Well Trajectory Design
Description of calculation methodology related to well trajectory design and design primitives

Wellbores and Related Quantities

The coordinate system used throughout this text is the well known NEV - coordinate system, i.e. North-East-Vertical. The vertical axes has positive direction downwards. As a right-handed coordinate system, i.e. X −Y −Z, a possible identification is N = X, E = Y, and V = Z. We remark that every coordinate related consideration below is with respect to NEV-coordinates.

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Definitions

Definition 1: A Wellbore Section between two points
$\mathbf x_1, \mathbf x_2\in \mathbb R^3$
, is defined as a smooth curve
$\gamma : [a,b] \mapsto \mathbb R^3, \quad a,b \in \mathbb R \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)$
such that
$\gamma(a) = \mathbf x_1 \text{ and }\ \gamma(b) = \mathbf x_2$
.
Definition 2: A wellbore is a piecewise differentiable curve
$\Gamma : [a,b]\mapsto \mathbb R^3$
,
$a,b \in\mathbb R$
, such that each piece is described by a wellbore section. Further, given a point
$\mathbf x_0$
on the curve, i.e.
$\mathbf x_0 = \Gamma(t_0)$
for
$t_0\in [a,b]$
, the tangent vector to the curve at
$\mathbf x_0$
is given by
$\mathbf T(\mathbf x_0) = \frac{d\Gamma}{dt}(t_0) := \Gamma'(t_0)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2)$
Definition 3: Given a wellbore
$\Gamma$
and a point
$\mathbf x_0$
on the curve. The inclination
$\alpha$
, to the wellbore at
$\mathbf x_0$
is defined as the angle between the tangent at
$\mathbf x_0$
and the vertical.
Definition 4: Given a wellbore
$\Gamma$
and a point
$\mathbf x_0$
on the curve. The azimuth
$\epsilon$
, to the wellbore at
$\mathbf x_0$
is defined as the angle between the tangent at
$\mathbf x_0$
and north.
Definition 5: Curvature, aka Dogleg severity, and radius of curvature. Given a wellbore section
$\gamma$
. Assume also that the tangent vector
$\mathbf T$
is nowhere zero, and that the curve is parametrized with respect to curve length, t, such that
$\|T(t)\| = 1, \forall t$
. Then the curvature, also known as dogleg severity is defined as
$\kappa(t) = \|\mathbf T'(t)\| = \|\gamma''(t)\|\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(3)$
The radius of curvature is defined as
$R(t) = \frac{1}{\kappa(t)}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(4)$
We remark that points on a wellbore where the curve is not twice differentiable, have no well-defined curvature and radius of curvature.
Definition 6: Build- & Turn- rates, Br and Tr . Build- and Turn- rates are defined as rate of change of inclination and azimuth along the wellbore, respectively. Formally, let
$\alpha(t)$
and
$\epsilon(t)$
denote the inclination and azimuth along the wellbore, parametrized with respect to curve length, t, Then the Build- and Turn- rates are defined as
$Br(t) = \frac{d\alpha}{dt}(t), \qquad Tr(t) = \frac{d\epsilon}{dt}(t) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(5)$
Definition 7: Toolface. Given a wellbore, and two points
$\mathbf x_0 \neq \mathbf x_1$
on it, with inclinations,
$\alpha_0, \alpha_1$
, and azimuths,
$\epsilon_0$
,
$\epsilon_1$
. Assume that the arc between
$\mathbf x_0$
and
$\mathbf x_1$
spans the angle
$\theta$
. The associated toolface angle
$\gamma$
is defined, either by
$\cos(\gamma) = \frac{\cos(\alpha_0)\cos(\theta) - \cos(\alpha_1)}{\sin(\alpha_0)\sin(\theta)} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(6)$
or equivalently
$\sin(\gamma) = \frac{\sin(\alpha_1)\sin(\epsilon_1 - \epsilon_0)}{\sin(\theta)}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(7)$

Identities

From equation 6 we can express the inclination associated with
$\mathbf x_1$
as
$\cos(\alpha_1) = \cos(\alpha_0)\cos(\theta) - \sin(\alpha_0)\sin(\theta)\cos(\gamma) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(8)$
From [1] we get the following relation between change in azimuth,
$\epsilon_1 - \epsilon_0$
, toolface,
$\gamma_{10}$
, start inclination,
$\alpha_0$
, and total angle change,
$\theta_{10}$
$\tan(\epsilon_1 - \epsilon_0) = \frac{\sin(\gamma_{10})\tan(\theta_{10})}{\sin(\alpha_0) + \cos(\alpha_0)\cos(\gamma_{10})\tan(\theta_{10})}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(9)$

