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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Oliasoft Technical Docs - Well Trajectory Design

Definitions

Definition 1: A Wellbore Section between two points
x1,x2R3\mathbf x_1, \mathbf x_2\in \mathbb R^3
, is defined as a smooth curve
γ:[a,b]R3,a,bR                                                    (1)\gamma : [a,b] \mapsto \mathbb R^3, \quad a,b \in \mathbb R \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(1)
such that
γ(a)=x1 and  γ(b)=x2\gamma(a) = \mathbf x_1 \text{ and }\ \gamma(b) = \mathbf x_2
.
Definition 2: A wellbore is a piecewise differentiable curve
Γ:[a,b]R3\Gamma : [a,b]\mapsto \mathbb R^3
,
a,bRa,b \in\mathbb R
, such that each piece is described by a wellbore section. Further, given a point
x0\mathbf x_0
on the curve, i.e.
x0=Γ(t0)\mathbf x_0 = \Gamma(t_0)
for
t0[a,b]t_0\in [a,b]
, the tangent vector to the curve at
x0\mathbf x_0
is given by
T(x0)=dΓdt(t0):=Γ(t0)                                                    (2)\mathbf T(\mathbf x_0) = \frac{d\Gamma}{dt}(t_0) := \Gamma'(t_0)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(2)
Definition 3: Given a wellbore
Γ\Gamma
and a point
x0\mathbf x_0
on the curve. The inclination
α\alpha
, to the wellbore at
x0\mathbf x_0
is defined as the angle between the tangent at
x0\mathbf x_0
and the vertical.
Definition 4: Given a wellbore
Γ\Gamma
and a point
x0\mathbf x_0
on the curve. The azimuth
ϵ\epsilon
, to the wellbore at
x0\mathbf x_0
is defined as the angle between the tangent at
x0\mathbf x_0
and north.
Definition 5: Curvature, aka Dogleg severity, and radius of curvature. Given a wellbore section
γ\gamma
. Assume also that the tangent vector
T\mathbf T
is nowhere zero, and that the curve is parametrized with respect to curve length, t, such that
T(t)=1,t\|T(t)\| = 1, \forall t
. Then the curvature, also known as dogleg severity is defined as
κ(t)=T(t)=γ(t)                                                    (3)\kappa(t) = \|\mathbf T'(t)\| = \|\gamma''(t)\|\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(3)
The radius of curvature is defined as
R(t)=1κ(t)                                                    (4)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
α(t)\alpha(t)
and
ϵ(t)\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)=dαdt(t),Tr(t)=dϵdt(t)                                                    (5)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
x0x1\mathbf x_0 \neq \mathbf x_1
on it, with inclinations,
α0,α1\alpha_0, \alpha_1
, and azimuths,
ϵ0\epsilon_0
,
ϵ1\epsilon_1
. Assume that the arc between
x0\mathbf x_0
and
x1\mathbf x_1
spans the angle
θ\theta
. The associated toolface angle
γ\gamma
is defined, either by
cos(γ)=cos(α0)cos(θ)cos(α1)sin(α0)sin(θ)                                                    (6)\cos(\gamma) = \frac{\cos(\alpha_0)\cos(\theta) - \cos(\alpha_1)}{\sin(\alpha_0)\sin(\theta)} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(6)
or equivalently
sin(γ)=sin(α1)sin(ϵ1ϵ0)sin(θ)                                                    (7)\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
x1\mathbf x_1
as
cos(α1)=cos(α0)cos(θ)sin(α0)sin(θ)cos(γ)                                                    (8)\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,
ϵ1ϵ0\epsilon_1 - \epsilon_0
, toolface,
γ10\gamma_{10}
, start inclination,
α0\alpha_0
, and total angle change,
θ10\theta_{10}
tan(ϵ1ϵ0)=sin(γ10)tan(θ10)sin(α0)+cos(α0)cos(γ10)tan(θ10)                                                    (9)\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
    xA=(NA,EA,VA)R3\mathbf x_A = (N_A, E_A, V_A)\in \mathbb R^3
    Two unit tangent vectors,
    TA\mathbf T_A
    at
    xA\mathbf x_A
    , and
    TB\mathbf T_B
    at
    xBR3\mathbf x_B\in \mathbb R^3
    , where
    xB\mathbf x_B
    to be determined.
    The curve length, dM , i.e. the difference in measured depth between
    xA\mathbf x_A
    and
    xB\mathbf x_B
The tangent vectors are decomposed as
Ti=(dNi,dEi,dVi),i=A,B,                                                    (10)\mathbf T_i = (\text{d} N_i, \text{d}E_i, \text{d}V_i), \quad i = A, B, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(10)
where
dNi=sin(αi)cos(ϵi)                                                                                              (11)\text{d}N_i = \sin(\alpha_i)\cos(\epsilon_i) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(11)
dEi=sin(αi)sin(ϵi)                                                                                              (12)\text{d}E_i = \sin(\alpha_i)\sin(\epsilon_i) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(12)
dVi=cos(αi)                                                                                                                      (13)\text{d}V_i = \cos(\alpha_i) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(13)
