Microresistivity tools and flushed-zone resistivity

Quick refresher

The flushed zone can be only a few centimetres thick. So we measure $R_{xo}$ with pad-mounted microresistivity tools whose depth of investigation (how far into the rock they look) is 2–10 cm. The current reaches the formation through the mudcake.
The microlog is made of two unfocused sondes. The microgradient sonde sees mainly the mudcake. The micropotential sonde also sees part of the flushed zone. A positive separation points to mudcake, and so to a permeable bed, but it does not give the size of the permeability.
A focused microresistivity sonde (the microlaterolog and its descendants) forces the current through the mudcake into the formation. Its reading depends mainly on $R_{xo}$ if the mudcake is thinner than about 10 mm and the invasion front (the outer edge of the zone that the filtrate has reached) is more than 8–10 cm from the wall.
A thick mudcake of low resistivity lowers the reading. Shallow invasion mixes $R_t$ into the reading. The corrections come from the tool maker's charts.
We find the water saturation of the flushed zone, $S_{xo}$ , from Archie's equation with $R_{xo}$ and $R_{mf}$ at formation temperature. The difference $S_{xo} - S_w$ ( $S_w$ is the water saturation of the uninvaded formation) is the share of the pores that holds moveable hydrocarbons, and $1 - S_{xo}$ is the share that holds residual hydrocarbons.
The vertical resolution of microresistivity tools is 5–10 cm. With them we pick out thin permeable beds.

Key relationships

R \approx J_{xo}\,R_{xo} + (1 - J_{xo})\,R_t

reading of a focused microresistivity sonde with a thin mudcake

S_{xo} = \left(\dfrac{a\,R_{mf}}{\phi^m R_{xo}}\right)^{1/n}

water saturation of the flushed zone (the water in it is filtrate)

V_{hm} = \phi\,(S_{xo} - S_w),\quad V_{hr} = \phi\,(1 - S_{xo})

moveable and residual hydrocarbons in a unit volume of rock

R_2 = R_1\,\dfrac{T_1 + 21.5}{T_2 + 21.5}

bringing $R_{mf}$ and $R_{mc}$ to formation temperature (°C)

Common pitfalls

Taking the sonde reading as $R_{xo}$ without a mudcake correction and without checking that the pad touches the wall.
Using the filtrate resistivity at the surface measurement temperature in the $S_{xo}$ calculation.
Judging the permeability from the size of the positive separation.
Taking $1 - S_{xo}$ as the exact residual oil saturation when calculating oil recovery.
Expecting readings from the microlog and the microlaterolog in a well drilled with oil-based mud (mud made with oil instead of water).

After this lesson you can

explain why measuring $R_{xo}$ needs a pad tool that looks only centimetres into the rock, and how unfocused (microlog) and focused (microlaterolog) sondes differ
pick out permeable beds by the positive separation on the microlog, and recognise false signs
list what distorts the reading of a microresistivity sonde (mudcake, shallow invasion, washouts, which are sections where the hole is wider than the bit), say whether each one pushes the reading up or down, and find $R_{xo}$ with corrections
calculate the water saturation of the flushed zone from $R_{mf}$ and Archie's equation (the basic link between resistivity, porosity and saturation), and estimate the moveable and residual hydrocarbons (the oil and gas that can and cannot be pushed out of the pores)
use the high vertical resolution of microresistivity tools for thin beds and net thickness

Why we measure the resistivity of the flushed zone

In the flushed zone (Borehole environment: invaded zone and mudcake) the mud filtrate has pushed out almost all the formation water and the moveable part of the hydrocarbons. Its resistivity is $R_{xo}$ . In Russian-language notation it is written $\rho_\text{pp}$ , with the subscript spelled in Latin letters. It depends on the filtrate resistivity $R_{mf}$ , the porosity (the share of the rock volume that is pore space) and the residual hydrocarbons. $R_t$ is hidden behind the invaded zone (the ring of rock that the filtrate has entered). $R_{xo}$ lies right at the wall. But the flushed zone can be only a few centimetres thick.

This sets the requirements for the tool. It must see a few centimetres into the rock, and it must not see the mud in the hole. An ordinary sonde that stands off the wall would measure mainly the mud. So the electrodes sit on a pad: an insulating cushion that arms and springs press against the wall. The pad cuts off the mud. The current from the electrodes goes into the formation and crosses only the mudcake on the way.