Minimum Curvature Method and Constant Build and Turn Method

Minimum Curvature Method

Minimum curvature method (MCM) is the preferred parametrization of the section between adjacent survey points in a wellbore. This is consistent with the reasonable assumption that the wellbore between two such survey points has constant curvature.
In MCM, the following inputs are given
Start point
$\mathbf x_A = (N_A, E_A, V_A)\in \mathbb R^3$
Two unit tangent vectors,
$\mathbf T_A$
at
$\mathbf x_A$
, and
$\mathbf T_B$
at
$\mathbf x_B\in \mathbb R^3$
, where
$\mathbf x_B$
to be determined.
The curve length, dM , i.e. the difference in measured depth between
$\mathbf x_A$
and
$\mathbf x_B$
The tangent vectors are decomposed as
$\mathbf T_i = (\text{d} N_i, \text{d}E_i, \text{d}V_i), \quad i = A, B, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(10)$
where
$\text{d}N_i = \sin(\alpha_i)\cos(\epsilon_i) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(11)$
$\text{d}E_i = \sin(\alpha_i)\sin(\epsilon_i) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(12)$
$\text{d}V_i = \cos(\alpha_i) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(13)$
where
$\alpha_i$
, and
$\epsilon_i$
denote the inclination and azimuth of the tangent vectors, respectively. Since the curve is assumed to have a constant curvature, it is part of a circular arc, hence, the angle spanned by the arc of length dM,
$\theta$
, also known as dogleg, is the same as the angle spanned by the two tangent vectors, i.e.
$\cos(\theta) = \mathbf T_A \cdot \mathbf T_B = \text{d}N_A\text{d}N_B + \text{d}E_A\text{d}E_B + \text{d}V_A\text{d}V_B \;\;\;\;\;\;\;\;(14)$
From this, the radius of the circular arc is
$R = \frac{\text d M}{\theta} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(15)$
and the coordinates of point
$\mathbf x_B = (N_B, E_B, V_B)$
are given by
$N_B = N_A + R\tan\left(\frac{\theta}{2}\right)(\text dN_A + \text dN_B) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(16a)$
$E_B = E_A + R\tan\left(\frac{\theta}{2}\right)(\text dE_A + \text dE_B) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(16b)$
$V_B = V_A + R\tan\left(\frac{\theta}{2}\right)(\text dV_A + \text dV_B) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(16c)$
To actually parametrize this arc between
$\mathbf x_A$
and
$\mathbf x_B$
, we do as follows. Let
$MD_A$
and
$MD_B$
be the measured depths at
$\mathbf x_A$
and
$\mathbf x_B$
respectively, d
$M= MD_A - MD_B$
, and let t
$\in [0, dM]$
represent curve length. The parametrization will be with respect to t. Also, given the inclinations,
$\alpha_A, \alpha_B$
, the azimuths
$\epsilon_A, \epsilon_B$
, and the angle
$\theta$
we can calculate the toolface,
$\gamma_{AB}$
, from either equation 6, equation 7, or a combination. Then, by using the identities 8 and 9 from the previous section, we can parametrize the inclination,
$\alpha$
, and azimuth
$\epsilon$
along the arc by
$\cos(\alpha(t)) = \cos(\alpha_A)\cos\left(\frac{\theta\, t}{\text dM}\right) - \sin(\alpha_A) \cos(\gamma_{AB}) \sin\left(\frac{\theta\, t}{\text dM}\right), \quad t \in [0, \text dM] \;\;\;\;\;\;\;\;\;\;\;\;(17)$
and
$\tan\left(\epsilon(t) - \epsilon_A\right) = \frac{\sin(\gamma_{AB})\tan\left(\frac{\theta\, t}{\text dM}\right)}{\sin(\alpha_A) + \cos(\alpha_A)\cos(\gamma_{AB})\tan\left(\frac{\theta\, t}{\text dM}\right)}, \quad t\in[0, \text dM] \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(18)$

