Electrical, Electromagnetic and Well Logging Methods

Resistivity electrode arrays

A resistivity electrode array is the set of two current electrodes (C1, C2) and two potential electrodes (P1, P2) used to inject current into the ground and measure the resulting potential difference \Delta V. The apparent resistivity is \rho_a = K\,\Delta V / I, where K is the geometric factor, which depends only on the electrode geometry. Once the array is fixed, every field reading is converted to \rho_a with this single relation.

In the Wenner array, four equally spaced electrodes with spacing a are placed in the order C1–P1–P2–C2, and K = 2\pi a, so \rho_a = 2\pi a\,\Delta V / I. The outer current electrodes are therefore 3a apart, which matters when a problem gives the total current-electrode separation rather than a. Equally spaced electrodes in the order C–C–P–P, with the gap between the current dipole and the potential dipole equal to one electrode interval, form a dipole–dipole array with separation factor n=1. Identifying an array therefore depends on the order of the four electrodes along the line.

In a double-dipole (dipole–dipole) array, the current dipole and the potential dipole each have length a and are separated by na, and \rho_a = \pi n(n+1)(n+2)\,a\,\Delta V / I. For n=1 the factor becomes 6\pi a, so the dipole–dipole array has the same geometric factor as the Wenner array when its dipole length is one-sixth of the Wenner spacing. Telluric currents are natural background fields. They are not part of the array geometry or the apparent resistivity expression, so they do not change \rho_a, although they add noise to the measured potentials.

The measurement level n controls how many positions fit along a profile. The four electrodes of a dipole–dipole set at level n span n+2 electrode intervals, so for N equally spaced electrodes there are N-(n+2) positions at that level. The pole–dipole array, in which one current electrode is effectively at a remote location, allows the remaining electrodes to be combined in more positions and separations than the Wenner or Schlumberger arrays, which fix the electrode spacing ratio. It therefore gives the largest number of data points among these arrays.

The self-potential between potential electrodes is a natural constant offset that adds to the measured voltage. Its sign is unchanged when the current direction is reversed, whereas the IR potential changes sign. Averaging the direct and reverse readings therefore cancels the IR term, and the self-potential equals half the sum of the two readings. In practice the potential is read before current injection and subtracted from the later reading.

Noise can reverse the sign of a small potential difference. Dipole–dipole arrays measure very weak potentials at large separations, so noise produces the most negative apparent resistivity values in that array, whereas Wenner and Wenner–Schlumberger arrays give stronger signals. The two-electrode array and the Wenner array have the same geometric factor K = 2\pi a, but the remote electrode in a two-electrode array spreads its response laterally. The compact, symmetric Wenner layout therefore gives the better lateral resolution.

A horizontal fraction of the input current crossing a vertical plane between two depths is obtained by integrating over depth. The integral reduces to differences of arctangents, giving a dimensionless fraction. For depths from L/2 to \sqrt{3}L/2, the fraction is (2/\pi)(\pi/3-\pi/4) = 1/6.

Electromagnetic methods principles

Electromagnetic (EM) prospecting uses a time-varying primary field that induces eddy currents in conductive bodies. A low-frequency alternating current source is used to generate the primary field, because low frequencies penetrate conductive ground to useful depths, whereas a steady direct current produces no induction.

The depth of investigation is controlled by the skin depth, \delta \approx 503\sqrt{\rho/f} m, where \rho is resistivity in \Omega·m and f is frequency in Hz. Magnetotellurics, which uses natural sources at very low frequencies, probes the greatest depth. AFMAG probes less, VLF less still, and ground penetrating radar, which uses high-frequency waves strongly attenuated in conductive ground, sees only the shallow subsurface. The order from deepest to shallowest is therefore MT, AFMAG, VLF, GPR.

Because \delta \propto \sqrt{\rho/f}, keeping a fixed target depth while the host resistivity changes requires the frequency to scale in direct proportion to resistivity, so f_2/f_1 = \rho_2/\rho_1. A less resistive host therefore needs a proportionally lower frequency.

Each EM method operates in a characteristic frequency band. VLF uses transmitters in the band of about 15–30 kHz, controlled-source magnetotellurics uses 1 Hz–20 kHz, radio magnetotellurics uses about 100 kHz–1 MHz, and ground penetrating radar uses about 10 MHz–1 GHz. Matching each method to its band is a standard recall task.

In VLF surveys the primary field is horizontal, so the total horizontal field is the primary field plus the horizontal secondary component, while the vertical field is purely secondary. The measured tilt angle with respect to the horizontal is then \theta = \arctan(H_z / H_x).

EM waves in the Earth are transverse, so they are not analogous to longitudinal seismic P waves. They are attenuated by conduction and dielectric losses, they are diffracted at boundaries and around bodies, and highly conductive ore bodies can be detected because they strongly distort the field.

In a coil system, the induced secondary current obeys L\,dI/dt + RI = -M\,dI_P/dt, so it lags the primary by \phi = \arctan(\omega L / R). The phase difference thus depends on the inductance L and resistance R of the receiving conductor and on the angular frequency \omega. The primary current amplitude only scales both fields and does not change the phase.

