Comparing formation-tester, core and well-test data

Quick refresher

Three measurements are three different questions. The first is the volume: cubic centimetres, centimetres at the probe, or tens and hundreds of metres around the well. The second is the fluid: gas or water in the laboratory; filtrate with residual oil (the oil that the filtrate cannot push out); reservoir oil with irreducible water (the water that cannot be pushed out of the pores). The third is the direction: along the axis of the sample; a mix of $k_h$ and $k_v$ at the probe; horizontal in a well test.
The common basis of comparison is the absolute horizontal permeability to liquid at reservoir conditions, averaged over the volume that the larger measurement sees. For a well test it is the flow capacity $kh$ of the tested interval.
Drawdown mobility is turned into permeability with the viscosity of the filtrate. We divide it by the end-point relative permeability of the filtrate in the flushed zone (the rock near the borehole that the filtrate has washed) and by the anisotropy factor of the probe. In the oil-bearing part, with a water-based mud, these corrections alone change the number by several times: in the lesson's example by a factor of 4.7–6.6.
We compare ratios in logarithms. The mean of the logarithm is the systematic bias (the average shift of one method against the other). Its standard error shows whether the bias is significant, and the scatter is the random part. The share of points “within a factor of two” is easy to see, but it mixes the bias with the scatter.
We compare with the well test through $kh$ , not permeability. The thickness that actually flows is known only roughly, and the $kh$ of continuous layers (layers that run unbroken over the whole area) adds up. A thin high-permeability layer may give half of $kh$ and fall between the pretest points.
The disagreement that remains is information about the formation. If the well test gives more than core, think of fractures or thin layers that were missed. If it gives less, think of discontinuous layers, boundaries or a second phase around the well.

Key relationships

k_h = \dfrac{(k/\mu)_d\,\mu_f}{k_{rf}'\,f_A},\qquad f_A = \dfrac{k_d}{k_h} = \dfrac{\pi}{2K(1 - 1/A)}

drawdown mobility to horizontal absolute permeability

k_b = c\,k_a^{\,n}

from the gas permeability of core to the permeability to formation water under reservoir stress (a relation found from pairs of samples)

(kh)_{\text{abs}} = \dfrac{2.149\,qB\mu}{m\,k_{ro}'}

flow capacity from the slope of the Horner line (the straight-line part of a build-up plot), converted to absolute permeability; metric units

\bar d = \dfrac1n\sum_i \lg\dfrac{k_{\text{pt},i}}{k_{\text{core},i}},\qquad s_{\bar d} = \dfrac{s}{\sqrt n}

systematic bias of pretests relative to core, and its standard error

Common pitfalls

Comparing the gas permeability of a dry sample with a well test without bringing it to reservoir conditions.
Turning pretest mobility into permeability with the viscosity of reservoir oil, and forgetting the end-point relative permeability of the filtrate.
Dividing the $kh$ of a well test by an arbitrarily chosen thickness and comparing permeabilities instead of flow capacities.
Averaging core samples in one way for every comparison, whatever the geometry of the flow.
Judging the agreement from one or two points, or from the share of points within a factor of two, without estimating the bias and its error.

After this lesson you can

explain which quantity, in what volume of rock, for which fluid and in which direction is measured by a core sample, a pretest (drawdown and build-up), a dual-packer module and a well test
name the causes of disagreement and say in which direction each of them shifts each measurement. The causes are the conditions of the test, phase permeability (permeability to one fluid when others are present), anisotropy (permeability that differs with direction), heterogeneity and scale, damage, fractures, and a second phase at the well
bring core and drawdown mobility to a common basis: reservoir conditions, the averaging window (the stretch of core that is averaged), the viscosity of the filtrate, the end-point relative permeabilities (the relative permeability to a fluid at the end of a displacement) and a correction for anisotropy
compare sets of measurements in logarithms: separate the systematic bias from the scatter, and judge whether there are enough points for a conclusion
bring together the flow capacity of an interval from core and from pretests with $kh$ from the pressure build-up curve, and interpret the disagreement that remains

What each method measures

The bare number “permeability” is incomplete. To compare numbers, we need to know four things. First, what volume of rock is averaged. Second, what fluid flowed, and at what saturation (how much of the pore space each fluid filled). Third, in which direction the flow went. Fourth, under what conditions: at what stress and temperature, and in what state the clays were. On each of these points, core, formation tester and well test differ.

What core, the formation tester and a well test measure

Method: Core sample, standard analysis
Volume: 10–100 cm³
Fluid: nitrogen, helium or air; more rarely formation water
Direction: along the axis of the sample
Result: absolute permeability under the conditions of the test

Method: Pretest: drawdown
Volume: centimetres at the probe
Fluid: filtrate with residual oil
Direction: a mix of $k_h$ and $k_v$
Result: mobility $(k/\mu)_d$

Method: Pretest: build-up
Volume: tens of centimetres to metres
Fluid: filtrate, farther out a mixture
Direction: spherical
Result: $k_s = (k_h^2k_v)^{1/3}$

Method: Observation probe, dual-packer module
Volume: metres
Fluid: filtrate and formation fluid
Direction: $k_h$ and $k_v$ ; radial flow to the interval
Result: $k_h$ , $k_v$ , $kh$ of the interval

Method: Well test (DST, build-up test)
Volume: tens to hundreds of metres in radius
Fluid: reservoir oil with irreducible water
Direction: horizontal when the well penetrates the whole interval
Result: $kh$ , skin, $p^*$

Method	Volume	Fluid	Direction	Result
Core sample, standard analysis	10–100 cm³	nitrogen, helium or air; more rarely formation water	along the axis of the sample	absolute permeability under the conditions of the test
Pretest: drawdown	centimetres at the probe	filtrate with residual oil	a mix of $k_h$ and $k_v$	mobility $(k/\mu)_d$
Pretest: build-up	tens of centimetres to metres	filtrate, farther out a mixture	spherical	$k_s = (k_h^2k_v)^{1/3}$
Observation probe, dual-packer module	metres	filtrate and formation fluid	$k_h$ and $k_v$ ; radial flow to the interval	$k_h$ , $k_v$ , $kh$ of the interval
Well test (DST, build-up test)	tens to hundreds of metres in radius	reservoir oil with irreducible water	horizontal when the well penetrates the whole interval	$kh$ , skin, $p^*$

Orders of magnitude. The volume that the record covers grows with time (radius of investigation). The scale of volumes, from a thin section to an accumulation, is in the topic on core. DST is the drill stem test. The build-up test records the pressure build-up curve, the rise of pressure after the well is shut in (Pressure build-up analysis). Estimates from pretests are covered in the topic on supercharging and permeability. Skin is a number that shows how much the zone around the well helps or hinders flow.

