RESERVOIR

BACKGROUND OF THE INVENTION

The present invention relates to the determination of hydrocarbon production in reservoirs for oil/gas industry.

The approaches described in this section could be pursued, but are not necessarily approaches that have been previously conceived or pursued. Therefore, unless otherwise indicated herein, the approaches described in this section are not prior art to the claims in this application and are not admitted to be prior art by inclusion in this section. Furthermore, all embodiments are not necessarily intended to solve all or even any of the problems brought forward in this section.

For the oil/gas company, the determination of the expected production of a reservoir is a key indicator in order to determine where to drill a well and in order to assess the economic value of a reservoir.

In the prior art, dynamic simulator is used to determine the expected production.

From a mathematical point of view, the continuity equation cannot be solved with an analytical solution (in the general case). Therefore, to solve this problem, numerical solutions are sought, by discretizing the space (i.e. gridding the space into a plurality of adjacent cells) and time (time steps). To determine the numerical solutions, many methods propose solving the following equation (combination of mass conservation equation and Darcy’s Law) for each component:

with dM _k the mass accumulated in cell k during the current time step dt for said component, F _ki the net flow rate from cell k into the neighboring cell i during dt, and Q _k-W the net flow rate from cell k into well w during dt.

Although this equation has a linear expression, it is, in fact, non-linear with respect to the physical unknowns (pressure, hydrocarbon component compositions, temperatures...) that are needed to compute the net flow rates. Thus at each time step, one must find the unknown values in cell k so that the non-linear residual (R _{ƒ l} ) _k is nullified:

The method for the resolution (non-linear Newton iterations) is summarized in Figure 1. For a given time step, this time step is considered (step 101 ).

For the time step t ₁ corresponding to the considered time step, the simulator chooses, a priori, a new solution X1 and starts looking at which correction Dx is needed to get the solution R(X ₁ + Dx) = 0.

As it is often considered that Dx is small, it is possible to linearize the equation (step 102) into (X ₁ ) + J _x1 Dx = 0.

To solve the equation (X ₁ ) + J _x1 Dx = 0 (step 103), it is required to invert the Jacobian matrix J _X1 : Dx = - J _x1 ^-1 R(X ₁ .

Once Dx determined, the non-linear residual R(X ₁ + Dx) is determined (step 104). If the non-linear residual R(X ₁ + Dx) is lower than a predetermined threshold e (step 105), the time step is considered as solved and X1 is outputted (step 106).

Otherwise, step 102 is reiterated with XI = C1' with X1 ' = X1 + Dx if the maximum number of nonlinear iterations is not reached (test 107, a non- linear iteration being the iteration triggered by test 107: e.g. steps 103®104®105®107®102®103).

If the maximum number of non-linear iterations is reached, the time step may be reduced (step 108) and the non-linear iteration restarted with the reduced time step (step 101 ). Due to this algorithm, it is determined that most of the computation time is spent on the step 103, i.e. inversion of the Jacobian matrix J _X1 .

The Jacobian J _X1 is a sparse matrix: indeed, each equation of the Jacobian only involves unknowns related to cells of the gridded model that are connected. The entries of the Jacobian matrix that are related to non- adjacent cells are 0.

The method for the resolution of the above flow problem involves an iterative computation, at each time step and for each cell of the gridded representation of the reservoir, of an equation of the type Dx = b which links the values of physical parameters involved in the flow problem in the cell to the values of these parameters in adjacent cells.

In classical reservoir simulation, the solving of equation of the type Dx = b (where A is a Jacobian matrix or a matrix derived from a Jacobian matrix, b a vector and x the solution vector to find) uses iterative methods (e.g. GMRES, generalized minimal residual method, Conjugate Gradient, BICGStab, see Y. Saad, Iterative Methods for Sparse Linear Systems (2 ^nd edition), SIAM, 2003 for details).The iterative methods classically rely on finding an approximation of the solution in a Krylov subspace such that it satisfies a Petrov-Galerkin condition (or variant of Petrov-Galerkin condition, see Y. Saad, Iterative Methods for Sparse Linear Systems (2 ^nd edition), SIAM, 2003 for more details).

The convergence of said method is function of the ratio between the greatest eigenvalue and the smallest eigenvalue of the matrix A. This ratio is called the condition number of the matrix. The larger is the condition number, the higher will be the number of iterations to solve the linear system Dx = b at a prescribed accuracy.

Instead of solving Dx = b, it is possible to use a preconditioner M ^-1 and to solve: M ^-1 Dx M ^-1 b (left preconditioner providing the same solution) - AM ^-1 x' = b (right preconditioner, used after a variable change)

Using a preconditioner generally increases the convergence speed of the iterative resolution algorithm of the linear system.