where
αi\alpha_i
, and
ϵi\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,
θ\theta
, also known as dogleg, is the same as the angle spanned by the two tangent vectors, i.e.
cos(θ)=TATB=dNAdNB+dEAdEB+dVAdVB                (14)\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=dMθ                                                                                                                                                            (15)R = \frac{\text d M}{\theta} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(15)
and the coordinates of point
xB=(NB,EB,VB)\mathbf x_B = (N_B, E_B, V_B)
are given by
NB=NA+Rtan(θ2)(dNA+dNB)                                (16a)N_B = N_A + R\tan\left(\frac{\theta}{2}\right)(\text dN_A + \text dN_B) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(16a)
EB=EA+Rtan(θ2)(dEA+dEB)                                (16b)E_B = E_A + R\tan\left(\frac{\theta}{2}\right)(\text dE_A + \text dE_B) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(16b)
VB=VA+Rtan(θ2)(dVA+dVB)                                        (16c)V_B = V_A + R\tan\left(\frac{\theta}{2}\right)(\text dV_A + \text dV_B) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(16c)
To actually parametrize this arc between
xA\mathbf x_A
and
xB\mathbf x_B
, we do as follows. Let
MDAMD_A
and
MDBMD_B
be the measured depths at
xA\mathbf x_A
and
xB\mathbf x_B
respectively, d
M=MDAMDBM= MD_A - MD_B
, and let t
[0,dM]\in [0, dM]
represent curve length. The parametrization will be with respect to t. Also, given the inclinations,
αA,αB\alpha_A, \alpha_B
, the azimuths
ϵA,ϵB\epsilon_A, \epsilon_B
, and the angle
θ\theta
we can calculate the toolface,
γAB\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,
α\alpha
, and azimuth
ϵ\epsilon
along the arc by
cos(α(t))=cos(αA)cos(θtdM)sin(αA)cos(γAB)sin(θtdM),t[0,dM]                        (17)\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(ϵ(t)ϵA)=sin(γAB)tan(θtdM)sin(αA)+cos(αA)cos(γAB)tan(θtdM),t[0,dM]                                                      (18)\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
    xA=(NA,EA,VA)R3\mathbf x_A = (N_A, E_A, V_A)\in \mathbb R^3
    Two unit tangent vectors,
    TA\mathbf T_A
    at
    xA\mathbf x_A
    , and
    TB\mathbf T_B
    at
    xBR3\mathbf x_B\in \mathbb R^3
    , where
    xB\mathbf x_B
    to be determined
    The curve length, dM i.e. the difference in measured depth between
    xA\mathbf x_A
    and
    xB\mathbf x_B
Since both build and turn rates are assumed constant, these can immediately be calculated to be
Br=αBαAdM                                                      (19)Br = \frac{\alpha_B - \alpha_A}{\text dM} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(19)
Tr=ϵBϵAdM                                                      (20)Tr = \frac{\epsilon_B - \epsilon_A}{\text dM} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(20)
Then using the differential expression for the tangent vectors
dNdt=sin(α)cos(ϵ)                                          (21)\frac{d N}{dt} = \sin(\alpha)\cos(\epsilon) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(21)
dEdt=sin(α)sin(ϵ)                                          (22)\frac{d E}{dt} = \sin(\alpha)\sin(\epsilon) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(22)
dVdt=cos(α)                                                          (23)\frac{d V}{dt} = \cos(\alpha) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(23)
where we have parametrized the curve with regards to curve length, t, the coordinates at
xB\mathbf x_B
are calculated to be
NB=NA+1Tr2Br2[Tr(sin(αB)sin(ϵB)sin(αA)sin(ϵA))+Br(cos(αB)cos(ϵB)cos(αA)cos(ϵA))]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] \\
EB=EA+1Tr2Br2[Tr(sin(αB)cos(ϵB)sin(αA)cos(ϵA))+Br(cos(αB)sin(ϵB)cos(αA)sin(ϵA))]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] \\
VB=VA+1Br(sin(αB)sin(αA))                                (24)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
α(t)=αA+tBrϵ(t)=ϵA+tTr,t[0,dM]                                (25)\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
xA=(NA,EA,VA)\mathbf x_A = (N_A, E_A, V_A)
, a unit tangent vector
TA\mathbf T_A
at
xA\mathbf x_A
, and the curve length dM. The end coordinates are given by
NB=NA+dMsin(αA)cos(ϵA)                                                    (26)N_B = N_A + \text dM\sin(\alpha_A)\cos(\epsilon_A) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(26)
EB=EA+dMsin(αA)sin(ϵA)                                                    (27)E_B = E_A + \text dM\sin(\alpha_A)\sin(\epsilon_A) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(27)
VB=VA+dMcos(αA)                                                                            (28)V_B = V_A + \text dM\cos(\alpha_A) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(28)