Each of these numbers is a correct answer to its own question. The sample gives the absolute permeability (the permeability of the rock itself) of a small piece of rock under the conditions of the test. The pretest drawdown gives the mobility of the filtrate in the flushed zone at the probe: an effective permeability (the permeability to one fluid when another is present) divided by the viscosity. The well test gives the effective flow capacity of the formation for oil, averaged over the whole volume that the pressure disturbance has reached. So the question “which number is correct” makes no sense. The right question is what is left of the disagreement when we bring all three to the same quantity, and what that remainder tells us.

Scale effect

The larger the volume of rock that a measurement covers, the more often the permeability comes out higher. This is the scale effect. A large volume includes fractures, large cavities and continuous high-permeability thin layers. These do not get into a small sample, or they are lost when the core is taken. The reverse also happens: samples see discontinuous permeable lenses (small lens-shaped bodies of rock), while a well test over hundreds of metres does not.
Above a certain size, the averaged property stops changing with the volume. This size is the representative elementary volume. For a uniform sandstone, a single sample can be that volume. For a layered or fractured rock it is larger than the volume that the probe sees, and sometimes larger than a full-size core. The larger this volume is compared with the volume of the measurement, the greater the scatter of the estimates at single points.

Why the numbers differ

There are not many causes of disagreement, and each has a sign: some make one measurement too low, others make another too high. If we know the signs, the direction of the disagreement lets us guess the cause.

Causes of disagreement and their direction

Cause: Conditions of the test: gas, low stress, dry clays
Core: too high, more so in tight and shaly rocks
Pretest: drawdown: no effect in the formation
Well test: no effect in the formation
What to do: bring the core to reservoir conditions

Cause: Phase and saturation
Core: one phase: absolute permeability
Pretest: drawdown: filtrate with residual oil: lower by several times
Well test: oil with irreducible water: a little below the absolute
What to do: convert with the end-point relative permeabilities

Cause: Anisotropy
Core: a horizontal sample gives $k_h$
Pretest: drawdown: between $k_s$ and $k_h$
Well test: $k_h$ when the whole interval is open
What to do: a correction for $A = k_h/k_v$

Cause: Heterogeneity and scale
Core: points; friable thin layers are lost
Pretest: drawdown: points; the probe is set in the better places
Well test: an average over the volume that the well drains
What to do: an averaging window; compare $kh$

Cause: Damage at the wall
Core: none
Pretest: drawdown: makes it lower; cannot be separated
Well test: goes into the skin, does not change $kh$
What to do: a repeat pretest, neighbouring points

Cause: Fractures and cavities
Core: hardly get into the samples
Pretest: drawdown: occasionally too high: the probe is on a fracture
Well test: raise $kh$
What to do: borehole images, core recovery

Cause: A second phase at the well
Core: none
Pretest: drawdown: none
Well test: gas or water in the formation around the well lowers $kh$ for oil
What to do: bubble point pressure, water cut

Cause	Core	Pretest: drawdown	Well test	What to do
Conditions of the test: gas, low stress, dry clays	too high, more so in tight and shaly rocks	no effect in the formation	no effect in the formation	bring the core to reservoir conditions
Phase and saturation	one phase: absolute permeability	filtrate with residual oil: lower by several times	oil with irreducible water: a little below the absolute	convert with the end-point relative permeabilities
Anisotropy	a horizontal sample gives $k_h$	between $k_s$ and $k_h$	$k_h$ when the whole interval is open	a correction for $A = k_h/k_v$
Heterogeneity and scale	points; friable thin layers are lost	points; the probe is set in the better places	an average over the volume that the well drains	an averaging window; compare $kh$
Damage at the wall	none	makes it lower; cannot be separated	goes into the skin, does not change $kh$	a repeat pretest, neighbouring points
Fractures and cavities	hardly get into the samples	occasionally too high: the probe is on a fracture	raise $kh$	borehole images, core recovery
A second phase at the well	none	none	gas or water in the formation around the well lowers $kh$ for oil	bubble point pressure, water cut

The directions are for the usual cases: a well drilled with water-based mud, a water-wet rock (a rock that water clings to more than oil), pretests in the oil-bearing part.

Three corrections to the pretest (phase, anisotropy and damage) all act in one direction. So the raw mobility multiplied by the viscosity of the filtrate is often 5–7 times lower than the core permeability of the same rock. Core that has not been brought to reservoir conditions, on the other hand, is too high (laboratory and reservoir conditions). A comparison of two uncorrected numbers can be off by an order of magnitude (a factor of ten) without any cause in the formation itself.

Why the phase matters for a pretest. In the oil-bearing part of a well drilled with water-based mud, the filtrate has pushed the moveable oil away from the wall, down to the residual saturation. What flows to the probe is the filtrate, with residual oil still in the rock. Its permeability is the absolute permeability multiplied by the end-point relative permeability to water $k_{rw}'$ (end-point relative permeability: the relative permeability at the end of the displacement). In a water-wet rock $k_{rw}'$ is usually 0.1–0.3, so this correction alone is a factor of 3–10. In the water-bearing part there is no oil, and the factor is one. In a well drilled with oil-based mud, a hydrocarbon filtrate flows to the probe, and irreducible water stays in the rock. Then we use the end-point relative permeability to oil $k_{ro}'$ and the viscosity of that filtrate (relative permeability curves).

Bringing the measurements to a common basis

The common basis is the absolute horizontal permeability to liquid at reservoir conditions, averaged over the volume of the larger of the compared measurements. For a comparison with a well test, the basis is the flow capacity $kh$ of the tested interval. The way to it is different for each kind of data.