The choice of the preconditioner M ^-1 is very important; it must be chosen such that the condition number of the linear operator M ^-1 A (or AM ^-1 ) is lower than the condition number of A. The ideal choice to lower the condition number is M ^-1 = A ^-1 but of course this has no interest as a preconditioner since the problem is precisely to determine A ^-1 x. Thus a good preconditioner is a linear operator such that the condition number of the preconditioned system M ^-1 Dx M ^-1 b (or AM ^-1 x' = b), i.e. the condition number of M ^-1 A (or AM ^-1 ), is much lower than the original system Dx = b, i.e. the condition number of A, and such that applying M ^-1 on a vector is much cheaper in number of arithmetic operations than applying directly A ^-1 . It is noticeable that a solver can play the role of preconditioner for another solver, this has to be kept in mind to avoid any confusion since many preconditioner techniques are also solver on their own (but of course when used as preconditioner the convergence tolerance is looser than when they are used as plain solver). As a direct consequence of the condition number definition, an efficient preconditioner can be constructed from the knowledge of the eigenvectors corresponding to the extremum values of the eigenvalues (for example by ensuring that M ^-1 Av = v for v an eigenvector corresponding to a large or a small eigenvalue of A, and M ^-1 Av = Av for the other eigenvalues of A). In practice, computing eigenvectors is usually much more costly in number of arithmetic operations than solving a linear system. Nevertheless, in some problems some inexpensive approximations of the eigenvalues can be computed either by using some physical knowledge of the problem from which the Jacobian system is derived or by exploiting some properties of the matrix A and an efficient preconditioner can be devised from those approximations. In industrial reservoir simulators, the usual numerical discretization scheme is fully implicit thus each equation of the Jacobian system involves a set of unknowns. For instance, the unknowns involved in reservoir simulators may correspond to pressure, or fluid (e.g. oil, water, gas) saturation and concentration in each cell of the gridded model. In a reservoir, the fluid motion is driven by the pressure field, this fact is reflected on the fact that the conditioning of the Jacobian system is mainly dependent from the conditioning of the pressure subsystem obtained after elimination of the other types of unknowns.

According to CPR (Constrained Pressure Residual) preconditioning method, the problem can be reduced to a system that depends solely on the pressure at the preconditioner. Such method is presented for instance in J.R. Wallis, R.P. Kendall T.E. Little:“Constrained Residual Acceleration of Conjugate Residual Methods”, SPE 13563, presented at the 8th Symposium on Reservoir Simulation, Dallas, Feb 10-13, 1985, and in Cao, H., Tchelepi, FI.A., Wallis, J., Yardumian, H.:“Parallel Scalable Unstructured CPR-Type Linear solver for Reservoir simulation”, SPE 96809, Proceedings of the SPE Annual Technical Conference, Dallas, Oct. 9-12, 2005. The CPR method is based on this property: a Jacobian preconditioner is built as a multistep preconditioner which main operation consists in preconditioning an approximation of the reduced pressure linear system. It is noticeable that other numerical schemes such as IMPES (IMplicit Pressure Explicit Saturation) can lead directly to a Jacobian system where there are only pressure unknowns in the Jacobian system. The method of preconditioning that we present in this document is not dependent from the fully implicit numerical scheme nor the CPR method, it applies to any type of numerical scheme discretization used in a reservoir simulation as long as the preconditioner or part of the preconditioner of the Jacobian concerns a pressure unknowns subsystem which matrix is Symmetric Positive Definite (SPD - see Y. Saad, Iterative Methods for Sparse Linear Systems (2 ^nd edition), SIAM, 2003, for mathematical definition of a SPD matrix) or close to a SPD matrix (in the sense that is small compared to ||A|| and that

eigenvectors of A ¹ A can be considered as good approximations of those of A). Due to the nature of the physical equations governing the fluid flows in porous media at any scale, the method is likely to be applied for any fluid flow simulation in a porous medium: simulation of fluid flows at the scale of a reservoir (several kilometers) or at the scale of a pore network in a rock (up to a few centimeters). As a consequence, in the rest of the document, the linear system considered for the preconditioning method described will not necessarily refer to the full Jacobian system of a reservoir simulator; it can be a system or subsystem resulting from any algebraic transformation of the Jacobian system of a reservoir simulator or system issued from any fluid flow simulator used as a pre or post stage of a reservoir simulator: the typical example being simulations at pore scale of the rocks that are used to numerically determine some rock properties needed in the reservoir simulator (those kind of simulation are usually referred to as Digital Rock Physics simulations).

The CPR method is known to be optimal when the pressure matrix A is an M-matrix, i.e. a matrix such as its inverse A ^-1 has positive (or null) coefficients. In reservoir simulators, A is generally an M-matrix or is“close” to an M-matrix (in the sense described above).

The only applicability condition for the preconditioning method is that the considered system matrix is SPD or close to a SPD matrix (in the sense described above). There exists a number of methods to determine a proper preconditioner (or optimal preconditioner) on SPD matrix, e.g. Multi-Grid, Algebraic Multi-Grid (AMG), or multi levels Domain Decomposition methods (see V. Dolean, P. Jolivet and F. Nataf, An Introduction to Domain Decomposition Methods: algorithms, theory and parallel implementation, SIAM bookstore , 2015; and Stiiben K. Algebraic Multigrid (AMG) : An Introduction with Applications. Nov 10. 1999). A preconditioning method is qualified as purely algebraic when it only requires the matrix A of the linear system for the construction of the preconditioner (non purely algebraic method are those that require additional mathematical data such as the mesh geometry, some derivative compute from the continuous equation of the initial problem, etc.). The purely algebraic nature of the preconditioner is important in reservoir simulation because the system is not necessarily directly obtained by a discretization of continuous equations: for example this is the case in the fully implicit scheme when one needs to apply a preconditioner for the pressure system produced by the CPR method. Another advantage of a purely algebraic preconditioner is that it is simpler to implement and maintain in a reservoir simulator software. The usual method used to precondition the pressure subsystem in the CPR method is the Algebraic Multi-Grid (see Stiiben K. Algebraic Multigrid (AMG): An Introduction with Applications. Nov 10. 1999), thus the CPR preconditioner is usually called the CPR-AMG method. The pressure system or subsystem preconditioner is the performance bottleneck in a reservoir simulator running on a massively parallel supercomputer (a computer that connects a very large number of CPU units). The invention presented in this document allows building a purely algebraic preconditioning method that can replace AMG in the CPR method or be used as part of a preconditioner for a pressure linear system in a reservoir simulator.

