Directional wells: applications and profile

Quick refresher

Inclined wells are drilled when it is impossible or not worth putting the rig above the target: from well pads (sites from which many wells are drilled), platforms and islands; under cities, rivers and the sea; around salt domes (rising columns of salt) and around damaged parts of a well; and to kill blowouts (uncontrolled flows of oil or gas).
The position of a point of the well is given by its true vertical depth and its displacements to the north and to the east from the wellhead. A depth is always measured from some zero: the rotary table (the turning table on the rig floor), sea level or the wellhead. Without saying which zero, a depth means nothing.
The measured depth (MD, the length along the hole) is never less than the true vertical depth (TVD, the depth straight down). The driller measures the first, and the pressure and the geology are tied to the second. The profile gives the conversion between them.
Three quantities define the profile: the kick-off depth, the build-up rate and the inclination of the tangent section. At a constant rate the build section is an arc of a circle with the radius $R = 180\,l_0/(\pi i)$ .
The inclination of the tangent section is found by calculation, from the condition that the straight section touches the build arc and passes through the target. The deeper the kick-off and the farther the target, the greater the angle and the longer the well.
An S-shaped profile enters the formation vertically, but it pays for this with a greater angle, a longer well and one more curved section.

Key relationships

R = \dfrac{180}{\pi}\cdot\dfrac{l_0}{i}

radius of curvature; $i$ is degrees per the reference length $l_0$ (30 m, 100 ft or 10 m)

A = \sqrt{\Delta N^2 + \Delta E^2}, \qquad \tan\varphi = \dfrac{\Delta E}{\Delta N}

departure and azimuth of the target from the difference of coordinates

\theta = \arctan\dfrac{A_{\text{T}} - R}{\Delta H} + \arcsin\dfrac{R}{\sqrt{(A_{\text{T}} - R)^2 + \Delta H^2}}

inclination of the tangent section; $\Delta H$ is the vertical distance from the kick-off point to the target

L = H_{\text{KOP}} + \dfrac{l_0\,\theta}{i} + \sqrt{(A_{\text{T}} - R)^2 + \Delta H^2 - R^2}

length of the well: the vertical section ( $H_{\text{KOP}}$ is the depth of the kick-off point, KOP), the build and the tangent section ( $\theta$ in degrees)

Common pitfalls

Naming a depth without saying whether it is along the hole or vertical, and from which zero it is measured.
Mixing up a rate in degrees per 30 m, per 100 ft and per 10 m.
Putting degrees where a program or a spreadsheet expects radians, and the other way round.
Calculating the azimuth with the arctangent without checking the quadrant. A target in the south-west would end up in the north-east.
Taking the angle of the inclined section from a drawing. An error of one degree on a long hole means tens of metres of miss.

After this lesson you can

name the tasks that directional wells solve, from pad drilling to relief wells
tell the measured depth (the length along the hole) from the true vertical depth (the depth straight down), and the rotary table elevation (the height of the rotary table above sea level) from the elevation of a point (its height above sea level), and convert depths between zero datums
find the departure and the azimuth of the target from the coordinates of the wellhead and the target
explain the parameters of the profile (the kick-off point, the build-up and drop-off rates, the inclination of the tangent section, which is the straight inclined part) and convert a rate from degrees per 30 m to degrees per 100 ft and per 10 m, and to a radius of curvature
calculate a build-and-hold profile and an S-shaped profile: the angle, the coordinates of the key points and the length of the well
list the constraints that are taken into account when choosing a profile: the target and its tolerance, neighbouring wells, the geological section, the casing programme and the drilling mud

Why we deviate the well

There is no such thing as a strictly vertical well. Inclined layers of different hardness and the bending of the drill string push the bit sideways, and even in a “vertical” well the bottom can be many metres away from the wellhead. Directional drilling turns this drift from a nuisance into a tool: we deviate the well on purpose and steer it to the target along a calculated path (How a well is drilled: the process and the people involved).

Tasks that an inclined well solves

Task: Pad drilling, offshore platforms, artificial islands
How the inclined well helps: dozens of wells are drilled from one site, and their bottoms spread over the whole accumulation: at sea this is how a field is developed from a platform, and on land it saves on sites, roads and utilities

Task: An inaccessible surface
How the inclined well helps: the target lies under a city, a river, the sea near the shore, a mountain or a protected area, and the rig is put to one side

Task: Crossing a fault (a break in the rock along which the two sides have slipped)
How the inclined well helps: the well is led along the fault and crosses it at a chosen place, so that movements along the fault do not damage the casing string

Task: Sidetrack (a new hole drilled out of an existing well)
How the inclined well helps: we bypass tools left in the well, move away from the old hole to a new target, and correct a hole that has drifted from the plan

Task: Salt dome
How the inclined well helps: to reach an accumulation under the steep flank of a dome, the well runs beside the salt and not through it: salt can have caverns, can flow, and can contain layers with high pressure

Task: Relief well
How the inclined well helps: in a blowout, an inclined well is steered to the failed well at depth, and the failed well is killed by pumping in a heavy fluid

Task	How the inclined well helps
Pad drilling, offshore platforms, artificial islands	dozens of wells are drilled from one site, and their bottoms spread over the whole accumulation: at sea this is how a field is developed from a platform, and on land it saves on sites, roads and utilities
An inaccessible surface	the target lies under a city, a river, the sea near the shore, a mountain or a protected area, and the rig is put to one side
Crossing a fault (a break in the rock along which the two sides have slipped)	the well is led along the fault and crosses it at a chosen place, so that movements along the fault do not damage the casing string
Sidetrack (a new hole drilled out of an existing well)	we bypass tools left in the well, move away from the old hole to a new target, and correct a hole that has drifted from the plan
Salt dome	to reach an accumulation under the steep flank of a dome, the well runs beside the salt and not through it: salt can have caverns, can flow, and can contain layers with high pressure
Relief well	in a blowout, an inclined well is steered to the failed well at depth, and the failed well is killed by pumping in a heavy fluid

A special case is horizontal wells, where the well is steered along the formation to have a larger area of contact with it, and extended-reach wells, which reach distant parts of an accumulation (Horizontal wells).

