Spontaneous potential (SP) log

Quick refresher

The SP is a passive method: the tool sends out nothing. It only records the potential difference (the voltage) between an electrode in the borehole and an electrode at the surface. The curve has no absolute zero. We measure deflections from the shale baseline.
The main source is the electrochemical electromotive force (EMF, a voltage that can drive a current). It has two parts. One is the membrane potential at shales, which pass Na⁺ cations (positive sodium ions) but hold back anions. The other is the liquid-junction potential at the contact of the filtrate with the formation water, which is about five times smaller. At 25 °C together they give about 71 mV for each tenfold difference in the activities (effective concentrations) of the solutions.
The water contrast sets the sign. If $R_{mf} > R_w$ (fresh mud, saline water), the deflection is negative. If $R_{mf} < R_w$ , it is positive. When the salinities are close, there is almost no deflection.
The static SP (SSP) is the total EMF that the curve would show if no currents flowed. The recorded deflection reaches it only in thick, clean, water-bearing beds of low resistivity.
A thin bed, a bed of high resistivity, shale in the bed and hydrocarbons all reduce the deflection. Streaming potentials (voltages made by fluid flowing through the rock) are usually small. But with a high overbalance (mud pressure well above the formation pressure) and in tight beds (beds of very low permeability), they distort the curve.
We find $R_w$ from the SSP, the filtrate resistivity and the formation temperature. All resistivities must be taken at formation temperature.

Key relationships

\mathrm{SSP} = -K\lg\dfrac{R_{mfe}}{R_{we}}

static SP of a clean bed; equivalent resistivities at formation temperature, $\lg$ is the base-10 logarithm

K \approx 65 + 0.24\,T

coefficient in mV with $T$ in °C; about 71 mV at 25 °C

R_{we} = R_{mfe}\cdot 10^{\,\mathrm{SSP}/K}

equivalent formation water resistivity from the static SP (SSP with its own sign)

Common pitfalls

Taking the deflection of a thin, shaly or oil-bearing bed for the static amplitude. The $R_w$ you get will be too high.
Using $R_{mf}$ at the surface measuring temperature instead of at formation temperature.
Taking a lack of deflection as a sign of an impermeable bed. When the filtrate and the water have the same salinity, the SP stays “silent” opposite a good reservoir too.
Confusing the equivalent resistivity of a strong brine (very salty water) with the true $R_w$ .

After this lesson you can

explain how membrane and liquid-junction potentials arise and why their sum depends on the ratio of the resistivities of the filtrate and the formation water
predict the sign (positive or negative) and the approximate size of the SP deflection from the resistivities $R_{mf}$ (mud filtrate) and $R_w$ (formation water) and the temperature
draw the shale baseline, pick permeable beds and their boundaries, and distinguish the static amplitude from the recorded deflection
list what reduces the SP amplitude (a thin bed, a bed of high resistivity, shale in the bed, hydrocarbons) and the wells in which the method does not work
calculate the formation water resistivity from the static SP and estimate how errors in the amplitude and the temperature turn into an error in $R_w$

What the SP tool measures

The measuring circuit is very simple. One electrode is lowered into the borehole on a cable, usually as part of a tool string (the set of logging tools). The second electrode is grounded at the surface, in a tank of drilling mud or in the soil. We record the potential difference between them in millivolts. The tool sends out nothing and applies no current to the rock. So the method, like the gamma ray log, is called passive.

The potential of the surface electrode is constant, so all the changes in the curve come from the borehole. The absolute level means nothing. It depends on the electrodes and can slowly “drift”. Only the deflections matter, measured from the level that the curve keeps opposite thick shales. This level is the shale baseline. The curve is drawn in the left track of the log (a column of the log display), with negative values to the left. We set the scale and the shift during recording so that the curve does not leave the track.

In Russian-language practice, the SP log and one more curve together make up the standard logging suite. The second curve is the apparent resistivity of the standard electrode array (the resistivity that the tool reads before corrections). The suite is recorded over the whole borehole. We write the deflection from the shale baseline as $\Delta U_\text{SP}$ . The method needs a conducting mud and an open hole. So the SP is not recorded in wells drilled with oil-based mud or with air, or behind casing (Borehole environment: invaded zone and mudcake).