Such technical problem has been addressed in the international application PCT/IB2017/001566. In this application, a two-level preconditioner based on a domain decomposition of the set of unknowns is considered. In the following, W denotes the entire domain and {W _i, i Î l}, where l is a set of indexes, is a partition of W in a set of subdomains W _i . Such type of two-level preconditioner can be written in a general additive form as:

where the matrix R _i corresponds to a restriction operator from the global set of unknowns toward the subset of unknowns of the subdomain

corresponds to the prolongation operator), and the matrix A _i corresponds to the restriction of the global matrix A to the unknowns of subdomain i (it is noted that can be interpreted as the local solvers in the subdomains

and Z(Z ^t AZ) ^-1 Z ^t can be interpreted as the coarse solver (or “coarse correction”). Note that other variants (such as the multiplicative form) can be written as a two-level preconditioner based on a domain decomposition (see V. Dolean, P. Jolivet and F. Nataf, An Introduction to Domain Decomposition Methods: algorithms, theory and parallel implementation SIAM bookstore , 2015 for a detailed overview of those types of preconditioners).

The main ingredient in a two-level preconditioner as formulated above is the linear operator Z.

Z is classically named a projector (or algebraic projector): it is difficult to identify a proper projector Z such that the coarse problem solution contains enough information so that the condition number of the preconditioned system can be bounded independently of the number of mesh cells used in the simulation and of the physical heterogeneity (such as rock permeability heterogeneity that influences a lot the condition number of the Jacobian system). In addition, it is important that the determination of the projector Z may be parallelized easily (i.e. that a plurality of processors may compute it without important communications between processors). Finally, in order to obtain a purely algebraic preconditioner, Z needs to be constructed only from the matrix of the linear system (we will also denote this matrix by A in the following).

In the previous system used in reservoir simulation (see AMG method), the determination of the coarsening (i.e. the projector Z) implies many communications between processors in charge of said determination. It is well known that communication between processors in parallelized tasks is a real bottleneck that should be avoided.

Nevertheless, and even if the scientists are well aware of said bottlenecks for linear systems coming from reservoir simulation, no method for the purely algebraic determination of projectors is proposed in which the communication between processors are limited to the maximum or simply avoided.

The method described in PCT/IB2017/001566 allows determining an adequate projector Z in an efficient algebraic way, which can be parallelized. In particular, the projector Z defined in this application allows projecting the matrix A on a coarser grid: Z ^t AZ = A _c . In other words, the projector Z allows projecting the original linear system Ax = b on a smaller set of unknowns. However, this method is fully efficient only in the case of a two-level preconditioner. The method of PCT/IB2017/001566 evokes a generalization to more than two levels, by recursively applying the same process to A _c until the size of A _c is correct (i.e. adequate for the use of the users computing said matrix). However, such generalization is not optimal, in particular because A _c is not close to an M-matrix. Actually, the larger the size of A _c is, the less the method is optimal. This is all the more problematic because in many cases, for instance in case of processing large blocks of data in parallel by a graphics processing unit (GPU), the size of A _c may indeed be very large.

There is thus a need for a method for determining an adequate projector Z in an efficient algebraic way, which can be parallelized, as part of a multi- level approach.

SUMMARY OF THE INVENTION

The invention relates to a method implemented by computer means for determining hydrocarbon production of a reservoir, wherein the method comprises:

- modeling the reservoir with a gridded model W comprising a plurality of cells, said gridded model having:

- a coarse partition P ₂ , said coarse partition having coarse subdomains {W _2,j } _j - a fine partition P-i, said fine partition having fine subdomains {W _1,i } _i , each coarse subdomain being composed of a subset of fine subdomains;

- determining a first matrix A based on a Jacobian matrix function of the gridded model; wherein each fine subdomain has a respective first order value, said first order value being function of an index of a line in a subset of consecutive lines of the first matrix corresponding to said fine subdomain, wherein each coarse subdomain has a respective second order value, said second order value being function of an index of a line in a subset of consecutive lines of the first matrix corresponding to said coarse subdomain; wherein the method further comprises:

- splitting the first matrix into subsets of consecutive lines, each subset of consecutive lines corresponding to a respective fine subdomain W _1, , each subset of consecutive lines being received by one dedicated processor; - for each subset of consecutive lines:

- creating a first respective square matrix A _pp based on said subset;

- creating a second respective square matrix B _p based on said subset;

- determining extended generalized eigenvectors x _i and respective eigenvalues l _i of the first square matrix A _pp and the second square matrix B _p ;

- determining relevant eigenvectors, the relevant eigenvector being the determined eigenvectors having respective eigenvalues below a first predetermined threshol

- determining a first projector matrix Z ₁ as a concatenation of the relevant eigenvectors ordered according to:

- firstly, the respective first order value of the fine subdomain W _1,i corresponding to the subset of consecutive lines for which the relevant eigenvector is determined; and

- secondly, the respective eigenvalue of the relevant eigenvector; characterized in that the method further comprises:

- for each coarse subdomain W _2,p :

- extracting a submatrix Z _1[p] from the first projector matrix Z _{1 ;} said submatrix Z _1[p] corresponding to a subset of fine subdomains {W _1,i } _i composing said coarse subdomain W _2,p :