For Kazakhstan many rows of the table are everyday practice: well pads on land and artificial islands in the shallow water of the North Caspian, the many salt domes (rising columns of salt) of the Pre-Caspian Basin, and sidetracks drilled out of old wells in mature fields.

Calculating the profile is only the beginning: it must still be followed during drilling. On the rig this is done by directional drillers and MWD (telemetry) engineers, most often specialists of an oilfield service company. They choose the deflection tools (Deflection tools and downhole motors), the equipment at the bottom of the string that turns the well. They watch the measurements of inclination and azimuth (Directional survey calculations) and correct the path when the well drifts away from the plan.

Reference systems: depths and coordinates

The position of any point of the well is given by three numbers: the true vertical depth, and two horizontal displacements from the wellhead, to the north and to the east. For these numbers to mean anything, we must agree what they are measured from. We must choose a zero for the depths (the datum) and a system of coordinates in the plan view (the view from above).

SchematicTwo projections of an inclined well (a sketch). On the left is the vertical projection on a plane through the wellhead and the target: the true vertical depth

H

, the length along the hole

L

from the wellhead to the target, the departure

A

(the horizontal displacement) and the inclination

\theta

. On the right is the horizontal projection (the plan view): the displacements of the target to the north

\Delta N

and to the east

\Delta E

, the departure

A

and the azimuth

\varphi

, measured from north clockwise.

Every point of an inclined well has two depths. The measured depth (MD, also called along-hole depth, AHD) is the path that the bit has travelled from the zero datum. The driller always knows it from the length of the pipe run into the hole, and the lengths of strings, the volumes and the time are tied to it. The true vertical depth (TVD) shows how far below the same zero the point lies. The geology and the pressures are tied to it (Pore pressure and overburden pressure). It cannot be measured directly. We get it by calculation from the measurements of inclination and azimuth along the well. Below the point where the well has left the vertical, the measured depth is always greater than the true vertical depth.

What depth is measured from

Zero datum: Sea level (mean sea level, MSL)
What is good about it: constant and common to all wells, maps and sections; geologists and reservoir engineers measure the depths of formations from it
What is awkward about it: it cannot be “seen” on the rig: we need the height of the rotary table above sea level

Zero datum: Rotary table (RT, the turning table on the rig floor); the zeros close to it are the drill floor (DF) and the rotary kelly bushing (RKB, the insert in the table that turns the kelly, the square pipe at the top of the string)
What is good about it: the working zero of the driller: everything run into the well is measured from it
What is awkward about it: it exists only while the rig stands on the spot; another rig has a different height

Zero datum: Wellhead (the casing head, the fitting on top of the casing; for wells with a subsea wellhead, the body of the wellhead housing on the sea floor)
What is good about it: it remains after the rig has gone: the zero for production and for workovers (repairs of the well)
What is awkward about it: it appears only after the first strings are run; when the well is abandoned, it is removed

Zero datum	What is good about it	What is awkward about it
Sea level (mean sea level, MSL)	constant and common to all wells, maps and sections; geologists and reservoir engineers measure the depths of formations from it	it cannot be “seen” on the rig: we need the height of the rotary table above sea level
Rotary table (RT, the turning table on the rig floor); the zeros close to it are the drill floor (DF) and the rotary kelly bushing (RKB, the insert in the table that turns the kelly, the square pipe at the top of the string)	the working zero of the driller: everything run into the well is measured from it	it exists only while the rig stands on the spot; another rig has a different height
Wellhead (the casing head, the fitting on top of the casing; for wells with a subsea wellhead, the body of the wellhead housing on the sea floor)	it remains after the rig has gone: the zero for production and for workovers (repairs of the well)	it appears only after the first strings are run; when the well is abandoned, it is removed

In the CIS countries (the Commonwealth of Independent States, which includes Kazakhstan) heights are measured from the zero of the Kronstadt tide gauge (the Baltic height system). The level of the Caspian Sea lies almost thirty metres below this zero, and on the Caspian coast the ground surface and the rotary table often have negative elevations.

The height of the rotary table above sea level is called the rotary table elevation (RTE). The height of a point underground relative to sea level is its elevation (its height above sea level, which is negative below sea level). In English-language documents the same quantity with the opposite sign is written as the depth below sea level (true vertical depth subsea, TVDSS). The geologist gives the target as an elevation, and the driller needs the depth from the rotary table.

True vertical depth from the rotary table, from the elevation

H = h_{\text{RT}} - z_{\text{el}}

where:

$H$ true vertical depth of the point from the rotary table, m
$h_{\text{RT}}$ rotary table elevation, its height above sea level; negative if the rotary table is below sea level, m
$z_{\text{el}}$ elevation of the point; negative below sea level, m

Units: metres (or feet), the same for all the quantities

Holds when

Both heights are measured from the same sea level (the same height system).
The rotary table elevation equals the elevation of the ground plus the height of the rotary table above the ground; at sea it is the height of the rotary table above sea level.

The same point of a formation has a different depth from the rotary table for different rigs. So the rotary table elevation is written in the well documents, and the zero datum is stated next to every depth.

Worked exampleFrom elevation to the driller's depth

Given (assumed data)

Target, the top of a formation: elevation −2415 m
Ground surface at the site: elevation −24 m (below sea level)
Rotary table of the drilling rig: 9 m above the ground
Rotary table of the workover rig that will come for a repair in years to come: 6 m above the ground

Solution

The rotary table elevation of the drilling rig: $h_{\text{RT}} = -24 + 9 = -15$ m.
The true vertical depth of the target from the rotary table: $H = -15 - (-2415) = 2400$ m.
For the workover rig: $h_{\text{RT}} = -24 + 6 = -18$ m, and the same top of the formation will be at a depth of $-18 + 2415 = 2397$ m.
From the ground surface: $-24 + 2415 = 2391$ m.