Constant Build- and Turn- Method

The constant build- and turn- method describes a curve where the build- and turn- rates are assumed to be constants. The input to this method is the following
Start point
$\mathbf x_A = (N_A, E_A, V_A)\in \mathbb R^3$
Two unit tangent vectors,
$\mathbf T_A$
at
$\mathbf x_A$
, and
$\mathbf T_B$
at
$\mathbf x_B\in \mathbb R^3$
, where
$\mathbf x_B$
to be determined
The curve length, dM i.e. the difference in measured depth between
$\mathbf x_A$
and
$\mathbf x_B$
Since both build and turn rates are assumed constant, these can immediately be calculated to be
$Br = \frac{\alpha_B - \alpha_A}{\text dM} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(19)$
$Tr = \frac{\epsilon_B - \epsilon_A}{\text dM} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(20)$
Then using the differential expression for the tangent vectors
$\frac{d N}{dt} = \sin(\alpha)\cos(\epsilon) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(21)$
$\frac{d E}{dt} = \sin(\alpha)\sin(\epsilon) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(22)$
$\frac{d V}{dt} = \cos(\alpha) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(23)$
where we have parametrized the curve with regards to curve length, t, the coordinates at
$\mathbf x_B$
are calculated to be
$N_B =N_A+ \frac{1}{Tr^2 - Br^2}\Big[Tr\Big(\sin(\alpha_B)\sin(\epsilon_B) - \sin(\alpha_A)\sin(\epsilon_A)\Big) + Br\Big(\cos(\alpha_B)\cos(\epsilon_B) - \cos(\alpha_A)\cos(\epsilon_A)\Big)\Big] \\$
$E_B =E_A+ \frac{1}{Tr^2 - Br^2}\Big[-Tr\Big(\sin(\alpha_B)\cos(\epsilon_B) - \sin(\alpha_A)\cos(\epsilon_A)\Big) + Br\Big(\cos(\alpha_B)\sin(\epsilon_B) - \cos(\alpha_A)\sin(\epsilon_A)\Big)\Big] \\$
$V_B = V_A +\frac{1}{Br}\Big(\sin(\alpha_B) - \sin(\alpha_A) \Big) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(24)$
Since both Build- and Turn- rates are assumed constants, the curve is parametrized using curve length, t, with inclination and azimuth parametrized as
$\alpha(t) = \alpha_A + t\cdot Br \qquad \epsilon(t) = \epsilon_A + t\cdot Tr, \qquad t\in[0, \text d M] \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(25)$

Parametric Build Functions

Oliasoft offers a range of parametric build functions, described below. We subdivide them into three categories, straight lines (1), parametrizations defined by the Minimum Curvature method (2), and parametrizations defined by the constant build- and turn- method (3). Some of the parametric build functions have versions both described by the minimum curvature method and constant build- and turn- method.

Straight Lines

As the name suggests, these are straight lines, and comes in two flavors, Line MD and Line TVD.

Line MD

Input to this function is the start point
$\mathbf x_A = (N_A, E_A, V_A)$
, a unit tangent vector
$\mathbf T_A$
at
$\mathbf x_A$
, and the curve length dM. The end coordinates are given by
$N_B = N_A + \text dM\sin(\alpha_A)\cos(\epsilon_A) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(26)$
$E_B = E_A + \text dM\sin(\alpha_A)\sin(\epsilon_A) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(27)$
$V_B = V_A + \text dM\cos(\alpha_A) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(28)$

Line TVD

Input to this function is the start point
$\mathbf x_A = (N_A, E_A, V_A)$
, a unit tangent vector
$\mathbf T_A$
at
$\mathbf x_A$
and final vertical depth,
$V_B$
. Then the curve length is given
$by^1$
$\text dM = \frac{V_B - V_A}{\cos(\alpha_A)} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(29)$
and the remaining end coordinates are given by
$N_B = N_A + \text dM\sin(\alpha_A)\cos(\epsilon_A) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(30)$
$E_B = E_A + \text dM\sin(\alpha_A)\sin(\epsilon_A) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(31)$
This method is not well defined for inclination equal to 90 degrees

Dogleg & Toolface Methods

These are methods where the dogleg severity (angle change per meter) is assumed constant, hence are described by the minimum curvature method. The initial toolface is applied throughout the curve, and the end coordinates are given by equation 16.

INC AZI MD

Inputs to this function are the start point
$\mathbf x_A$
in
$\mathbb R^3$
, initial and final inclination and azimuth,
$\alpha_A,$
$\alpha_B,$
$\epsilon_A,$
$\epsilon_B$
and curve length dM. The minimum curvature method applies directly.

INC AZI TVD

Inputs to this function are the start point
$\mathbf x_A\in \mathbb R^3$
, initial and final inclination and azimuth,
$\alpha_A,$
$\alpha_B,$
$\epsilon_A,$
$\epsilon_B$
and final vertical depth
$V_B$
. The curve length dM is calculated by inverting the equation for the vertical coordinate in equation 16.

