Forced-gradient technique and multi-well analysis

Quick refresher

A free gradient from two or three points over a few metres is known with an error of tenths of a kilopascal per metre. A gradient from the density of a sample is known to hundredths. While there are few points and the interval is short, the set gradient is more accurate.
A line with a set gradient passes through the mean point of the chosen measurements. The uncertainty of its level is the scatter of a point divided by the square root of their number. Far from the points, the error of the density multiplied by the distance is added.
Supercharging only raises the pressure. So in a tight rock the line is drawn along the left-hand boundary of the reliable points. It goes through the mean of the points with high mobility, not through the leftmost point and not through the middle of the whole cloud. The residual supercharging of the best points shifts the line to the right, and the contact by fractions of a metre or by metres.
Within one well, a constant error of the gauge does not affect its contact, and a common shift of depths moves the contact together with the log of the same well. Between wells these errors remain: ±15–20 kPa, and metres of depth. A gauge error shifts the oil and water points equally, a depth error shifts them in proportion to their gradients, and a separate system shifts only one phase.
On a level diagram (a plot with one point for each well) each point is “depth of the FWL, pressure at it”. A gauge error moves it horizontally, and a depth error vertically. A shift of the water line moves it along the oil line, and a shift of the oil line moves it along the water line.
The levels of the wells are combined by a weighted mean with weights $1/\sigma^2$ , and consistency is checked by $\chi^2$ (chi-squared, a measure of how well the values agree with each other). A well that does not fit the others is checked in order: depths, gauge, supercharging of one of the lines, the date, a separate block. Only after that do we consider a tilted or perched contact (water held up above the main water level).

Key relationships

p(z) = \bar p + \rho g\,(z - \bar z),\quad \sigma^2(z) = \dfrac{s^2}{n} + (g\sigma_\rho)^2 (z - \bar z)^2

a line with a set gradient and its uncertainty

\delta_o = \delta + G_o\,\Delta z + \Delta_o,\quad \delta_w = \delta + G_w\,\Delta z + \Delta_w

shifts of the oil and water points of a well from the field lines: gauge, depth, the system itself

\delta z^* = \dfrac{\delta_o - \delta_w}{G_w - G_o}

shift of the FWL of a well from the shifts of its lines (plus is downward)

\bar z^* = \dfrac{\sum z_i/\sigma_i^2}{\sum 1/\sigma_i^2},\quad \chi^2 = \sum\left(\dfrac{z_i - \bar z^*}{\sigma_i}\right)^2

weighted mean level of the FWL over the wells, and a check of consistency

Common pitfalls

Drawing a free line through two or three points of a thin zone and believing the density that results.
Drawing the line through the middle of a cloud of supercharged points, or through the one leftmost point.
Setting the gradient from a sample taken in another part of the accumulation, where the fluid has another composition, or taking the density of gas without allowing for pressure.
Taking a shift of the lines of two wells by 20–30 kPa as proof of separate systems, without comparing it with the accuracy of comparing wells and with the shift of the second phase.
Explaining a contact level that does not fit the others by a tilt or by perched water without checking the rotary table elevation (the height of the rig floor), the directional survey and the gauge calibration.

After this lesson you can

explain when a free line through pretests is unreliable, and why the density of a fluid from a sample is often more accurate than the gradient from the points
draw a line with a set gradient through reliable points, estimate its uncertainty with the error of the density, and find the FWL and the gas–oil contact (GOC) from such lines
allow for supercharging: draw the line along the left-hand boundary of the reliable points, and estimate the residual supercharging of the best points and its effect on the contact
combine the surveys of several wells: a common water line, a plot of deviations from the field lines, the absolute accuracy of gauges and the depth matching
tell from the shifts of the lines a gauge error, an error in matching the depths, supercharging of the water-bearing part and a separate fluid system
process the contact levels of the wells statistically: a weighted mean, a check of consistency, and the search for and checking of a well that does not fit the others

When a line through the points will not do

A free line through pretests needs enough points over a long enough interval. With one or two degrees of freedom the scatter of the points cannot be estimated (the degrees of freedom are the number of points minus the number of fitted parameters). A slope from points over a few metres is uncertain (how many points we need). In practice this is not an exception but a common case. An oil zone on the flank of a structure is a few metres thick. In a tight rock some of the pretests are “dry” (no fluid flows in) or supercharged, and the points that remain are scattered (how supercharging distorts lines). And in the water-bearing part there may be only a single valid point.

At the same time, the density of a fluid in the formation is usually known independently, and more accurately. The density of oil comes from a PVT analysis of a sample and from a mass balance (an account of the masses of oil and dissolved gas; density of reservoir oil). The density of water comes from analysis of a sample and the water formation volume factor (water line from analysis). The density of gas comes from its composition and the gas deviation factor (how far the gas departs from an ideal gas) at reservoir pressure and temperature (Ideal and real gas: the gas deviation factor (Z)). Then we set the slope of the line from the density, $G = \rho g$ , and move the line on the plot until it takes the position that the reliable points give.