Answer: The target lies at a true vertical depth of 2400 m from the rotary table of the drilling rig, 2397 m from the rotary table of the workover rig, and 2391 m from the ground surface.

What it means. The formation has stayed in place, but its “depth” has changed by 3 m, because the zero datum has changed. In a vertical well the depth we found is also the length of the hole. In an inclined well the length along the hole is greater, and the profile calculation gives it. Later in the lesson, a depth of 2400 m from the rotary table is taken for this target. The data in this example are assumed for illustration.

In the plan view there is one main rule: the coordinates of all the wells of a field must be in one system, or the distance between neighbouring wells cannot be estimated. Usually these are rectangular coordinates in metres, in the Gauss–Krüger projection that is used in the surveying of the CIS countries, or in the UTM projection (Universal Transverse Mercator). Latitude and longitude are converted into them. The profile itself is calculated in local coordinates, the displacements to the north and to the east from the wellhead (on a pad or a platform, from a point common to all the wells). The seven-digit numbers of the national grid are not convenient for this.

Departure and azimuth of the target from the coordinates

A = \sqrt{\Delta N^2 + \Delta E^2}, \qquad \varphi = \arctan\frac{\Delta E}{\Delta N}

where:

$A$ departure, the distance from the wellhead to the projection of the point on the horizontal plane, m
$\Delta N,\ \Delta E$ displacements of the point from the wellhead to the north and to the east, the differences of the northing and easting coordinates; to the south and to the west they are negative, m
$\varphi$ azimuth, the angle in the horizontal plane from the direction to the north to the direction to the point, clockwise, from 0 to 360°, degrees

Units: metres; the angle in degrees (the arctan function in spreadsheets and programming languages returns radians: multiply by $180/\pi$ )

Holds when

The arctangent gives an angle from −90° to +90°, so we check the quadrant from the signs: for $\Delta N < 0$ we add 180° to the result, and for $\Delta N > 0$ and $\Delta E < 0$ we add 360°.
For $\Delta N = 0$ the azimuth is 90° (the target is to the east) or 270° (to the west).
The azimuth is measured from the same north as the coordinate grid (see the box below).

The departure and the azimuth turn a three-dimensional problem into a flat one. We calculate the profile in the vertical plane that passes through the wellhead and the target at the azimuth $\varphi$ , as if the well were drawn on a sheet of paper.

Worked exampleDeparture and azimuth of the target

Given (assumed data)

Wellhead (rectangular coordinates): northing $X = 5{,}187{,}420$ m, easting $Y = 512{,}360$ m
Target: $X = 5{,}188{,}080$ m, $Y = 513{,}240$ m

Solution

Displacements: $\Delta N = 5{,}188{,}080 - 5{,}187{,}420 = 660$ m, $\Delta E = 513{,}240 - 512{,}360 = 880$ m.
Departure: $A = \sqrt{660^2 + 880^2} = 1100$ m.
Azimuth: $\varphi = \arctan(880/660) = 53.1^\circ$ . Both displacements are positive, so the target is in the north-east quadrant, and no correction is needed.

Answer: The target lies 1100 m from the wellhead at an azimuth of 53.1°.

What it means. In the surveying of the CIS countries the $X$ axis points north and the $Y$ axis points east, the other way round from school mathematics. English-language documents write Northing and Easting. If the axes are mixed up, the well is turned in the wrong direction. The data in this example are assumed for illustration.

Types of profiles

A profile is made of sections of three kinds: vertical, curved at a constant rate (a build or a drop of the inclination) and straight inclined. There are three typical combinations.

Three typical profiles

Profile: “Build and hold”, also called tangential or three-interval
What it consists of: a vertical section, a build of the inclination and a straight inclined section to the target, the tangent section
When it is chosen: by default: there is only one curved section, and this profile is the easiest to follow in drilling
What it costs: the well enters the formation at an angle, and the farther the target, the greater the angle

Profile: S-shaped
What it consists of: the same three sections and a section where the angle drops, often down to vertical
When it is chosen: when the formation must be penetrated vertically or almost vertically: in exploration and appraisal wells it is then easier to judge the productivity of the formation and the results of its treatments
What it costs: a greater angle in the tangent section, a longer well and a second curved section (example below)

Profile: Deep kick-off
What it consists of: a long vertical section and the build of the angle at a great depth
When it is chosen: to pass under an obstacle, for example under the flank of a salt dome, and in horizontal wells
What it costs: the well has to be deviated in deep, dense and hard rocks, which is the most difficult variant

Profile	What it consists of	When it is chosen	What it costs
“Build and hold”, also called tangential or three-interval	a vertical section, a build of the inclination and a straight inclined section to the target, the tangent section	by default: there is only one curved section, and this profile is the easiest to follow in drilling	the well enters the formation at an angle, and the farther the target, the greater the angle
S-shaped	the same three sections and a section where the angle drops, often down to vertical	when the formation must be penetrated vertically or almost vertically: in exploration and appraisal wells it is then easier to judge the productivity of the formation and the results of its treatments	a greater angle in the tangent section, a longer well and a second curved section (example below)
Deep kick-off	a long vertical section and the build of the angle at a great depth	to pass under an obstacle, for example under the flank of a salt dome, and in horizontal wells	the well has to be deviated in deep, dense and hard rocks, which is the most difficult variant

CalculatedCalculated profiles of two wells to the same target, from the examples of the lesson (depth 2400 m, departure 1100 m, kick-off at 500 m, build 3° per 30 m): the build-and-hold profile, and the S-shaped one with a drop of 2° per 30 m and vertical for the last 100 m. The scales of the axes are different, so the slope in the picture is exaggerated.