DLS INC AZI

Inputs to this function are the start point
$\mathbf x_A\in \mathbb R^3$
, initial and final inclination and azimuth,
$\alpha_A,$
$\alpha_B,$
$\epsilon_A,$
$\epsilon_B$
, and the dogleg severity,
$\beta$
(constant). The curve length, dM, is calculated using equation 14, and the identity
$\theta = \text dM\cdot \beta$
.

DT1 MD

Inputs to this function are the start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A,$
dogleg severity,
$\beta$
, toolface,
$\gamma$
, and curve length, dM. Final inclination and azimuth are calculated using equation 8 and 9.

DT1 TVD

Inputs to this function are the start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A,$
dogleg severity,
$\beta$
, toolface,
$\gamma$
, and final vertical depth,
$V_B$
. Curve length is found by inverting the equation for the vertical coordinate in equation 16 , using equation 17 for the final inclination and the identity
$\theta = \text dM\cdot \beta$

DT1 Inc

Inputs to this function are the start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A,$
dogleg severity,
$\beta$
, toolface,
$\gamma$
, and final inclination,
$\alpha_B$
. Curve length is found by solving equation 8 with respect to
$\theta$
, and using the relation
$\theta = \text dM\cdot \beta$
.

DT1 Azi

Inputs to this function are the start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A,$
dogleg severity,
$\beta$
, toolface,
$\gamma$
, and final azimuth,
$\epsilon_B$
. Curve length is found by solving equation 9 with respect to
$\theta$
, and using the relation
$\theta = \text dM\cdot \beta$
.

Tangent to Point

Inputs to this function are the start point ,
$\mathbf x_A\in\mathbb R^3$
initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A,$
a dogleg severity,
$\beta$
, and a target
$\mathbf x_T\in \mathbb R^3$
, including inclination and azimuth,
$\alpha_T,$
$\epsilon_T$
. Then, two sections are constructed, either a dogleg (constant curvature) section followed by a hold section to target, or reversed. For the dogleg section, the initial toolface is calculated, and for the hold section, the curve length is calculated.

Plan to Point

Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A$
, and a point
$\mathbf x\in \mathbb R^3$
. A single minimum curvature method curve is constructed to reach
$\mathbf x$
.

Online by TVD

Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A, \epsilon_A$
, a point
$\mathbf x\in \mathbb R^3$
, and a vertical depth where you want to be aligned with
$\mathbf x$
,
$TVD_0$
. Then, two sections are constructed, a constant curvature section which is aligned with the point
$\mathbf x$
at
$TVD_0$
, followed by a hold section to reach
$\mathbf x$
.

Hold Curve Hold

Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A$
, a point
$\mathbf x\in\mathbb R^3$
, a dogleg severity
$\beta$
, and the inclination
$\alpha$
to hit
$\mathbf x$
with. Then, three sections are constructed, a hold curve, a constant curvature section to reach the required inclination aligned with
$\mathbf x$
, and a final hold section to reach
$\mathbf x$
.

Constant Build and Turn Methods

These methods assume a constant build- and turn- rate. The end coordinates are given by equation 24.

INC AZI MD

Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial and final inclination and azimuth,
$\alpha_A$
,
$\alpha_B$
,
$\epsilon_A$
,
$\epsilon_B$
, and curve length dM. The build- and turn- rates are found from equation 19 and 20, respectively.

Inc Azi TVD

Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial and final inclination and azimuth,
$\alpha_A,$
$\alpha_B,$
$\epsilon_A,$
$\epsilon_B,$
and final vertical depth,
$V_B$
. The curve length is found by solving the equation for the vertical coordinate in equation 24 with regards to dM.

BT3 Inc

Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A$
, build- and turn- rates,
$Br$
and
$Tr$
, and final inclination,
$\alpha_B$
. Curve length is found solving equation 19 with regards to dM, and final azimuth is found by solving equation 20 with regards to
$\epsilon_B$
.

BT3 Azi

Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A$
, build- and turn- rates,
$Br$
and
$Tr$
, and final azimuth,
$\epsilon_B$
. Curve length is found solving equation 20 with regards to dM, and final inclination is found by solving equation 19 with regards to
$\alpha_B$
.

BT3 MD

Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A$
, build- and turn- rates,
$Br$
and
$Tr$
, and curve length dM. Final inclination and azimuth,
$\alpha_B,$
$\epsilon_B$
, are found using equations 19 and 20, respectively.

BT3 TVD

Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A$
, build- and turn- rates,
$Br$
and
$Tr$
, and final vertical depth
$V_B$
. Final inclination,
$\alpha_B$
, is found by solving the equation for the vertical coordinate in equation 24 with regards to
$\alpha_B$
, and curve length, dM, is found from equation 19.