Where the potential comes from

Take a permeable sandstone between two thick shales, drilled with fresh mud (mud of low salinity). Near the borehole is the invaded zone, where the filtrate has pushed into the rock. There the pores are filled with filtrate, which contains little salt. Beyond the invasion front, the pores hold saline formation water. For simplicity, we assume that both solutions contain only dissolved sodium chloride (common salt). In the water this means Na⁺ cations (positive ions) and Cl⁻ anions (negative ions). The ions tend to move from the saline solution into the fresh one. At two different boundaries this gives two different potentials.

Membrane potential: The surfaces of the clay mineral layers carry a negative charge. So a shale lets Na⁺ cations through but repels Cl⁻ anions. It works as an ion-selective membrane (a barrier that passes only some ions). Through the shales above and below the bed, cations move from the saline formation water into the fresh mud in the borehole. The mud opposite the shales becomes positively charged. This is the main component: about 59 mV for each tenfold difference in the activities (effective concentrations) of the solutions at 25 °C. In Russian-language literature it is also called the adsorption potential.
Liquid-junction potential: At the invasion front, the filtrate and the formation water touch directly, and salt diffuses (spreads by itself) from the saline water into the fresh water. Cl⁻ anions are about one and a half times more mobile than Na⁺ cations, so they move ahead. The fresh side, which is the filtrate, becomes negatively charged, and so does the mud opposite the sandstone. This potential is about five times smaller than the membrane potential: about 12 mV per decade (per tenfold difference) at 25 °C.

Both EMFs (electromotive forces) act in the same direction. They are connected in series in a closed circuit: formation water → shale → mud in the borehole → invaded zone → formation water. Their sum is called the electrochemical potential (in Russian-language literature, the diffusion-adsorption potential). The SP current flows around this circuit. In the mud, it flows from the parts opposite the shales towards the part opposite the sandstone. The mud in the narrow borehole is the “thinnest” link of the circuit, so most of the EMF falls across it as a voltage drop. The electrode records exactly this drop.

SchematicHow the SP arises opposite a sandstone between shales, with a fresh mud (

R_{mf} > R_w

). Na⁺ cations pass through the shales into the borehole (membrane potential). Cl⁻ anions outrun the cations at the invasion front (liquid-junction potential). The SP current closes through the mud. On the left is the SP curve, which deflects to the left of the shale baseline. A schematic drawing, not to scale.

Sign of the SP deflection opposite a clean bed

Mud and formation water: Fresh mud, saline formation water (the most common case)
Resistivities: $R_{mf} > R_w$
Deflection: negative, to the left of the shale baseline

Mud and formation water: Salinity of the filtrate and the water close
Resistivities: $R_{mf} \approx R_w$
Deflection: almost none: the bed does not stand out on the SP

Mud and formation water: Saline mud or fresh formation water
Resistivities: $R_{mf} < R_w$
Deflection: positive, to the right

Mud and formation water: Oil-based mud, air, casing
Resistivities: —
Deflection: the SP is not recorded: the circuit is not closed

Mud and formation water	Resistivities	Deflection
Fresh mud, saline formation water (the most common case)	$R_{mf} > R_w$	negative, to the left of the shale baseline
Salinity of the filtrate and the water close	$R_{mf} \approx R_w$	almost none: the bed does not stand out on the SP
Saline mud or fresh formation water	$R_{mf} < R_w$	positive, to the right
Oil-based mud, air, casing	—	the SP is not recorded: the circuit is not closed

We compare the resistivities at the same temperature. Strictly speaking, the ratio of the activities of the solutions sets the sign and size. The ratio of resistivities is a convenient substitute for it (see below).

Static amplitude and water resistivities

A battery is a good way to understand the static amplitude. While the circuit is open, a voltmeter across the battery's terminals shows the full EMF. When the circuit is closed, part of the EMF is spent on the internal resistance, and the voltmeter shows less. In the borehole, the mud opposite the shales and the mud opposite the sandstone play the role of the terminals. Suppose the SP current could not flow along the borehole. Then the potential difference between them would equal the full electrochemical EMF. This quantity is called the static SP (SSP), or static amplitude. In a thick, clean, water-bearing bed of low resistivity, the recorded deflection almost reaches it. In other cases it is smaller (see the section on the reduction of the amplitude).