Parameters of the profile

The kick-off point (KOP) is the depth below which the well stops being vertical: here the build of the inclination starts, and the well is turned to the azimuth of the target. Its choice is a compromise between geometry and geology. Geometry calls for starting higher up the hole (at a shallower depth): the more height is left to the target, the smaller the inclination needed, so for distant targets the kick-off is made as early as possible. Geology sets its own conditions: in the upper part of the section the rocks are softer and it is easier to deviate the well, but in loose, uncompacted clays the build of the angle goes badly, and the kick-off needs a stable interval.

The build-up rate (BUR) and the drop-off rate (DOR) show by how many degrees the inclination changes per unit of length of the well. In international practice they are given in degrees per 30 m or per 100 ft, and in Russian-language documents often in degrees per 10 m. At a constant rate the build section is an arc of a circle. In ordinary inclined wells the build is most often 1–3° per 30 m. In projects in the fields of the CIS countries one also finds 1–1.5° per 10 m, that is, 3–4.5° per 30 m. It is not worth bending the well more than needed: the sharper the bend, the harder it is to run strings and instruments, and the faster the pipes wear.

Radius of curvature of a section with a constant rate

R = \frac{180}{\pi}\cdot\frac{l_0}{i}

where:

$R$ radius of curvature of the axis of the well, m
$i$ build-up or drop-off rate: degrees per the reference length $l_0$ , °/30 m
$l_0$ reference length of the well to which the rate is related: 30 m, 100 ft or 10 m, m

Units: $i$ in degrees per the reference length; $R$ in the same units of length as $l_0$

Holds when

The rate is constant and the azimuth does not change: the section is an arc of a circle in the vertical plane.
The factor $180/\pi$ appears because the rate is given in degrees, while the length of an arc is the radius times the angle in radians (1 rad $= 180/\pi \approx 57.3^\circ$ ).

The higher the rate, the smaller the radius and the sharper the turn. In a flat profile (one that lies in a single plane) the build-up rate is the same as the dogleg severity of the well, which shows how sharply the well bends. In the general case the dogleg severity also takes the change of the azimuth into account (Directional survey calculations).

In other unit systems (3)

Degrees per 30 m

R\,[\text{m}] = \frac{1718.9}{i\,[^\circ/30\ \text{m}]}

$i$ in degrees per 30 m, $R$ in metres.

$1718.9 = 180\cdot 30/\pi$ .

Degrees per 100 ft (oilfield units)

R\,[\text{ft}] = \frac{5729.6}{i\,[^\circ/100\ \text{ft}]}

$i$ in degrees per 100 ft, $R$ in feet.

$5729.6 = 180\cdot 100/\pi$ . Since 100 ft is 30.48 m and not 30 m, $i\,[^\circ/30\ \text{m}] = 0.984\,i\,[^\circ/100\ \text{ft}]$ .

Degrees per 10 m

R\,[\text{m}] = \frac{573.0}{i\,[^\circ/10\ \text{m}]}

$i$ in degrees per 10 m, $R$ in metres.

$573.0 = 180\cdot 10/\pi$ ; $i\,[^\circ/10\ \text{m}] = i\,[^\circ/30\ \text{m}]/3$ .

Worked exampleA rate of 3° per 30 m: the radius and other units

Given (assumed data)

Build-up rate: 3° per 30 m
Inclination to build: 32°

Solution

Radius of curvature: $R = 180\cdot 30/(\pi\cdot 3) = 573.0$ m.
In degrees per 100 ft: $3\cdot 30.48/30 = 3.048$ . In degrees per 10 m: $3/3 = 1$ . In degrees per 100 m: $3\cdot 100/30 = 10$ .
Length of the build section up to 32°: $30\cdot 32/3 = 320$ m of the hole.
For comparison: a rate of 3° per 100 ft is 2.953° per 30 m, and the radius is 1910 ft, that is about 582 m.

Answer: A rate of 3° per 30 m has a radius of 573.0 m, and building 32° takes 320 m of the hole.

What it means. Degrees per 30 m and per 100 ft differ by only 1.6%, and in practice they are often not distinguished. But on a long build section even this difference gives metres. Degrees per 10 m are three times smaller in value: if we mix them up with degrees per 30 m, we get a well that is three times too steep or three times too gentle. The reference length is always stated. The data in this example are assumed for illustration.

Radius of curvature at different rates

Degrees per 30 m: 1
Degrees per 10 m: 0.33
Radius, m: 1719

Degrees per 30 m: 2
Degrees per 10 m: 0.67
Radius, m: 859

Degrees per 30 m: 3
Degrees per 10 m: 1.0
Radius, m: 573

Degrees per 30 m: 4.5
Degrees per 10 m: 1.5
Radius, m: 382

Degrees per 30 m: 6
Degrees per 10 m: 2.0
Radius, m: 286

Degrees per 30 m: 10
Degrees per 10 m: 3.3
Radius, m: 172

Degrees per 30 m	Degrees per 10 m	Radius, m
1	0.33	1719
2	0.67	859
3	1.0	573
4.5	1.5	382
6	2.0	286
10	3.3	172

The high rates in the last rows belong to special profiles, for example to the build of the angle before the horizontal section of a well.

The inclination of the tangent section (the tangent angle, or drift angle) is the angle between the vertical and the straight inclined section that runs from the end of the build to the target, touching the arc. This angle is not assigned. It is found by calculation, and we check whether it falls into a convenient range: for ordinary wells, about 10 to 60°. At smaller angles it is hard to keep the well on course: in an almost vertical well the azimuth is poorly defined, and the well “wanders”. At 60–70° the instruments on a wireline (a cable) stop going down under their own weight (Open-hole logging: why, how and when), and the friction of the strings against the walls becomes a serious limit. We must take care of cuttings removal (carrying the rock fragments out of the hole) already at medium angles.