What we need

R_{xo}

for

Task: True formation resistivity
How $R_{xo}$ is used: the third curve, to correct the deep sondes for invasion
More detail: three curves of different depths of investigation

Task: Saturation of the flushed zone
How $R_{xo}$ is used: $S_{xo}$ from Archie's equation with $R_{mf}$
More detail: this lesson

Task: Moveable oil
How $R_{xo}$ is used: comparing $S_{xo}$ with $S_w$
More detail: this lesson; without porosity, see Resistivity ratio methods and movable oil

Task: Porosity correction for hydrocarbons
How $R_{xo}$ is used: the residual hydrocarbon saturation of the flushed zone
More detail: Gas, shale, carbonate and pressure effects on porosity logs

Task: Checking $R_{mf}$ and the formation resistivity factor $F$ (the resistivity of water-filled rock divided by that of the water)
How $R_{xo}$ is used: water-bearing beds, where $S_{xo} = 1$
More detail: advanced section; Resistivity–porosity crossplots

Task: Permeable and thin beds
How $R_{xo}$ is used: the mudcake and the high vertical resolution
More detail: this lesson

Task	How $R_{xo}$ is used	More detail
True formation resistivity	the third curve, to correct the deep sondes for invasion	three curves of different depths of investigation
Saturation of the flushed zone	$S_{xo}$ from Archie's equation with $R_{mf}$	this lesson
Moveable oil	comparing $S_{xo}$ with $S_w$	this lesson; without porosity, see Resistivity ratio methods and movable oil
Porosity correction for hydrocarbons	the residual hydrocarbon saturation of the flushed zone	Gas, shale, carbonate and pressure effects on porosity logs
Checking $R_{mf}$ and the formation resistivity factor $F$ (the resistivity of water-filled rock divided by that of the water)	water-bearing beds, where $S_{xo} = 1$	advanced section; Resistivity–porosity crossplots
Permeable and thin beds	the mudcake and the high vertical resolution	this lesson

How microresistivity tools are built

All microresistivity tools share the same main design. Small electrodes are set into a rubber or plastic pad. They are only centimetres apart (2.5 cm in the microlog). The closer the electrodes are, the shallower the sonde “looks” into the formation (its depth of investigation), and the thinner the beds it can tell apart. The arm that presses the pad is also a caliper (a device that measures the hole diameter). We need this curve to estimate the mudcake thickness and to see how well the pad touches the wall.

The tools differ in how their current passes through the mudcake. In the unfocused sondes of the microlog the current spreads freely. If the mudcake conducts well, a large part of the current runs along the mudcake and into the mud. In focused sondes (the microlaterolog and its descendants) a guard current (an auxiliary screening current) stops the measuring current from spreading. It pushes the measuring current as a beam through the mudcake into the flushed zone.

SchematicCurrent paths of pad sondes (schematic, not to scale): (a) an unfocused microlog sonde: a large part of the current leaves along the low-resistivity mudcake into the mud; (b) a focused sonde: the guard current (grey lines) forces the measuring beam through the mudcake into the flushed zone; (c) the same beam with shallow invasion reaches the uninvaded part of the formation. The dashed line is the invasion front.

Microresistivity tools and what they can do (approximate)

Tool: Microlog (microgradient and micropotential sondes)
Depth of investigation: about 4 and 5–10 cm
Mudcake: thicker than 12 mm: $R_{xo}$ cannot be found
Use: permeable beds

Tool: Microlaterolog (MLL)
Depth of investigation: 7–8 cm
Mudcake: up to 10 mm: almost no effect
Use: $R_{xo}$

Tool: Wide-pad sonde (obsolete)
Depth of investigation: deeper than the microlaterolog
Mudcake: up to 19 mm: almost no effect
Use: $R_{xo}$ with deep invasion

Tool: Spherical and cylindrical focusing
Depth of investigation: a few centimetres
Mudcake: minimal effect
Use: $R_{xo}$ together with the deep sondes; mudcake parameters

Tool	Depth of investigation	Mudcake	Use
Microlog (microgradient and micropotential sondes)	about 4 and 5–10 cm	thicker than 12 mm: $R_{xo}$ cannot be found	permeable beds
Microlaterolog (MLL)	7–8 cm	up to 10 mm: almost no effect	$R_{xo}$
Wide-pad sonde (obsolete)	deeper than the microlaterolog	up to 19 mm: almost no effect	$R_{xo}$ with deep invasion
Spherical and cylindrical focusing	a few centimetres	minimal effect	$R_{xo}$ together with the deep sondes; mudcake parameters

These are rough values. The characteristics of a particular tool are in its documentation. Logs of obsolete tools are found in the records of old wells.

The microlog: two curves and positive separation

On the pad of the microlog, three small electrodes stand in a line, 2.5 cm apart: the current electrode $A$ and the measuring electrodes $M$ and $N$ . They form two sondes. One is the microgradient sonde A0.025M0.025N. The other is the micropotential sonde A0.05M. (The numbers are the distances between the electrodes, in metres.) In English-language literature they are the microinverse 1″×1″ and the micronormal 2″. Both curves are recorded at the same time. The gradient sonde sees a layer about 4 cm deep, counted from the pad. This is mainly the mudcake. The potential sonde sees 5–10 cm deep, so the flushed zone adds noticeably to its reading.

Opposite a permeable bed, the wall is covered by mudcake: the solid part of the mud, soaked with filtrate. It conducts current almost as well as the mud itself, and much better than the flushed zone behind it. The shallow gradient sonde takes most of its signal from the mudcake and shows a low resistivity. The potential sonde looks deeper. It gets a noticeable contribution from the flushed zone and shows more. This difference is the positive separation, the main sign of a permeable bed on the microlog. The readings themselves are not large: usually 2–10 times the resistivity of the mud. Opposite tight rock (rock with almost no open pores) the filtrate has nowhere to go, so there is no mudcake, and the picture is different. Both curves are high and coincide, or the gradient sonde reads higher than the potential sonde. That is a negative separation. In shales (clay-rich rocks) both curves are low and close together. In washouts, where the hole is wider and the pad lifts off the wall, both curves move towards the resistivity of the mud.

