Uncertainty Modelling
Description of methodology related to uncertainty modelling

Introduction

In this section we describe the content and functionality available in Oliasoft’s Uncertainty modelling module. We begin with a description of the error modelling framework we use, based on the standard developed by The Industry Steering Committee on Wellbore Survey Accuracy (ISCWSA). Then we discuss minimum distance calculations between wellbores, both 3D closest approach, and the earlier horizontal- and perpendicular- scan methods. Finally, we present separation factor calculations based on framework developed by ISCWSA and associates.
Oliasoft’s implementation is based on the ISCWSA standard, and a detailed description can be found here [1] [2].

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Oliasoft Technical Docs - Uncertainty Modelling

Uncertainty Modelling

In the following, a simple description is given. The framework is as follows. A single wellbore is under consideration, and a set of sensors are used to measure the position along the wellbore in NEV-coordinates. All sensors are assumed to be statistically independent, i.e. a measurement by one does not affect any of the others.
This makes the model linear as a function of sensors, and one can add/subtract sensors as needed.
Furthermore, the wellbore is assumed to consist of a set of survey legs, and each survey leg consists of a set of survey stations. To make it formal, denote the wellbore by W, the survey legs by
{l}lL\{l\}_{l\in L}
and the survey stations by
{k}kK\{k\}_{k\in K}
. Then,
W=l=1Ll=k=1Kk                                            (1)W = \bigcup_{l = 1}^{|L|}l = \bigcup_{k = 1}^{|K|} k \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1)
where
L|L|
and
K|K|
are the cardinality of
LL
and
KK
, respectively. Also, given a survey station,
kKk\in K
, there exists a unique survey leg
lLl\in L
, such that
klk\in l
, i.e.
ll=l \cap l' = \emptyset
if
lll\neq l'
. The wellbore as a continuous curve is described by a parametrisation between the survey stations, usually using the minimum curvature method.
For every sensor
ii
, at a given survey station
kk
, in survey leg
ll
, there is an associated so-called error vector,
ei,l,k\mathbf e_{i,l,k}
. This error vector is given by
ei,l,k=σi,l(dΔrkdpk+dΔrk+1dpk)pkϵi,                                            (2)\mathbf e_{i,l,k} = \sigma_{i,l} \left(\frac{\text d\Delta\mathbf r_k}{\text d\mathbf p_k} + \frac{\text d\Delta\mathbf r_{k+1}}{\text d\mathbf p_k}\right)\frac{\partial\mathbf p_k}{\partial \epsilon_i}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (2)
where
Δr\Delta\mathbf r
is given by the parametrization used of the wellbore between survey stations,
p\mathbf p
is the measurement vector,
σi,l\sigma_{i,l}
is the magnitude of the error source over the survey leg, usually quoted at 1
σ\sigma
, and finally
pkϵi\frac{\partial\mathbf p_k}{\partial \epsilon_i}
is the weighting function for the error source. The error summation terminates at the station of interest, and the vector error at this station is given by
ei,l,k=σi,ldΔrKdpKpKϵi,                                                                                      (3)\mathbf e_{i,l,k}^* = \sigma_{i,l} \frac{\text d\Delta\mathbf r_K}{\text d\mathbf p_K} \frac{\partial\mathbf p_K}{\partial \epsilon_i}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (3)
Usually, the wellbore is parametrized by the minimum curvature method between survey stations. However, there is no significant loss of accuracy in using the balanced tangential method [ref Williamson], and this is the preferred parametrisation in uncertainty modelling. From these error vectors, a covariance matrix,
CC
, is constructed, effectively the outer product of such. This covariance matrix can furthermore be used to calculate uncertainty in position of the wellbore in any direction.
Sensors come in four flavors, depending on how the associated error/uncertainty propagate down the wellbore. Assume we have
LL
survey legs, and the survey station of interest is
KK
in leg
LL
. A short description of the different types of sensors/errors follows.

