Method of estimating the fracture density in a rock medium
The invention is a method of constructing a three-dimensional fracture density log of a porous formation traversed by fractures and a borehole. A number N of intersections between the fractures and the borehole is measured over a section of the borehole of length L located at a depth d, from observations of the wall of this section. A conditional probability law of the three-dimensional fracture density is then estimated, knowing the number N of intersections. The value of the three-dimensional fracture density at depth d is determined by calculating the mean of this conditional probability law. Finally, the three-dimensional fracture density log is constructed by repeating the previous stages for different depths. A three-dimensional fracture density uncertainty log can also be associated by calculating quantiles of the conditional probability law.
1. A computer implemented method of developing an underground formation including constructing a fracture density log of the underground formation traversed by a series of fractures and at least one borehole, comprising:
a) measuring a number of intersections between fractures and the at least one borehole over a section of the at least one borehole of a length located at a depth from observations of a wall of the section;
b) expressing a conditional probability law of a three-dimensional fracture density with knowledge of the number of intersections and the three-dimensional fracture density corresponding to a sum of fracture surface areas per volume unit;
c) determining with a computer a three-dimensional fracture density value at the depth calculated from the expression of the conditional probability law;
d) constructing a three-dimensional fracture density log by repeating a), b) and c) for different depths;
e) constructing a three-dimensional fracture density log and at least one three-dimensional fracture density uncertainty log;
f) constructing a three-dimensional fracture density map and a map of uncertainties of fracture density values from at least one of the logs; and
g) optimizing with a computer development of the underground formation by accounting for the density map and the map of the uncertainties to extract flows between the fractures; and wherein
at least one three-dimensional fracture density uncertainty log is associated with the three-dimensional fracture density log by calculating quantiles of the conditional probability law.
2. A method as claimed in claim 1 , wherein the conditional probability law is expressed by considering the fractures to have barycenters distributed in the formation according to Poisson's law.
3. A method as claimed in claim 2 , wherein the conditional probability law is expressed as a function of a probability law of an angle defined between a fracture and the at least one borehole, the number of intersections and the length of the section.
4. A method as claimed in claim 3 , wherein:
the fractures have an elliptical shape and the borehole section is cylindrical and further comprising:
defining a fracture length probability law and a fracture width probability law; and
estimating a conditional probability function by accounting for the fracture length and width probability laws.
5. A method as claimed in claim 2 , wherein the at least one borehole has a smaller cross section in comparison with an extension of the fractures.
6. A method as claimed in claim 5 , wherein the conditional probability law is expressed with a Gamma law of parameters N+1 and k o L, with k o being a mean of the angle calculated from the probability law of the angle wherein N is the number of intersections and L is the length of the section.
7. A method as claimed in claim 6 , wherein
the fractures have an elliptical shape and the borehole section is cylindrical and further comprising:
defining a fracture length probability law and a fracture width probability law; and
estimating a conditional probability function by accounting for the fracture length and width probability laws.
8. A method as claimed in claim 5 , wherein:
the fractures have an elliptical shape and the borehole section is cylindrical and further comprising:
defining a fracture length probability law and a fracture width probability law; and
estimating a conditional probability function by accounting for the fracture length and width probability laws.
9. A method as claimed in claim 2 , wherein:
the fractures have an elliptical shape and the borehole section is cylindrical and further comprising:
defining a fracture length probability law and a fracture width probability law; and
estimating a conditional probability function by accounting for the fracture length and width probability laws.
10. A method as claimed in claim 1 , wherein the conditional probability law is expressed as a function of a probability law of an angle defined between a fracture and the at least one borehole, the number of intersections and the length of the section.
11. A method as claimed in claim 10 , wherein the at least one borehole has a smaller cross section in comparison with an extension of the fractures.
12. A method as claimed in claim 11 , wherein the conditional probability law is expressed with a Gamma law of parameters N+1 and k o L, with k o being a mean of the angle calculated from the probability law of the angle wherein N is the number of intersections and L is the length of the section.
13. A method as claimed in claim 12 , wherein:
the fractures have an elliptical shape and the borehole section is cylindrical and further comprising:
defining a fracture length probability law and a fracture width probability law; and
estimating a conditional probability function by accounting for the fracture length and width probability laws.
14. A method as claimed in claim 11 , wherein:
the fractures have an elliptical shape and the borehole section is cylindrical and further comprising:
defining a fracture length probability law and a fracture width probability law; and
estimating a conditional probability function by accounting for the fracture length and width probability laws.
15. A method as claimed in claim 10 , wherein:
the fractures have an elliptical shape and the borehole section is cylindrical and further comprising:
defining a fracture length probability law and a fracture width probability law; and
estimating a conditional probability function by accounting for the fracture length and width probability laws.
16. A method as claimed in claim 1 , wherein the at least one borehole has a smaller cross section in comparison with an extension of the fractures.
17. A method as claimed in claim 16 , wherein the conditional probability law is expressed with a Gamma law of parameters N+1 and k o L, with k o being a mean of the angle calculated from the probability law of the angle wherein N is the number of intersections and L is the length of the section.
18. A method as claimed in claim 17 , wherein:
the fractures have an elliptical shape and the borehole section is cylindrical and further comprising:
defining a fracture length probability law and a fracture width probability law; and
estimating a conditional probability function by accounting for the fracture length and width probability laws.
19. A method as claimed in claim 16 , wherein:
the fractures have an elliptical shape and the borehole section is cylindrical and further comprising:
defining a fracture length probability law and a fracture width probability law; and
estimating a conditional probability function by accounting for the fracture length and width probability laws.
20. A method as claimed in claim 1 , wherein:
the fractures have an elliptical shape and the borehole section is cylindrical and further comprising:
defining a fracture length probability law and a fracture width probability law; and
estimating a conditional probability function by accounting for the fracture length and width probability laws.
21. A method as claimed in claim 20 , wherein the number of intersections is replaced by a number of partial intersections corresponding to fractures that do not completely intersect a cross section of the borehole.