Static SP of a clean bed

\mathrm{SSP} = -K\lg\frac{a_w}{a_{mf}} = -K\lg\frac{R_{mfe}}{R_{we}}

where:

$\mathrm{SSP}$ static amplitude: the deflection from the shale baseline opposite a clean water-bearing bed when no currents flow; negative when the formation water is more saline than the filtrate, mV
$K$ electrochemical coefficient, proportional to the absolute temperature: $K \approx 65 + 0.24\,T$ with $T$ in °C, about 71 mV at 25 °C, mV
$a_w,\ a_{mf}$ chemical activities of the formation water and the filtrate at formation temperature, the “effective” ion concentrations, —
$R_{mfe},\ R_{we}$ equivalent resistivities of the filtrate and the formation water at formation temperature, which are inversely proportional to the activities, Ω·m

Units: mV; resistivities in any consistent units; $\lg$ is the base-10 logarithm

Holds when

A clean (shale-free) water-bearing bed between thick shales that behave as ideal cation membranes.
The solutions are close in composition to an NaCl solution; with a large share of other salts the coefficient at 25 °C differs from 71 mV.
The streaming potentials are small or cancel each other.
Both resistivities and the coefficient $K$ refer to formation temperature.

Each tenfold difference in activities gives about 71 mV at 25 °C and about 89 mV at 100 °C. The amplitude depends on the ratio of the resistivities, not on their absolute values: the pairs 1.0 and 0.1 Ω·m or 0.5 and 0.05 Ω·m give the same SSP.

In other unit systems (1)

Temperature in degrees Fahrenheit

K \approx 61 + 0.133\,T\,[^\circ\text{F}]

$T$ in °F; at the same temperature it gives practically the same value (within 0.3 mV) as $65 + 0.24\,T$ with $T$ in °C.

Both forms say that $K$ is proportional to the absolute temperature, with the reference point of 71 mV at 25 °C: $71\cdot(T + 273.15)/298.15 = 65.0 + 0.238\,T$ .

Why do we call the resistivities “equivalent”? The activities of the solutions set the EMF. In dilute solutions the activity is proportional to the concentration, and so also to the electrical conductivity. So we can replace the ratio of the activities by the inverse ratio of the resistivities. In strong brines the activity grows faster than the conductivity, and the equivalent resistivity comes out smaller than the true one. The practical rules are these:

For formation waters of moderate salinity, up to about 50 g/L of NaCl (a resistivity at 24 °C of not less than 0.13 Ω·m), $R_{we}$ and $R_w$ differ by no more than about 10%.
For brines, we go from $R_{we}$ to $R_w$ with charts or correlations. The difference grows with salinity. For brines close to saturation (as salty as water can get) it reaches several times.
For the filtrate of ordinary fresh NaCl-based muds (resistivity above about 0.1 Ω·m at 24 °C), take $R_{mfe} \approx 0.85\,R_{mf}$ . This is an empirical rule, which comes from experience and not from theory. For saline and gypsum muds, use charts.
Fresh formation waters with a noticeable share of calcium, magnesium, sulfates and bicarbonates also depart from the “sodium” relationship. For them the estimate of $R_w$ from the SP is rough.

CalculatedStatic SP against the ratio of the equivalent resistivities

R_{mfe}/R_{we}

at 25, 85 and 150 °C, calculated from the formula above. The dot is the bed of the example below: a ratio of about 5 at 85 °C gives −60 mV.

How to read the SP curve

We read the SP curve together with the gamma ray log (GR), the caliper log (a record of the hole diameter) and the resistivity curves. The procedure is as follows.

Marking up the curve

1
Shale baseline
Draw a line through the readings opposite thick, uniform shales. It is rarely perfectly vertical. The electrodes drift, and the salinity of the waters changes with depth. So we refine the line interval by interval and mark abrupt shifts separately.
2
Clean-bed line
Draw a line through the largest deflections in thick, clean, water-bearing beds. The distance between this line and the shale baseline is the static amplitude for this interval of the section.
3
Bed boundaries
The slope of the curve is proportional to the density of the SP current in the mud (how much current flows through each unit of area). This current density is greatest opposite the bed boundary. So we place the boundaries at the inflection points, where the curve is steepest. In a thick bed this is about the middle of the amplitude.
4
Permeable beds
A deflection from the shale baseline means that ions pass through the bed. So the bed is permeable, even if only weakly. The size of the deflection says nothing about how permeable the bed is. The water contrast, the shale volume and the resistivity of the bed set it.
5
Check
Compare the beds you picked with three other records. The GR shows shales. The caliper log shows mudcake (a layer of mud solids on the hole wall) opposite permeable beds. The third check is the separation between resistivity curves of different depths of investigation (Borehole environment: invaded zone and mudcake).