SchematicAn assumed microlog example (schematic): on the left the caliper log and the bit diameter, on the right the curves of the microgradient and micropotential sondes on a linear scale. The positive separation (shaded) marks permeable sandstones with mudcake. Tight limestone gives high, close readings. Opposite the washout in the shales both curves fall to the resistivity of the mud.

We can find $R_{xo}$ from the microlog only in favourable conditions. The ratio $R_{xo}/R_{mc}$ (the flushed-zone resistivity divided by the mudcake resistivity) must be less than about 15 (porosity above about 15%). The mudcake must be no thicker than about 12 mm. And the invasion front must be more than about 10 cm from the wall, or $R_t$ will enter the reading. Then we take four inputs: the two readings, the resistivity of the mudcake (measured, or estimated from the resistivity of the mud) and the mudcake thickness from the caliper log. With the tool maker's chart we find $R_{xo}$ (Interpretation charts: how to read them). In all other cases we use the microlog only qualitatively, and we measure $R_{xo}$ with focused sondes.

Focused microresistivity sondes

A focused microresistivity sonde is a laterolog in miniature. (A laterolog is a resistivity method in which guard electrodes force the current into the rock.) In the centre of the pad is a small current electrode $A_0$ . Around it are concentric rings: monitor rings and an outer guard ring. A constant current goes through $A_0$ . The automatic control sets the current of the guard ring so that there is no potential difference between the monitor rings. Then the measuring current cannot spread along the wall. It goes into the formation in a narrow beam, which widens only a few centimetres from the pad. In Russian-language practice the method is called the microlaterolog (MLL).

Why focusing matters. When the ratio $R_{xo}/R_{mc}$ is large, the current of an unfocused sonde takes the path along the low-resistivity mudcake and through it into the mud. The mudcake acts as a short circuit around the formation, and the reading almost stops depending on $R_{xo}$ . A focused current crosses the mudcake and the flushed zone one after the other, so their resistivities add up in the reading. A thin mudcake adds little, and the reading depends mainly on $R_{xo}$ .

Reading of a focused microresistivity sonde in the step model

R_\text{pad} = J_{mc}\,R_{mc} + J_{xo}\,R_{xo} + \left(1 - J_{mc} - J_{xo}\right) R_t

where:

$R_\text{pad}$ reading of the sonde (apparent resistivity), Ω·m
$J_{mc},\ J_{xo}$ pseudo-geometric factors of the mudcake and the flushed zone: their shares in the reading. The remainder $1 - J_{mc} - J_{xo}$ belongs to the uninvaded part of the formation, —
$R_{mc},\ R_{xo},\ R_t$ resistivities of the mudcake, the flushed zone and the uninvaded part of the formation, Ω·m

Units: any identical units of resistivity

Holds when

A focused sonde: the measuring current crosses the zones one after the other, so the resistivities add up. The model does not suit the unfocused sondes of the microlog.
A step profile: mudcake of thickness $h_{mc}$ , the flushed zone up to the invasion front, and beyond it the uninvaded formation.
$J_{mc}$ grows with the thickness of the mudcake, and the share of the formation grows as the front comes closer to the wall. The factors depend on the design of the tool and, to some extent, on the resistivity contrast, so real corrections come from the charts of the tool maker.

This is the same model as for the laterolog in the topic on finding Rt from three curves, with a mudcake term added. With a thin mudcake and deep invasion, $J_{xo} \approx 1$ , and the sonde reads $R_{xo}$ . A low-resistivity mudcake lowers the reading more when it is thicker and when $R_{xo}/R_{mc}$ is larger. Shallow invasion mixes in $R_t$ . The reading is too high if $R_t > R_{xo}$ (an oil-bearing bed drilled with fresh mud, which has little salt) and too low if $R_t < R_{xo}$ .

Here are the typical limits of a focused sonde of the microlaterolog type. If the invasion front is more than 8–10 cm from the wall, the uninvaded part of the formation has practically no effect on the reading. A mudcake thinner than about 10 mm hardly matters, but its effect grows quickly as it gets thicker. The wide-pad sonde tolerated mudcake up to 19 mm, but it saw deeper. Its reading equalled $R_{xo}$ only when the invasion diameter was about 1 m. At a smaller diameter it lay between $R_{xo}$ and $R_t$ , usually closer to $R_{xo}$ . Modern sondes with spherical focusing are designed to react little to mudcake while staying shallow. Synthetic microlog curves (curves that imitate the microlog) are also calculated from their measurements. Sondes with cylindrical focusing give, besides $R_{xo}$ , the resistivity and the thickness of the mudcake.