Calculating the build-and-hold profile

The positions of the wellhead and the target are known, that is, the true vertical depth of the target and its departure. The designer chooses the kick-off depth and the build-up rate. We have to find the inclination of the tangent section, the coordinates of the end of the build, and the lengths of the sections along the hole. The driller sees only the length along the hole. By this length the driller must know where to end the build and where to expect the target.

SchematicThe geometry of the build-and-hold profile (a sketch). K is the kick-off point, O is the centre of the build arc, E is the end of the build, and T is the target. The straight section E–T touches the arc, so it is perpendicular to the radius O–E, and the triangle O–E–T is right-angled. The inclination

\theta

is the sum of the angle

\beta

between the vertical and the line T–O and the angle

\gamma

between the lines T–O and T–E. In the sketch the departure of the target is greater than the radius of the arc. If it is smaller, the centre O lies farther out horizontally than the target, the angle

\beta

is negative, and

\theta = \gamma - |\beta|

Inclination of the tangent section

\theta = \beta + \gamma, \qquad \beta = \arctan\frac{A_{\text{T}} - R}{\Delta H}, \qquad \gamma = \arcsin\frac{R}{\sqrt{(A_{\text{T}} - R)^2 + \Delta H^2}}

where:

$\theta$ inclination of the tangent section, degrees
$A_{\text{T}}$ departure of the target from the wellhead, m
$R$ radius of the build arc, m
$\Delta H$ difference between the true vertical depths of the target and of the kick-off point, $H_{\text{T}} - H_{\text{KOP}}$ , m
$\beta$ angle between the vertical and the line that joins the target with the centre of the arc; it is negative if the departure is smaller than the radius, degrees
$\gamma$ angle between this line and the tangent section; under the root is the square of the distance from the centre of the arc to the target, degrees

Units: lengths in the same units; angles in degrees (if the arctangent and arcsine return radians, multiply the result by $180/\pi$ )

Holds when

The profile lies in one vertical plane, which passes through the wellhead and the target; the azimuth is constant.
The upper section is vertical. The build goes at a constant rate from zero to $\theta$ , and then the well is straight right up to the target.
The solution exists if the distance from the centre of the arc to the target is not less than the radius. Otherwise the target lies inside the build circle, and we need a higher rate or a shallower kick-off.
The formula is written for angles up to 90°.

The angle is decided by three lengths: the departure, the height from the kick-off to the target, and the radius. When the radius is small compared with them, the second term tends to zero, and what is left is the obvious $\tan\theta \approx A_{\text{T}}/\Delta H$ , the straight line from the kick-off point to the target.

Points and lengths of the build-and-hold profile

\begin{aligned} L_{\text{build}} &= \frac{\pi R\,\theta}{180} = \frac{l_0\,\theta}{i}, \qquad L_{\text{tan}} = \sqrt{(A_{\text{T}} - R)^2 + \Delta H^2 - R^2},\\ H_{\text{E}} &= H_{\text{KOP}} + R\sin\theta, \qquad A_{\text{E}} = R\,(1 - \cos\theta),\\ L &= H_{\text{KOP}} + L_{\text{build}} + L_{\text{tan}} \end{aligned}

where:

$L_{\text{build}}$ length of the build section along the hole, m
$L_{\text{tan}}$ length of the tangent section, m
$H_{\text{E}},\ A_{\text{E}}$ true vertical depth and departure of the end of the build section, m
$H_{\text{KOP}}$ depth of the kick-off point (KOP); the upper section is vertical, so it is also the length along the hole, m
$L$ length of the well from the zero datum to the target, m
$\theta$ inclination of the tangent section; in the first formula in degrees, degrees
$i,\ l_0$ build-up rate and its reference length, °/30 m; m
$R,\ A_{\text{T}},\ \Delta H$ radius of the build arc, departure of the target and the difference between the depths of the target and of the kick-off point, as in the formula for the angle, m

Units: lengths in metres; $\theta$ in the formula for the length of the arc in degrees ( $\pi\theta/180$ is the same angle in radians)

Holds when

The same as for the angle. The length of the tangent section is a leg of the right triangle O–E–T.
A check: $L_{\text{tan}} = (H_{\text{T}} - H_{\text{E}})/\cos\theta = (A_{\text{T}} - A_{\text{E}})/\sin\theta$ .

The length along the hole at the end of the build, $H_{\text{KOP}} + L_{\text{build}}$ , tells the driller where to stop building and go over to the tangent section. The full length $L$ shows how much pipe, time and mud the well will need.

Worked exampleThe build-and-hold profile to the target of the lesson

Given (assumed data)

Target: true vertical depth 2400 m from the rotary table, departure 1100 m
Kick-off depth $H_{\text{KOP}}$: 500 m, below the shoe of the surface casing (the first casing string)
Build-up rate $i$: 3° per 30 m

Solution

Radius: $R = 180\cdot 30/(\pi\cdot 3) = 572.96$ m. Height from the kick-off to the target: $\Delta H = 2400 - 500 = 1900$ m.
The angle $\beta = \arctan\dfrac{1100 - 572.96}{1900} = 15.5^\circ$ .
The distance from the centre of the arc to the target: $\sqrt{(1100 - 572.96)^2 + 1900^2} = 1971.74$ m. The angle $\gamma = \arcsin(572.96/1971.74) = 16.9^\circ$ .
The inclination of the tangent section: $\theta = 15.5 + 16.9 = 32.4^\circ$ .
The build section: $L_{\text{build}} = 30\cdot 32.4/3 = 324$ m. It ends at a length along the hole of $500 + 324 = 824$ m, at a depth of $H_{\text{E}} = 500 + 572.96\cdot\sin 32.4^\circ = 807$ m, with a departure of $A_{\text{E}} = 572.96\cdot(1 - \cos 32.4^\circ) = 89$ m.
The tangent section: $L_{\text{tan}} = \sqrt{1971.74^2 - 572.96^2} = 1887$ m. The check: $(2400 - 807)/\cos 32.4^\circ = 1887$ m.
The length of the well to the target: $L = 500 + 324 + 1887 = 2711$ m, which is 311 m more than the true vertical depth.