- determining generalized eigenvectors wf of a third square matrix and a fourth square matrix and respective eigenvalues if, said third square matrix and fourth square matrix being function of said submatrix Z _1[p] ; - determining relevant second eigenvectors, the relevant second eigenvectors being the determined eigenvectors of the third square matrix and the fourth square matrix having respective second eigenvalues below a second predetermined threshold 2 ·,

- determining a projector matrix Z based on a concatenation of vectors, each vector being function of a respective relevant second eigenvector, said vectors being ordered according to:

- firstly, the second order value of the coarse subdomain W ₂ , _p for which the respective relevant second eigenvector is determined; and

- secondly, the respective eigenvalue of the relevant second eigenvector;

- determining a preconditioner operator M ^-1 based on the projector matrix Z;

- determining hydrocarbon production for the reservoir based on the preconditioner operator M ^-1 .

The Jacobian matrix (or a transformation of the Jacobian matrix as in the first stage matrix of the CPR method) may be determined by classical methods such as methods described above (i.e. using non-linear Newton iterations).

“Consecutive lines” of a matrix are lines that have a consecutive index (i.e. line number of the matrix) in said matrix: most of the time the index of a matrix is comprised between 1 (i.e. the first line of the matrix) and the number of lines (i.e. the last line of the matrix).

The order of the subsets may be function of the index of the lines that are in said subset: therefore, the order value of a subset that comprises lines 1 - 10 may be 1 , the order value of a subset that comprises lines 11 -15 may be 2, the order value of a subset that comprises lines 16-25 may be 3, etc.

This definition is equivalent to the following: the order of a given subset may be equal or function of the number of subsets having at least one line with a respective index lower than any index of lines that are in said given subset.

By creating a square matrix for each subset (the dimension of the square matrix is equal to the number of lines in the given subset), it is possible to allocate a specific task to a plurality of processors (i.e. determining the eigenvalues and eigenvectors). This task may be performed independently to any other tasks given to other processors. Therefore, the processors do not need to communicate and avoid any bottleneck as described above.

When each processor ends its respective tasks, the projector may be determined by simply extending the eigenvector to the dimension of the global system matrix by adding zeros at indexes of lines not in the subset and then concatenating the extended eigenvectors (horizontal concatenation, i.e. if k vectors should be concatenated, the final matrix has a width of k and a height of the height of the vectors).

In one or several embodiments, the determining of the first matrix A may comprise:

- permuting a Jacobian matrix function of the gridded model according to an ordering of unknowns involved in a computation of flow rates in the gridded model, each unknown having a respective index in said permutation, the permutation being performed so that all unknowns related to a same partition among the coarse partition and fine partition have contiguous indexes in the new ordering and so that inside each contiguous set of indexes, the unknowns of cells that are connected have an index greater than the indexes of unknowns of cells that have no connection; - determining a first matrix A based on the permutation of the Jacobian matrix.

The method requires that the lines of the Jacobian matrix J _X1 corresponding to the division into subdomains be contiguous. When it is not the case, it is possible to use a permutation to do so. Furthermore, ordering unknowns corresponding to a same subdomain so that the unknowns of cells located on an edge of the subdomain connected to another subdomain correspond to lines of higher indexes compared to other unknowns (i.e. unknowns that are not on an edge) may optimize the solving of the eigenvalue problem.

In one or several embodiments, each line of the first matrix A may belong to only one subset among the subsets of consecutive lines.

In other words, each line of the matrix A is in a subset, and no line is more than once in a subset.

In one or several embodiments, for a partition among the fine partition and the coarse partition, the order value of the first subdomain may be greater than the order value of the second subdomain if: - a first subdomain of the partition corresponds to a first subset of consecutive lines of the first matrix A,

- a second subdomain of the partition corresponds to a second subset of consecutive lines of the first matrix A, and

- the first subset has a line subsequent to a line of the second subset. Therefore, the respective order value of the subsets increases while the index of the lines comprised in said subset increases. In one or several embodiments, the method may further comprise:

- for each determined relevant second eigenvector determining a

respective extended relevant second eigenvector

and the determining of the projector matrix Z may comprise: - determining a second projector matrix Z ₂ as a concatenation of the extended relevant second eigenvectors ordered according to:

- firstly, the second order value of the coarse subdomain W ₂ , _p for which the respective relevant second eigenvector is determined; and - secondly, the respective eigenvalue of the relevant second eigenvector;

- determining the projector matrix Z based on the first projector matrix

Z- _L and the second projector matrix Z ₂ .

In addition, the projector matrix Z may be a product of the first projector matrix Z ₁ and the second projector matrix Z ₂ .

In one or several embodiments, the method may comprise:

- for each determined relevant second eigenvector , determining a

respective vector as a function of a product of the relevant

second eigenvector and the submatrix Z _1[p] corresponding to the

coarse subdomain W ₂ , _p for which the respective relevant second eigenvector is determined; wherein the projector matrix Z is a concatenation of the determined vector ordered according to:

- firstly, the second order value of the coarse subdomain W ₂ , _p for which the respective relevant second eigenvector is determined; and

- secondly, the respective eigenvalue of the relevant second eigenvector.

In addition, each vector may be an extended vector derived from a

product of the submatrix Z _1[p] and the relevant second eigenvector

In one or several embodiments, the first predetermined threshold q ₁ may be greater than or equal to the second predetermined threshold ₂ .