In the quasi-static approximation used for induction in the Earth, charge neutrality makes the current density solenoidal, \nabla\cdot\mathbf{J} = 0. The magnetic field is defined by \mathbf{B} = \nabla\times\mathbf{A}, so it is also solenoidal, while the vector potential is not irrotational because its curl equals \mathbf{B}. Faraday's law then gives \nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t = -\partial(\nabla\times\mathbf{A})/\partial t.

Earth induction is analysed under several assumptions. The inducing field from distant sources is approximated as a plane wave. Displacement currents are neglected because conduction dominates, so \sigma \gg \omega\varepsilon. The Earth is treated as an Ohmic conductor with \mathbf{J} = \sigma\mathbf{E}, and charge conservation keeps the current divergence-free. Since the Earth contains no internal electromagnetic sources, it can only absorb and dissipate the incident energy.

Well logging tool responses

The spontaneous potential (SP) log records electrochemical potentials that arise at shale–sand boundaries, mainly from membrane effects of shale and from salinity contrast between mud filtrate and formation water. In dispersed shale, clay particles fill the pore throats and act as a negatively charged membrane that repels anions. Chloride ion movement is impeded, which lowers the membrane potential and the SP deflection.

Sonic logs measure interval transit time, and cracks or altered zones near the borehole wall delay and distort the first arrival. Short-spacing measurements are most affected because their path stays in the disturbed zone. A long-spacing sonic tool samples the formation deeper, beyond the damaged region, so it is the correct procedure for reducing this effect.

Nuclear magnetic resonance (NMR) logging measures the signal from hydrogen protons in the pore fluids. Its response gives porosity and the distribution of relaxation times, which separates bound (irreducible) water from free fluid. Mineral composition affects the response through matrix and surface relaxation, whereas pore fluid pressure does not change proton density or relaxation times and therefore does not influence the measurement.

Logging tools differ in their depth of investigation. Shallow devices with small electrode spacing, such as the microlaterolog and the microspherically focused log, measure the resistivity of the flushed zone R_{xo}. Deep devices, such as the lateral log and the long normal log, measure the resistivity of the uninvaded formation R_t.

Laminar, structural and dispersed shale change the combined hydrogen and bulk-density responses in different ways, so their distribution type is recognised on a neutron–density crossplot, where each type plots in a different region. Crossplots of SP against water resistivity, or of two resistivity curves, do not separate these shale types.

A caliper log records borehole diameter with spring-loaded or multi-arm mechanical sensors pressed against the wall. Sonic logs respond to acoustic travel time, density logs to bulk density, and neutron logs to hydrogen content, so none of these measures the borehole size directly.

The neutron tool responds mainly to hydrogen, which in most formations resides in liquid-filled pores, so its reading is chiefly a measure of liquid-filled porosity. Gas has a much lower hydrogen density than water or oil, so a gas-bearing clean sandstone reads a neutron porosity below its true porosity. A low neutron porosity indicates a low hydrogen index, and overpressured shale, which has anomalously high porosity, does not give a very low neutron reading.

A crossover is the separation of the neutron and density porosity curves in a gas zone, where the neutron reads low and the density reads high. Among common log pairs, only neutron and density show this behaviour. Caliper with resistivity, self-potential with neutron, and caliper with gamma-ray do not produce a gas crossover.

Archie's law water saturation

Archie's law links the resistivity of a clean, porous rock to its porosity, the resistivity of the pore water and the fraction of pore space filled with water. Its water-saturation form is S_w=\left(\dfrac{aR_w}{\phi^{m}R_t}\right)^{1/n}, where a is the tortuosity factor, m the cementation exponent, n the saturation exponent, \phi the fractional porosity, R_w the formation-water resistivity and R_t the true formation resistivity read by a deep-investigating log. The hydrocarbon saturation is the complement, S_h = 1 - S_w, so a high deep resistivity relative to the water-bearing value indicates hydrocarbons. The exponents must be kept apart: m controls how resistivity rises as pores become less connected, through the formation factor F=a/\phi^{m}, while n controls how resistivity rises as hydrocarbon displaces water. Both are often given as 2 in exam problems, so the symbol in each term must be checked rather than assumed. For a fully water-saturated rock, S_w = 1 and the law reduces to R_o = aR_w\phi^{-m}, where R_o is the resistivity of the rock. With a=1 and m=2 this gives \phi=\sqrt{aR_w/R_o}; for R_w = 0.5\ \Omegam and R_o = 50\ \Omegam the porosity is 10%. The bulk volume of water is the product of porosity and water saturation, BVW = \phi S_w, and it expresses the share of total rock volume occupied by water, not the share of pore space. If \phi = 0.20 and S_w = 0.30, then BVW = 0.06, that is 6% of the rock. In the flushed zone the mud filtrate displaces part of the pore water, so the water saturation S_{xo} near the borehole is computed with R_{xo} and R_{mf} in the same Archie form. The movable hydrocarbon saturation is the difference S_{xo} - S_w, the fraction of pore volume pushed out by the filtrate. For S_{xo}=0.6 and S_w=0.3 the movable hydrocarbon is 30% of pore volume. A worked check: with a=1, m=n=2, \phi=0.2, R_w=0.05 and R_t=10, S_w=\sqrt{0.05/(0.04\times 10)}\approx 0.35 and S_h \approx 0.65.

