Computer estimation method, and method for oil exploration and development using such a method
The invention relates to a method for estimating a set of variables associated with sites of an area, the method comprising the following steps: a) a set of observation data is supplied, b) a variable associated with a site of a first sub-area is estimated by krigeing, using the precision matrix of the first sub-area, c) a second adjacent sub-area is determined, d) a variable associated with a site of the second sub-area is estimated by krigeing, on the basis of the first sub-space, the second sub-space, and the precision matrix determined for the first sub-space.
1. A computerized method for estimating a suite of quantities associated with locations of a space, wherein said space is a subsurface envisaged for oil exploitation, wherein said quantities are suitable to model a hydrocarbon reservoir in said space, the method comprises the following steps:
a) a set of observation data comprising quantities associated with certain locations is provided,
b) a quantity associated with at least one location of a first sub-space included in said space is estimated by kriging, using a precision matrix of the first sub-space,
c) a second sub-space included in said space and neighboring the first sub-space is determined,
d) a quantity associated with at least one location of the second sub-space is estimated by kriging on the basis of the first sub-space, of the second sub-space, and of the precision matrix determined for the first sub-space,
e) confirming the presence of hydrocarbons to be extracted from the space based on the quantity estimated at d).
2. The method as claimed in claim 1 in which, in step d), a precision matrix of the second sub-space is determined on the basis of the precision matrix of the first sub-space, and of a Schur complement of a covariance matrix of one of the first and second sub-spaces.
3. The method as claimed in claim 2 , in which the second sub-space is obtained by adding locations to the first sub-space, and in which, in step d), we solve:
C
AUB
,
AUB
-
1
=
(
C
A
,
A
-
1
+
UR
t
-
R
-
R
t
T
)
,
Where
A denotes the set of locations of the first sub-space,
B denotes the set of added locations,
U denotes the set union operation,
C X,Y denotes the covariance matrix of the sets X and Y,
−1denotes the matrix inversion operation,
t denotes the matrix transposition operation,
U=C −1 A,A C A,B ,
S=C B,B −C t A,B U, is the Schur complement of C AUB,AUB ,
T=S −1 ,
R=UT.
4. The method as claimed in claim 2 , in which the second sub-space is obtained by removing locations from the first sub-space, and in which, in step d), we solve:
C
AUB
,
AUB
-
1
=
(
C
A
,
A
-
1
+
UR
t
-
R
-
R
t
T
)
,
Where
A denotes the set of locations of the second sub-space,
B denotes the set of removed locations,
U denotes the set union operation,
C X,Y denotes the covariance matrix of the sets X and Y,
−1denotes the matrix inversion operation,
t denotes the matrix transposition operation,
U=C −1 A,A C A,B ,
S=C B,B −C t A,B U, is the Schur complement of C AUB,AUB ,
T=S −1 ,
R=UT.
5. The method as claimed in claim 2 , in which the second sub-space is obtained at one and the same time by adding locations to and removing locations from the first sub-space, and wherein the second sub-space is obtained by adding locations to the first sub-space, and in which, in step d), we solve:
C
AUB
,
AUB
-
1
=
(
C
A
,
A
-
1
+
UR
t
-
R
-
R
t
T
)
,
where
A denotes the set of locations of the first sub-space,
B denotes the set of added locations,
U denotes the set union operation,
C X,Y denotes the covariance matrix of the sets X and Y,
−1 denotes the matrix inversion operation,
t denotes the matrix transposition operation,
U=C −1 A,A C, A,B ,
S=C B,B −C t A,B U, is the Schur complement of C AUB,AUB ,
T=S −1 ,
R=UT; and
in which the second sub-space is obtained by removing locations from the first sub-space, and in which, in step d), we solve:
C
CU
′
D
,
CU
′
D
-
1
=
(
C
C
,
C
-
1
+
U
′
R
″
-
R
′
-
R
″
T
)
,
Where
C denotes the set of locations of the second sub-space,
D denotes the set of removed locations,
U′denotes the set union operation,
C X,Y denotes the covariance matrix of the sets X and Y,
−1 denotes the matrix inversion operation,
t denotes the matrix transposition operation,
U′=C −1 C,C C C,D ,
S′=C D,D −C t C,D U′, is the Schur complement of C CUD,CUD ,
T′=S′ 31 1 ,
R′=U′T′.
6. The method as claimed in claim 1 in which, in step b), a suite of quantities associated with a plurality of locations of the first sub-space is estimated by kriging when the first sub-space is sufficiently near to said plurality of locations.
7. The method as claimed in claim 1 in which, in step c), the second sub-space is determined on the basis of the first sub-space.
8. The method as claimed in claim 7 in which, in step c), one and/or the other of the following operations are/is implemented:
addition of locations to the first sub-space, said added locations being near to said location for which step d) is implemented,
removal of locations from the first sub-space, said removed locations being far from said location for which step d) is implemented.
9. The method as claimed in claim 1 in which:
after step a), and before step c), a step z) is implemented in the course of which a tree kD-tree is constructed of the locations for which quantities obtained in step a) are available, and
in step c), the second sub-space is determined on the basis of said tree.
10. The method as claimed in claim 9 , in which, in step z), the tree comprises a plurality of levels each defining a partition of the set of locations determined in step a), each node of a level comprising a group of locations, said group being partitioned into at least one node of the directly lower level branched to this node, disregarding the groups associated with terminal nodes.
11. The method as claimed in claim 9 , in which, in step b), the first subset is constructed by sweeping the tree in search of the locations nearest to the location for which it is desired to estimate the quantity.
12. The method as claimed in claim 11 , in which, in step c), the second subset is constructed by using the first subset and the distance between the locations for which it is desired to determine the quantities in step b) and in step d).
13. The method as claimed in claim 11 , where, in step c), to impose a distribution of the nearest neighbors, the search for the nearest neighbors is constrained in accordance with several sectors and then the searches are merged.
14. The method as claimed in claim 9 , in which the rows and the columns of the precision matrix are ordered according to the distance of the locations of the second sub-space from the location considered on the basis of said tree.
15. The method as claimed in claim 1 , in which steps c) and d) are repeated by taking account of a previous sub-space.
16. An oil exploration method, in which the space is a subsurface, in which the quantities are geological quantities of said subsurface, and in which the method as claimed in claim 1 is implemented by using in step a) quantities obtained by seismic imaging.
17. An oil exploitation method in which a method as claimed in claim 16 is implemented, and in which a step y) is implemented in the course of which an oil exploitation rig is constructed for said space.
18. A computer program product suitable for implementing the steps of the methods as claimed in claim 1 , when it is implemented on a programmable machine.