In one or several embodiments, the preconditioner operator M ^-1 may be function of Z(Z ^t AZ) ^-1 Z ^t , Z being the determined projector matrix and A being the first matrix.

Another object of the invention relates to a non-transitory computer readable storage medium, having stored thereon a computer program comprising program instructions, the computer program being loadable into a data-processing unit and adapted to cause the data-processing unit to carry out the steps of any of the above methods when the computer program is run by the data-processing device.

Yet another object of the invention relates to a device for determining hydrocarbon production for a reservoir, wherein the device comprises an interface to receive information of the reservoir,

wherein the device comprises a first processor adapted for: - modeling the reservoir with a gridded model W comprising a plurality of cells, said gridded model having:

- a coarse partition P ₂ , said coarse partition having coarse subdomains {W _2,j } _j , - a fine partition P ₁ , said fine partition having fine subdomains {W _1,i } _i , each coarse subdomain being composed of a subset of fine subdomains;

- determining a first matrix A based on a Jacobian matrix function of the gridded model; wherein each fine subdomain has a respective first order value, said first order value being function of an index of a line in a subset of consecutive lines of the first matrix A corresponding to said fine subdomain, wherein each coarse subdomain has a respective second order value, said second order value being function of an index of a line in a subset of consecutive lines of the first matrix A corresponding to said coarse subdomain; wherein the method further comprises:

- splitting the first matrix A into subsets of consecutive lines, each subset of consecutive lines corresponding to a respective fine subdomain W _1,j , each subset of consecutive lines being received by one dedicated processor; wherein the device further comprises a plurality of processors, each subset of consecutive lines being received by one dedicated processor in said plurality of processors; - wherein, for each subset of consecutive lines, a processor in the plurality of processor is configured for: - creating a first respective square matrix A _pp based on said subset;

- creating a second respective square matrix B _p based on said subset;

- determining extended generalized eigenvectors x _i and respective eigenvalues l _i of the first square matrix A _pp and the second square matrix B _p ;

- determining relevant eigenvectors, the relevant eigenvector being the determined eigenvectors having respective eigenvalues below a first predetermined threshold wherein the first processor is further configured for:

- determining a first projector matrix Z ₁ as a concatenation of the relevant eigenvectors ordered according to:

- firstly, the respective first order value of the fine subdomain W _1,i corresponding to the subset of consecutive lines for which the relevant eigenvector is determined; and

- secondly, the respective eigenvalue of the relevant eigenvector; characterized in that the first processor is further configured for:

- for each coarse subdomain W ₂ , _p :

- extracting a submatrix Z _1[p] from the first projector matrix Z ₁ said submatrix Z _1[p] corresponding to a subset of fine subdomains {W _1,i } _i composing said coarse subdomain W _2,p ;

- determining generalized eigenvectors wf of a third square matrix and a fourth square matrix and respective eigenvalues if, said third square matrix and fourth square matrix being function of said submatrix Z _1[p] ; - determining relevant second eigenvectors, the relevant second eigenvectors being the determined eigenvectors of the third square matrix and the fourth square matrix having respective second eigenvalues below a second predetermined threshold 2 ·,

- determining a projector matrix Z based on a concatenation of vectors, each vector being function of a respective relevant second eigenvector, said vectors being ordered according to:

- firstly, the second order value of the coarse subdomain W ₂ , _p for which the respective relevant second eigenvector is determined; and

- secondly, the respective eigenvalue of the relevant second eigenvector;

- determining a preconditioner operator M ^-1 based on the projector matrix Z;

- determining hydrocarbon production for the reservoir based on the preconditioner operator M ^-1 .

Other features and advantages of the method and apparatus disclosed herein will become apparent from the following description of non-limiting embodiments, with reference to the appended drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is illustrated by way of example, and not by way of limitation, in the figures of the accompanying drawings, in which like reference numerals refer to similar elements and in which :

- Figure 1 is a diagram describing a process of solving a flow problem ; - Figures 2a to 2g represent a method of the prior art for determining a projector Z;

- Figure 3 represents two nested partitions of the domain in a possible embodiment of the present invention; - Figure 4 is a representation of the part of the projector Z ₁ corresponding to the p-th subdomain of the coarse partition;

- Figure 5 represents the reduction of the dimension of the generalized eigenvalue problem in a possible embodiment of the present invention; - Figure 6 is a flow chart describing a possible embodiment of the present invention;

- Figure 7 is a possible embodiment for a device that enables the present invention;

- Figure 8 is a representation of submatrices corresponding to the p- th subdomain of the coarse partition.

DESCRIPTION OF PREFERRED EMBODIMENTS

For the purpose of simplification, the Jacobian matrix J _X1 or a transformation of the Jacobian matrix (for instance, by permuting lines or columns of J _X1 as detailed below) is noted in the following description A. It is assumed that lines or columns of the Jacobian matrix J _X1 may be beforehand permuted according to an ordering of unknowns. Indeed, the Jacobian matrix J _X1 obtained by using an arbitrary ordering of unknowns involved in the computation of flow rates in the gridded model is generally not adapted to the method for determining hydrocarbon production of the reservoir, which requires that the lines (or the column, depending on the equations considered) of the matrix corresponding to the division into subdomains be contiguous. Therefore, a permutation of the lines/columns of Jx 1 may be used to ensure that the unknowns of each subdomain of a partition have contiguous indexes, while maintaining (if possible and if the original is symmetric) the symmetry of the matrix. Furthermore, it may be interesting, in terms of optimization of the method, to impose an ordering of unknowns corresponding to a same subdomain so that the unknowns of cells located on an edge of the subdomain connected to another subdomain correspond to lines of higher indexes compared to other unknowns (i.e. unknowns that are not on an edge). Such an ordering optimizes the calculations involved in solving the eigenvalue problem relating to this subdomain. The purpose of determining the projector Z is to determine a proper preconditioner matrix M ^-1 and thus to ease the determination of hydrocarbon production in reservoirs for oil/gas industry.