Worked exampleShallow invasion: what a shallow and a deeper microresistivity sonde will show

Given (assumed data)

Oil-bearing bed (fresh mud): $R_{xo} = 10.2$ Ω·m, $R_t = 21$ Ω·m
Water-bearing bed of the same section: $R_{xo} = 6.4$ Ω·m, $R_t = 1.0$ Ω·m
Invasion: the front is a few centimetres from the wall; the mudcake is thin, so we ignore its contribution
Share of the flushed zone in the reading (assumed): $J_{xo} = 0.95$ for a shallow focused sonde, 0.70 for a sonde with a greater depth of investigation

Solution

Oil-bearing bed, shallow sonde: $0.95\cdot 10.2 + 0.05\cdot 21 = 10.74$ Ω·m, which is 5.3% more than $R_{xo}$ .
Oil-bearing bed, deeper sonde: $0.70\cdot 10.2 + 0.30\cdot 21 = 13.44$ Ω·m, which is 32% more.
Water-bearing bed: $0.95\cdot 6.4 + 0.05\cdot 1.0 = 6.13$ and $0.70\cdot 6.4 + 0.30\cdot 1.0 = 4.78$ Ω·m, which are 4.2% and 25% below $R_{xo}$ .
The water saturation of the flushed zone is inversely proportional to the square root of $R_{xo}$ . So from the reading of the deeper sonde in the oil-bearing bed it would come out too low, by a factor of $\sqrt{13.44/10.2} = 1.15$ .

Answer: The shallow sonde is wrong by no more than 5.3%, the deeper one by 25–32%. In the oil-bearing bed a sonde reads

R_{xo}

too high, and in the water-bearing bed too low.

What it means. Errors of opposite sign shrink the contrast between oil-bearing and water-bearing beds. In the oil-bearing bed, the too-high $R_{xo}$ makes the residual oil come out larger than the true value and the moveable oil smaller. So where the invasion is shallow, we choose the shallowest focused sonde, or we find $R_{xo}$ , $R_t$ and the invasion diameter together from three curves. Invasion is shallow in high-porosity rocks, and most of all when the log is recorded soon after the bed was drilled. The values of $J$ here are assumed. Real ones depend on the tool and come from its specification. The data in this example are assumed for illustration.

How to get $R_{xo}$ : checks and corrections

From the sonde reading to

R_{xo}

1
Contact with the wall
Compare the hole diameter from the caliper of the pad tool with the bit diameter. In washouts and on a rough wall the pad lifts off. Mud gets between the pad and the rock, and the reading falls towards $R_m$ (the resistivity of the mud). We exclude such intervals.
2
Resistivity of the mud and the mudcake
We take $R_m$ , $R_{mf}$ and $R_{mc}$ from the log heading (the block of data at the top of the log). Then we bring them to formation temperature with the Arps formula (the rule for converting the resistivity of salty water from one temperature to another; Formation water and formation temperature). If $R_{mc}$ was not measured, we estimate it from $R_m$ with the charts of the tool maker.
3
Mudcake thickness
From the caliper log, $h_{mc} = (d_{bit} - d_{cal})/2$ , where $d_{bit}$ is the bit diameter and $d_{cal}$ is the caliper diameter (checking the hole). Mudcake can be anisotropic: it conducts better along the wall than across it. Then its electrical thickness is larger than the caliper shows.
4
Mudcake correction
Use the chart for the specific tool. The inputs are the ratio of the reading to $R_{mc}$ and the mudcake thickness. The output is a multiplier for the reading (Interpretation charts: how to read them).
5
Check for invasion
If the invasion may be shallow, compare the reading with the other curves. Or find $R_{xo}$ , $R_t$ and the invasion diameter together from three curves (Resistivity logs: electrical and induction tools).
6
Check against a water-bearing bed
In a clean water-bearing bed the calculated $S_{xo}$ should be close to 1. If it is not, review $R_{mf}$ , the corrections and the coefficients (advanced section).

What distorts the reading of a microresistivity sonde

Condition: Thick low-resistivity mudcake with a large $R_{xo}/R_{mc}$
Reading: too low
Calculated $S_{xo}$: too high: the residual oil is underestimated

Condition: Shallow invasion, $R_t > R_{xo}$
Reading: too high
Calculated $S_{xo}$: too low: the residual oil is overestimated

Condition: Shallow invasion, $R_t < R_{xo}$
Reading: too low
Calculated $S_{xo}$: too high; in a water-bearing bed it can come out above 1

Condition: Washout, the pad lifts off the wall
Reading: moves towards $R_m$
Calculated $S_{xo}$: far too high: the interval is excluded

Condition: Filtrate mixed with formation water
Reading: lower than expected
Calculated $S_{xo}$: too high when calculated with $R_{mf}$

Condition: Oil-based mud
Reading: no reading
Calculated $S_{xo}$: the microlog and the microlaterolog do not work

Condition	Reading	Calculated $S_{xo}$
Thick low-resistivity mudcake with a large $R_{xo}/R_{mc}$	too low	too high: the residual oil is underestimated
Shallow invasion, $R_t > R_{xo}$	too high	too low: the residual oil is overestimated
Shallow invasion, $R_t < R_{xo}$	too low	too high; in a water-bearing bed it can come out above 1
Washout, the pad lifts off the wall	moves towards $R_m$	far too high: the interval is excluded
Filtrate mixed with formation water	lower than expected	too high when calculated with $R_{mf}$
Oil-based mud	no reading	the microlog and the microlaterolog do not work

The directions are for a water-based mud and for a calculation from the reading without the proper correction.