Tangent to Point

Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A$
, build- and turn- rates, in addition to a point,
$\mathbf x\in \mathbb R^3$
, and a unit vector at
$\mathbf x$
with specified inclination and azimuth. Then, three sections are constructed, first a build- and turn- section is constructed to either the final inclination or azimuth is reached, then either a build- or a turn- section is constructed to reach the final inclination or azimuth, and finally a hold section is constructed to reach the point
$\mathbf x$
.

Plan to Point

Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A$
, and a point
$\mathbf x\in \mathbb R^3$
. A single build- and turn- curve is constructed to hit the point
$\mathbf x$
, i.e. equation 24 is solved for final inclination, azimuth, and curve length.

Online by TVD

Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A$
, a point
$\mathbf x\in \mathbb R^3$
, and a vertical depth where you want to be aligned with
$\mathbf x, TVD_0$
. Then, two sections are constructed, a build- and turn- section which is aligned with the point
$\mathbf x at TVD_0$
, followed by a hold section to reach
$\mathbf x$
.

Line up Target

Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A$
, and a point
$\mathbf x\in \mathbb R^3$
. A single turn- curve is constructed, aligning the azimuth with the azimuth between the two points,
$\mathbf x_A$
and
$\mathbf x$
.

Build-to-Target

Build-to-target methods come in two flavors, two dimensional Slant-wells and S-wells, and three dimensional Optimum align. For each of these methods, one gives a set of input parameters, in addition to the target coordinates. The total parameter space is bigger than the number of parameters given (i.e. some follows from the others), hence, each of the methods consists of several possibilities.

Slant Wells

A Slant-well consists of three sections, a vertical/hold section, a build section, and a hold section. Since this is a two dimensional method, the azimuth is given by the start and end coordinates. This also implies that if the start inclination is different from zero, the start azimuth has to be aligned with the azimuth angle between the start and end coordinates. Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination (usually zero making the azimuth redundant) and azimuth,
$\alpha_A,$
$\epsilon_A$
and a point
$\mathbf x_T\in \mathbb R^3$
in addition to two of the following four
Kickoff / First hold
Build rate
Max angle
Second hold
The remaining two, follows as a consequence of the others.

S-Wells

An S-well consists of five sections, a vertical/hold section, a build section, a hold section, a drop/build section, and a final hold section. As for Slant-wells, the azimuth is given by the start and end coordinates, implying that the start azimuth has to be aligned with the target, if start inclination is different from zero.
Inputs to this method are a start point
$\mathbf x_A\in \mathbb R^3$
initial inclination (usually zero making the azimuth redundant) and azimuth,
$\alpha_A,$
$\epsilon_A$
and a point
$\mathbf x_T\in \mathbb R^3$
in addition to five of the following seven
Kickoff / First hold
First buid rate
Max angle
Second hold
Second build rate
Final inclination
Last hold
The remaining two, follows as a consequence of the others.

Optimum Align

Optimum align is a three dimensional build-to-target method, which can be used if one wants to hit a target with a specific azimuth different from the direction between start- and end- coordinates. Optimum align comes in two varieties, either as curve-curve, or as a curve-hold-curve wellbore from start to target. This further decides which input to give.
If the curve-curve alternative is chosen, the inputs are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A,$
a target
$\mathbf x_T\in \mathbb R^3$
, final inclination and azimuth to hit the target with,
$\alpha_T$
and
$\epsilon_T$
, in addition to one of the following three
Dogleg severity of the first curve
Dogleg severity of the second curve
Balanced, i.e same dogleg severity for both curves
If the curve-hold-curve alternative is chosen, the inputs are a start point
$\mathbf x_A\in \mathbb R^3$
, initial inclination and azimuth,
$\alpha_A,$
$\epsilon_A$
, a point
$\mathbf x_T\in \mathbb R^3$
, final inclination and azimuth to hit the target with,
$\alpha_T$
and
$\epsilon_T$
, in addition to one of the following three
Dogleg severities for both the curves (not necessarily the same value
True vertical depth at the start and end of the hold section,
Tangent length, i.e. the length of the hold section.
Instead of giving the final inclination and azimuth to hit the target with, it is possible to give a second target,
$\mathbf x_{T'}\in \mathbb R^3$
, which the wellbore should be aligned with at
$\mathbf x_T$
.

References

[1] Bourgoyne Jr. et. al. Applied Drilling Engineering, volume 2. SPE Textbook Series, 1991.