Directional drilling with stochastic path optimization of operating parameters

1. A drilling method that comprises:

obtaining a formation model representing formation properties to be encountered by a drilling assembly being steered towards a target;

identifying at least one path-dependent drilling model for predicting the drilling assembly's response to one or more operating parameters, wherein each of the at least one path-dependent drilling models is based on formation properties of each formation bed identified by the formation model;

characterizing uncertainties associated with said formation model and said at least one drilling model, said characterizing yielding a probability density function for each uncertainty; and

representing an acceptable range for each of said one or more operating parameters as a probability density function.

2. The drilling method of claim 1 , further comprising:

employing the probability density functions to determine random samples of said uncertainties and of said one or more operating parameters; and

applying a cost function to the random samples to determine an expected cost as a function said one or more operating parameters.

3. The drilling method of claim 2 , further comprising:

displaying the randomly sampled operating parameters having a minimum expected cost as optimized operating parameters.

4. The method of claim 3 , further comprising using the optimized operating parameters to steer the drilling assembly towards the target.

5. The method of claim 3 , wherein said one or more operating parameters include weight on bit and rotations per minute.

6. The method of claim 3 , wherein said one or more operating parameters include hook load and top drive torque.

7. The method of claim 3 , wherein said one or more operating parameters further include toolface orientation and dogleg severity.

8. The method of claim 1 , wherein at least some of said uncertainties are associated with:

position of at least one formation bed boundary;

formation rock hardness; and

bit wear.

9. The method of claim 1 , wherein at least some of said uncertainties are associated with:

formation heterogeneity; and

position of one or more formation anomalies.

10. The method of claim 1 , wherein at least some of said uncertainties are associated with:

sensor measurements of said one or more operating parameters;

bit walk rate; and

accuracy of the at least one path-dependent drilling model.

11. The method of claim 3 , wherein the cost function is a weighted sum of optimization merits including one or more of: energy dissipation ratio, bit wear, mud motor wear, cutting efficiency, rate of penetration, financial cost, vibration excitation, and quantified path geometry.

12. The method of claim 3 , further comprising:

monitoring drilling data to refine the formation model and the drilling model; and

repeating said identifying, characterizing, representing, employing, applying, and displaying.

13. A drilling system that comprises:

a drilling assembly including:

a bottomhole assembly with a drill bit and a steering assembly; and

a drill string that connects the bottomhole assembly to a drilling rig; and

a processing system that provides optimized values of operating parameters for steering the drilling assembly to a target, by:

obtaining a formation model representing formation properties to be encountered by the drilling assembly;

identifying at least one path-dependent drilling model for predicting the drilling assembly's response to the operating parameters, wherein each of the at least one path-dependent drilling models is based on formation properties of each formation bed identified by the formation model;

characterizing uncertainties associated with said formation model and said at least one drilling model, said characterizing yielding a probability density function for each uncertainty; and

representing an acceptable range for each of said operating parameters as a probability density function.

14. The drilling system of claim 13 , wherein the processing system further provides optimized values by:

employing the probability density functions to determine random samples of said uncertainties and of said operating parameters; and

applying a cost function to the random samples to determine an expected cost as a function said operating parameters.

15. The drilling system of claim 14 , wherein the processing system provides optimized values by further:

displaying the randomly sampled operating parameters having a minimum expected cost as optimized values for the operating parameters.

16. The system of claim 13 , wherein the drilling assembly uses the optimized values of operating parameters to steer the drilling assembly towards the target.

17. The system of claim 13 , wherein said one or more operating parameters include weight on bit and rotations per minute.

18. The system of claim 13 , wherein said one or more operating parameters include hook load and top drive torque.

19. The system of claim 18 , wherein said one or more operating parameters further include toolface orientation and dogleg severity.

20. The system of claim 13 , wherein at least some of said uncertainties are associated with:

position of at least one formation bed boundary;

formation rock hardness; and

bit wear.

21. The system of claim 13 , wherein at least some of said uncertainties are associated with:

formation heterogeneity; and

position of one or more formation anomalies.

22. The system of claim 13 , wherein at least some of said uncertainties are associated with:

sensor measurements of said one or more operating parameters;

bit walk rate; and

accuracy of the at least one path-dependent drilling model.

23. The system of claim 14 , wherein the cost function is a weighted sum of optimization merits including one or more of: energy dissipation ratio, bit wear, mud motor wear, cutting efficiency, rate of penetration, financial cost, vibration excitation, and quantified path geometry.

24. The system of claim 15 , wherein the processing system further provides the optimized values by:

monitoring drilling data to refine the formation model and the drilling model; and

repeating said identifying, characterizing, representing, employing, applying, and displaying.