Where to take the gradient from, and how accurate it is

Source of the gradient: Oil from a PVT analysis of a sample
Density error, g/cm³: 0.004–0.01
Gradient error, kPa/m: 0.04–0.1
What to watch for: the sample is not from this zone; a change of composition with depth

Source of the gradient: Water from analysis and $B_w$
Density error, g/cm³: about 0.005
Gradient error, kPa/m: about 0.05
What to watch for: a different salinity or temperature in another part of the formation

Source of the gradient: Gas from composition and the gas deviation factor
Density error, g/cm³: 0.003–0.006
Gradient error, kPa/m: 0.03–0.06
What to watch for: gas density grows with pressure: in a tall column we calculate by sections

Source of the gradient: Two points over 3 m with a scatter of 1 kPa
Density error, g/cm³: about 0.05
Gradient error, kPa/m: about 0.47
What to watch for: any error of one point goes entirely into the slope

Source of the gradient: Five points over 20 m with a scatter of 1 kPa
Density error, g/cm³: about 0.006
Gradient error, kPa/m: about 0.06
What to watch for: supercharging and unnoticed outliers (points far from the rest)

Source of the gradient	Density error, g/cm³	Gradient error, kPa/m	What to watch for
Oil from a PVT analysis of a sample	0.004–0.01	0.04–0.1	the sample is not from this zone; a change of composition with depth
Water from analysis and $B_w$	about 0.005	about 0.05	a different salinity or temperature in another part of the formation
Gas from composition and the gas deviation factor	0.003–0.006	0.03–0.06	gas density grows with pressure: in a tall column we calculate by sections
Two points over 3 m with a scatter of 1 kPa	about 0.05	about 0.47	any error of one point goes entirely into the slope
Five points over 20 m with a scatter of 1 kPa	about 0.006	about 0.06	supercharging and unnoticed outliers (points far from the rest)

Approximate values. The error of the density from a sample depends first of all on how representative the sample is. The error of the gradient from points is the standard error of the slope for evenly spaced points.

We may set the gradient only if the fluid of the line is the same as in the sample. In tall accumulations the composition of the oil can change with depth, and the density of a sample from one depth does not suit another (change of composition with depth). The density of gas depends on pressure. So in a gas column hundreds of metres tall the line is curved, and we calculate it by sections (pressure in a gas column). If there are enough points, we compare the free gradient with the set one. Their difference, divided by its standard error, is tested in the same way as the deviation of a gradient from the hydrostatic one when fluid flows vertically (significance of the deviation). In the new well of the lesson's field, the free oil gradient is $7.065 \pm 0.018$ kPa/m and the one set from the sample is $7.073 \pm 0.039$ . The difference of −0.008 has an error of 0.043, so it is one fifth of the error. The sample and the line agree.

A line with a set gradient, and the contact from it

If the slope is set, the line has one free parameter left: its level. Least squares with one parameter gives a simple answer: the line passes through the mean point of the chosen measurements. The uncertainty of the level is the scatter of a point divided by the square root of the number of points. Far from the points, the error of the set slope, multiplied by the distance of extrapolation, is added.

A pressure line with a set gradient and its uncertainty

p(z) = \bar p + \rho g\,(z - \bar z),\qquad \sigma^2(z) = \frac{s^2}{n} + (g\,\sigma_\rho)^2\,(z - \bar z)^2

where:

$\bar z,\ \bar p$ mean depth and mean pressure of the chosen reliable points, m, Pa
$n$ number of these points, —
$\rho,\ \sigma_\rho$ density of the fluid in the formation from the sample, and its standard error, kg/m³
$s$ scatter of a valid point about the line: the repeatability and the depth errors, known from other lines of the survey or from repeat pretests, Pa
$\sigma(z)$ standard error of the pressure on the line at depth $z$ , Pa

Units: SI (in practice MPa, kPa/m and m); $g$ = 9.81 m/s²

Holds when

All the chosen points belong to one continuous phase, are valid, and have no noticeable supercharging.
The density of the sample refers to the fluid of this line, and it is constant over the interval of extrapolation.
The errors of the points and the error of the density are independent. With one point, the line passes through it, and $s$ is taken from other measurements.

Compared with a free line, the error of the slope $s/\sqrt{\sum(z - \bar z)^2}$ is replaced by the error of the density $g\sigma_\rho$ , which is usually several times smaller. But a forced line cannot detect its own error. A wrong density shifts the contact, and the scatter of the points will not show it.

We find a contact from forced lines in the same way as from free ones: by the crossing of two lines (depth of the crossing of lines). The error is the uncertainty of both lines at the depth of the crossing, divided by the difference of the gradients. The only difference is that for a forced line we calculate the uncertainty by the formula above. We may set the gradient for one line, for both, or for neither, depending on where there are few points.

Pretests of well 4 on the lower flank

Depth, m: 2532.5
Pressure, MPa: 27.9253
Mobility, mD/(mPa·s): 0.5
Where: oil-bearing part

Depth, m: 2534.0
Pressure, MPa: 27.9125
Mobility, mD/(mPa·s): 22
Where: oil-bearing part

Depth, m: 2535.5
Pressure, MPa: 27.9342
Mobility, mD/(mPa·s): 1.1
Where: oil-bearing part

Depth, m: 2537.0
Pressure, MPa: 27.9347
Mobility, mD/(mPa·s): 14
Where: oil-bearing part

Depth, m: 2545.0
Pressure, MPa: 28.0053
Mobility, mD/(mPa·s): 38
Where: water-bearing part

Depth, m: 2551.0
Pressure, MPa: 28.0669
Mobility, mD/(mPa·s): 85
Where: water-bearing part

Depth, m: 2557.0
Pressure, MPa: 28.1309
Mobility, mD/(mPa·s): 24
Where: water-bearing part