Line TVD

Input to this function is the start point
xA=(NA,EA,VA)\mathbf x_A = (N_A, E_A, V_A)
, a unit tangent vector
TA\mathbf T_A
at
xA\mathbf x_A
and final vertical depth,
VBV_B
. Then the curve length is given
by1by^1
dM=VBVAcos(αA)                                                                            (29)\text dM = \frac{V_B - V_A}{\cos(\alpha_A)} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(29)
and the remaining end coordinates are given by
NB=NA+dMsin(αA)cos(ϵA)                                                                            (30)N_B = N_A + \text dM\sin(\alpha_A)\cos(\epsilon_A) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(30)
EB=EA+dMsin(αA)sin(ϵA)                                                                            (31)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
xA\mathbf x_A
in
R3\mathbb R^3
, initial and final inclination and azimuth,
αA,\alpha_A,
αB,\alpha_B,
ϵA, \epsilon_A,
ϵB \epsilon_B
and curve length dM. The minimum curvature method applies directly.

INC AZI TVD

Inputs to this function are the start point
xAR3\mathbf x_A\in \mathbb R^3
, initial and final inclination and azimuth,
αA,\alpha_A,
αB, \alpha_B,
ϵA,\epsilon_A,
ϵB\epsilon_B
and final vertical depth
VBV_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
xAR3\mathbf x_A\in \mathbb R^3
, initial and final inclination and azimuth,
αA,\alpha_A,
αB, \alpha_B,
ϵA,\epsilon_A,
ϵB\epsilon_B
, and the dogleg severity,
β\beta
(constant). The curve length, dM, is calculated using equation 14, and the identity
θ=dMβ\theta = \text dM\cdot \beta
.

DT1 MD

Inputs to this function are the start point
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA,\epsilon_A,
dogleg severity,
β\beta
, toolface,
γ\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
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA,\epsilon_A,
dogleg severity,
β\beta
, toolface,
γ\gamma
, and final vertical depth,
VBV_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
θ=dMβ\theta = \text dM\cdot \beta

DT1 Inc

Inputs to this function are the start point
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA,\epsilon_A,
dogleg severity,
β\beta
, toolface,
γ\gamma
, and final inclination,
αB\alpha_B
. Curve length is found by solving equation 8 with respect to
θ\theta
, and using the relation
θ=dMβ\theta = \text dM\cdot \beta
.

DT1 Azi

Inputs to this function are the start point
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA,\epsilon_A,
dogleg severity,
β\beta
, toolface,
γ\gamma
, and final azimuth,
ϵB\epsilon_B
. Curve length is found by solving equation 9 with respect to
θ\theta
, and using the relation
θ=dMβ\theta = \text dM\cdot \beta
.

Tangent to Point

Inputs to this function are the start point ,
xAR3\mathbf x_A\in\mathbb R^3
initial inclination and azimuth,
αA,\alpha_A,
ϵA,\epsilon_A,
a dogleg severity,
β\beta
, and a target
xTR3\mathbf x_T\in \mathbb R^3
, including inclination and azimuth,
αT,\alpha_T,
ϵT\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
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA\epsilon_A
, and a point
xR3\mathbf x\in \mathbb R^3
. A single minimum curvature method curve is constructed to reach
x\mathbf x
.