Figures 2a to 2g represent a method described in PCT/IB2017/001566 for determining a projector Z, in the case where a single partition {W _i , ί e /} of the domain W is considered.

Figure 2a represents an example of a square matrix A of dimensions n x n. The matrix A is usually a sparse matrix, in which many values (blank zones in the representation of matrix A) are null. This property is mainly due to the fact that the interactions between cells are limited to the neighboring cells.

In order to parallelize the task, a slice of the matrix A is determined for each processor p: the first processor receives the first ki lines (e.g. the order number of said subset comprising the first ki lines may be 1 ), the second processor receives the next k ₂ subsequent lines (e.g. the order number of said subset comprising the next k ₂ subsequent lines may be 2), the third processor receives the next k ₃ subsequent lines (e.g. the order number of said subset comprising the next k ₃ subsequent lines may be 3), etc. The number k ₁ , k ₂ , k ₃ , etc. may be identical but it is not mandatory: it is possible to have a different number of lines if the load / capacity / performance / number of available flops / etc. are different for each processor.

This slicing is a splitting of the Jacobian matrix into subsets of consecutive lines. The subset of line corresponding to the restriction of the matrix A _p to the received lines is denoted A _p* .

Each processor p then determines a square matrix A _p with the received lines (at the same position that in the original matrix). The new matrix A _p may then be regarded as (the following assertions are equivalent): - the matrix A where the non-received lines are set to zero;

- a null matrix where the received lines are copied at a position identical to their original position in the matrix A\ or

- a matrix where:

- line having index equal to the index of a second line in the received lines is identical to said second line; and

- line having no index equal to the index of a second line in the received lines is a line with 0-values.

Then, a square matrix A _pp is obtained by considering the restriction of A _p* to the columns which indexes are in the p ^th subset of indexes, as represented in Figure 2a.

A square matrix B _p is then obtained by adding all the entries A _p* (i,j) that are outside the square matrix A _pp in the diagonal coefficient A _pp (i, i ) of A _pp (the indexing X(i,j) refers to a coefficient in the i ^th line and j ^th column the considered matrix X). It is then possible to compute the eigenvalues l and eigenvectors v of the generalized eigenvalue problem: B _p v = lA _pp n such that l < with e [0, 1] (it can be proven that all eigenvalues A are in [0, 1 ]).

When A is a symmetric positive definite (SPD) matrix, then A _pp is also an SPD matrix. Thus, we can write the Cholesky decomposition of A _pp = LI}, where L is a lower triangular matrix (see https://en.wikipedia.org/wiki/Cholesky_decomposition) to transform the generalized eigenvalue problem into:

Thus it is noticeable that the eigenvectors v can be equivalently computed as with w being the eigenvector solution to the eigenvalue problem

When A is not strictly an SPD matrix, the adaptation of the projector Z construction relies upon this reformulated eigenvalue problem. When A is not a symmetric positive definite matrix (which is the case if we want to apply our method as a more parallel replacement for AMG in the CPR-AMG method where the matrix is not perfectly symmetric) we adapt the method as follows:

- instead of computing the Cholesky decomposition, we compute the L.D.U decomposition of the square matrix A _pp (as defined in https://en.wikipedia.org/wiki/LU_decomposition). In practice, for reservoir simulation, we assume that this decomposition is always possible without considering a permutation of A _pp as in the general formula. But if such a case was encountered, a simple line permutation of the A _p* matrix would also allow writing the L.D.U decomposition without having to use a permutation matrix.

In this decomposition L is a lower triangular matrix with unitary coefficients on the diagonal, D is a diagonal matrix and U is an upper triangular matrix with unitary coefficients on the diagonal;

- we consider the matrix S that is the diagonal matrix obtained from D such that the diagonal coefficient of S are computed as S(U) = sqrt(abs(D(U)) ) where S(i, i ) and D(i, i ) are respectively the coefficient in line i and column i of S and D, abs(x) is the absolute value of x and sqrt(x) is the square root of x;

- we compute the square matrix M _p = S ^_1 L ^_1 B _p U ^_1 S ^_1 ;

- we build the square symmetric matrix K _p = M _p

- given the chosen threshold > 0, we select all the eigenvectors w solution of K _p w = Aw such that A < ² ·

- finally we obtain the set of eigenvectors v needed by computing v = U ^_1 S ^_1 w for any w computed at the previous step.

One can notice that if we apply these more general computations in the case where A is SPD, we end up with the same eigenvectors as with the computations for the previous method dedicated to the purely SPD case (but in a less optimized manner since we do at least some extra computations to obtain K _p ).

As mentioned above, it may be interesting to impose a prior ordering of lines or columns of the matrix A so that, for unknowns corresponding to a same subdomain, the unknowns of cells located on an edge of the subdomain connected to another subdomain correspond to lines of higher indexes compared to other unknowns (i.e. unknowns that are not on an edge). This ordering allows optimizing the solving of the generalized eigenvalue problem B _p v = lA _rr n corresponding to each subdomain.