Depth, m: 2563.0
Pressure, MPa: 28.1946
Mobility, mD/(mPa·s): 60
Where: water-bearing part

Depth, m: 2569.0
Pressure, MPa: 28.2560
Mobility, mD/(mPa·s): 31
Where: water-bearing part

Depth, m	Pressure, MPa	Mobility, mD/(mPa·s)	Where
2532.5	27.9253	0.5	oil-bearing part
2534.0	27.9125	22	oil-bearing part
2535.5	27.9342	1.1	oil-bearing part
2537.0	27.9347	14	oil-bearing part
2545.0	28.0053	38	water-bearing part
2551.0	28.0669	85	water-bearing part
2557.0	28.1309	24	water-bearing part
2563.0	28.1946	60	water-bearing part
2569.0	28.2560	31	water-bearing part

The same field as in the topics Reservoir pressure and temperature and Pressure gradients, fluid contacts and capillary equilibrium. The top of the formation in the well is 2532 m, and the OWC from logs is 2538.3 m. The mud is water-based, with a low fluid loss (it gives off little filtrate), and all the records have stabilised. The pressures are absolute, and the depths are vertical. The data are assumed for illustration.

Worked exampleFWL in a well with a thin oil zone

Given (assumed data)

Pretests: the table above: four in the oil-bearing part over 4.5 m, and five in the water-bearing part
Density of oil in the formation: from the PVT analysis of the sample from 2503.5 m in the new well: $0.721 \pm 0.004$ g/cm³ (the error is assumed)
Scatter of a valid point: 1.0 kPa (from the lines of the new well)
Mud: filtration through the cake five times lower than in the examples of the topic Supercharging and permeability from formation-tester pretests: 0.144 mm/h during circulation and 0.06 mm/h after it stops; the supercharging model is from there

Solution

Choosing the points. Mobilities of 0.5 and 1.1 mD/(mPa·s) are candidates for supercharging, and 22 and 14 are reliable. The threshold for this survey is 5 mD/(mPa·s). By the supercharging model for its mud, at such a mobility the excess is about 3 kPa. With another mud the threshold is different.
A free line through all four points: $G = 3.33$ kPa/m, a “density” of $3.33/9.81 = 0.34$ g/cm³. There is no such oil: the supercharged points set the slope. Through the two reliable points: $G = (27.9347 - 27.9125)/3.0 = 7.40$ kPa/m, but with an error of $\sqrt2\cdot 1.0/3.0 = 0.47$ kPa/m, which means a density of $0.75 \pm 0.05$ g/cm³. The FWL from such a line is $2541.4 \pm 1.0$ m.
The forced oil line: $G = 9.81\cdot 0.721 = 7.073$ kPa/m, with an error of $9.81\cdot 0.004 = 0.039$ kPa/m. The mean point of the reliable measurements is $\bar z = 2535.5$ m, $\bar p = 27.9236$ MPa. At 2541 m the line gives $27.9236 + 0.007073\cdot 5.5 = 27.96250$ MPa.
The water line (free, five points): $G_w = 10.485$ kPa/m, $s = 0.87$ kPa, $\bar z_w = 2557.0$ m, $\sum(z - \bar z)^2 = 360$ m², $\bar p_w = 28.13074$ MPa. At 2541 m it gives $28.13074 - 0.010485\cdot 16 = 27.96298$ MPa.
The crossing: $z^* = 2541 + (27.96250 - 27.96298)\cdot 1000/(10.485 - 7.073) = 2541 + (-0.48)/3.412 = 2540.86$ m.
The error. The oil line at the FWL: $\sqrt{1.0^2/2 + (0.039\cdot 5.36)^2} = 0.74$ kPa, where 5.36 m is the distance from the mean point. The water line: $0.87\sqrt{1/5 + (2540.86 - 2557)^2/360} = 0.84$ kPa. The FWL: $\sqrt{0.74^2 + 0.84^2}/3.412 = 0.33$ m.
Checking the suspect points. At 2532.5 and 2535.5 m they are to the right of the forced line by 22.9 and 10.6 kPa. Each deviation multiplied by the mobility gives 11.45 and 11.66 kPa·mD/(mPa·s). The two products are almost the same: the deviations are inversely proportional to the mobility, as supercharging should be.
Residual supercharging of the reliable points by the supercharging model for this mud: 0.70 and 1.08 kPa, on average 0.89 kPa. The oil line is to the right of the true one by the same amount, and the FWL is lower than the true one by $0.89/3.412 = 0.26$ m. With the correction: $2540.86 - 0.26 = 2540.60$ m. We take the correction as known to within half of its size, and the error becomes $\sqrt{0.33^2 + 0.13^2} = 0.35$ m.

Answer: The FWL in well 4 is 2540.60 ± 0.35 m (with the correction for residual supercharging; without it 2540.86 ± 0.33 m). In the new well it is 2540.23 ± 0.33 m. The difference is 0.37 m, with an error of 0.48 m.

What it means. Two reliable points over three metres would have given the oil density with an error of ±0.05 g/cm³, and the sample gives ±0.004. The forced line is ten times more accurate in slope, and the FWL is found three times more accurately. The supercharged points did not spoil the result, because they were recognised by their mobility and not by their position. The residual supercharging of the “valid” points, about a kilopascal, shifted the contact by a quarter of a metre. In a tight rock with a mud of higher fluid loss, such a shift reaches metres. Then the correction must be estimated, or the level must be checked against other wells (the next sections). The data in this example are assumed for illustration.