Order of the conversion

1
Match the depths
The core is shifted to the log depths using the gamma ray log (the log of the natural radioactivity of the rock, core depth matching). The depths of the pretests are already log depths. The remaining error of the match is 0.1–0.3 m, so the averaging window is not made narrower than a few tenths of a metre.
2
Bring the core to reservoir conditions
The gas permeability of dry samples is converted into permeability to formation water under reservoir stress. We use a relation found on pairs of samples that were measured both ways (the equation below). Each rock type has its own relation.
3
Average the core over the volume of the measurement
For a pretest, we take the samples within a few tenths of a metre of the point, with the geometric mean (the n-th root of the product of n values). The flow to the probe is three-dimensional, and the laminae (very thin layers) in it work both in parallel and in series (advanced section). For a well test, we take the whole flowing interval: the sum of $k_ih_i$ over the continuous layers.
4
Convert mobility into permeability
Multiply by the viscosity of the filtrate at reservoir temperature. Divide by its end-point relative permeability in the flushed zone and by the anisotropy factor of the probe $f_A$ .
5
Convert the well test to absolute permeability
Divide the $kh$ for oil by $k_{ro}'$ . If gas has come out of solution around the well or water is flowing, such a conversion is wrong: the permeability to oil is lowered by the second phase.

Relation between the permeability of core to gas and to formation water at reservoir conditions

k_b = c\,k_a^{\,n}\qquad\Longleftrightarrow\qquad \lg k_b = \lg c + n\lg k_a

where:

$k_a$ permeability of the sample to gas in a standard analysis (low confining stress, dry sample), mD
$k_b$ permeability of the same sample to formation water under the reservoir effective stress, mD
$c,\ n$ constants of the relation, found by least squares from pairs of samples in logarithms, —

Units: mD; the constant $c$ refers specifically to millidarcies

Holds when

The relation is found on samples of the same rock type, and it is applied within the permeabilities of these pairs.
It already includes gas slippage (gas slipping along the pore walls, which makes gas permeability read too high; Klinkenberg correction), the stress and the swelling of clays. There is no need to separate these corrections if the pairs were measured in this way.

The ratio is $k_b/k_a = c\,k_a^{\,n - 1}$ . At $n > 1$ it is smaller for tight rocks: gas slippage, the opening of microfractures without confining stress and dry clays overstate small permeabilities more. The scatter of samples around such a relation is often 15–25%. It is part of the error of any comparison with core.

Horizontal absolute permeability from the drawdown mobility

k_h = \frac{(k/\mu)_d\,\mu_f}{k_{rf}'\,f_A},\qquad f_A = \frac{k_d}{k_h} = \frac{\pi}{2K(1 - 1/A)}

where:

$(k/\mu)_d$ drawdown mobility of the pretest (how we find it), mD/(mPa·s)
$\mu_f$ viscosity of the filtrate at reservoir temperature, mPa·s
$k_{rf}'$ end-point relative permeability of the filtrate in the flushed zone: $k_{rw}'$ for a water-based filtrate in the oil-bearing part, 1 in the water-bearing part, $k_{ro}'$ for a hydrocarbon filtrate, —
$f_A$ anisotropy factor: the drawdown permeability divided by the horizontal one, —
$A$ anisotropy $k_h/k_v$ of the rock at the probe, —
$K(m)$ complete elliptic integral of the first kind with the parameter $m = 1 - 1/A$ , —

Units: $k_h$ in mD, for $(k/\mu)_d$ in mD/(mPa·s) and $\mu_f$ in mPa·s

Holds when

The flushed zone is at the residual saturation of the displaced phase, and there is no extra resistance at the opening of the probe (cake, fine particles). Otherwise $k_h$ comes out too low.
$f_A$ comes from the model of a round probe on a flat impermeable wall (derivation). Solutions with an ellipsoidal sink give values about 10% smaller.
$A$ is taken from pairs of horizontal and vertical samples of the same layer. At the scale of the probe, lamination (thin layering) can give a larger anisotropy than that of the samples.

We multiply the mobility by the viscosity of the filtrate, not of the oil, and divide by two factors, each less than one: the end-point relative permeability and the anisotropy factor. At $A$ = 2, 3, 5 and 10, the factor $f_A$ is 0.847, 0.774, 0.696 and 0.609. The probe loses the horizontal permeability slowly, much more slowly than the spherical permeability does: $k_s/k_h = A^{-1/3}$ .

Flow capacity from the Horner line, converted to absolute permeability

(kh)_{\text{abs}} = \frac{(kh)_o}{k_{ro}'} = \frac{2.149\,qB\mu}{m\,k_{ro}'}

where:

$(kh)_o$ flow capacity to oil from the slope of the Horner line (formula), mD·m
$q,\ B,\ \mu$ oil rate before shut-in at standard conditions, formation volume factor, and oil viscosity at reservoir conditions, m³/d, m³/m³, mPa·s
$m$ slope of the line per decimal logarithmic cycle (per tenfold change on the logarithmic axis), MPa
$k_{ro}'$ end-point relative permeability to oil at irreducible water, —

Units: metric oilfield units; $2.149 = \ln 10/(4\pi\cdot 0.0853)$

Holds when

Only oil flows in the formation around the well, with irreducible water (the pressure is above the bubble point pressure, and the water is immobile).
The slope is taken from the period of radial flow in an infinite reservoir (Diagnosis with the pressure derivative). The whole interval is open, and the layers are continuous within the radius of investigation.

The test gives the sum of the flow capacities of the flowing layers for oil. Division by $k_{ro}'$ brings it to the same basis as the core. An error in $k_{ro}'$ (±0.05–0.1 at a value of about 0.8, which is 6–12%) passes in full into $kh$ .