For instance, when matrix A _pp is purely SPD, a Choleski decomposition of A _pp may be performed: A _pp = LLK Thus, the above generalized eigenvalue problem may be reduced to a simpler eigenvalue problem by posing v = L ^_t w (where L ^_t = (L ^t ) ^-1 ):

B _p is equal to matrix A _pp everywhere except for lines that have connections with unknowns in other subdomains. When all the unknowns that are on the edges of the subdomain p are ordered last by the prior ordering, it follows that the matrix B _p differs from A _pp only on the diagonal coefficients which are all grouped at the end of the matrix B _p . B _p may thus be written:

where D is a diagonal matrix having a dimension equal to the number of unknowns on the edges of the subdomain p. Since L ^_1 4 _p L ^_i: = /, then: where a matrix having a dimension much

smaller than B _p , because K has a dimension equal to the number of unknowns on the edges of the subdomain p.

Hence, the problem (L ¹ B _p L ^f )w = Aw may be simplified into a problem of smaller dimension. Kw _boundary where w _boundary is the restriction of w to the unknowns of edges. By filling w _boundary filled with zero values, the eigenvectors w of the complete problem on all the unknowns of the subdomains. Then, the following change of variable may be applied: v = L ^~t w.

Once the eigenvectors have been computed each vector v is prolonged into a vector x of the same dimension than the A matrix by completing with zero values for the indexes that do not correspond to a received line. With reference to Figure 2b, this means that, in the vector x, the values that are not in the hatched zone (i.e. zone corresponding to the received lines) may be set to zero.

When the n eigenvalues and the n eigenvectors are

determined, it is possible to order the eigenvectors {x _k } _ken according to the order of the corresponding l _k (ascending order). In the example of Figure

As represented in Figure 2c, the eigenvectors

having respective eigenvalues greater than a predetermined threshold may be set to zero. Then, all the non-identified eigenvectors, for each processor having received lines from the matrix A, may be concatenated, in the order of the sequence of lines (i.e. order number of the subset of lines) that has been received by the processor:

- if the processor Prod has received the first lines, the identified eigenvectors for said processor is retrieved and forms the first columns of a new matrix Z (in the example of Fig. 2d, three eigenvectors have been identified for Prod , said processor Prod having received the first lines of the matrix A);

- if the processor Proc2 has received a number of immediate subsequent lines (i.e. after the lines of Prod ), the identified eigenvectors for said processor is retrieved and forms the immediate subsequent columns of the new matrix Z (in the example of Fig. 2d, two eigenvectors have been identified for Proc2, said processor Proc2 having received the immediate subsequent lines of the matrix A, after the lines of Prod );

- if the processor Proc3 has received a number of immediate subsequent lines (i.e. after the lines of Proc2), the identified eigenvectors for said processor is retrieved and forms the immediate subsequent columns of the new matrix Z (in the example of Fig. 2d, four eigenvectors have been identified for Proc3, said processor Proc3 having received the immediate subsequent lines of the matrix A, after the lines of Proc2).

This is equivalent to determining eigenvectors having respective eigenvalues below a predetermined threshold (0) as relevant eigenvectors and concatenating said relevant eigenvectors according:

- firstly, the respective order number of the subset for which the relevant eigenvector is determined;

- secondly, the respective eigenvalue of the relevant eigenvector. It is noted that there are two criteria for the ordering: it means that, first, the groups of eigenvectors (i.e. one group per processor) are ordered according to respective order number of the subset for which the relevant eigenvectors are determined, and then each group of eigenvectors are ordered according to the respective eigenvalue. As represented in Fig. 2b, the eigenvectors have null values in zones that do not correspond to the received lines. Therefore, and as shown in Fig 2e, the Z matrix is formed of diagonal blocks, outside said block the value of Z is zero (see white zones of the vectors x _£ ,y _£ , z _£ in Figure 2e). The Z matrix has a height of n, the width of matrix Z is m, m being lesser or equal to n (depending of the value of the threshold 0 : the lesser 0 is, the lesser m is). Said matrix Z allows projecting the matrix A (LEV1 ) on a coarser grid, Z ^t AZ = A _c (see LEV2, Fig. 2f).

Once matrix Z have been determined, the preconditioner M may be calculated. Figure 2g is a flow chart of a method described in PCT/IB2017/001566 for determining a projector Z.

As described above, the matrix A may first be splitted 201 into a plurality of subsets of consecutive lines. Then, for each subset of consecutive lines, the respective square matrix A _pp may be determined 202. The respective square matrix B _p may then be determined 203 by adding all the entries A _p* (i,j) that are outside the square matrix A _pp in the diagonal coefficient A _pp (i, i) of A _pp . Then, the eigenvalues l and eigenvectors v of the generalized eigenvalue problem: may be found 204. It is possible to keep

only the “relevant” eigenvectors, i.e. the eigenvectors associated with eigenvalues l below a predefined threshold θ (λ < ). The relevant eigenvectors v may then be prolonged into vectors x of the same dimension than the A matrix by completing with zero values for the indexes that do not correspond to a received line.

Finally, the projector Z may be determined 205 based on the prolonged relevant eigenvectors x determined for all subsets of consecutive lines. For instance, the projector Z may be a concatenation of the relevant eigenvectors ordered according to multiple criteria:

- firstly, the respective order number of the subset for which the relevant eigenvector is determined; and

- secondly, the respective eigenvalue of the relevant eigenvector.

The preconditioner M may then be calculated based on the projector Z, for instance according to the following formula: with {W _Ϊ } _i a set of local domain/space. Of course, other formula linking M and Z may be used.