CalculatedWell 4 on a plot of excess over its water line: the forced oil line through two reliable points with a ±2σ band, the free line through all four points of the oil-bearing part, and the free water level. The empty circles are supercharged points, labelled with their mobility.

Supercharging: a line along the left-hand boundary of the points

The main use of the technique is a tight rock, where many pretests are supercharged. Supercharging only raises the pressure: a supercharged point lies to the right of the true line, and never to the left. So the forced line must go not through the middle of the cloud of points, but along its left-hand boundary. The reliable points must lie on it, and the others to the right. We label each point with its mobility: the lower it is, the greater the excess that we expect (signs of supercharging).

The left-hand boundary is not the leftmost point. Each point also has a random error. So the leftmost of several points is almost always shifted to the left by its own noise: on average by 0.56 of the scatter with two points, by 1.16 with five and by 1.54 with ten. A line through the leftmost of ten points with a scatter of 1 kPa would be 1.5 kPa to the left of the true one, and the FWL would shift by almost half a metre. So we take the level from the mean of the points with high mobility. Then we check that no point, of any mobility, lies to the left of the line by more than two or three times the scatter. Such a point means that the line is wrong: the set density is not the right one, or the point is from a depleted or separate layer, or the depth is wrong.

What remains is the residual supercharging of the best points. In a tight rock the excess is present even in the points with the highest mobility, and the forced line is shifted to the right by their mean supercharging. A shift of the oil line to the right lowers the FWL that we find, and a shift of the water line raises it (distortion of the contact by supercharging). We can estimate the residual supercharging by the model of filtrate injection, for the mobility of the best points and the properties of the mud, as in the example above. We can also estimate it from the dependence of the deviations of the less reliable points on the inverse mobility. An independent check is the comparison with other wells: the residual supercharging differs from well to well, and it shows as scatter in their contacts.

Surveys of several wells on one plot

The surveys of all the wells of an accumulation are drawn on one “pressure–depth” plot. In one hydrodynamic system in equilibrium, their lines must coincide. But they coincide only to the accuracy of comparing wells. Each survey has its own gauge with its own absolute error, and its own depth matching. Within one well, a constant error of the gauge does not affect its contact: both lines shift together. A common shift of depths moves the contact together with the log of the same well (where the formula for the contact error comes from). Between wells these errors remain. In the example of the topic on contacts, it is ±16 kPa on the pressure difference of two wells, with a depth error of about a metre in each.

Depths cause errors more often than gauges do. Depths along the hole are converted to vertical depths with the directional survey. Vertical depths are converted to absolute elevations (heights above sea level) by subtracting the height of the rotary table (the rig floor) above sea level (rotary table elevation). An error in the elevation, or a wrong zero level, shifts all the depths of the well. A survey error in a deviated well grows with depth (Directional survey calculations, depth errors on a cable). A stated depth accuracy of a few tenths of a metre often turns out in practice to be metres, especially in deviated wells. So when contacts disagree between wells, the depth matching is the first thing to check.

The oil line of a well that did not reach water, and the water line of a well that penetrated only the water-bearing part, cross at a contact that neither of them saw. From the first two wells we estimate the contact and the volume of the accumulation before any well has penetrated the contact. The price is the errors of comparing two wells. In the example of the topic on contacts, they gave ±4.7 m instead of ±0.3 m in one well (FWL from the regional water line).

It is most convenient to compare wells on an excess pressure plot extended to both phases. From each oil measurement we subtract the pressure on the oil line of the field at the same depth, and from each water measurement the pressure on the common water line. The lines of the field are built from the best well, or from all the consistent wells together. The points of each well fall on verticals shifted by its deviations $\delta_o$ and $\delta_w$ , and the combination of the two shifts shows the cause.

What the shifts of the lines of a well are made of

\delta_o = \delta + G_o\,\Delta z + \Delta_o,\qquad \delta_w = \delta + G_w\,\Delta z + \Delta_w

where:

$\delta_o,\ \delta_w$ deviations of the oil and water points of a well from the oil and water lines of the field, at the recorded depths, Pa
$\delta$ constant error of the gauge in this survey, Pa
$\Delta z$ constant error of the depths of the well: by how much the recorded depths are less than the true ones, m
$G_o,\ G_w$ gradients of oil and water, Pa/m
$\Delta_o,\ \Delta_w$ real differences of pressure in the oil and in the water at this well: a separate system, supercharging, depletion (a pressure drop caused by production) by the date of the measurement, Pa

Units: SI (in practice kPa, kPa/m and m)

Holds when

The lines of the field and of the well are single-phase, in equilibrium, with the same gradients.
The errors are constant over the well. A depth error that grows with depth changes the slope of the lines too.

A gauge error shifts both phases equally. A depth error shifts them in proportion to the gradients, so the oil by about two thirds of the shift of the water. A separate system of oil with a common aquifer shifts only the oil. Supercharging of the water-bearing part, or a separate water-bearing block, shifts only the water. A well that penetrated one phase gives one equation with three unknowns, so from it alone the cause of the shift cannot be found.