Saturation of the flushed zone and moveable hydrocarbons

The rock in the flushed zone is the same as in the formation. It has the same porosity and the same coefficients of Archie's equation (Archie's equation and saturation of clean reservoirs). The water and the saturation are different. The pores hold filtrate with the resistivity $R_{mf}$ , and residual hydrocarbons. So we find the water saturation of the flushed zone with the same equation, putting in $R_{xo}$ and $R_{mf}$ .

Water saturation of the flushed zone

S_{xo} = \left(\frac{a\,R_{mf}}{\phi^{m}\,R_{xo}}\right)^{1/n}

where:

$S_{xo}$ water saturation of the flushed zone (the water in it is filtrate), fraction
$R_{mf}$ resistivity of the filtrate at formation temperature, Ω·m
$R_{xo}$ resistivity of the flushed zone after corrections, Ω·m
$\phi$ porosity, fraction
$a,\ m,\ n$ coefficients of Archie's equation for this rock, —

Units: dimensionless; resistivities in the same units

Holds when

A clean formation (one without shale). In a shaly one we use models of shaly reservoirs with the same substitutions (Water saturation in shaly sands).
The water in the flushed zone is practically only filtrate. If the filtrate is mixed with formation water, the resistivity of that water is lower than $R_{mf}$ , and a calculation with $R_{mf}$ gives $S_{xo}$ too high.
A water-based mud: the filtrate conducts current.
$R_{mf}$ is brought to formation temperature.

In a water-bearing bed $S_{xo} = 1$ . In a productive bed $S_{xo} < 1$ : the filtrate has not pushed out all the hydrocarbons, and the remainder $1 - S_{xo}$ is the residual hydrocarbon saturation of the flushed zone, $S_{hr}$ .

Moveable and residual hydrocarbons

V_{hm} = \phi\,(S_{xo} - S_w),\qquad V_{hr} = \phi\,(1 - S_{xo})

where:

$V_{hm}$ volume of moveable hydrocarbons in a unit volume of rock, fraction of rock volume
$V_{hr}$ volume of residual hydrocarbons in a unit volume of rock, fraction of rock volume
$S_w$ water saturation of the uninvaded formation (from $R_t$ and $R_w$ ), fraction
$S_{xo}$ water saturation of the flushed zone, fraction

Units: fraction of rock volume

Holds when

$S_w$ and $S_{xo}$ are calculated with the same porosity and the same coefficients.
The filtrate displaced the formation fluids as in waterflooding (water pushing oil out of the rock). The hydrocarbons it moved are counted as moveable.
Before the invasion, the flushed zone was the same as the uninvaded formation: the same bed and the same saturation.

The difference $S_{xo} - S_w$ is the share of the pores from which the filtrate pushed out hydrocarbons. The ratio $S_w/S_{xo}$ (in English-language practice, the moveable hydrocarbon index) near 1 means that the hydrocarbons did not move. This happens when the bed is water-bearing, or the oil is viscous (thick) and will not move, or the rock is tight. By an empirical rule, values of about 0.7 and below point to moveable hydrocarbons (more in Resistivity ratio methods and movable oil).

The flushed zone is a kind of small waterflood at the borehole wall. The share of the oil that the filtrate pushed out, $(S_{xo} - S_w)/(1 - S_w)$ , is close in meaning to the displacement efficiency (the share of the oil that water pushes out of the rock; Why water is injected: pressure maintenance and oil displacement). And $1 - S_{xo}$ is close to the residual oil saturation. But this is an estimate, and it does not replace experiments on core (rock samples taken from the well). The invasion happens under large pressure differences and over a short time. The filtrate may contain surfactants (additives that change how oil and water behave at their contact surface). And the error in $R_{xo}$ passes directly into $1 - S_{xo}$ .

CalculatedAn assumed interval calculated with Archie's equation: on the left the resistivities

R_t

and

R_{xo}

, on the right the volumes in the rock: the formation water

\phi S_w

, the water of the flushed zone

\phi S_{xo}

and the porosity

\phi

. Between

\phi S_w

and

\phi S_{xo}

lies the moveable oil, and between

\phi S_{xo}

and

\phi

the residual oil. Below the free water level (FWL, 1860 m; the level below which the rock holds only water) the three curves merge. The label “example” marks the depth of the example below.