Random Errors

These are randomly propagating errors, and the contribution to survey station uncertainty from such a source
ii
over leg
lLl \neq L
, is
Ci,lR=k=1Kl(ei,l,k)(ei,l,k)T,                                                                                  (4)C_{i,l}^R = \sum_{k=1}^{K_l} (\mathbf e_{i,l,k})(\mathbf e_{i,l,k})^T, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (4)
where
KlK_l
is the number of survey stations in leg
ll
. The total contribution is
Ci,KR=l=1L1Ci,lR+k=1K1(ei,L,k)(ei,L,k)T+(ei,L,k)(ei,L,k)T.                                                            (5)C_{i,K}^R = \sum_{l=1}^{L-1}C_{i,l}^R + \sum_{k=1}^{K - 1} (\mathbf e_{i,L,k})(\mathbf e_{i,L,k})^T + (\mathbf e_{i,L,k}^*)(\mathbf e_{i,L,k}^*)^T. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (5)

Systematic Errors

These are systematically propagating errors, and the contribution to survey station uncertainty from such a source
ii
over leg
lLl \neq L
, is
Ci,lS=(k=1Klei,l,k)(k=1Klei,l,k)T,                                                                                                            (6)C_{i,l}^S = \left(\sum_{k=1}^{K_l} \mathbf e_{i,l,k}\right) \left(\sum_{k=1}^{K_l} \mathbf e_{i,l,k}\right)^T, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\; (6)
where
KlK_l
is the number of survey stations in leg
ll
. The total contribution is
Ci,KS=l=1L1Ci,lS+(k=1K1ei,L,k+ei,L,k)(k=1K1ei,L,k+ei,L,k)T                             (7)C_{i,K}^S = \sum_{l=1}^{L-1}C_{i,l}^S + \left( \sum_{k=1}^{K - 1} \mathbf e_{i,L,k} + \mathbf e_{i, L,k}^* \right) \left( \sum_{k=1}^{K - 1} \mathbf e_{i,L,k} + \mathbf e_{i, L,k}^* \right)^T \;\;\;\;\;\;\;\;\;\;\;\;\;\;\ (7)

Well by Well and Global Errors

These are error types systematic among all the stations in the well. Hence, we can construct a total vector error from beginning to station
KK
, i.e.
Ei,K=l=1L1(k=1Klei,l,k)+k=1K1ei,l,k+ei,L,K,                                                                        (8)\mathbf E_{i, K} = \sum_{l = 1}^{L-1} \left(\sum_{k = 1}^{K_l} \mathbf e_{i,l,k} \right) + \sum_{k = 1}^{K-1} \mathbf e_{i,l,k} + \mathbf e_{i,L,K}^*, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (8)
where
KlK_l
is the number of survey stations in leg
ll
. The total contribution to the uncertainty at station
KK
is
Ci,KW/G=Ei,KEi,KT.                                                                                                                                                  (9)C_{i, K}^{W/G} = \mathbf E_{i, K} \mathbf E_{i, K} ^T. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (9)
Well-by-Well errors are (perfectly) correlated within a well, while Global errors are (perfectly) correlated between all wellbores.

The Position Covariance

The total position covariance at survey station
KK
is the sum over all error sources, i.e.
CK=iRCi,KR+iSCi,KS+iW/GCi,KW/G.                                                                                    (10)C_K = \sum_{i\in R} C_{i,K}^R + \sum_{i\in S} C_{i,K}^S + \sum_{i\in W/G} C_{i,K}^{W/G}. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (10)

Remarks

A. We observe that for Random , Well-by-Well and Global error sources, the contribution to the covariance matrix is independent of survey legs, and only dependent on the survey stations. This is not the case for Systematic error sources, which is seen from equation 6 and 7 and the fact that the contribution is not 'linear' in survey stations.
B. When comparing the positional uncertainty between wellbores, the contribution from global error sources should be excluded. If in addition, the wellbores are in the same well, also the well-by-well sources should be omitted.
C. Details about uploading of error/uncertainty- models to the app can be found here (link to decription of IPM/manually defined error models).

Minimum Distance

An important quantity in its own right, and also in separation factor calculations is the distance between two wellbores. Precisely, given two wellbores
W1W_1
and
W2W_2
, and denote
W1W_1
as the reference wellbore, and
W2W_2
as the offset wellbore. Then, given a survey station
kW1k\in W_1
, calculate the minimum distance between
kk
and
W2W_2
. Oliasoft offers three different methods for calculating/approximating this distance
    1.
    3D Closest Approach
    2.
    Horizontal Scan
    3.
    Perpendicular Scan
The methods are implemented under the assumptions that the wellbore trajectories are described by the minimum curvature method between survey stations. The last two are included for completeness, and are approximations to the true minimum distance, found by 3D closest approach.