As mentioned above, this algorithm is fully efficient in the case of a single partition of the domain W (i.e. a two-level representation of the space), but it does not provide good results in the case of multilevel (i.e. a number of levels strictly higher than 2) representation of space. The present invention proposes an alternative efficient multilevel algorithm.

In the following, it is considered two nested partitions of the reservoir gridblocks. Of course, the proposed method may be used for more than two nested partitions.

Figure 3 represents two nested partitions of the domain (i.e. the set of unknowns) W. The domain W is divided into a first partition

called“coarse” partition, represented in full lines in Figure 3. The domain W may also be divided into a second partition called “fine”

partition, represented in dashed lines in Figure 3. The fine partition P ₁ is defined such that each subdomain W _2,1 of the coarse partition may be decomposed into a union of at least one subdomain W _{1 ;·} of the fine partition (i.e. for each subdomain W _2,1 of the coarse partition, there is a subset of at least one subdomain W _{1 ;·} which is a partition of W _2,1 ). Such partitions are called“nested” partitions.

Each partition of the domain comprises a plurality of subdomains, and each subdomain of a partition comprises (or“covers”) a plurality of cells of the gridded model. For each partition, it is assumed that each subdomain has a respective order value.

As mentioned above, the matrix A may be splitted into a plurality of subsets of consecutive lines, each subset corresponding to a subdomain of a given partition. For instance, the first ki lines of A may correspond to the first subdomain W _{1 1} of the fine partition, the k ₂ lines of A immediately subsequent to the k-i-th line may correspond to the second subdomain W _1,2 of the fine partition, the k ₃ lines of A immediately subsequent to the k ₂ -th line may correspond to the third subdomain W _1,3 of the fine partition, etc.

Similarly, the first r ₁ lines of A may correspond to the first subdomain W _2,1 of the coarse partition, the r ₂ lines of A immediately subsequent to the n-th line may correspond to the second subdomain W _2,2 of the coarse partition, the r ₃ lines of A immediately subsequent to the r ₂ -th line may correspond to the third subdomain W _2,3 of the coarse partition, etc.

The order value of a subdomain may be function of the index of the lines that are in a subset of lines corresponding to the subdomain. For instance, the order value of the first subdomain W _{1 ,1} of the fine partition may be equal to 1 , the order value of the second subdomain W _1,2 of the fine partition may be equal to 2, etc. Similarly, the order value of the first subdomain W _2,1 of the coarse partition may be equal to 1 , the order value of the second subdomain W _{2 2} of the coarse partition may be equal to 2, etc.

The preconditioner M ^-1 may be decomposed on these two nested partitions. For instance, the following additive formula may be considered:

Let’s introduce a few notations that will be used in the following and also to explicit this formula:

- The matrices R _{1 ,i} correspond to a restriction operator from the global set of unknowns toward the subset of unknowns of the i-th subdomain of the partition P ₁ .

The matrices A _{1 ,i} correspond to the restriction of the global matrix A to the unknowns of the i-th subdomain of the partition P ₁ - The matrices R _2,p correspond to a restriction operator from the global set of unknowns toward the subset of unknowns of the p-th subdomain of the partition P ₂ ;

Z _1[p] denotes the restriction of Z ₁ corresponding to the subdomains W _{1 ,i} of the partition P ₁ that compose the subdomain W ₂ , _p , as represented in Figure 4 (the black parts of Z are the blocks obtained by applying the two-level method to the coarse partition P ₂ , and the striped parts of Z ₁ are the blocks obtained by applying the two-level method to the fine partition - The matrix A _{2 [p]} is the submatrix corresponding to the restriction of the global coarse matrix to the coarse unknowns representing the subdomain p in the coarse partition P ₂ . We have

Note that in figure 8, corresponds to the

matrix A _{2 [p]} with this notation and in figure 5 A _2,p is represented by App .

Of course, the present invention is not limited to the above formula. Other expressions are possible, such as a multiplicative form of the three-level preconditioner based on a decomposition of the domain on two nested partitions. Another possible formulation consists in considering a recursive formulation of the method in PCT/IB2017/001566 where Z ₁ is used to build the preconditioner and is used in

the same manner to build a preconditioner for the matrix in

order to solve the inverse problem in the formula of by a preconditioned iterative method (we consider the partition on the set of coarse unknowns induced by the partition P ₂ ).

Z ₁ corresponds to the projector associated to the fine partition P ₁ and may be computed according to the method of PCT/IB2017/001566 recalled above. A first threshold may be chosen for determining the“relevant” eigenvectors.

As represented in Figure 4, Z ₁ is a block diagonal matrix where the k ^th diagonal block is of size (n _k , m _k ) where n _k is the number of unknowns in the subdomain k and m _k is the number of relevant eigenvectors.

The present invention aims to determine a projector Z ₂ such that the product Z ₁ Z ₂ is equivalent, in terms of convergence, to the projector Z that would be obtained by applying the two-level method (i.e. the method of PCT/IB2017/001566) to the coarse partition P ₂ . In other words, Z _1[p] is composed of the eigenvectors v corresponding to the subdomains W _{1 ,i} of the partition P ₁ that compose the subdomain W ₂ , _p , wherein each relevant eigenvectors v has been prolonged into a vector x' of the same height than the number of lines of the restriction of Z ₁ to the subdomain W ₂ , _p by completing with zero values for the indexes that do not correspond to a received line.

For determining Z ