Fusion of wellbore surveys and ranging measurements

1 . A method for combining wellbore surveying measurements and ranging measurements to obtain an improved wellbore trajectory, the method comprising:

obtaining first and second wellbore trajectories for first and second corresponding wellbores and first and second positional uncertainties for the first and second wellbore trajectories;

obtaining a set of ranging measurements and a corresponding ranging uncertainty, the ranging measurements including at least a distance between the first and second wellbores, the ranging measurements made using a bottom hole assembly (BHA) deployed in the second wellbore;

defining a probability function including the first and second wellbore trajectories, the first and second positional uncertainties, the ranging measurements, and the ranging uncertainty, the probability function determining a probability of observing the obtained first and second wellbore trajectories and the obtained ranging measurements for various actual first and second wellbore trajectories;

optimizing the probability function to compute an improved wellbore trajectory of at least one of the first and second wellbores, the improved wellbore trajectory having an improved positional uncertainty;

determining a direction of drilling the second wellbore from the improved wellbore trajectory of the at least one of the first and second wellbores; and

causing a BHA to drill the second wellbore in the determined direction of drilling.

2 . The method of claim 1 , wherein:

the first and second wellbore trajectories are computed from first and second corresponding sets of wellbore survey measurements; and

the first and second positional uncertainties comprise first and second covariance matrices that are computed from corresponding error models for the first and second sets of wellbore survey measurements.

3 . The method of claim 1 , wherein the ranging uncertainty is a ranging covariance computed from a ranging error model.

4 . The method of claim 1 , wherein the probability function comprises a likelihood function and optimizing the probability function comprises selecting the actual first and second wellbore trajectories that are most likely based on the likelihood function.

5 . The method of claim 4 , wherein the defined likelihood function comprises a joint likelihood including a product of first, second, and third independent probabilities, the first probability being a probability of observing the obtained first wellbore trajectory for the various actual first wellbore trajectories, the second probability being a probability of observing the obtained second wellbore trajectory for the various actual second wellbore trajectories, and the third probability being a probability of observing the obtained ranging measurements for the various actual first and second wellbore trajectories.

6 . The method of claim 1 , wherein the ranging measurements comprise a vector quantity including the distance and a direction between the first and second wellbores.

7 . The method of claim 1 , wherein the ranging measurements comprise a plurality of unique sets of ranging measurements and corresponding ranging uncertainties, the unique sets of ranging measurements made using corresponding distinct ranging measurement techniques.

8 . The method of claim 1 , further comprising computing an improved ranging uncertainty from the improved wellbore trajectory of the at least one of the first and second wellbores obtained in the optimizing.

9 . The method of claim 8 , further comprising displaying a plot of the improved wellbore trajectory of the at least one of the first and second wellbores and at least one of the improved positional uncertainties or the improved ranging uncertainty.

10 . The method of claim 1 , wherein the determining a direction of drilling the second wellbore further comprises computing an improved ranging uncertainty from the improved wellbore trajectory of the at least one of the first and second wellbores obtained in the optimizing and determining the direction of drilling from the ranging measurements and the improved ranging uncertainty.

11 . A system for combining wellbore surveying measurements and ranging measurements to obtain an improved wellbore trajectory, comprising:

one or more processors;

memory, accessible by the one or more processors, and storing instructions that, when executed by the one or more processors, cause the one or more processors to perform operations comprising:

receiving first and second wellbore trajectories for first and second corresponding wellbores and first and second positional uncertainties for the first and second wellbore trajectories;

receiving a set of ranging measurements and a corresponding ranging uncertainty, the ranging measurements including at least a distance between the first and second wellbores;

optimizing a probability function to compute the improved wellbore trajectory of at least one of the first and second wellbores, the probability function including the first and second wellbore trajectories, the first and second positional uncertainties, the ranging measurements, and the ranging uncertainty, the probability function determining a probability of observing the received first and second wellbore trajectories and the received ranging measurements for various actual first and second wellbore trajectories; and

determining a direction of drilling the second wellbore from the improved wellbore trajectory of the at least one of the first and second wellbores; and

a bottom hole assembly (BHA) configured for deployment in the second wellbore on a drill string, the BHA configured to make the set of ranging measurements in the second wellbore, the BHA still further configured to receive the determined direction of drilling from the one or more processors and to drill the second wellbore along the determined direction of drilling.

12 . The system of claim 11 , wherein the BHA is yet further configured to make survey measurements from which the second wellbore trajectory is computed.

13 . The system of claim 11 , wherein the determining a direction of drilling the second wellbore further comprises:

computing an improved ranging uncertainty from the improved wellbore trajectory of the at least one of the first and second wellbores obtained in the optimizing; and

determining the direction of drilling the second wellbore from the ranging measurements and the improved ranging uncertainty.

14 . The system of claim 11 , wherein the probability function comprises a likelihood function and optimizing the probability function comprises selecting the actual first and second wellbore trajectories that are most likely based on the likelihood function.

15 . The system of claim 14 , wherein the likelihood function comprises a joint likelihood including a product of first, second, and third independent probabilities, the first probability being a probability of observing the obtained first wellbore trajectory for the various actual first wellbore trajectories, the second probability being a probability of observing the obtained second wellbore trajectory for the various actual second wellbore trajectories, and the third probability being a probability of observing the obtained ranging measurements for the various actual first and second wellbore trajectories.

16 . A method for drilling a subterranean wellbore, the method comprising:

obtaining a first wellbore trajectory and corresponding first positional uncertainties for a first wellbore;

drilling a second wellbore in proximity to the first wellbore using a bottom hole assembly (BHA) deployed in the second wellbore;

making wellbore surveying measurements and ranging measurements in the second wellbore using the BHA, the ranging measurements measuring at least a distance between the first wellbore and the second wellbore;

computing a second wellbore trajectory and corresponding second positional uncertainties for the second wellbore;

computing a ranging uncertainty;

defining a probability function including the first and second wellbore trajectories, the first and second positional uncertainties, the ranging measurements, and the ranging uncertainty, the probability function determining a probability of observing the obtained first and second wellbore trajectories and the obtained ranging measurements for various actual first and second wellbore trajectories;

optimizing the probability function to compute an improved wellbore trajectory of at least one of the first and second wellbores;

computing an improved ranging uncertainty from the improved wellbore trajectory of the at least one of the first and second wellbores obtained in the optimizing;

determining a direction of drilling or a change in the direction of drilling of the second wellbore from the ranging measurements and the improved ranging uncertainty; and

using the BHA to drill the second wellbore in the determined direction of drilling.

17 . The method of claim 16 , wherein the probability function comprises a likelihood function and optimizing the probability function comprises selecting the actual first and second wellbore trajectories that are most likely based on the likelihood function.

18 . The method of claim 17 , wherein the defined likelihood function comprises a joint likelihood including a product of first, second, and third independent probabilities, the first probability being a probability of observing the obtained first wellbore trajectory for the various actual first wellbore trajectories, the second probability being a probability of observing the obtained second wellbore trajectory for the various actual second wellbore trajectories, and the third probability being a probability of observing the obtained ranging measurements for the various actual first and second wellbore trajectories.

19 . The method of claim 16 , wherein the ranging measurements comprise a vector quantity including the distance and a direction between the first and second wellbores.

20 . The method of claim 16 , wherein the computing an improved ranging uncertainty comprises computing a refined covariance.