Worked exampleOne point: the pretest at 2845.5 m and the core next to it

Given (assumed data)

Pretest: drawdown mobility 65.8 mD/(mPa·s); the oil-bearing part, water-based mud; filtrate 0.40 mPa·s
Rock: $k_{rw}' = 0.25$ at residual oil (Relative permeability); from the pairs of samples of the layer, $A \approx 2$
Core within ±0.3 m: four horizontal samples, measured with gas: 230; 310; 265 and 205 mD
Relation “gas to formation water” for this rock: $k_b = 0.42\,k_a^{1.06}$ (from 16 pairs of samples)

Solution

Core at reservoir conditions: $0.42\cdot 230^{1.06} = 133.9$ mD; in the same way 183.7, 155.6 and 118.5 mD.
The average over the window is the geometric mean: $k_G = (133.9\cdot183.7\cdot155.6\cdot118.5)^{1/4} = 145.9$ mD. The mean of the gas values would be 249 mD, which is 1.71 times more.
The mobility multiplied by the viscosity of the filtrate: $65.8\cdot 0.40 = 26.32$ mD. This is 0.18 of the core.
End-point relative permeability: $26.32/0.25 = 105.3$ mD (0.72 of the core).
Anisotropy: $f_A = 0.847$ at $A = 2$ ; $k_h = 105.3/0.847 = 124.3$ mD. The ratio to the core is $124.3/145.9 = 0.85$ .

Answer: The pretest gives a horizontal permeability of about 124.3 mD against 145.9 mD from core: a ratio of 0.85 instead of 0.18 before the corrections.

What it means. Almost nothing is left of the fivefold disagreement. It was caused by the end-point relative permeability (a factor of 4) and the anisotropy (18%). The remainder of 15% is within the errors of the “gas to water” relation and of the anisotropy itself. If the mobility had been multiplied by the oil viscosity of 1.4 mPa·s, the result would have been 92.1 mD, which is 0.63 of the core. That is much closer than the raw 0.18, but for the wrong reason (the previous section). The data in this example are assumed for illustration.

How to compare a set of points

One point proves nothing: both the core and the pretest at it can be wrong by tens of per cent for random reasons. We compare a set of points, and we do it in logarithms. The errors of permeability are multiplicative (they are a share of the value: not “±10 mD”, but “±30%”). The permeabilities of different layers differ by orders of magnitude. And the mean of ratios on an ordinary scale is biased: the ratios 2 and 0.5 are disagreements of the same size in opposite directions, but their mean is 1.25, not 1.

Systematic bias and scatter of the ratios

d_i = \lg\frac{k_{\text{pt},i}}{k_{\text{core},i}},\qquad \bar d = \frac{1}{n}\sum_i d_i,\qquad s = \sqrt{\frac{\sum_i (d_i - \bar d)^2}{n - 1}},\qquad \bar d \pm t\,\frac{s}{\sqrt n}

where:

$k_{\text{pt},i},\ k_{\text{core},i}$ permeabilities of point $i$ from the pretest (after the corrections) and from the core in the window, mD
$d_i$ decimal logarithm of their ratio: 0 means agreement, 0.301 means twice as large, −0.301 means half as large, —
$\bar d,\ s$ mean and standard deviation of the logarithms, —
$t$ Student's coefficient for $n - 1$ degrees of freedom (about 2 for large $n$ ), —
$n$ number of points, —

Units: dimensionless (decimal logarithms)

Holds when

The deviations of the points are independent and close to lognormal (their logarithms follow the usual bell-shaped normal distribution). Points with a known defect of the measurement are excluded beforehand, not because of the size of their deviation.
$10^{\bar d}$ is the geometric mean of the ratios, a systematic multiplier. $10^{s}$ is the typical scatter of one point, as “by how many times”.

The bias is significant if the interval $\bar d \pm t\,s/\sqrt n$ does not contain zero. The share of points within a factor of two depends on both the bias and the scatter. So it is good as an illustration, but it does not answer the question whether a correction is needed.

Twelve pretests of an appraisal well and core in windows of ±0.3 m

Point, m: 2826.5
$A$: 3
Core $k_G$ , mD: 37.5
Mobility, mD/(mPa·s): 23.9
Ratio after corrections: 1.32

Point, m: 2830.0
$A$: 5
Core $k_G$ , mD: 9.0
Mobility, mD/(mPa·s): 2.46
Ratio after corrections: 0.63

Point, m: 2833.5
$A$: 2
Core $k_G$ , mD: 86.6
Mobility, mD/(mPa·s): 51.5
Ratio after corrections: 1.12

Point, m: 2841.5
$A$: 3
Core $k_G$ , mD: 62.8
Mobility, mD/(mPa·s): 25.3
Ratio after corrections: 0.83

Point, m: 2843.0
$A$: 3
Core $k_G$ , mD: 52.3
Mobility, mD/(mPa·s): 37.4
Ratio after corrections: 1.48

Point, m: 2845.5
$A$: 2
Core $k_G$ , mD: 146
Mobility, mD/(mPa·s): 65.8
Ratio after corrections: 0.85

Point, m: 2848.0
$A$: 2
Core $k_G$ , mD: 165
Mobility, mD/(mPa·s): 111
Ratio after corrections: 1.27

Point, m: 2852.0
$A$: 10
Core $k_G$ , mD: 22.4
Mobility, mD/(mPa·s): 4.59
Ratio after corrections: 0.54

Point, m: 2854.5
$A$: 10
Core $k_G$ , mD: 18.7
Mobility, mD/(mPa·s): 8.20
Ratio after corrections: 1.15

Point, m: 2858.0
$A$: 5
Core $k_G$ , mD: 4.2
Mobility, mD/(mPa·s): 1.29
Ratio after corrections: 0.71

Point, m: 2864.0
$A$: 3
Core $k_G$ , mD: 40.1
Mobility, mD/(mPa·s): 32.2
Ratio after corrections: 1.66

Point, m: 2869.5
$A$: 5
Core $k_G$ , mD: 13.2
Mobility, mD/(mPa·s): 4.55
Ratio after corrections: 0.79

Point, m	$A$	Core $k_G$ , mD	Mobility, mD/(mPa·s)	Ratio after corrections
2826.5	3	37.5	23.9	1.32
2830.0	5	9.0	2.46	0.63
2833.5	2	86.6	51.5	1.12
2841.5	3	62.8	25.3	0.83
2843.0	3	52.3	37.4	1.48
2845.5	2	146	65.8	0.85
2848.0	2	165	111	1.27
2852.0	10	22.4	4.59	0.54
2854.5	10	18.7	8.20	1.15
2858.0	5	4.2	1.29	0.71
2864.0	3	40.1	32.2	1.66
2869.5	5	13.2	4.55	0.79

Core is the geometric mean of the permeability to formation water under reservoir stress in the window. $A$ is the median (middle value) of the pairs of samples of the layer. The ratio is $k_h$ from the pretest (filtrate viscosity 0.40 mPa·s, $k_{rw}' = 0.25$ , $f_A$ ) divided by the core. All the pretests are valid, in the oil-bearing part, with water-based mud. The data are assumed for illustration.