Deviations of wells from the field lines, kPa

Well: 5
Oil points: 34.4
Water points: 35.8
What this means: the same shift of both phases: gauge calibration

Well: 6
Oil points: 21.1
Water points: 31.5
What this means: shifts in the ratio of the oil and water gradients: depth matching

Well: 3
Oil points: −100.2
Water points: −0.5
What this means: only the oil is shifted: a separate system

Well: 2
Oil points: −0.1
Water points: from 2.9 to 17.6
What this means: only the water is shifted, more at tight points: supercharging

Well	Oil points	Water points	What this means
5	34.4	35.8	the same shift of both phases: gauge calibration
6	21.1	31.5	shifts in the ratio of the oil and water gradients: depth matching
3	−100.2	−0.5	only the oil is shifted: a separate system
2	−0.1	from 2.9 to 17.6	only the water is shifted, more at tight points: supercharging

The field lines are the lines of the new well. Well 3 is a block behind a fault (separate systems). Well 2 has a water line from supercharged points without a correction (tight water-bearing formation). For a depth error of 3.0 m the expected shifts are $7.065\cdot 3.0 = 21.2$ and $10.49\cdot 3.0 = 31.5$ kPa. The data are assumed for illustration.

CalculatedDeviations of the pretests of four wells from the field lines: filled points are oil, empty ones are water. Each cause has its own pattern: the same shift of both phases, shifts in the ratio of the gradients, a shift of only the oil, and a shift of only the water that grows at the tight points.

Contact levels in the wells: statistics

Each well that penetrated both phases gives its own FWL level and the pressure at it. We put these pairs on a level diagram (one point for each well): “pressure at the FWL against depth of the FWL”. For one fluid system, the scatter of points on it is created only by errors. So it serves as a measure of the quality of the data, and the direction in which a single well departs from the rest suggests the cause.

How shifts of the lines move the FWL of a well

\delta z^* = \frac{\delta_o - \delta_w}{G_w - G_o},\qquad \delta p_{\text{FWL}} = \delta_o + G_o\,\delta z^*

where:

$\delta z^*$ shift of the depth of the FWL of a well (plus is downward) caused by the shifts of its lines in pressure, m
$\delta p_{\text{FWL}}$ shift of the pressure at its FWL, Pa
$\delta_o,\ \delta_w$ shifts of the oil and water lines in pressure (the formula above without the depth error), Pa

Units: SI (in practice kPa, kPa/m and m)

Holds when

The gradients of the phases at the well are the same as in the field.
A depth error $\Delta z$ shifts the level vertically by $-\Delta z$ at the same pressure. It is allowed for separately.

The same shift of both lines (a gauge error) does not change the depth but changes the pressure: the point moves horizontally. A depth error moves the point vertically. A shift of the water line alone moves the FWL along the oil line, upwards if the water is supercharged. A shift of the oil line alone moves it along the water line. A well behind a fault with separate oil and a common aquifer moves along the water line.

If there are several wells and their levels agree, we find the common level as a weighted mean: an accurate level weighs more. We check consistency with $\chi^2$ (chi-squared), the sum of the squared deviations, expressed in the wells' own errors. For consistent data it is on average equal to the number of degrees of freedom, which is one less than the number of wells. A value above the 95% limit means that at least one well does not fit the others, or that the errors are underestimated (Distributions and sample size).

Weighted mean level and a check of consistency

\bar z^* = \frac{\sum_i z_i^*/\sigma_i^2}{\sum_i 1/\sigma_i^2},\qquad \sigma_{\bar z^*} = \frac{1}{\sqrt{\sum_i 1/\sigma_i^2}},\qquad \chi^2 = \sum_i\left(\frac{z_i^* - \bar z^*}{\sigma_i}\right)^2

where:

$z_i^*,\ \sigma_i$ FWL level of well $i$ and its standard error (from its own lines), m
$\bar z^*,\ \sigma_{\bar z^*}$ weighted mean level and its standard error, m
$\chi^2$ a measure of consistency; it is compared with the limit for $n - 1$ degrees of freedom (for four degrees of freedom, 9.49), —

Units: any consistent system

Holds when

The errors of the wells are independent and correctly estimated. Errors common to all wells (the fluid density, systematic errors of one type of tool) are not reduced by this method.
All the wells belong to one system. A well that does not fit the others is excluded only after the cause is explained.

A simple mean and a standard deviation over the wells are good when the errors of the wells are unknown and about equal. If they can be estimated, the weighted mean is more accurate, and $\chi^2$ at once says whether the scatter is explained by the errors or something else is in the data.

FWL levels in the wells of the field

Well: Discovery
FWL, m: 2540.29
Error, m: 0.61
Pressure at the FWL, MPa: 27.9561
FWL minus OWC from logs, m: —

Well: New
FWL, m: 2540.23
Error, m: 0.33
Pressure at the FWL, MPa: 27.9557
FWL minus OWC from logs, m: 2.03

Well: 2
FWL, m: 2540.30
Error, m: 0.51
Pressure at the FWL, MPa: 27.9562
FWL minus OWC from logs, m: —

Well: 4
FWL, m: 2540.60
Error, m: 0.35
Pressure at the FWL, MPa: 27.9588
FWL minus OWC from logs, m: 2.30

Well: 5
FWL, m: 2539.88
Error, m: 0.51
Pressure at the FWL, MPa: 27.9879
FWL minus OWC from logs, m: 1.78

Well: 6
FWL, m: 2537.45
Error, m: 0.44
Pressure at the FWL, MPa: 27.9573
FWL minus OWC from logs, m: 2.15

Well	FWL, m	Error, m	Pressure at the FWL, MPa	FWL minus OWC from logs, m
Discovery	2540.29	0.61	27.9561	—
New	2540.23	0.33	27.9557	2.03
2	2540.30	0.51	27.9562	—
4	2540.60	0.35	27.9588	2.30
5	2539.88	0.51	27.9879	1.78
6	2537.45	0.44	27.9573	2.15

The FWL depth of each well uses that well's own depth matching. The discovery well: two points per zone, with the scatter taken as 1 kPa. Well 2: with the correction of the water line for supercharging. Well 4: from the forced oil line with the correction for residual supercharging (the example above). Well 5: a survey with a gauge calibration error. Well 6: as recorded. The data are assumed for illustration.

