figure 2 The SP anomaly of a water table model. (a) The SP anomaly, and (b) the depth to the water
table.

Near Surface Geoscience
Turin, Italy, 6-10 September 2015
Figure 3 Scaling function analysis. (a) 2nd order derivative of the SP anomaly and its ridges, (b)
scaling function of the ridge at distance 500 m and (c) scaling function of the ridge at distance 100 m.
Figure 4 Application of DEXP method and geometrical method for the estimation of water table
depth. (a) DEXP image; white stars indicate the estimated depths (b) the depth to the water table
estimated by the geometrical method.
Real Example
The DEXP imaging is applied to the streaming potential data collected near a pumping well from
Bogoslovsky and Oglivy (1973). Figure 5 shows the SP anomaly over the pumping well (K-1) and
also the piezometric depths. We note a positive anomaly with a symmetric form over the well location
and other two small negative anomalies related to the infiltration from the surface drainage ditches.
Figure 5 The streaming potential data near a pumping well (a) and the piezometric depths (b) after
Bogoslovsky & Oglivy, 1973.
First, the 1st derivative of SP anomaly is continued from 0 to 30 m with a 0.2 m vertical step (Figure
6a). Then the scaling exponent is estimated by analyzing the scaling function along vertical ridges at
x={95, 163, 270} m (Figure 6d,e,and f). We found that the central positive anomaly over the pumping
well and the negative anomaly related to the infiltration of the water in the subsurface at x= 270 m
need both a 0.9 scaling exponent, whereas the negative anomaly related to the infiltration of water in
the subsurface at x= 95 m needs a 1.15 scaling exponent. Finally, we obtained two DEXP images with
the two estimated values of scaling exponents (Figure 6b and c). The estimated depths to the water
table, indicated by the white stars, agree well with the measured piezometric depths.

Near Surface Geoscience
Turin, Italy, 6-10 September 2015
Figure 6 Application of the DEXP method to SP data near a pumping well. a) first-order derivative of
SP at different scales and ridges (cyan solid lines); b) DEXP image with 0.9 scaling exponent
overlaid by piezometric depths (white solid line); c) DEXP image with a 1.15 scaling exponent
overlaid by piezometric depths (white solid line); d) scaling function of the rightmost (d) central ridge
(e) and leftmost ridge (f). White stars in b&c indicate the estimated depth to water table.
Conclusions
We applied the DEXP method to the streaming potential data related to groundwater flow. DEXP is a
fast imaging method transforming the field data, or its derivatives, into a quantity proportional to the
source distribution. It depends on the knowledge of the structural index of the source which may be a
priori determined by applying related multiscale methods based on the scaling function. The depth to
water table is estimated by the extreme point of the DEXP image and does not need a priori estimate
of the hydraulic coupling coefficient. We showed the usefulness of DEXP method to SP dataset near a
pumping well. The estimated depths agree well with the known information.
References
Baranov, V.I. [1975] Potential fields and their transformations in applied geophysics (No. 6).
Lubrecht & Cramer Limited.
Bogoslovsky, V.V. and Ogilvy, A.A. [1973] Deformations of natural electric fields near drainage
structures. Geophysical Prospecting, 21, 716-723.
Fedi, M. [2007] DEXP: A fast method to determine the depth and the structural index of potential
fields sources. Geophysics, 72(1), I1-I11.
Fedi, M. and Abbas, M.A. [2013] A fast interpretation of self-potential data using the depth from
extreme points method. Geophysics, 78(2), E107-E116.
Fedi, M. and Pilkington, M. [2012] Understanding imaging methods for potential field data.
Geophysics, 77(1), G13-G24.
Fournier, C. [1989] Spontaneous Potentials and Resistivity Surveys Applied To Hydrogeology In A
Volcanic Area: Case History Of The Chaîne Des Puys (Puy-De-Dôme, France). Geophysical
Prospecting, 37, 647-668.
Jackson, D.B. and Kauahikaua, J. [1987] Regional self-potential anomalies at Kilauea volcano. USGS
Professional paper, 1350(40), 947-959.
Revil, A., Ehouarne, L., and Thyreault, E. [2001] Tomography of self-potential anomalies of
electrochemical nature. Geophysical Research Letters, 28, 4363-4366.
Revil, A., Naudet, V. and Meunier, J.D. [2004] The hydroelectric problem of porous rocks: Inversion
of the water table from self-potential data. Geophysical Journal International, 159, 435-444.
Rizzo, E., Suski, B., Revil, A., Straface, S. and Troisi, S. [2004] Self-potential signals associated with
pumping-tests experiments. Journal of Geophysical Research, 109, B10203.
Zablocki, C.J. [1978] Streaming potentials resulting from the descent of meteoric water. A possible
source mechanism for Kilauean self-potential anomalies. Trans. Geotherm. Resour. Council, 2, 747-
748.