Worked exampleBias and scatter: twelve points before and after the corrections

Given (assumed data)

Points: the table above
Corrections: filtrate viscosity 0.40 mPa·s; $k_{rw}' = 0.25$ ; $f_A$ = 0.847, 0.774, 0.696 and 0.609 at $A$ = 2, 3, 5 and 10
Student's coefficient: $t$ = 2.201 for 11 degrees of freedom (95%)

Solution

Without corrections, with the mobility multiplied only by the viscosity of the filtrate: the ratios to the core are from 0.082 to 0.321, and the mean of the logarithm is −0.745. So on average the pretests are 5.6 times lower than the core.
After the corrections, the logarithms of the ratios in the table are: 0.121; −0.201; 0.049; −0.081; 0.170; −0.071; 0.104; −0.268; 0.061; −0.149; 0.220; −0.102. The sum is −0.147, the mean is $\bar d = -0.012$ , and the standard deviation is $s = 0.155$ .
The standard error is $s/\sqrt{12} = 0.045$ . The 95% interval $-0.012 \pm 2.201\cdot0.045$ runs from −0.111 to 0.087, which is a multiplier from 0.77 to 1.22. The bias is not significant.
The scatter of one point: $10^{0.155} = 1.43$ . All 12 points lie within a factor of two ( $|d| \le 0.301$ ). The largest deviations are in the finely layered rock with $A = 10$ (2852.0 m) and at the point 2864.0 m.
If the anisotropy had not been allowed for (only the viscosity and the relative permeability), the mean would have been −0.143, a multiplier of 0.72. This is a systematic shortfall of 28%.

Answer: After the corrections, the pretests and the core agree: the systematic multiplier is 0.97 (95%: 0.77–1.22), and the scatter of one point is a factor of 1.43 either way.

What it means. The raw numbers differed by a factor of 5.6, but two known corrections, the end-point relative permeability and the anisotropy of the probe, explained the whole disagreement. The random part ( $s$ = 0.155) is about what the depth matching, the choice of the window and the variation of the rock at the scale of the probe would give. In real sets the scatter is usually larger, 0.25–0.4 in the logarithm. Points that fall outside the band are first checked against the pretest record (a drawdown that did not steady, a blocked probe, a repeat test), and only then are they excluded. If a significant bias downwards remained after the corrections, the first suspects would be damage at the probe, an underestimated anisotropy or a wrong $k_{rw}'$ . For a bias upwards, the suspect would be core that does not represent the thin permeable layers. The multiplier and the scatter that we found are a calibration of the pretests for this rock type. In wells without core, $k_h$ from a pretest carries an error of about a factor of 1.43 per point. The data in this example are assumed for illustration.

CalculatedRatio of the permeability from the pretest to that from core, for the twelve points of the table. Empty circles: the mobility multiplied only by the viscosity of the filtrate. Filled circles: after the corrections for end-point relative permeability and anisotropy. The solid line marks agreement, and the dashed lines show a factor of two.

Flow capacity of an interval: core, pretests and the build-up test

We compare a well test not with permeability but with flow capacity $kh$ . There are two reasons. The test gives the $kh$ of the flowing thickness, and the thickness itself is known only roughly. Three thicknesses are in use, and they all differ. The net thickness from logs is the thickness of the rock that can flow. The perforated thickness is the part opened by holes through the casing. The flowing thickness comes from production logging (PLT), a survey of where the fluid enters the well. And in radial flow to a vertical well, continuous layers work in parallel, so their flow capacities add up, as in the arithmetic mean weighted by thickness. So from core and pretests we build the sum $\sum k_ih_i$ over the layers of the interval, and we compare it with $kh$ from the pressure build-up curve, converted to absolute permeability.

Core gives a value for each layer where there are samples. Pretests are points, and each is assigned the part of the layer that it represents on the logs. A layer with no pretest drops out of the sum from pretests. If it is a thin high-permeability layer, a noticeable part of the flow capacity drops out.

Tested interval 2840–2860 m: layers, core and pretests

Layer: 1
Depths, m: 2840.0–2844.0
$h$ , m: 4.0
Core, mD: 58.4
Pretests $k_h$ , mD: 64.8

Layer: 2
Depths, m: 2844.0–2849.5
$h$ , m: 5.5
Core, mD: 152
Pretests $k_h$ , mD: 167.0

Layer: 3
Depths, m: 2849.5–2850.3
$h$ , m: 0.8
Core, mD: 1180
Pretests $k_h$ , mD: no pretest

Layer: 4
Depths, m: 2850.3–2856.0
$h$ , m: 5.7
Core, mD: 21.6
Pretests $k_h$ , mD: 16.8

Layer: 5
Depths, m: 2856.0–2860.0
$h$ , m: 4.0
Core, mD: 4.4
Pretests $k_h$ , mD: 2.97

Layer	Depths, m	$h$ , m	Core, mD	Pretests $k_h$ , mD
1	2840.0–2844.0	4.0	58.4	64.8
2	2844.0–2849.5	5.5	152	167.0
3	2849.5–2850.3	0.8	1180	no pretest
4	2850.3–2856.0	5.7	21.6	16.8
5	2856.0–2860.0	4.0	4.4	2.97

Core is the arithmetic mean of the permeability of the samples of the layer to formation water at reservoir conditions. Pretests are the mean $k_h$ of the points of the layer after the corrections (from the table of the previous section: 52.3 and 77.3; 124.3 and 209.7; 12.1 and 21.5; 2.97 mD). Layer 3 is a thin layer of coarse-grained rock. The logs pick it out, but there is no pretest in it. The data are assumed for illustration.