Worked exampleThe common FWL of the field, and the well that does not fit

Given (assumed data)

FWL levels: the table above
Deviations of the lines of wells 5 and 6 from the field lines: the table in the previous section
Depth check: in well 6, in converting to absolute elevations, the rotary table elevation taken was 3.0 m larger than the real one

Solution

Weighted mean over five wells without well 6 (weights $1/\sigma^2$ : 2.687; 9.183; 3.845; 8.163; 3.845): $\bar z^* = 2540.31$ m, with an error $1/\sqrt{27.72} = 0.19$ m.
Consistency: $\chi^2 = 1.5$ with four degrees of freedom, and the limit is 9.49. The five wells agree. The plain standard deviation of the levels is 0.26 m, within their own errors.
Well 5: the pressure at the FWL is higher than in the others by 32 kPa at almost the same depth. Its oil and water points are shifted equally (34.4 and 35.8 kPa): this is a gauge calibration. Its level from its own lines is right, and it goes into the mean.
Well 6: $(2537.45 - 2540.31)/\sqrt{0.44^2 + 0.19^2} = -6.0$ , far beyond the error. The pressure at its FWL differs from the others by only 1.6 kPa: the point has moved vertically. The shifts of its points are 21.1 kPa for oil and 31.5 for water, and the ratio 0.67 ≈ $G_o/G_w$ . For each phase the depth error is $21.1/7.065 = 2.99$ m and $31.5/10.49 = 3.00$ m.
An independent check: the FWL minus the OWC from logs in well 6 is 2.15 m, as in the other wells (1.78–2.30 m). The contact level is shifted together with the log, so all the depths of the well are shifted.
With the corrected elevation: 2537.45 + 3.0 = 2540.45 m, and the deviation is 0.3 of the error. Over six wells: $\bar z^* = 2540.33 \pm 0.17$ m.

Answer: The FWL of the field is 2540.33 ± 0.17 m over six wells. Well 6 did not fit because of an elevation error of 3 m. Well 5 had a gauge shift, which does not affect its level.

What it means. Six wells gave the level twice as accurately as the best of them. The well that did not fit did not become a “tilted contact”. What helped to sort it out was the direction on the level diagram, the ratio of the shifts of the two phases, and the check against the OWC from the logs of the same well, which share its shifted depths. A weighted mean does not reduce the errors that are common to all wells, for example a wrong oil density in the forced lines. These are estimated separately. In all the wells the OWC from logs is about two metres above the FWL. This is the capillary height of the entry pressure of the rock (FWL and OWC). The data in this example are assumed for illustration.

CalculatedLevel diagram: the depth of the FWL of each well against the pressure at it, with its error in depth. The directions of the oil and water gradients are drawn through the weighted mean level. Well 5 moved horizontally (gauge), and well 6 vertically (depths). The empty circle is well 2 with a water line from supercharged points: it moved up along the oil line.

What to check if the level of a well does not fit the others

1
Depths
The matching of the formation tester's depth to the log, the directional survey, the rotary table elevation and the zero level. The sign of this is a shift in the vertical direction at the same pressure, and the same distance from the FWL to the OWC from logs.
2
Gauge
Calibration and drift. The sign is the same shift of the oil and water points, and a shift of the level horizontally.
3
Supercharging of one of the lines
More often of the water-bearing part. The sign is deviations that grow with the inverse mobility, and a shift of the level along the other line.
4
Date of the measurement
If production is already going on in the accumulation or in the common aquifer, pressures of different years are compared only after bringing them to one date (Pressure measurements in producing reservoirs: barriers and accuracy).
5
A separate block
A shifted line of one phase while the other coincides, and faults on the seismic data (separate systems).
6
Capillary shift and non-equilibrium
The OWC from logs taken instead of the FWL, or another rock with another entry pressure. Only after that, flow in the aquifer and perched water (water held up above the main water level; Perched and tilted contacts, trapped water).

Separate systems, or an error

The decision “a separate system” is expensive: it divides the accumulation into volumes with their own contacts and changes the development scheme. So we compare the shift of a line not with the scatter of the points but with the accuracy of comparing wells. A shift of less than two such errors proves nothing. A shift of several errors, with the second phase coinciding, is an argument for a separate system. An example is the block behind the fault in the topic on contacts: a shift of 100 kPa against an error of comparison of ±16 kPa (block behind a fault). A deviation of the line in a well that penetrated one phase is a reason to repeat the measurement with another tool or to penetrate the second phase, not a basis to divide the accumulation.