SchematicAn assumed SP curve in a sand–shale section with a fresh mud (

R_{mf} > R_w

). Only a thick, clean, water-bearing sandstone gives the full deflection. Shaly, thin and oil-bearing beds deflect the curve less. A schematic drawing, not a calculation.

What the SP shows in different beds at

R_{mf} > R_w

Bed: Thick shale
Deflection: shale baseline
Why: the reference level

Bed: Thick clean water-bearing sandstone
Deflection: full, close to the static amplitude
Why: the full EMF; the resistivity of the bed is low

Bed: Shaly sandstone
Deflection: less than the static amplitude
Why: the shale in the bed creates an opposing membrane EMF

Bed: Thin bed
Deflection: smaller, with a rounded shape
Why: only a small share of the EMF falls across the short column of mud opposite the bed

Bed: Oil- or gas-bearing sandstone
Deflection: usually less than in the water-bearing part
Why: the resistivity of the bed is higher, the current is weaker

Bed: Tight high-resistivity rocks
Deflection: the curve changes smoothly
Why: the current bypasses high-resistivity beds, the boundaries are blurred

Bed	Deflection	Why
Thick shale	shale baseline	the reference level
Thick clean water-bearing sandstone	full, close to the static amplitude	the full EMF; the resistivity of the bed is low
Shaly sandstone	less than the static amplitude	the shale in the bed creates an opposing membrane EMF
Thin bed	smaller, with a rounded shape	only a small share of the EMF falls across the short column of mud opposite the bed
Oil- or gas-bearing sandstone	usually less than in the water-bearing part	the resistivity of the bed is higher, the current is weaker
Tight high-resistivity rocks	the curve changes smoothly	the current bypasses high-resistivity beds, the boundaries are blurred

A qualitative picture: the real amplitude depends on a combination of these factors.

In Russian-language practice, the deflection of a bed is divided by the static amplitude. This gives the relative SP amplitude $\alpha_\text{SP} = \Delta U_\text{SP}/\Delta U_{\text{SP},\max}$ , which varies from zero opposite shales to one opposite clean beds. We use it to pick reservoirs (with a cut-off value set for the area), to estimate shale volume (Determining shale volume), and to build relationships with porosity and permeability from core. The shape of the SP curve, like that of the GR, repeats in neighbouring wells, and we use it to correlate sections (Well correlation).

Why the recorded deflection is smaller than the static amplitude

The SP current flows in a closed circuit made of three sections in series: the mud in the borehole, the bed with its invaded zone, and the enclosing shales. The EMF is shared between them in proportion to their resistances. The electrode “sees” only the part that falls across the mud.

Recorded SP deflection in the equivalent circuit

\mathrm{SP} = \mathrm{SSP}\cdot\frac{r_m}{r_m + r_{sd} + r_{sh}}

where:

$\mathrm{SP}$ recorded deflection from the shale baseline opposite the bed, mV
$\mathrm{SSP}$ static amplitude, the full EMF of the circuit, mV
$r_m,\ r_{sd},\ r_{sh}$ resistances of the sections of the current path: the mud in the borehole, the bed with its invaded zone, the enclosing shales (resistances of bodies, not resistivities), Ω

Units: mV; resistances of the sections in ohms

Holds when

This is a qualitative equivalent circuit. The currents in the bed and in the shales spread in a complicated way, and we do not measure $r_{sd}$ and $r_{sh}$ directly.
The electrochemical EMF is concentrated at the shale–mud and filtrate–formation water boundaries.

The deflection is close to the static one when the mud has the largest resistance in the circuit. This happens when the bed is thick, the path of the current in the rock is wide and short, and the rock itself conducts well. In a thin bed, a bed of high resistivity, or a bed flushed with fresh filtrate, the resistance of the current path through the rocks grows compared with $r_m$ , and the recorded deflection falls. We correct for the thickness and resistivity of the bed with charts (Interpretation charts: how to read them).