Worked exampleMoveable oil from the microresistivity sonde and the deep sonde

Given (assumed data)

Formation: clean sandstone, $\phi = 0.24$ from the density and neutron logs; $a = 1$ , $m = n = 2$ from core
Uninvaded part of the formation: $R_t = 16$ Ω·m after corrections; $R_w = 0.070$ Ω·m at 68 °C
Filtrate: $R_{mf} = 0.62$ Ω·m at 22 °C (log heading); formation temperature 68 °C
Focused microresistivity sonde: reading 8.8 Ω·m; mudcake 8 mm from the caliper log, $R_{mc} = 0.60$ Ω·m at 68 °C; the correction multiplier from the chart of the tool (assumed) is 1.15

Solution

The filtrate at formation temperature: $R_{mf} = 0.62\cdot\dfrac{22 + 21.5}{68 + 21.5} = 0.301$ Ω·m.
The resistivity of the flushed zone: $R_{xo} = 1.15\cdot 8.8 = 10.12$ Ω·m (the inputs to the chart are the ratio $8.8/0.60 = 14.7$ and the mudcake thickness of 8 mm).
The formation resistivity factor is $F = 1/0.24^2 = 17.36$ . The water saturation of the formation is $S_w = \sqrt{17.36\cdot 0.070/16} = 0.276$ .
The water saturation of the flushed zone: $S_{xo} = \sqrt{17.36\cdot 0.301/10.12} = 0.719$ .
Moveable oil is $S_{xo} - S_w = 0.443$ of the pore volume, and residual oil is $1 - S_{xo} = 0.281$ . The ratio is $S_w/S_{xo} = 0.38$ . The filtrate pushed out $0.443/(1 - 0.276)$ , that is 61% of the initial oil.
Volumes in the rock: moveable oil $0.24\cdot 0.443 = 0.106$ , residual oil $0.24\cdot 0.281 = 0.067$ . In a bed 1 m thick over an area of 1 hectare (10,000 m²), this is about 1060 m³ of moveable oil at reservoir conditions.
Without the mudcake correction ( $R_{xo} = 8.8$ Ω·m): $S_{xo} = 0.771$ . The residual oil is 0.229, which is 18.5% less, and the moveable oil is 0.495, which is 11.7% more.

Answer: The oil of the formation is moveable. The filtrate has pushed out oil that filled 0.443 of the pore volume of the flushed zone, the residual oil saturation is about 0.281, and the ratio

S_w/S_{xo} = 0.38

is well below 0.7.

What it means. If we had skipped the mudcake correction, the residual oil would have come out almost a fifth too low. That is far more than a rounding error. The quantity $1 - S_{xo}$ is fine as a first estimate of the residual oil saturation for displacement by water. But for designing a waterflood we check it against core experiments (Relative permeability). Suppose that, with the same $R_t$ , the ratio $S_w/S_{xo}$ came out close to 1. Then we would suspect viscous oil that will not move, or shallow invasion. The data in this example are assumed for illustration.

A sign of permeability, and thin beds

Mudcake and an invaded zone form only where the filtrate can enter the rock. So microresistivity tools are among the most sensitive indicators of permeable beds. In Russian-language practice, a positive separation on the microlog and mudcake on the caliper log count as direct qualitative signs of a reservoir (a rock that can hold and give up oil or gas). Another reliable sign is filtrate invasion that shows up as a difference between sondes of different depths of investigation (Resistivity logs: electrical and induction tools), or in repeat measurements. A deflection of the spontaneous potential curve (SP, the natural voltage in the borehole; Spontaneous potential (SP) log) is an indirect sign. None of these signs measures permeability. They all say only that filtration took place. The reverse is not true: if there are no signs, the bed is not necessarily tight. The mudcake may have been washed away, and invasion can be very shallow.

The second advantage of microresistivity tools is the vertical resolution of 5–10 cm (the thinnest bed a tool can show separately). Together with the photoelectric factor curve (a log curve that responds to the minerals), this is the best resolution among the standard open-hole methods. In a thinly layered sequence, deep sondes and nuclear methods (logs that use radioactivity) average the layers over an interval of 0.3 to 2 m (the effect of enclosing rocks). Microresistivity tools show every layer that is at least a few centimetres thick.

Net thickness. From the microresistivity curves we pick the boundaries of permeable layers and add up their thicknesses.
Thinly layered reservoirs. Models of laminated shaly reservoirs need the share of sand layers from the microresistivity tools (Water saturation in shaly sands). We also need it to split the readings of the deep sondes between the layers.
Tying the core and choosing test points. A detailed curve helps to match core depths to the log. It also helps to choose points for sidewall cores (small rock samples cut from the wall of the hole) and for a formation tester (a tool that tests a bed through the wall of the hole).
Bed dip and image logs. Several microresistivity pads on different arms make a dipmeter (a tool that finds the tilt of the beds). Dozens of button electrodes make an electrical imager (borehole image log).

Terms and symbols

Microresistivity tool: An electrode sonde with electrodes a few centimetres apart, on a pad pressed against the wall.
Microlog; microgradient and micropotential sondes: A record from two unfocused microresistivity sondes: the microgradient sonde A0.025M0.025N and the micropotential sonde A0.05M.
Microlaterolog (MLL): A record from a focused microresistivity sonde. Its reading depends mainly on the resistivity of the flushed zone.
Positive separation: The reading of the micropotential sonde is higher than that of the microgradient sonde opposite a permeable bed with mudcake.
$R_{xo}$ ( $\rho_\text{pp}$ ): The resistivity of the flushed zone.
$S_{xo}$: The water saturation of the flushed zone. $1 - S_{xo}$ is the residual hydrocarbon saturation $S_{hr}$ .
Pseudo-geometric factor $J$: The share of a zone (the mudcake, the flushed zone, the formation) in the reading of the sonde.
$S_w/S_{xo}$: An index of hydrocarbon mobility. About 1 means that the hydrocarbons did not move. About 0.7 and below means, by an empirical rule, that they are moveable.