The forced gradient technique is especially useful when there are several systems. Suppose different sand bodies have a common aquifer (the water-bearing formation) but their own gas or oil caps, and each body has few points. Lines with the density of gas or oil from samples are drawn through the few reliable points of each body. The crossing of each of them with the common water line gives the contact of that body. The error of the contact of each body then includes the errors of comparing wells, because the water line is common and built from other wells.

The reverse case is a common oil or gas line with different water lines. This happens when part of the aquifer is closed off and separated from the rest, and each well sees its own water level. Such perched contacts, and contacts tilted by the flow of edge water (water of the aquifer outside the accumulation), are covered in the topic Perched and tilted contacts, trapped water. Whether there is a gas cap over the accumulation is checked by the bubble point pressure of a sample and the oil line (GOC from bubble point pressure).

Terms and symbols

Forced gradient technique: Drawing a pressure line with the slope $\rho g$ from the density of the fluid in a sample. From the measurements we find only its level.
Left-hand boundary of the points: The position of a forced line when there is supercharging: the reliable points lie on the line, and the others to the right. The level comes from the mean of the reliable points.
Residual supercharging: The excess pressure at the best, most mobile points of a tight rock. It shifts the forced line to the right.
Accuracy of comparing wells: The error of the difference of pressures and depths of two surveys: the absolute accuracy of the gauges and the depth matching, tens of kilopascals and metres.
Level diagram: The depths of the FWL of wells against the pressure at them, with one point for each well. The direction in which a well departs from the rest points to the cause.
Rotary table elevation: The height of the zero level of the depths of a well above sea level. An error in it shifts all the depths of the well.
$\chi^2$ of the levels: The sum of the squared deviations of the levels of wells from the weighted mean, expressed in their errors. It is a check of consistency.

Common mistakes and misconceptions

Wrong

“A free line through all the points is more honest than an imposed gradient.”

In fact

Through two or three points over a few metres, the slope is known ten times worse than the density from a sample. A set gradient with a check is better: if there are enough points, the free and the set gradients must agree.

Wrong

“We draw the forced line through the middle of the cloud of points, so that the errors cancel.”

In fact

Supercharging only raises the pressure and does not cancel. In a tight rock the line goes along the left-hand boundary of the reliable points, and the points with low mobility lie to the right.

Wrong

“The leftmost point is the least supercharged, so we draw the line through it.”

In fact

The leftmost point is shifted to the left by its own noise: out of five points, on average by 1.16 of the scatter. We take the level from the mean of the reliable points, and we check that nothing lies significantly to the left of the line.

Wrong

“The oil lines of two wells differ by 25 kPa, so they are different systems.”

In fact

The error of comparing wells is about ±16 kPa. Did the water lines differ in the same way? The same shift of both phases is the gauge, and a shift in the ratio of the gradients is the depths. A separate system is confirmed by a shift of one phase by several errors.

Wrong

“The FWL of a well is three metres above the others, so the contact is tilted.”

In fact

First the depths: the elevation, the directional survey, the matching to the log. If the distance from the FWL to the OWC from logs is the same as in the other wells, the depths have shifted, not the contact.

In depth: advanced detail

When a set gradient is more accurate than a fitted one

A free line minimises the sum of the squared deviations over two parameters, and a forced line over one: $\sum (p_i - a - Gz_i)^2$ over $a$ at a set $G$ . The derivative with respect to $a$ is zero at $a = \overline{p - Gz}$ . So the line passes through the mean point, and the variance of its level there is $s^2/n$ . The errors of the slope of the two methods are $s/\sqrt{\sum(z_i - \bar z)^2}$ and $g\sigma_\rho$ . The set gradient is more accurate as long as $\sum(z_i - \bar z)^2 < (s/g\sigma_\rho)^2$ .

With a scatter of 1 kPa and an oil density from a sample with an error of 0.004 g/cm³, the limit is $(1/0.039)^2 \approx 657$ m². For evenly spaced points this is an interval of 36 m with three points, 32 m with five and 25 m with ten. A free line beats a set gradient only over intervals of tens of metres. If both estimates are available, they can be combined by a weighted mean, with weights $1/s_G^2$ and $1/(g\sigma_\rho)^2$ , provided that they agree.

A systematic error (one that always pushes the same way) is not reduced by the number of points. Suppose the true density of the oil differs from the one taken from the sample (0.721 g/cm³) by 0.01 g/cm³. Then the slope of the line is wrong by 0.098 kPa/m. Over an extrapolation of 10 m this is about 1 kPa, and the contact shifts by 0.3 m, however many points are on the line. So we take the density for a forced line from a sample of the same zone, and we check it against the free gradient where there are enough points. The error of the sample itself, ±0.004 g/cm³ in the lesson's example, is already in the formula for the uncertainty of the line.

Where the patterns of the shifts come from

Let the recorded depths of a well be less than the true ones by $\Delta z$ , and let the gauge read too high by $\delta$ . A water point at the true depth $z$ has the pressure $p_w(z) + \delta$ . On the plot it falls at the depth $z - \Delta z$ , where the water line of the field passes $G_w\Delta z$ to the left. The deviation is $\delta + G_w\Delta z$ , and for oil it is $\delta + G_o\Delta z$ . When we add the real differences of the systems, we get the formula of the main section.

The lines of a well cross where $p_o + \delta_o = p_w + \delta_w$ . The pressure difference of the phases changes with depth at the rate $G_w - G_o$ , so the crossing shifts by $(\delta_o - \delta_w)/(G_w - G_o)$ . The pressure at it shifts by $\delta_o$ plus the change of the oil pressure along this path. A depth error moves the level by $-\Delta z$ with no change of pressure. Hence the directions on the level diagram: the gauge moves a point horizontally ( $\delta_o = \delta_w$ ), the depth moves it vertically, and a shift of one phase moves it along the line of the other.