Small bed thickness. Opposite a thin bed the column of mud is short. Its resistance is small compared with the resistance of the current path through the rocks, so the curve does not have time to reach the full deflection. In beds a few metres thick or less, the correction is usually noticeable, especially when the resistivity of the invaded zone is high.
High resistivity of the bed and the invaded zone. Fresh filtrate, hydrocarbons and tight rock increase $r_{sd}$ .
Shale volume. Shale in the bed itself acts as a membrane of the opposite sign. It reduces the EMF itself. So the static amplitude of a shaly bed, called the pseudo-static SP (PSP), is smaller than the SSP of a clean bed. The estimate of shale volume from the SP is based on this (Determining shale volume).
Hydrocarbons. In the oil- or gas-bearing part of a bed, the deflection is usually smaller than in the water-bearing part with the same water. The resistivity of the bed is higher. Also, a shaly reservoir that holds hydrocarbons has less water in it, so the shale has a stronger effect.
The borehole. In a wide hole with a saline mud, the resistance $r_m$ is small, so a smaller share of the EMF falls across the mud.

Formation water resistivity from the SP

From the formula for the static amplitude, we can write the equivalent resistivity of the formation water directly:

Equivalent formation water resistivity from the static SP

R_{we} = R_{mfe}\cdot 10^{\,\mathrm{SSP}/K}

where:

$R_{we}$ equivalent formation water resistivity at formation temperature, Ω·m
$R_{mfe}$ equivalent filtrate resistivity at formation temperature; for ordinary fresh muds $\approx 0.85\,R_{mf}$ , Ω·m
$\mathrm{SSP}$ static amplitude with its own sign (negative at $R_{mf} > R_w$ ), mV
$K$ electrochemical coefficient at formation temperature, $65 + 0.24\,T$ , mV

Units: Ω·m; SSP and $K$ in mV

Holds when

The SSP was taken in a thick clean water-bearing bed or corrected for the thickness and resistivity of the bed.
$R_{mfe}$ and $K$ refer to formation temperature.
The water and the filtrate are close to NaCl solutions, and the streaming potentials are small.

For a negative SSP the factor $10^{\mathrm{SSP}/K}$ is less than one: the formation water conducts better than the filtrate. At $K \approx 85$ mV an error of 5 mV in the amplitude changes $R_{we}$ by about 15%.

Calculation procedure

1
Choose a bed
Pick a thick, clean, water-bearing bed near the interval of interest, with the same formation water. Check from the GR and the resistivity that it is clean and water-bearing.
2
Find the formation temperature
Use the geothermal gradient (how fast the temperature rises with depth) and the corrected bottom-hole temperature (Formation water and formation temperature).
3
Convert the filtrate resistivity
Take $R_{mf}$ from the log heading (the block of data at the top of the log). Convert it to formation temperature and find $R_{mfe}$ .
4
Read the static amplitude
Measure from the shale baseline to the clean-bed line. If necessary, correct for the thickness and resistivity of the bed.
5
Calculate $R_{we}$ and move to $R_w$
For waters of moderate salinity, $R_w \approx R_{we}$ . For brines, use a chart or a correlation.
6
Check
Compare the result with water sample analyses, the water catalogue of the area and the apparent water resistivity $R_{wa}$ of water-bearing beds (Determining formation water resistivity).

Worked exampleFormation water resistivity from the static amplitude

Given (assumed data)

Static amplitude in a thick clean water-bearing sandstone: −60 mV
Formation temperature $T$: 85 °C
Filtrate resistivity (measured on a sample): 0.95 Ω·m at 22 °C
Mud: fresh, water-based, salts mainly NaCl

Solution

Filtrate at formation temperature (we convert with the Arps formula, see Formation water and formation temperature): $R_{mf} = 0.95\cdot\dfrac{22 + 21.5}{85 + 21.5} = 0.388$ Ω·m. At 24 °C it would be 0.91 Ω·m, which is more than 0.1. So the rule $R_{mfe} = 0.85\,R_{mf} = 0.330$ Ω·m applies.
Coefficient: $K = 65 + 0.24\cdot 85 = 85.4$ mV.
Resistivity ratio: $R_{mfe}/R_{we} = 10^{60/85.4} = 5.04$ , so $R_{we} = 0.330/5.04 = 0.065$ Ω·m at 85 °C.
Salinity check: at 24 °C this is 0.153 Ω·m, that is, about 42 g of NaCl per kilogram of solution. This is a water of moderate salinity, so $R_w \approx R_{we} \approx 0.065$ Ω·m.
Sensitivity: suppose we took the deflection of a thin bed, −55 mV, instead of the static amplitude. The result would be $R_w = 0.075$ Ω·m, which is 14% higher. Now suppose we forgot to convert $R_{mf}$ to formation temperature. The result would be 0.16 Ω·m, which is 2.4 times higher.