Online by TVD

Inputs to this method are a start point
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,ϵA\alpha_A, \epsilon_A
, a point
xR3\mathbf x\in \mathbb R^3
, and a vertical depth where you want to be aligned with
x\mathbf x
,
TVD0TVD_0
. Then, two sections are constructed, a constant curvature section which is aligned with the point
x\mathbf x
at
TVD0TVD_0
, followed by a hold section to reach
x\mathbf x
.

Hold Curve Hold

Inputs to this method are a start point
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA\epsilon_A
, a point
xR3\mathbf x\in\mathbb R^3
, a dogleg severity
β\beta
, and the inclination
α\alpha
to hit
x\mathbf x
with. Then, three sections are constructed, a hold curve, a constant curvature section to reach the required inclination aligned with
x\mathbf x
, and a final hold section to reach
x\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
xAR3\mathbf x_A\in \mathbb R^3
, initial and final inclination and azimuth,
αA\alpha_A
,
αB\alpha_B
,
ϵA\epsilon_A
,
ϵB\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
xAR3\mathbf x_A\in \mathbb R^3
, initial and final inclination and azimuth,
αA,\alpha_A,
αB,\alpha_B,
ϵA,\epsilon_A,
ϵB,\epsilon_B,
and final vertical depth,
VBV_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
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA\epsilon_A
, build- and turn- rates,
BrBr
and
TrTr
, and final inclination,
αB\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
ϵB\epsilon_B
.

BT3 Azi

Inputs to this method are a start point
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA\epsilon_A
, build- and turn- rates,
BrBr
and
TrTr
, and final azimuth,
ϵB\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
αB\alpha_B
.

BT3 MD

Inputs to this method are a start point
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA\epsilon_A
, build- and turn- rates,
BrBr
and
TrTr
, and curve length dM. Final inclination and azimuth,
αB,\alpha_B,
ϵB\epsilon_B
, are found using equations 19 and 20, respectively.

BT3 TVD

Inputs to this method are a start point
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA\epsilon_A
, build- and turn- rates,
BrBr
and
TrTr
, and final vertical depth
VBV_B
. Final inclination,
αB\alpha_B
, is found by solving the equation for the vertical coordinate in equation 24 with regards to
αB\alpha_B
, and curve length, dM, is found from equation 19.

Tangent to Point

Inputs to this method are a start point
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA\epsilon_A
, build- and turn- rates, in addition to a point,
xR3\mathbf x\in \mathbb R^3
, and a unit vector at
x\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
x\mathbf x
.

Plan to Point

Inputs to this method are a start point
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA\epsilon_A
, and a point
xR3\mathbf x\in \mathbb R^3
. A single build- and turn- curve is constructed to hit the point
x\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
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA\epsilon_A
, a point
xR3\mathbf x\in \mathbb R^3
, and a vertical depth where you want to be aligned with
x,TVD0\mathbf x, TVD_0
. Then, two sections are constructed, a build- and turn- section which is aligned with the point
xatTVD0\mathbf x at TVD_0
, followed by a hold section to reach
x\mathbf x
.

Line up Target

Inputs to this method are a start point
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA\epsilon_A
, and a point
xR3\mathbf x\in \mathbb R^3
. A single turn- curve is constructed, aligning the azimuth with the azimuth between the two points,
xA\mathbf x_A
and
x\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
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination (usually zero making the azimuth redundant) and azimuth,
αA,\alpha_A,
ϵA\epsilon_A
and a point
xTR3\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
xAR3\mathbf x_A\in \mathbb R^3
initial inclination (usually zero making the azimuth redundant) and azimuth,
αA,\alpha_A,
ϵA\epsilon_A
and a point
xTR3\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
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA,\epsilon_A,
a target
xTR3\mathbf x_T\in \mathbb R^3
, final inclination and azimuth to hit the target with,
αT\alpha_T
and
ϵT\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
xAR3\mathbf x_A\in \mathbb R^3
, initial inclination and azimuth,
αA,\alpha_A,
ϵA\epsilon_A
, a point
xTR3\mathbf x_T\in \mathbb R^3
, final inclination and azimuth to hit the target with,
αT\alpha_T
and
ϵT\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,
xTR3\mathbf x_{T'}\in \mathbb R^3
, which the wellbore should be aligned with at
xT\mathbf x_T
.

References

[1] Bourgoyne Jr. et. al. Applied Drilling Engineering, volume 2. SPE Textbook Series, 1991.
Last modified 3mo ago