Common mistakes and misconceptions

Wrong

“A microresistivity sonde shows $R_{xo}$ .”

In fact

It shows an apparent resistivity, which includes the mudcake and, with shallow invasion, the uninvaded formation. We get $R_{xo}$ after checking the contact, correcting for the mudcake and estimating the invasion.

Wrong

“A large positive separation means high permeability.”

In fact

The separation says only that there is mudcake, that is, that filtration took place. Its size depends on the thickness and resistivity of the mudcake and on $R_{xo}$ , not on the permeability.

Wrong

“We take $R_{mf}$ straight from the log heading.”

In fact

The heading gives it at the surface measurement temperature. Before calculating $S_{xo}$ we bring it to formation temperature. At 22 and 68 °C the filtrate resistivities differ by about a factor of two.

Wrong

“Since $S_{xo}$ is much larger than $S_w$ , the bed will give oil with no water.”

In fact

Moveable hydrocarbons do not exclude moveable water. We judge the water inflow by comparing $S_w$ with the irreducible water saturation (the water that stays in the pores and does not flow), from core or from the NMR log (nuclear magnetic resonance; Nuclear magnetic resonance (NMR) logging), and by well tests.

Wrong

“The residual oil saturation of the bed equals $1 - S_{xo}$ .”

In fact

It is an estimate from a small waterflood at the wall in special conditions: large pressure differences, a short time, additives in the filtrate. It also carries the error of $R_{xo}$ . For design we check it against core experiments.

In depth: advanced detail

Formation resistivity factor from

R_{xo}

in a water-bearing bed

In a clean water-bearing bed $S_{xo} = 1$ , and Archie's equation for the flushed zone gives the formation resistivity factor without any porosity method. We use it to check $R_{mf}$ , the corrections to the microresistivity sonde and the cementation exponent. If these are reliable, we can also check the porosity itself.

Formation resistivity factor from the resistivity of the flushed zone

F = \frac{R_{xo}}{R_{mf}} = \frac{a}{\phi^{m}}

where:

$F$ formation resistivity factor, —
$R_{xo}$ resistivity of the flushed zone of a water-bearing bed after corrections, Ω·m
$R_{mf}$ resistivity of the filtrate at formation temperature, Ω·m
$a,\ m$ coefficients of Archie's equation, —

Units: dimensionless

Holds when

A clean water-bearing bed completely flushed by the filtrate: $S_{xo} = 1$ , and the water in the flushed zone is filtrate.
The invasion front is deeper than the depth of investigation of the microresistivity sonde, and the mudcake correction has been applied.
In shaly beds, the conductivity of the shales makes $F$ too low.

If the filtrate is mixed with formation water, the resistivity of the water in the flushed zone is lower than $R_{mf}$ . Then the formula gives $F$ too low, and the porosity comes out too high.

Worked exampleCheck against a water-bearing bed

Given (assumed data)

Water-bearing sandstone of the same section: $R_{xo} = 7.4$ Ω·m after the mudcake correction; density porosity 0.205
Filtrate: $R_{mf} = 0.301$ Ω·m at formation temperature (from the example on moveable oil)

Solution

The formation resistivity factor: $F = 7.4/0.301 = 24.58$ .
The porosity with $a = 1$ , $m = 2$ : $\phi = 1/\sqrt{24.58} = 0.202$ , which is 0.3 porosity units (percentage points of porosity) below the density porosity.
The cementation exponent at which the two estimates agree: $m = \ln 24.58/\ln(1/0.205) = 2.02$ .
If by mistake we take $R_{mf} = 0.62$ Ω·m at 22 °C, then $F = 11.94$ and $\phi = 0.289$ , which is 8.4 porosity units above the density porosity.

Answer: The estimates agree. The porosity from

R_{xo}

is 0.202, against the density porosity of 0.205, and the cementation exponent is about 2.02. So

R_{mf}

, the mudcake correction and

m = 2

are confirmed.

What it means. A difference of a fraction of a porosity unit is within the errors. A difference of several units points to an error in $R_{mf}$ (the temperature, or mixing with formation water), to an error in the mudcake correction, or to a different cementation exponent. The last step shows what a temperature error costs. A filtrate resistivity taken at the surface temperature halves $F$ , and the porosity comes out absurdly high. The data in this example are assumed for illustration.