Answer: The formation water resistivity is about 0.065 Ω·m at 85 °C.

What it means. In Archie's equation with $n = 2$ , water saturation is proportional to $\sqrt{R_w}$ (Archie's equation and saturation of clean reservoirs). So an error of 5 mV in the amplitude gives an error in $S_w$ of about 7%. A forgotten temperature correction of the filtrate makes $S_w$ 1.56 times too high, and a productive bed could look water-bearing. We always check a result from the SP against sample analyses and against the $R_{wa}$ of water-bearing beds. The data in this example are assumed for illustration.

Streaming potentials, noise and limitations

Besides the electrochemical EMF, there is an electrokinetic, or streaming, potential. It arises when a pressure difference pushes a solution through a porous medium. In a borehole, the filtrate passes through the mudcake opposite permeable beds and, in small amounts, through the shales. Both components shift the reading in the negative direction, opposite the bed and opposite the shales alike. So in the deflection from the shale baseline they largely cancel each other out. With an ordinary overbalance and saline formation water, we neglect them.

Their contribution grows with the pressure difference. These cases are dangerous. One is a bed with reduced (depleted) pressure, or a weighted mud (a heavy mud): the overbalance is then large. Another is a very tight rock, where almost no mudcake forms and the pressure drop falls on the rock itself. A third is a high-resistivity mud with brackish formation water (slightly salty water). The streaming potential can then reach tens or even hundreds of millivolts, and the estimate of $R_w$ from the SP loses its meaning. On the curve itself, we cannot tell such a contribution from the electrochemical one. So any of these conditions is a reason to check $R_w$ by other means.

Shifts of the shale baseline. A shale may be an imperfect membrane, or it may separate beds with waters of different salinity. Then the shale baseline shifts abruptly. We measure the deflection of each bed from its “own” baseline.
High-resistivity sections. In carbonates, anhydrites and tight sandstones, the SP current goes around the high-resistivity beds. The curve between permeable streaks (thin permeable layers) changes almost in a straight line, and we cannot see the boundaries on it.
Electrical noise. A magnetised winch drum (the drum of the winch that lowers the tool) makes the curve oscillate regularly. Stray currents from electrical equipment and from the cathodic protection of pipelines (a system that stops pipes from corroding) give false deflections. So do redox potentials (voltages from chemical reactions) at pyrite, coal and metal in the hole.
Where the SP is not recorded. Oil-based mud, air or foam, and a cased hole. In these cases the current circuit through the mud is not closed.

Terms and symbols

SP: Spontaneous potentials, and the curve that records them along the borehole.
Shale baseline: The level of the SP curve opposite thick shales. We measure deflections from it.
Static SP, SSP: The total electrochemical EMF of a clean water-bearing bed: the deflection that we would see if no currents flowed.
Pseudo-static SP, PSP: The static amplitude of a shaly bed. It is smaller than the SSP of a clean bed with the same water.
Membrane potential: The EMF at shales, which pass cations and hold back anions.
Liquid-junction potential: The EMF at the contact of solutions of different salinity. It arises because the ions move at different speeds.
Streaming (electrokinetic) potential: The EMF that arises when a solution flows through a porous medium, for example through a mudcake.
Equivalent resistivities $R_{we}$ , $R_{mfe}$: Resistivities that are inversely proportional to the activities of the solutions. For waters of moderate salinity they are close to the true ones.
Relative SP amplitude, $\alpha_\text{SP}$: The ratio of the deflection in a bed to the static amplitude.

Common mistakes and misconceptions

Wrong

“The SP curve does not deflect, so the bed is impermeable.”

In fact

There is no deflection even opposite a good reservoir if the filtrate and the formation water have the same salinity. We judge permeability together with the caliper log (mudcake), resistivity curves of different depths of investigation and the microlog (a shallow resistivity log made with small electrodes on a pad).

Wrong

“The larger the SP deflection, the higher the permeability.”