Answer: The inclination of the tangent section is 32.4°; the build ends at 824 m along the hole; the length of the well to the target is 2711 m.

What it means. An angle of about 32° lies in the convenient range of 10 to 60°. An error of only 1° on a tangent section 1887 m long would take the bottom 33 m off target. This is why the angle is calculated and not read from a drawing with a protractor, and why in drilling the path is checked all the time with survey measurements. The data in this example are assumed for illustration.

Calculating the S-shaped profile

An S-shaped profile adds to the build and the tangent section a section where the angle drops: a second arc that curves the other way. If the angle is dropped to zero, then at the end of the drop the well is already above the target and goes on vertically. The tangent section now touches both arcs, and the calculation comes down to the one we know. In the formula for the angle the radius is replaced by the sum of the radii of the two arcs, and the depth of the target by the depth at which the drop of the angle ends.

Inclination of the tangent section of an S-shaped profile

\begin{aligned} \theta &= \arctan\frac{A_{\text{T}} - R_1 - R_2}{\Delta H} + \arcsin\frac{R_1 + R_2}{D},\\ D &= \sqrt{(A_{\text{T}} - R_1 - R_2)^2 + \Delta H^2}, \qquad L_{\text{tan}} = \sqrt{D^2 - (R_1 + R_2)^2} \end{aligned}

where:

$\theta$ inclination of the tangent section, degrees
$R_1,\ R_2$ radii of the build arc and of the drop arc, m
$\Delta H$ difference between the true vertical depths of the end of the drop section and of the kick-off point, $H_{\text{drop}} - H_{\text{KOP}}$ , m
$A_{\text{T}}$ departure of the target; it is fully built by the end of the drop section, m
$D$ distance between the centres of the two arcs, m
$L_{\text{tan}}$ length of the tangent section, the segment of the common tangent of the two arcs, m

Units: lengths in the same units; angles in degrees

Holds when

The profile is in one vertical plane: a build from zero to $\theta$ , a tangent section, a drop from $\theta$ to zero, and then a vertical section to the target.
The profile can be built only if the distance between the centres of the arcs is not less than the sum of the radii, $D \ge R_1 + R_2$ . Otherwise the arcs do not have enough room.
If the angle is dropped not to zero but to a given angle of entry into the formation, the calculation is based on the same condition of tangency, but the formulas are longer.

The lengths of the build and the drop sections are $l_0\theta/i_1$ and $l_0\theta/i_2$ , where $i_1$ and $i_2$ are the build-up and drop-off rates and $\theta$ is in degrees. The coordinates of the end of the build are as in the build-and-hold profile, and the start of the drop lies at a depth of $H_{\text{drop}} - R_2\sin\theta$ at a departure of $A_{\text{T}} - R_2(1 - \cos\theta)$ . Compared with a build-and-hold profile to the same target, the angle is greater, because the whole departure has to be built earlier and over a smaller height.

Worked exampleAn S-shaped profile to the same target

Given (assumed data)

Target: depth 2400 m, departure 1100 m; the formation must be entered vertically
End of the drop section $H_{\text{drop}}$: 2300 m, 100 m before the target
Kick-off depth: 500 m
Rates: build 3° per 30 m, drop 2° per 30 m

Solution

Radii: $R_1 = 180\cdot 30/(\pi\cdot 3) = 573.0$ m, $R_2 = 180\cdot 30/(\pi\cdot 2) = 859.4$ m; their sum is 1432.4 m. Height: $\Delta H = 2300 - 500 = 1800$ m.
The first term: $\arctan\dfrac{1100 - 1432.4}{1800} = -10.5^\circ$ . It is negative, because the departure is smaller than the sum of the radii.
The distance between the centres of the arcs: $D = \sqrt{(1100 - 1432.4)^2 + 1800^2} = 1830.4$ m. The second term: $\arcsin(1432.4/1830.4) = 51.5^\circ$ .
The inclination: $\theta = -10.5 + 51.5 = 41.0^\circ$ .
The lengths of the sections: build $30\cdot 41.0/3 = 410$ m; tangent $\sqrt{1830.4^2 - 1432.4^2} = 1140$ m; drop $30\cdot 41.0/2 = 615$ m; vertical section 100 m.
The length of the well: $500 + 410 + 1140 + 615 + 100 = 2765$ m.
If both the build and the drop were slower, with rates of 2 and 1° per 30 m, the sum of the radii would be 2578 m, which is greater than the distance between the centres, 2329 m. Such a profile cannot be built.

Answer: The inclination of the tangent section is 41.0°, and the length of the well is 2765 m.

What it means. With the same kick-off and the same build-up rate, entering the formation vertically has cost about 9° of inclination and 54 m of well, compared with the build-and-hold profile to the same target (32.4° and 2711 m). To this we add the second curved section and the greater friction of the strings against the walls. The data in this example are assumed for illustration.

What limits the choice of the profile

Geometry is only a frame. A profile that looks good on paper must still pass through the real geological section, fit the casing programme and not come too close to neighbouring wells.

The target and its tolerance. Geologists and reservoir engineers give the target by coordinates and by an elevation, and the designer converts them into a departure, an azimuth and a depth from the rotary table. The target is not a point but an area. In Russian-language practice it is most often given as a tolerance circle, a circle at the planned depth that the well must hit. (The tolerance is the error in position that is allowed.) The size and shape of this area is dictated by the geology: the outlines of the structure and the faults. A tight tolerance means frequent surveys and corrections of the path, and so a more expensive well.
The geological section. The geologists give the depths of the tops of the formations. Intervals where the deflection tool works badly, for example loose, uncompacted clays, are drilled vertically, and the build of the angle starts in the next suitable bed.
The casing programme and the mud. The sections of the profile are tied to the depths at which the casing strings are set (Casing: purpose, types, properties and standards). The kick-off often starts just below the shoe of the surface casing, and in strongly inclined wells the build section is cased off before the long tangent section is drilled. We try to put the most difficult sections in stable rocks. For the build and the tangent section, a switch to an oil-based mud (a mud with oil as the base liquid) is common: it lowers the friction and holds the walls better (Drilling fluids: functions and properties).
The wellhead. The simplest thing is to put the rig above the target, but this is not always possible. A well pad or a platform is placed so that it can reach all parts of the accumulation at acceptable inclinations.
Neighbouring wells. On a pad or a platform the wellheads stand a few metres apart, and the wells are closest to each other in the upper part of the section. The planned path must pass all the neighbouring wells with a margin that takes into account the position uncertainty of both the new well and the old ones.

