# 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.

NOTE!
Oliasoft’s implementation is based on the ISCWSA standard, and a detailed description can be found here .

### Printable Version#

Oliasoft Technical Documentation - 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 $\{l\}_{l\in L}$ and the survey stations by $\{k\}_{k\in K}$. Then:

$W = \bigcup_{l = 1}^{|L|}l = \bigcup_{k = 1}^{|K|} k \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1)$

where $|L|$ and $|K|$ are the cardinality of $L$ and $K$, respectively. Also, given a survey station, $k\in K$, there exists a unique survey leg $l\in L$, such that $k\in l$, i.e. $l \cap l' = \emptyset$ if $l\neq l'$. The wellbore as a continuous curve is described by a parametrization between the survey stations, usually using the minimum curvature method.

For every sensor $i$, at a given survey station $k$, in survey leg $l$, there is an associated so-called error vector, $\mathbf e_{i,l,k}$. This error vector is given by:

$\mathbf e_{i,l,k} = \sigma_{i,l} \left(\frac{\text d\Delta\mathbf r_k}{\text d\mathbf p_k} + \frac{\text d\Delta\mathbf r_{k+1}}{\text d\mathbf p_k}\right)\frac{\partial\mathbf p_k}{\partial \epsilon_i}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (2)$

where $\Delta\mathbf r$ is given by the parametrization used of the wellbore between survey stations, $\mathbf p$ is the measurement vector, $\sigma_{i,l}$ is the magnitude of the error source over the survey leg, usually quoted at 1 $\sigma$ , and finally $\frac{\partial\mathbf p_k}{\partial \epsilon_i}$ 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:

$\mathbf e_{i,l,k}^* = \sigma_{i,l} \frac{\text d\Delta\mathbf r_K}{\text d\mathbf p_K} \frac{\partial\mathbf p_K}{\partial \epsilon_i}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (3)$

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 , and this is the preferred parametrization in uncertainty modelling. From these error vectors, a covariance matrix, ​$C$ , 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 $L$ survey legs, and the survey station of interest is $K$ in leg $L$. 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 $i$ over leg $l \neq L$, is:

$C_{i,l}^R = \sum_{k=1}^{K_l} (\mathbf e_{i,l,k})(\mathbf e_{i,l,k})^T, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (4)$

where ​$K_l$ is the number of survey stations in leg ​$l$. The total contribution is:

$C_{i,K}^R = \sum_{l=1}^{L-1}C_{i,l}^R + \sum_{k=1}^{K - 1} (\mathbf e_{i,L,k})(\mathbf e_{i,L,k})^T + (\mathbf e_{i,L,k}^*)(\mathbf e_{i,L,k}^*)^T. \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (5)$

### Systematic Errors#

These are systematically propagating errors, and the contribution to survey station uncertainty from such a source $i$ over leg $l \neq L$, is:

$C_{i,l}^S = \left(\sum_{k=1}^{K_l} \mathbf e_{i,l,k}\right) \left(\sum_{k=1}^{K_l} \mathbf e_{i,l,k}\right)^T,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;\;\;\;\; (6)$

where $K_l$ is the number of survey stations in leg ​$l$. The total contribution is:

$C_{i,K}^S = \sum_{l=1}^{L-1}C_{i,l}^S + \left( \sum_{k=1}^{K - 1} \mathbf e_{i,L,k} + \mathbf e_{i, L,k}^* \right) \left( \sum_{k=1}^{K - 1} \mathbf e_{i,L,k} + \mathbf e_{i, L,k}^* \right)^T\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ (7)$

### 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 $K$, i.e.:

$\mathbf E_{i, K} = \sum_{l = 1}^{L-1} \left(\sum_{k = 1}^{K_l} \mathbf e_{i,l,k} \right) + \sum_{k = 1}^{K-1} \mathbf e_{i,l,k} + \mathbf e_{i,L,K}^*,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (8)$

where $K_l$​ is the number of survey stations in leg ​$l$. The total contribution to the uncertainty at station ​$K$ is:

$C_{i, K}^{W/G} = \mathbf E_{i, K} \mathbf E_{i, K} ^T.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (9)$

NOTE!
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 $K$ is the sum over all error sources, i.e.:

$C_K = \sum_{i\in R} C_{i,K}^R + \sum_{i\in S} C_{i,K}^S + \sum_{i\in W/G} C_{i,K}^{W/G}.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (10)$

### 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.

## 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 $W_1$ and $W_2$, and denote $W_1$ as the reference wellbore, and $W_2$ as the offset wellbore. Then, given a survey station $k\in W_1$, calculate the minimum distance between $k$ and $W_2$. 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 $k\in W_1$ and the wellbore $W_2$, 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 $k\in W_1$
2. Parametrize the offset wellbore trajectory, $W_2$, using the minimum curvature method between survey stations
3. Minimize the distance function $d(k, W_2)$

### Horizontal Scan#

The horizontal scan method is an approximation to the true minimum distance between a survey station $k\in W_1$ and the wellbore $W_2$. 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 $k\in W_1$, at a vertical depth $V_1$
2. Find the point(s), $\mathbf x\in W_2$ on the offset wellbore trajectory at the same vertical depth, $V_1$ (such point(s) does not necessarily exist)
3. Calculate the distance between $k$ and $x$ (and minimize if more than one)

### Perpendicular Scan#

The perpendicular scan method is an approximation to the true minimum distance between a survey station $k \in W_1$ and the wellbore $W_2$. It works good between vertical/horizontal wells, but has limitations between deviated wells. The algorithm goes as follows:

1. Given a survey station $k\in W_1$
2. Calculate the normal vector to the wellbore at $k$, and prolong it to it hits the offset well at $\mathbf x \in W_2$ (this point does not necessarily exist)
3. Calculate the distance between $k$ and $x$

## Separation 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, $W_1$ and $W_2$ , 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, $K\in W_1$, the minimum centre-to-centre distance to the offset well is calculated. Then, the separation factor, at survey station $K\in W_1$ is defined as:

$SF_K = \frac{D_K - R_r - R_o - S_m}{k\sqrt{\sigma_{s, K}^2 + \sigma_{pa}^2}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (11)$

where $D_K$ is the minimum distance between the survey station $K$ and the offset wellbore, $W_2$, and $\sigma{s, K}$ is the relative uncertainty at one standard deviation between the survey station $K$ and the point on the offset well, giving the minimum distance. The other quantities involved are:

1. $R_r$ and $R_o$ are the open hole radius, at the point of interest, of the reference and offset wellbore respectively.

2. $S_m$ is a surface margin, which effectively increases the radius of the offset well. It is introduced to accommodate small, unidentified errors. Recommended value is $S_m = 0.3$ m.

3. $k$ is a dimensional scaling factor that determines the probability of well crossing. Recommended value is $k=3.5$.

4. $\sigma_{pa}$ is introduced to quantify, at one standard deviation, the uncertainty in the projection ahead of the current survey station. Recommended value is $\sigma_{pa}= 0.5$ 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 $SF_K$.

### 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, $D_K$, but also thorough the relative uncertainty $\sigma{s, K}$. Best policy is to use the 3D closest approach which gives the true minimum distance.