In fact

The water contrast, the temperature, and the thickness, resistivity and shale volume of the bed set the amplitude. Even a poorly permeable clean bed can give the full deflection, because for the EMF it is enough that ions can pass through it.

Wrong

“The deflection of any sandstone will do for calculating $R_w$ .”

In fact

You need a thick, clean, water-bearing bed or a corrected amplitude. Thin, shaly and oil-bearing beds give an amplitude that is too low and an $R_w$ that is too high.

Wrong

“ $R_{mf}$ from the log heading is substituted as it is.”

In fact

It is measured at surface temperature. Convert it to formation temperature before the calculation. Otherwise $R_w$ comes out several times too high.

Wrong

“The deflection of an oil-bearing bed is smaller, so it is shalier.”

In fact

Hydrocarbons also reduce the deflection. Check the shale volume with the GR, the neutron and density logs and core.

In depth: advanced detail

Where 71 mV comes from: membrane and diffusion in numbers

For an ideal cation membrane, the Nernst equation gives the EMF: $E_m = \dfrac{2.303\,RT}{F}\lg\dfrac{a_w}{a_{mf}}$ . The factor $2.303\,RT/F$ is 59.2 mV at 25 °C. The liquid-junction potential at the contact of two solutions of the same salt is proportional to the difference of the transport numbers of the anion and the cation, $t_- - t_+$ . A transport number is the share of the current that one kind of ion carries. We add the two EMFs and use $t_- + t_+ = 1$ . This gives the coefficient of the static SP:

Electrochemical coefficient in terms of the transport numbers

K = (1 + t_- - t_+)\,\frac{2.303\,RT}{F} = 2\,t_-\,\frac{2.303\,RT}{F}

where:

$t_-,\ t_+$ transport numbers of the Cl⁻ anion and the Na⁺ cation in an NaCl solution, $t_- + t_+ = 1$ ; for a dilute solution at 25 °C $t_- \approx 0.60$ , —
$R$ universal gas constant, 8.314 J/(mol·K), J/(mol·K)
$T$ absolute temperature, K
$F$ Faraday constant, 96,485 C/mol, C/mol

Units: SI: $K$ comes out in volts, multiply by 1000 for millivolts; $2.303 = \ln 10$ converts the natural logarithm to the base-10 one

Holds when

The shales are an ideal membrane: they pass only cations.
The bed is clean and is not itself a membrane.
Both solutions are solutions of the same salt, NaCl; the transport numbers are constant over the range of concentrations considered.

The first term is the membrane potential and the second is the liquid-junction potential. Since $t_- - t_+ \approx 0.2$ , the second is about five times smaller. The coefficient is proportional to the absolute temperature. This is why the formula $K \approx 65 + 0.24\,T$ is linear.

Worked exampleThe theoretical coefficient at 25 and 85 °C

Given (assumed data)

Transport number of Cl⁻ in a dilute NaCl solution: $t_- = 0.604$ , so $t_+ = 0.396$
Temperatures: 25 °C (298.15 K) and 85 °C (358.15 K)

Solution

Nernst factor: at 298.15 K $2.303\,RT/F = 59.2$ mV, at 358.15 K 71.1 mV.
At 25 °C the membrane part is 59.2 mV per decade. The liquid-junction part is $(0.604 - 0.396)\cdot 59.2 = 12.3$ mV, that is, 21% of the membrane part.
Coefficient: $K = 2\cdot 0.604\cdot 59.2 = 71.5$ mV at 25 °C and 85.8 mV at 85 °C. The linear formula gives 71.0 and 85.4 mV.

Answer: Theory gives about 71.5 mV per decade at 25 °C. This is practically the same as the 71 mV that we use in interpretation.

What it means. The agreement shows that the electrochemical EMF of the SP is mainly the membrane potential of shales, with a small addition of the liquid-junction potential. The transport numbers change slightly with temperature and concentration, and real shales are imperfect membranes. So in practice we use the empirical coefficient. The transport numbers are for a dilute NaCl solution at 25 °C.

Connections to other topics

Borehole environment: invaded zone and mudcake — the invaded zone, the filtrate and why the SP is not recorded in oil-based mud.
Formation water and formation temperature — how to convert $R_{mf}$ and $R_w$ to formation temperature and relate the resistivity of the water to its salinity.
Determining shale volume — shale volume from the relative SP amplitude.
Determining formation water resistivity — all the ways of finding $R_w$ and how they check each other.
Archie's equation and saturation of clean reservoirs — how $R_w$ enters the water saturation calculation.
Gamma ray log and natural gamma ray spectrometry — the second passive method of identifying shales and permeable beds.
Resistivity logs: electrical and induction tools — the resistivity curves that are recorded together with the SP.
Well correlation — matching the sections of wells by the shape of the GR and SP curves.