3D Closest Approach

The 3D closest approach method gives the true minimum distance between a survey station
kW1k\in W_1
and the wellbore
W2W_2
, under the assumption that the wellbore trajectory is described by the minimum curvature method between survey stations. The algorithm is conceptually simple, although computationally relatively heavy
    1.
    Given a survey station
    kW1k\in W_1
    .
    2.
    Parametrize the offset wellbore trajectory,
    W2W_2
    , using the minimum curvature method between survey stations.
    3.
    Minimize the distance function
    d(k,W2)d(k, W_2)
    .

Horizontal Scan

The horizontal scan method is an approximation to the true minimum distance between a survey station
kW1k\in W_1
and the wellbore
W2W_2
. It is a good approximation between vertical/near vertical well, and is useless between horizontal wells. The algorithm goes as follows
    1.
    Given a survey station
    kW1k\in W_1
    , at a vertical depth
    V1V_1
    .
    2.
    Find the point(s),
    xW2\mathbf x\in W_2
    on the offset wellbore trajectory at the same vertical depth,
    V1V_1
    (such point(s) does not necessarily exist).
    3.
    Calculate the distance between
    kk
    and
    xx
    (and minimize if more than one).

Perpendicular Scan

The perpendicular scan method is an approximation to the true minimum distance between a survey station
kW1k \in W_1
and the wellbore
W2W_2
. It works good between vertical/horizontal wells, but has limitations between deviated wells. The algorithm goes as follows
    1.
    Given a survey station
    kW1k\in W_1
    .
    2.
    Calculate the normal vector to the wellbore at
    kk
    , and prolong it to it hits the offset well at
    xW2\mathbf x \in W_2
    (this point does not necessarily exist).
    3.
    Calculate the distance between
    kk
    and
    x\mathbf x
    .

Seperation Factor

The separation factor, calculated for every survey station in a reference well against an offset well, is essentially the ratio between the minimum centre-to-centre distance between the two wellbores and the relative positional uncertainty between the two. Currently, Oliasoft offers the Pedal Curve Method as separation factor model which is described in details here. Below we describe the essentials.
Separation factor calculations involve two wellbores,
W1W_1
and
W2W_2
, referred to as reference well and offset well respectively. For both the wellbores, NEV-covariance matrices, at every survey station have been calculated based on uncertainty models, not necessarily the same model applied to both the wellbores. In addition, for every survey station in the reference well,
KW1K\in W_1
, the minimum centre-to-centre distance to the offset well is calculated. Then, the separation factor, at survey station
KW1K\in W_1
is defined as
SFK=DKRrRoSmkσs,K2+σpa2,                                                                                (11)SF_K = \frac{D_K - R_r - R_o - S_m}{k\sqrt{\sigma_{s, K}^2 + \sigma_{pa}^2}}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (11)
where
DKD_K
is the minimum distance between the survey station
KK
and the offset wellbore,
W2W_2
, and
σs,K\sigma{s, K}
is the relative uncertainty at one standard deviation between the survey station
KK
and the point on the offset well, giving the minimum distance. The other quantities involved are
    1.
    RrR_r
    and
    RoR_o
    are the open hole radius, at the point of interest, of the reference and offset wellbore respectively.
    2.
    SmS_m
    is a surface margin, which effectively increases the radius of the offset well. It is introduced to accommodate small, unidentified errors. Recommended value is
    Sm=0.3S_m = 0.3
    m.
    3.
    kk
    is a dimensional scaling factor that determines the probability of well crossing. Recommended value is
    k=3.5k=3.5
    .
    4.
    σpa\sigma_{pa}
    is introduced to quantify, at one standard deviation, the uncertainty in the projection ahead of the current survey station. Recommended value is
    σpa=0.5\sigma_{pa}= 0.5
    m.

Algorithm

    1.
    Calculate the NEV-covariance matrices for the reference and offset wellbore. Remember to exclude any common global error source, and also common well error sources if the wellbores originate from the same well.
    2.
    Calculate the minimum distance between the survey stations in the reference well and the offset well.
    3.
    Project the NEV-covariance matrices along the direction defined by the minimum distance, to get the relative uncertainty.
    4.
    Assemble the separation factor
    SFKSF_K
    .

Remarks

The minimum distance calculation depends on which scan method is used, hence, also the separation factor calculations depend on this. Not only through the minimum distance,
DKD_K
, but also thorough the relative uncertainty
σs,K\sigma{s, K}
. Best policy is to use the 3D closest approach which gives the true minimum distance.

References

Last modified 2yr ago