All these requirements are linked, and the design goes round in a circle. The profile changes the depths at which the casing strings are set and the bending of the pipe in a curved well (Casing design), and the strings and the mud limit the profile. We recalculate it until the geometry, the casing programme, the loads on the drill string and the possibility of running well logs all agree.

Terms and symbols

Well profile: The planned path of the well in the vertical plane, made up of vertical, curved and straight inclined sections.
Measured depth and true vertical depth: The length of the well from the zero datum to a point (MD) and the vertical distance to it (TVD). In an inclined well the first is greater.
Rotary table elevation, elevation of a point: The height of the rotary table and the height of a point underground relative to sea level. Below sea level they are negative.
Inclination $\theta$: The angle between the axis of the well and the vertical.
Azimuth $\varphi$: The angle in the horizontal plane between the direction to the north and the horizontal projection of the well, measured clockwise.
Horizontal departure $A$: The horizontal distance from the wellhead to the projection of a point of the well or of the target.
Kick-off point (KOP): The depth at which the well starts to deviate from the vertical.
Build-up rate $i$: The change of inclination per unit of length of the well: degrees per 30 m, per 100 ft or per 10 m.
Tangent section: A straight inclined section with a constant inclination (also called the hold section).
Tolerance circle: The area at the planned depth that the well must hit.

Common mistakes and misconceptions

Wrong

“The depth of a well is one number.”

In fact

An inclined well has at least two, along the hole and vertical, and each is measured from its own zero: the rotary table, sea level, the wellhead. Without these details the depths cannot be compared.

Wrong

“3° per 30 m and 3° per 100 ft are the same, and degrees per 10 m are just another way of writing.”

In fact

The first two differ by 1.6%, but 3° per 10 m is 9° per 30 m, a bend three times as sharp. Always state the reference length.

Wrong

“The deeper we start the build, the less inclined well there is, and the simpler the well.”

In fact

With a deep kick-off the same departure has to be built over a smaller height. The angle of the tangent section grows, the well gets longer, and it has to be deviated in harder rocks.

Wrong

“An S-shaped profile is always better: we enter the formation vertically.”

In fact

A vertical entry costs a greater inclination, a longer well, a second curved section and more friction of the strings. It is chosen when a vertical entry is really needed.

Wrong

“The azimuth is the arctangent of the ratio of the east displacement to the north displacement.”

In fact

Only in the north-east quadrant. In the others the arctangent must be corrected by the signs of the displacements: for a target south of the wellhead it gives a direction turned by 180°, and in the north-west it gives a negative angle to which 360° is added. And we must always know which north the azimuth is measured from.

In depth: advanced detail

Derivation of the formula for the inclination

Let us put the origin of the coordinates at the wellhead, the departure axis towards the target, and the depth axis downwards. The centre of the build arc lies at the depth of the kick-off point, at a distance $R$ from the well. The point of the arc where the inclination is $\theta$ has the departure $R(1 - \cos\theta)$ and the depth $H_{\text{KOP}} + R\sin\theta$ , and the tangent at it makes the same angle $\theta$ with the vertical. Now let the tangent pass through the target.

The condition of tangency: the straight section passes through the target

\frac{A_{\text{T}} - R\,(1 - \cos\theta)}{\Delta H - R\sin\theta} = \tan\theta \quad\Longrightarrow\quad \Delta H\sin\theta - (A_{\text{T}} - R)\cos\theta = R

where:

$A_{\text{T}},\ \Delta H$ departure of the target and its depth relative to the kick-off point, m
$R$ radius of the build arc, m
$\theta$ inclination at the end of the build and in the tangent section, degrees or radians

Units: lengths in the same units; the trigonometric functions of the angle $\theta$ in the units that the calculator needs

Holds when

A flat profile: a vertical, an arc of a circle, and a tangent to it.
The second equality is obtained by multiplying the first by $\cos\theta$ and by the denominator.

Let us divide the second equality by the distance from the centre of the arc to the target, $D = \sqrt{(A_{\text{T}} - R)^2 + \Delta H^2}$ , and introduce the angle $\beta$ for which $\cos\beta = \Delta H/D$ and $\sin\beta = (A_{\text{T}} - R)/D$ . On the left we get $\sin\theta\cos\beta - \cos\theta\sin\beta = \sin(\theta - \beta)$ , and on the right $R/D$ . From this, $\theta = \beta + \arcsin(R/D)$ . The same can be seen in the figure: in the right triangle O–E–T the leg O–E equals $R$ and the hypotenuse O–T equals $D$ , so the sine of the angle $\gamma$ at the vertex T is $R/D$ .

For an S-shaped profile the reasoning is the same, except that the tangent is common to two arcs that lie on opposite sides of it. The centre of the drop arc is at the depth of the end of the drop, at a distance $R_2$ from the vertical through the target, on the side of the wellhead. The distances from the two centres to the tangent are $R_1$ and $R_2$ , and they are laid off in opposite directions. So in the condition of tangency the sum $R_1 + R_2$ appears instead of $R$ , and the distance between the centres instead of the distance to the target.