Schlumberger sounding interpretation

A Schlumberger vertical electrical sounding (VES) records apparent resistivity as the current electrode spacing grows, and the shape of the curve is used to infer how resistivity changes with depth. Sounding curves are named by a letter code that gives the order of resistivity in three successive layers: A for an ascending curve with \rho_1<\rho_2<\rho_3, Q for a descending curve with \rho_1>\rho_2>\rho_3, H for a curve with a minimum in the middle (\rho_1>\rho_2<\rho_3) and K for a curve with a maximum in the middle (\rho_1<\rho_2>\rho_3). For a three-layer earth the shape follows directly: A rises monotonically, Q falls monotonically, K rises to a peak and then falls, and H falls to a minimum and then rises. A four-layer curve is read as two overlapping three-layer triples, (\rho_1,\rho_2,\rho_3) and (\rho_2,\rho_3,\rho_4), taken in that order. Because the middle pair is shared, the comparison of \rho_2 with \rho_3 must agree in both triples. A and H end with \rho_2<\rho_3, so they can only be followed by a type that starts with \rho_2<\rho_3 (A or K). Q and K end with \rho_2>\rho_3, so they can only be followed by Q or H. Pairs such as HQ, AQ, HH and KA therefore cannot form a four-layer model. The principle of equivalence governs the thickness-resistivity ambiguity. A thin resistive layer keeps its transverse resistance T=\rho h nearly unchanged, and a thin conductive layer keeps its longitudinal conductance S=h/\rho. An interpreter who assigns a different resistivity to a resistive layer must therefore change its thickness in inverse proportion, h' = T/\rho'. For example, a layer of 300 \Omegam and 40 m thickness has T=12000\ \Omegam^2, and an interpreted resistivity of 600 \Omegam gives h'=20 m.

Magnetotelluric impedance tensor

In magnetotellurics the impedance tensor Z relates the horizontal electric and magnetic fields at each frequency, E_x = Z_{xx}H_x + Z_{xy}H_y and E_y = Z_{yx}H_x + Z_{yy}H_y. The apparent resistivity is \rho_a = |Z|^2/(\mu_0\omega) and the impedance phase is \phi = \arg Z, so Z_{Re} = |Z|\cos\phi and Z_{Im} = |Z|\sin\phi. The ratio of imaginary to real part is therefore Z_{Im}/Z_{Re} = \tan\phi, which does not depend on frequency or on the magnitude |Z|. A phase of 30° gives a ratio of \tan 30^\circ \approx 0.58. Over a homogeneous half-space the impedance is Z=(1+i)\sqrt{\omega\mu/2\sigma}, so its real and imaginary parts are equal and the phase is 45° at every frequency. This is the reference value against which layered-earth phases are compared. For a horizontally stratified (one-dimensional) earth, Z_{xx}=Z_{yy}=0 and Z_{xy}=-Z_{yx}, so \rho_{xy}=\rho_{yx} at every site and frequency. Lateral variation breaks this equality. With the strike along x or y, the off-diagonal terms are the TE and TM modes, which generally differ in magnitude. Rotating a two-dimensional tensor to 45° from the strike makes the off-diagonal magnitudes equal, so \rho_{xy}=\rho_{yx} again. In regions of lateral conductivity contrast, charges accumulating at boundaries distort the local electric field by a real, frequency-independent factor. This galvanic (static) distortion produces a time-independent offset between the TE and TM apparent resistivity curves and does not change the impedance phase. For instance, scaling E by 0.5 scales \rho_a by 0.25 at all periods, shifting the curve vertically on a log plot.

EM skin depth

The skin depth is the distance over which the amplitude of an electromagnetic wave falls to 1/e of its surface value in a conducting medium, and it is also called the depth of penetration. For a plane wave, \delta=\sqrt{2/(\omega\mu\sigma)}=\sqrt{2\rho/(\omega\mu)}, where \omega=2\pi f and \mu is the magnetic permeability, which for earth materials is essentially \mu_0. The amplitude varies as e^{-z/\delta}, so at z=\delta the amplitude is reduced by 1/e, and at z=2\delta by e^{-2}. In practical units \delta \approx 503\sqrt{\rho/f} metres, with \rho in ohm-metres and f in hertz. Skin depth thus increases with resistivity and decreases with frequency, which is why high-frequency radar penetrates only a short distance into wet soil. A worked check: for \rho = 50\ \Omegam at 1 kHz, \delta \approx 503\sqrt{0.05}\approx 112 m. Working the formula in reverse, the resistivity follows from a known penetration depth as \rho = f(\delta/503)^2, and the conductivity from \sigma = 2/(\omega\mu\delta^2).