An example: a shift of the water line by +30 kPa with a gradient difference of 3.43 kPa/m raises the FWL by $30/3.43 = 8.75$ m. The pressure at it falls by $7.065\cdot 8.75 \approx 62$ kPa, so the point moves up and to the left along the oil line. A gauge shift of +35 kPa leaves the FWL in place, and increases the pressure at it by 35 kPa.

Connections to other topics

Pressure gradients, fluid contacts and capillary equilibrium — contacts from the crossing of lines, their error, a water line from analysis, a block behind a fault.
Supercharging and permeability from formation-tester pretests — supercharging, its model and a correction by the inverse mobility.
Pressure measurement with the wireline formation tester — pretests, classes of measurements, the accuracy of pressure and depth.
Pressure measurements in producing reservoirs: barriers and accuracy — pressure profiles after production starts, and a comparison of surveys of different years.
Density and compressibility of reservoir oil — density of reservoir oil from PVT data for the forced gradient.
Reservoir pressure and temperature — fluid gradients, pressure in a gas column, the datum depth.
Perched and tilted contacts, trapped water — perched and tilted contacts: when the levels of wells really differ.
Directional survey calculations — vertical depths from the directional survey, and their errors.
Gross rock volume from maps — how the level of a contact turns into the rock volume of an accumulation.

Check yourself

1
In an oil zone there are two reliable points 4 m apart in vertical depth, and the scatter of a point is 1 kPa. The oil density from a sample is known with an error of 0.005 g/cm³. Which gradient is more accurate, from the two points or from the sample, and by how many times?
Show the answer
The slope from two points has an error of $\sqrt2\cdot 1/4 = 0.35$ kPa/m. From the sample it is $9.81\cdot 0.005 = 0.049$ kPa/m, which is about seven times more accurate. A free line makes sense when there are many points and the interval is tens of metres.
2
In a tight water-bearing formation, six pretests with mobilities from 0.3 to 15 mD/(mPa·s) are scattered over 40 kPa. The density of water is known from a laboratory analysis. Where do we draw the forced water line?
1. A.Through the mean of all six points
2. B.Through the mean of the points with the highest mobility, after checking that nothing lies significantly to the left of the line
3. C.Through the rightmost point
4. D.Through the leftmost point, whatever its mobility
Show the answer
Correct answer: B. Through the mean of the points with the highest mobility, after checking that nothing lies significantly to the left of the line
Supercharging only raises the pressure, so the line goes along the left-hand boundary of the reliable points. The leftmost point is shifted to the left by its own noise, and the mean of all the points includes supercharging. The residual supercharging of the best points is estimated separately: it raises the FWL that we find.
3
The oil points of a well deviate from the field line by +18 kPa, and the water points by +27 kPa. The gradients of oil and water are 7.0 and 10.5 kPa/m. What most likely happened, and what will happen to its FWL?
Show the answer
The ratio of the shifts $18/27 = 0.67$ equals the ratio of the gradients $7.0/10.5$ . This is a depth error: the recorded depths are less than the true ones by about $27/10.5 \approx 2.6$ m. The FWL of the well, in its own depths, will be 2.6 m higher than the others, at the same pressure at it. We check the elevation, the directional survey and the matching to the log.
4
A well penetrated only oil. Its oil line is 40 kPa to the right of the field line, and the accuracy of comparing wells is ±16 kPa. What can we say?
1. A.The depths of the well are too small by 5–6 m
2. B.The gauge reads 40 kPa too high
3. C.From one phase the cause cannot be found: we need the second phase, another tool, or a check of the depths
4. D.The well is in a separate system with a higher pressure
Show the answer
Correct answer: C. From one phase the cause cannot be found: we need the second phase, another tool, or a check of the depths
The shift of one line is made of the gauge error, the depth error and a real difference of the system: one equation with three unknowns. The water points of the same well, a repeat measurement with another tool and a check of the depth matching help to tell them apart.
5
Three wells gave FWL levels of 2612.4 ± 0.4, 2611.8 ± 0.6 and 2613.9 ± 0.5 m. Find the weighted mean level and check consistency (the limit of $\chi^2$ for two degrees of freedom is 5.99).
Show the answer
The weights $1/\sigma^2$ are 6.25, 2.78 and 4.00, with a sum of 13.03. The weighted mean is $2612 + (0.4\cdot 6.25 - 0.2\cdot 2.78 + 1.9\cdot 4.00)/13.03 = 2612.73$ m, with an error of $1/\sqrt{13.03} = 0.28$ m. $\chi^2 = 0.69 + 2.42 + 5.45 = 8.6 > 5.99$ : the levels do not agree, and the third well does not fit the others. Before averaging, we check it: the depths, the gauge, supercharging, a separate block.
6
Why does a gauge calibration error of +35 kPa not change the FWL of the same well, but show itself on the level diagram?
Show the answer
A constant error shifts both lines of the well equally, and their crossing stays at the same depth. The pressure at the crossing is too high by the same 35 kPa. So on the level diagram the well moves horizontally away from the others. This is a sign of the gauge, not of the depths or of a separate system.

Updated: 28 October 2026