Worked exampleFlow capacity of the interval from core, pretests and the build-up test

Given (assumed data)

Layers: the table above
Build-up test after perforating the whole interval: oil rate 120 m³/d, $B = 1.28$ , oil viscosity 1.4 mPa·s; slope of the Horner line $m = 0.236$ MPa per cycle
Conversion to absolute: $k_{ro}' = 0.85$ ; the pressure is above the bubble point pressure, and the water is immobile

Solution

Core: $4.0\cdot 58.4 + 5.5\cdot 152 + 0.8\cdot 1180 + 5.7\cdot 21.6 + 4.0\cdot 4.4 = 233.6 + 836 + 944 + 123.1 + 17.6 = 2154$ mD·m.
Pretests: $4.0\cdot 64.8 + 5.5\cdot 167.0 + 5.7\cdot 16.8 + 4.0\cdot 2.97 = 259.2 + 918.5 + 95.8 + 11.9 = 1285$ mD·m (layer 3 is not in the sum).
Build-up test: $(kh)_o = 2.149\cdot 120\cdot 1.28\cdot 1.4/0.236 = 1958$ mD·m for oil. The absolute value is $1958/0.85 = 2304$ mD·m.
Comparison with the build-up test: core gives 0.93, and pretests give 0.56. Pretests together with thin layer 3 from core: $1285 + 944 = 2229$ mD·m, a ratio of 0.97.
The share of thin layer 3 in the flow capacity from core: $944/2154 = 44$ % for 4% of the thickness.

Answer: Core and the build-up test agree (0.93). Pretests give only 56% of the flow capacity, because they missed thin layer 3. With it, the ratio is 0.97.

What it means. A thin coarse-grained layer gives almost half of the flow capacity of the interval, and the pretests, set 1.5–4 m apart, did not touch it. Without core, the conclusion “the build-up test gives almost twice the pretests” is easily put down to fractures. In fact it was enough to set a point in the layer that the logs had picked out. Perforating and production logging will show that almost half of the inflow comes from 0.8 m. Core is lower than the build-up test by 7%. This is less than the errors of the “gas to water” relation (15–25% per sample), of $k_{ro}'$ (about 6%) and of the slope of the line (about 5%). If the build-up test had given noticeably less than the sum from core, we would check three things. The first is whether the layers are continuous within the radius of investigation. The second is which part of the line was used (as seen on the derivative). The third is the bubble point pressure. The data in this example are assumed for illustration.

CalculatedCumulative flow capacity of the interval from top to bottom, from core and from pretests. The vertical dashed line is the absolute

kh

from the build-up test. The band marks thin layer 3: there the core curve jumps, while the pretest curve goes vertically.

The remaining disagreement as information about the formation

When the known corrections have been made, the disagreement that remains is not “an error of one of the methods”. It is a sign that there is something in the formation that only one scale can see. We do not remove it with a multiplier until we have found the cause.

What the remaining disagreement may mean

What we see: A well test clearly above core
Likely causes: fractures; high-permeability thin layers that did not get into the samples or were lost with the core
How to check: borehole images, core recovery, production logging, signs of dual porosity (two kinds of pore space: fractures, and the pores of the rock itself) on the derivative

What we see: A well test clearly below core
Likely causes: discontinuous layers; boundaries within the radius of investigation; gas or water in the formation around the well; partial penetration (the well opens only part of the layer)
How to check: the pressure derivative, the bubble point pressure, the geological model

What we see: Pretests systematically below core
Likely causes: damage at the probe; $A$ or $k_{rf}'$ too low; a thick cake
How to check: a repeat pretest, the build-up after a pretest, special core analysis

What we see: Pretests above core
Likely causes: the probe is on a fracture or a coarse lamina; the core does not represent the layer; an error in matching the depths
How to check: neighbouring points, high-resolution logs

What we see: Build-up after a pretest below the drawdown
Likely causes: tool storage, anisotropy ( $k_s < k_d$ ), heterogeneity
How to check: late points of the build-up, a model with storage

What we see	Likely causes	How to check
A well test clearly above core	fractures; high-permeability thin layers that did not get into the samples or were lost with the core	borehole images, core recovery, production logging, signs of dual porosity (two kinds of pore space: fractures, and the pores of the rock itself) on the derivative
A well test clearly below core	discontinuous layers; boundaries within the radius of investigation; gas or water in the formation around the well; partial penetration (the well opens only part of the layer)	the pressure derivative, the bubble point pressure, the geological model
Pretests systematically below core	damage at the probe; $A$ or $k_{rf}'$ too low; a thick cake	a repeat pretest, the build-up after a pretest, special core analysis
Pretests above core	the probe is on a fracture or a coarse lamina; the core does not represent the layer; an error in matching the depths	neighbouring points, high-resolution logs
Build-up after a pretest below the drawdown	tool storage, anisotropy ( $k_s < k_d$ ), heterogeneity	late points of the build-up, a model with storage

The build-up after a pretest and tool storage are in the topic on the limits of the method. Partial penetration and spherical flow are in the topic Partial penetration and well deviation. Fractures are in the topic Fractures and naturally fractured reservoirs.

Pretests as a permeability profile. In wells with core, we find a calibration of the pretests for each rock type: a systematic multiplier and the scatter, as in the example above. In wells without core, the drawdown mobility, after the same corrections and divided by this multiplier, gives point estimates of $k_h$ with a known error. Between the points, the permeability is estimated from logs, from porosity and rock type (Reservoir quality: geological controls on porosity and permeability) or from the nuclear magnetic resonance log (permeability from NMR), and the pretest points serve as anchors for these curves. It is important to set points in the thin layers that the logs pick out as high-permeability: they most often carry a large share of the flow capacity.

The dual-packer module and the multi-probe tool give an intermediate scale: the $kh$ of a one-metre interval and the vertical permeability between the probes (observation probe). With them we check which core window and which anisotropy suit a given layer, before we extend the calibration to the whole unit.

In a reservoir model, the permeability of the layers is set from core and from calibrated logs, and the sum of the flow capacities in the well is checked against the well test. If the sum disagrees with it, the flow capacity of the layers in the model is corrected by a multiplier (History matching), but only after the cause is understood. Fractures call for a different flow model, and discontinuous layers for a different geometry, not just for a different number.

