25.1 Combined Surveys

For many years, we have taken MWD surveys then perhaps run a gyro afterwards and thrown away the MWD and replaced it with the gyro. There is no need to waste data like that. One of the simple rules of

statistics is that the standard deviation of a combined measurement is the square root of the sum of the squares of the individual measurements divided by the number of measurements.

In other words, if we had two similarly accurate instruments which we could demonstrate were uncorrelated in their error sources and we took an average of their observations, the uncertainty would reduce by 1/1.414 or approximately 30% reduction.

On certain high accuracy jobs, we can combine multiple gyro runs along with the MWD, (in-runs and out-runs) and fit a best fit curve through the data to produce a synthetic trajectory which is more accurate than any of the original surveys on their own.

Right now we don’t have a formal error model for combined surveys but it would not be difficult to derive as it would simply be a combination of the covariance matrices of the individual surveys. Nevertheless, it is often very worthwhile to combine the available surveys just to improve confidence in the well path position even if we do
not see the benefit in the calculated ellipse of uncertainty.

First of all, we need to understand how to curve fit smoothly through multiple observations.

This multiplication produces the following matrix equation from which the coefficients a,b,c and d can be easily
calculated.

This is a really useful tool for fitting curves through any data and if you need a higher ranking polynomial you just continue the pattern for extra parameters.

In order to combine surveys of different instruments and differing accuracies, we can create a rolling curve fit using say the nearest 6 observations of inclination and azimuth using the following procedure.
1. Choose a final survey interval of say 20 ft
2. From the first to the last measured depth value in the surveys in steps of 20ft:
a. Find the nearest 6 surveys
b. Let x = 0 at the Md of interest
c. For each survey set x = Md survey – Md of interest.
d. Let y = inclination of each survey
e. Fit the curve as above through all 6
f. The inclination required at Md of interest is just the value d (at x = 0)
g. Let y = azimuth of each survey (if passing through North add 360)
h. Fit the curve as above through all 6
i. The azimuth required at Md of Interest is just the value d (at x = 0)
3. When weighting the curve, the relative accuracy can guide the use of weights so all the additions should
be done with the appropriate weighting values. For example, if it is considered that a gyro survey is twice
as accurate as a raw MWD survey, you would add the gyro survey contributions to the matrix with a
weighting of 2 and the MWD with a weighting of 1.
In this example plot the various contributions to a relief well survey are superimposed in different colours with
inclination in the top graph and azimuth in the bottom graph. The grid on the left shows the disagreement survey
by survey in inclination and azimuth (all less than 0.5 degrees) and the length of the line is the relative weighting.

The final result is a complete synthetic survey that best fits all the input data.

In this case the uncertainty of the final well path was around 1 to 2 feet per thousand