Check yourself

1
A well is drilled with fresh mud, and the formation water is saline. How will the SP curve deflect opposite a thick clean water-bearing sandstone?
1. A.To the right, in the positive direction
2. B.To the left, in the negative direction
3. C.Not at all: deflection occurs only in oil-bearing beds
4. D.The direction depends on the permeability of the bed
Show the answer
Correct answer: B. To the left, in the negative direction
For $R_{mf} > R_w$ , the ratio $R_{mfe}/R_{we}$ is greater than one, so $\mathrm{SSP} = -K\lg(R_{mfe}/R_{we})$ is negative. In physical terms, the mud opposite the shales becomes positively charged, because cations pass through the shale from the saline water. Opposite the sandstone the mud becomes negatively charged, because anions outrun the cations at the invasion front.
2
The formation temperature is 60 °C, $R_{mfe} = 0.40$ Ω·m and the static amplitude is −80 mV. Find $R_{we}$ and say whether we can take it as equal to $R_w$ .
Show the answer
$K = 65 + 0.24\cdot 60 = 79.4$ mV; $10^{80/79.4} \approx 10.2$ ; $R_{we} = 0.40/10.2 \approx 0.039$ Ω·m. At 24 °C this is about $0.039\cdot 81.5/45.5 \approx 0.07$ Ω·m. That is already a brine, with about 100 g of NaCl per kilogram. For such waters the equivalent resistivity is smaller than the true one. We find $R_w$ from a chart or a correlation, and it will be somewhat larger than 0.039 Ω·m.
3
Opposite a sandstone that has a core permeability of 300 mD, the SP curve hardly deflects. Which explanation is most likely?
1. A.The bed is too thick
2. B.The bed is water-bearing
3. C.The salinities of the filtrate and the formation water are close
4. D.The bed is impermeable and the core result is wrong
Show the answer
Correct answer: C. The salinities of the filtrate and the formation water are close
The electrochemical EMF is proportional to the logarithm of the ratio of the activities of the solutions. When the filtrate and the water have the same salinity, it is close to zero, and even an excellent reservoir does not stand out on the SP. This happens with a saline mud or fresh formation water. We confirm permeability with the caliper log, the microlog and the separation between resistivity curves of different depths of investigation.
4
Why is the SP deflection opposite a sandstone 1 m thick smaller than opposite a thick sandstone with the same water?
Show the answer
The SP current flows around the circuit “mud – bed – shales”, and the electrode records only the voltage drop in the mud. Opposite a thin bed, the column of mud is short. Its resistance is small compared with the resistance of the current path through the rocks, so a smaller part of the EMF falls across the mud. For the $R_w$ calculation, we therefore correct such a deflection for thickness with a chart, or we take the static amplitude in a thick bed.
5
In which case is the estimate of $R_w$ from the SP the most reliable?
1. A.A thin oil-bearing bed in a carbonate section
2. B.A well drilled with oil-based mud
3. C.A depleted bed drilled with high overbalance
4. D.A thick clean water-bearing sandstone, fresh mud, water of moderate salinity
Show the answer
Correct answer: D. A thick clean water-bearing sandstone, fresh mud, water of moderate salinity
The formula for the static amplitude was derived for a clean water-bearing bed and for solutions close to NaCl. A thick bed gives the full amplitude, and at moderate salinity $R_w \approx R_{we}$ . In a thin oil-bearing bed the amplitude comes out too low. In a depleted bed the streaming potential interferes. With oil-based mud the SP is not recorded at all.
6
The SP curve opposite the oil-bearing top of a sandstone deflects 20% less than opposite its water-bearing base. Does this mean that the top is shalier?
Show the answer
Not necessarily. Hydrocarbons raise the resistivity of the bed, and so they reduce the SP deflection by themselves, especially in shaly reservoirs. Check the shale volume of the top with the GR, the neutron and density logs and core. The shale volume from the SP will come out too high for the oil-bearing part.

Updated: 19 October 2026