Terms and symbols

Common basis of comparison: The absolute horizontal permeability to liquid at reservoir conditions, averaged over the volume of the larger of the compared measurements. For a well test, the flow capacity $kh$ of the interval.
Averaging window: The interval of core around the point of a pretest whose samples are averaged for the comparison. Usually a few tenths of a metre, with the geometric mean.
Anisotropy factor $f_A$: The ratio of the drawdown permeability to the horizontal one, $\pi/(2K(1 - 1/A))$ . It is 0.85 at $A = 2$ and 0.61 at $A = 10$ .
End-point relative permeability of the filtrate $k_{rf}'$: The permeability of the flushed zone to the filtrate as a fraction of the absolute permeability: $k_{rw}'$ for a water-based filtrate in the oil-bearing part, 1 in the water-bearing part.
Logarithm of the ratio $d$: The decimal logarithm of the ratio, $\lg(k_{\text{pt}}/k_{\text{core}})$ . Its mean is the systematic bias, and its standard deviation is the scatter of the points.
Scale effect: The dependence of the measured permeability on the volume of rock that the measurement covers. Usually the permeability grows with the volume.

Common mistakes and misconceptions

Wrong

“Core shows 300 mD and the build-up test shows 90 mD: the near-wellbore zone is damaged.”

In fact

Damage raises the skin, but the $kh$ from the slope of the Horner line does not depend on it. First we bring the core to reservoir conditions and to oil ( $k_{ro}'$ ), and we compare $kh$ by layers, not permeabilities. Only then do we look for the cause in the formation: discontinuous layers, boundaries, a second phase.

Wrong

“The mobility is 40 mD/(mPa·s) and the oil viscosity is 2 mPa·s, so the permeability is 80 mD.”

In fact

Filtrate flows to the probe, not oil. We take the viscosity of the filtrate (0.3–0.6 mPa·s for a water-based mud), and we divide by its end-point relative permeability in the flushed zone and by the anisotropy factor. With $\mu_f = 0.40$ , $k_{rw}' = 0.25$ and $A = 3$ we get about 83 mD, which is close to 80 only by coincidence.

Wrong

“80% of the points lie within a factor of two, so the pretests and core agree, and no correction is needed.”

In fact

The share of points in the band depends on both the scatter and the bias. We need the mean of the logarithms of the ratios and its standard error. With a scatter of 0.15 and twenty points, a bias of a factor of 1.3 is already significant, although about 90% of the points stay in the band.

Wrong

“The build-up test gives twice the pretests, so there are fractures in the formation.”

In fact

Possibly, but first we check whether the pretests missed a thin high-permeability layer, and whether the thicknesses are weighted correctly. In the lesson's example, 0.8 m of a thin layer gave 44% of the flow capacity from core.

Wrong

“The build-up after a pretest covers more rock, so its permeability is closer to the well test.”

In fact

The build-up gives the spherical permeability, which is lower than the horizontal one in a layered rock. A slope taken from the early points is distorted by tool storage and comes out several times too low. For a comparison with a well test, the drawdown with the corrections is more reliable.

In depth: advanced detail

Which mean to take in the window

The three classical means are special cases of the power mean $k_\omega$ (power-law average). The arithmetic mean ( $\omega = 1$ ) is for layers working in parallel. The harmonic mean ( $\omega = -1$ ) is for layers working in series. The geometric mean ( $\omega \to 0$ ) is for two-dimensional flow in a random mosaic (a patchwork of blocks with random permeabilities). For three-dimensional flow in a random mosaic, the exponent is about $1/3$ . Fluid flows to a probe over a hemisphere: part of the path is along the laminae and part is across them. So the mean in the window lies between the harmonic and the arithmetic, and the geometric mean is a reasonable choice that is not sensitive to outliers (values far from the rest).

Worked exampleMeans in a uniform and in a layered window

Given (assumed data)

Window 2845.5 m (layer 2): permeability of the samples at reservoir conditions 133.9; 183.7; 155.6 and 118.5 mD (the example above)
Window 2852.0 m (layer 4, finely layered): 7.2; 48.3; 14.2 and 51.0 mD

Solution

Window 2845.5 m: the arithmetic mean is 147.9, the power mean with $\omega = 1/3$ is 146.6, the geometric mean is 145.9 and the harmonic mean is 144.0 mD. The difference between the extremes is 3%.
Window 2852.0 m: the arithmetic mean is 30.2, with $\omega = 1/3$ it is 25.0, the geometric mean is 22.4 and the harmonic mean is 16.0 mD. The extremes differ by a factor of 1.9.
The pretest at 2852.0 m after the corrections gave 12.1 mD, lower even than the harmonic mean of the window.

Answer: In a uniform window the way of averaging hardly matters (3%). In a layered one it changes the answer by a factor of 1.9.

What it means. The layered window explains why the points of layer 4 are more scattered than the others. The probe sees mostly the lamina it stands on, so its estimates fall now below the harmonic mean of the window, now above the geometric mean. This is not an error of the core or of the pretest: here the volume of the measurement is smaller than the representative elementary volume. In such units we compare not single points but means over several points of the layer. The data in this example are assumed for illustration.

How many points we need to conclude that there is a bias

Number of points needed to tell bias from scatter

n \approx \left(\frac{z\,s}{\lg F}\right)^2

where:

$n$ number of independent pairs “pretest and core”, —
$s$ standard deviation of the logarithms of the ratios, —
$F$ the systematic multiplier that we need to notice (for example 1.5), —
$z$ number of standard errors; about 2 for 95%, —

Units: dimensionless

Holds when

The condition is $|\bar d| > z\,s/\sqrt n$ at $\bar d = \lg F$ . For small $n$ , instead of $z$ we take Student's coefficient for $n - 1$ degrees of freedom. It is noticeably larger than two (3.18 for four points), and more points are needed.
This condition is for a mean equal to $\lg F$ . Because $\bar d$ itself scatters, a bias of exactly a factor of $F$ will be significant in only about half of the sets. To notice it in four sets out of five, we take $z$ as about 2.8, and about twice as many points are needed.
The points are independent: two points in one core window are in effect one.