Connections to other topics

Borehole environment: invaded zone and mudcake — the zones around the hole, the mudcake and the resistivity profile.
Resistivity logs: electrical and induction tools — deep sondes and finding $R_t$ from three curves, which needs $R_{xo}$ .
Archie's equation and saturation of clean reservoirs — Archie's equation and its coefficients.
Resistivity ratio methods and movable oil — saturation and moveable oil from the resistivity ratio, without porosity.
Formation water and formation temperature — bringing $R_{mf}$ and $R_{mc}$ to formation temperature.
Gas, shale, carbonate and pressure effects on porosity logs — how residual hydrocarbons in the flushed zone shift the density and neutron logs.
Interpretation charts: how to read them — charts for the mudcake and invasion corrections.
Nuclear magnetic resonance (NMR) logging — bound and free water in the same flushed zone from the NMR log.

Check yourself

1
Opposite a sandstone the micropotential sonde reads 6 Ω·m and the microgradient sonde reads 3.5 Ω·m. The caliper shows a diameter 10 mm smaller than the bit. What follows from this?
1. A.The sandstone is permeable: there is mudcake and a positive separation
2. B.The sandstone is tight, with no pores
3. C.The permeability of the sandstone is about 6 mD
4. D.The bed is water-bearing
Show the answer
Correct answer: A. The sandstone is permeable: there is mudcake and a positive separation
A positive separation and a diameter 10 mm smaller than the bit (mudcake about 5 mm thick) show that filtrate went into the formation, so the bed is permeable. These data give neither the size of the permeability nor the saturation.
2
Given $\phi = 0.25$ , $R_{xo} = 4.0$ Ω·m, $R_{mf} = 0.10$ Ω·m, $R_t = 30$ Ω·m, $R_w = 0.05$ Ω·m, $a = 1$ and $m = n = 2$ . Find $S_{xo}$ , $S_w$ and the volume of moveable hydrocarbons.
Show the answer
$F = 16$ ; $S_{xo} = \sqrt{16\cdot 0.10/4.0} \approx 0.63$ ; $S_w = \sqrt{16\cdot 0.05/30} \approx 0.16$ ; $V_{hm} = 0.25\cdot(0.63 - 0.16) \approx 0.12$ . The residual hydrocarbons are $0.25\cdot 0.37 \approx 0.09$ . The ratio $S_w/S_{xo} \approx 0.25$ , so the hydrocarbons are moveable.
3
The invasion is shallow, the bed is oil-bearing, and $R_t$ is three times $R_{xo}$ . How will a microresistivity sonde with a relatively large depth of investigation go wrong?
1. A.It will not go wrong: microresistivity sondes do not see $R_t$
2. B.It will show the resistivity of the mudcake
3. C.It will read $R_{xo}$ too low
4. D.It will read $R_{xo}$ too high, and $S_{xo}$ will come out too low
Show the answer
Correct answer: D. It will read $R_{xo}$ too high, and $S_{xo}$ will come out too low
The reading of a focused sonde includes a share $1 - J_{xo}$ of the uninvaded formation, which has a higher resistivity. So the reading is too high. And $S_{xo}$ , which is inversely proportional to the square root of $R_{xo}$ , is too low: the residual oil is overestimated.
4
Why does an unfocused microresistivity sonde measure $R_{xo}$ badly in low-porosity rocks?
Show the answer
In a low-porosity rock $R_{xo}$ is high, so the ratio $R_{xo}/R_{mc}$ is large. The current of an unfocused sonde takes the path along the low-resistivity mudcake and through it into the mud, and the reading almost stops depending on $R_{xo}$ . So we find $R_{xo}$ from the microlog only when $R_{xo}/R_{mc}$ is less than about 15. In tight rocks we use focused sondes.
5
In a water-bearing sandstone the calculated $S_{xo}$ came out as 1.3. What should we check first?
1. A.The saturation exponent $n$
2. B.Whether $R_{mf}$ was brought to formation temperature and the mudcake correction was applied
3. C.Nothing: this happens in water-bearing beds
4. D.The formation water resistivity $R_w$
Show the answer
Correct answer: B. Whether $R_{mf}$ was brought to formation temperature and the mudcake correction was applied
Water saturation cannot be above 1. A value of $S_{xo}$ that is too large means that $R_{mf}$ is too high (for example, taken at the surface temperature) or that $R_{xo}$ is too low (mudcake, washout, or shallow invasion with $R_t < R_{xo}$ ). $R_w$ is not part of the $S_{xo}$ calculation, and $n$ has almost no effect when the saturation is near 1.
6
A sequence is made of sand layers 15–20 cm thick that alternate with shales. Which method will show each layer, and why will the deep induction sonde not resolve them?
Show the answer
Microresistivity sondes with a resolution of 5–10 cm will show the boundaries of every layer. The deep induction sonde averages 1–2 m of the section. It gives the resistivity of the mixture, which is close to that of the shales. We need the share of sand layers from the microresistivity sondes for the net thickness and for models of layered beds.

Updated: 19 October 2026

Quick refresher

After this lesson you can

Why we measure the resistivity of the flushed zone

How microresistivity tools are built

The microlog: two curves and positive separation

Focused microresistivity sondes

How to get RxoR_{xo}Rxo​: checks and corrections

Saturation of the flushed zone and moveable hydrocarbons

A sign of permeability, and thin beds

Terms and symbols

Common mistakes and misconceptions

In depth: advanced detail

Connections to other topics

Check yourself

How to get $R_{xo}$ : checks and corrections