How the kick-off depth and the rate change the profile

CalculatedThe calculated inclination of the tangent section for the target of the lesson (depth 2400 m, departure 1100 m) against the kick-off depth, for build-up rates

i

of 2, 3 and 4.5° per 30 m. The dashed line is 60°, the guide for the upper limit for ordinary wells. Each curve stops where the angle reaches 90°: the build arc ends horizontally exactly at the depth of the target, and for a deeper kick-off there is no straight section to the target any more. The deeper the kick-off, the more the curves diverge.

Worked exampleKick-off at 250, 500 and 1400 m

Given (assumed data)

Target: depth 2400 m, departure 1100 m
Build-up rate: 3° per 30 m, radius 573.0 m
Kick-off depth: 250, 500 and 1400 m

Solution

Kick-off at 250 m: $\Delta H = 2150$ m, $\theta = 28.8^\circ$ , length of the well 2676 m.
Kick-off at 500 m (the example above): $\theta = 32.4^\circ$ , length of the well 2711 m.
Kick-off at 1400 m: $\Delta H = 1000$ m, $\theta = 58.2^\circ$ , length of the well 2957 m.
Another rate with the kick-off at 500 m: 2° per 30 m gives 33.9° and 2720 m, and 4.5° per 30 m gives 31.5° and 2705 m. The difference in the angle is about 2°.
The same rates with the kick-off at 1400 m: 70.2° and 53.8°. The difference is now about 16°.

Answer: If we move the kick-off from 250 down to 1400 m, we get an angle of 58.2° instead of 28.8° and a well 281 m longer.

What it means. The kick-off depth has the strongest effect: the deeper it is, the steeper and longer the well, and with a kick-off at 1400 m the angle is already close to the limit of 60°. With an early kick-off the build-up rate changes almost nothing (a couple of degrees for rates that differ by more than a factor of two). But with a deep kick-off it becomes decisive: with a slow build the angle goes beyond 70°. So for distant targets the kick-off is made as shallow as possible, as far as the rocks and the setting depth of the surface casing allow, and when a deep kick-off cannot be avoided, the angle is built at a higher rate. The data in this example are assumed for illustration.

Connections to other topics

How a well is drilled: the process and the people involved — the drilling programme, which includes the profile of the well, and the roles of the operator and the service companies.
Pore pressure and overburden pressure — why pressure is calculated from the true vertical depth and not from the length of the well.
Casing: purpose, types, properties and standards — the casing strings and the depths at which they are set, with which the sections of the profile are matched.
Casing design — the additional bending stresses in a string run into a curved well.
Drilling fluids: functions and properties — the choice of mud for inclined sections: friction, cuttings removal, stability of the walls.
Open-hole logging: why, how and when — how logging tools are delivered to wells with a large inclination.
Sucker-rod pumping — the wear of rods and tubing in a curved well, one of the consequences of the profile for production.
Deflection tools and downhole motors — what deviates the well: deflection tools, downhole motors, steerable systems.
Directional survey calculations — how the actual path is found from the measurements of the angle and azimuth and compared with the planned one.

Check yourself

1
A vertical well and an inclined well penetrate the same horizontal formation. Which of them has the greater depth of the top along the hole, and which the greater vertical depth (from the same zero datum)?
1. A.The inclined well has the greater depth along the hole; the vertical depths are the same
2. B.Both depths are the same
3. C.The depths along the hole are the same; the vertical depth is greater in the vertical well
4. D.Both depths are greater in the inclined well
Show the answer
Correct answer: A. The inclined well has the greater depth along the hole; the vertical depths are the same
The true vertical depth is the vertical distance from the zero datum to the point. For a horizontal formation it is the same for any well. The measured depth is the length of the path to this point, and in an inclined well it is greater.
2
The target lies 300 m south and 400 m east of the wellhead. Find the departure and the azimuth.
Show the answer
$\Delta N = -300$ m, $\Delta E = 400$ m. The departure is $\sqrt{300^2 + 400^2} = 500$ m. The arctangent: $\arctan(400/(-300)) = -53.1^\circ$ . The north displacement is negative, so we add 180°: the azimuth is $126.9^\circ$ , south-east, as it must be.
3
The build-up rate is 2.5° per 100 ft. How much is this in degrees per 30 m and per 10 m, and what is the radius of curvature?
Show the answer
In degrees per 30 m: $2.5\cdot 30/30.48 \approx 2.46$ ; per 10 m it is three times smaller, about 0.82. The radius is $5729.6/2.5 \approx 2292$ ft, that is about 699 m (the same is given by $1718.9/2.46$ ).
4
The departure of the target happens to equal the radius of the build arc: $A_{\text{T}} = R = 860$ m, and the height from the kick-off to the target is 1720 m. What is the inclination of the tangent section?
Show the answer
For $A_{\text{T}} = R$ the first term is zero, and the distance from the centre of the arc to the target equals $\Delta H$ . What is left is $\theta = \arcsin(R/\Delta H) = \arcsin(860/1720) = \arcsin 0.5 = 30^\circ$ .
5
The target is farther than planned: the departure has grown, and the depth is the same. Which of these will keep the inclination of the tangent section within its former limits?
1. A.Put the kick-off point deeper
2. B.Put the kick-off point shallower (higher up the hole)
3. C.Reduce the build-up rate
4. D.Change to an S-shaped profile
Show the answer
Correct answer: B. Put the kick-off point shallower (higher up the hole)
The shallower the kick-off, the more height there is to build the departure, and the smaller the angle we need. A deeper kick-off and an S-shaped profile increase the angle, and a smaller build-up rate stretches the arc and also increases the angle.
6
Why are S-shaped profiles often required for exploration and appraisal wells, although they are longer and more complicated?
Show the answer
In an S-shaped profile the well enters the formation vertically, at a right angle to the bedding. This makes it easier to evaluate the productivity of the formation and the results of treatments: the thickness penetrated is close to the true one, and the inflow conditions are the same as in a vertical well. The price is a greater inclination in the tangent section, a longer well and a second curved section.

Updated: 3 October 2026