The number of points grows as the square of the scatter. With the scatter of the lesson's example ( $s$ ≈ 0.155), a factor of 1.5 is noticeable already from 4 points, or from 6 with Student's coefficient. With $s$ = 0.35, which is usual for field data sets, it takes 16 points (18). So the calibration of pretests is built from all the wells with core of one rock type at once, not for each well separately.

What a well test sees in a heterogeneous formation

The slope of the semilog straight line reflects the flow capacity of the ring of formation that the disturbance has reached during the time of the line. It is not the near zone (that goes into the skin), and it is not the whole formation. If the layers are continuous over the whole of this distance, the flow capacity equals the sum $k_ih_i$ , whether the layers communicate with each other or not. Suppose, however, that the permeability changes randomly over the area, on a scale much smaller than the radius of investigation. Then the test sees a value close to the geometric mean over the area (random mosaic), and the sum from the core of one well may overstate it.

Discontinuous permeable bodies, such as channel sands (sand laid down by rivers) and lenses, give a high flow capacity from core and pretests. On a well test they first give a derivative plateau at this level, and then a rise of the derivative when the disturbance reaches the edges of the body (Several boundaries: parallel faults and wedge-shaped blocks). If the line was drawn through the late points, $kh$ will come out too low. So, before comparing $kh$ with core, we check on the derivative plot that the part of the record with radial flow was really chosen. We also use the formula for the radius of investigation to see what distance it covers.

Connections to other topics

Supercharging and permeability from formation-tester pretests — permeability from the build-up after a pretest, anisotropy, the observation probe and the limits of the method.
Pressure measurement with the wireline formation tester — the pretest, drawdown mobility and the classes of measurements.
Core analysis — laboratory and reservoir conditions, core depth matching and the scale of measurement.
Pressure build-up analysis — flow capacity $kh$ , skin and $p^*$ from the Horner line.
Reservoir heterogeneity and permeability averaging — mean permeabilities, the power mean and the random mosaic.
Relative permeability — end-point relative permeabilities, through which phase permeability is converted to absolute.
Partial penetration and well deviation — spherical flow and partial penetration at the scale of a well.
Nuclear magnetic resonance (NMR) logging — permeability from logs between the pretest points.
History matching — how the flow capacity of model layers is checked against well tests.
Two-phase flow and reservoir description from formation-tester data — pressure profiles from the formation tester when two phases move, and calibration of the reservoir model.

Check yourself

1
A pretest in the oil-bearing part (water-based mud): mobility 30 mD/(mPa·s), filtrate viscosity 0.40 mPa·s, $k_{rw}' = 0.25$ , anisotropy of the layer $A = 3$ ( $f_A$ = 0.774). Which horizontal absolute permeability is closest to the estimate from the pretest?
1. A.12 mD
2. B.48 mD
3. C.62 mD
4. D.15.5 mD
Show the answer
Correct answer: C. 62 mD
$k_h = 30\cdot 0.40/(0.25\cdot 0.774) = 12/0.1935 \approx 62$ mD. The answer 12 mD uses only the filtrate viscosity. The answer 15.5 mD leaves out the end-point relative permeability ( $12/0.774$ ). The answer 48 mD leaves out the correction for anisotropy ( $12/0.25$ ).
2
A build-up test of an interval: oil rate 90 m³/d, $B = 1.25$ , oil viscosity 2.0 mPa·s, slope of the Horner line 0.30 MPa per cycle, $k_{ro}' = 0.80$ . The sum $k_ih_i$ from core at reservoir conditions is 1100 mD·m. What can we say?
Show the answer
$(kh)_o = 2.149\cdot 90\cdot 1.25\cdot 2.0/0.30 \approx 1612$ mD·m for oil, and the absolute value is $1612/0.80 \approx 2015$ mD·m, which is 1.8 times the sum from core. This is more than the errors of the conversion. There is something in the formation that is not in the samples: fractures, or high-permeability thin layers that were missed when the samples were taken or lost with the core. We check core recovery, borehole images, production logging and the shape of the derivative (dual porosity).
3
Which mean of the permeability of samples of a layered interval do we compare with the $kh$ from the build-up test of a vertical well, if the layers are continuous over the area?
1. A.The median of the samples
2. B.The geometric mean
3. C.The arithmetic mean weighted by the thickness of the layers
4. D.The harmonic mean
Show the answer
Correct answer: C. The arithmetic mean weighted by the thickness of the layers
In radial flow to a vertical well, continuous layers work in parallel, and their flow capacities $k_ih_i$ add up. This is the arithmetic mean weighted by thickness, multiplied by the thickness. The harmonic mean describes flow across the layers, and the geometric mean describes a random mosaic.
4
From nine points, the mean of the logarithm of the ratio “pretest / core” after the corrections is −0.25, and the standard deviation is 0.30. Is there a systematic bias? Student's coefficient for 8 degrees of freedom is 2.306.
Show the answer
The standard error is $0.30/\sqrt 9 = 0.10$ . The interval $-0.25 \pm 2.306\cdot 0.10$ runs from −0.48 to −0.02 and does not contain zero. The pretests are systematically lower than core, by about $10^{0.25} \approx 1.8$ times (from 1.05 to 3 times). We check damage at the probe (repeat pretests), anisotropy at the scale of the probe, and $k_{rw}'$ .
5
A pretest is set in the water-bearing part of the formation, in a well with water-based mud. Which end-point relative permeability do we use when converting mobility to absolute permeability?
1. A. $k_{rw}'$ at residual oil
2. B.One
3. C. $k_{ro}'$ at irreducible water
4. D.None: the mobility is already absolute
Show the answer
Correct answer: B. One
There is no oil in the water-bearing part. The filtrate displaces the formation water, and one water phase flows, so the relative permeability factor is one. What remains is the filtrate viscosity and the correction for anisotropy.
6
Why can a thin layer 0.8 m thick give almost half of the flow capacity of a 20 m interval, and what follows for the spacing of the formation tester's points?
Show the answer
The flow capacity of a layer is proportional to the product $kh$ : 0.8 m at 1180 mD gives 944 mD·m, more than 5.5 m at 152 mD. Points set at equal spacing easily miss such a layer, and the sum from pretests comes out almost half as low as the well test. We set the points by the logs, and we must also set them in the thin layers that are marked as high-permeability. Their contribution is checked by production logging.

Updated: 28 October 2026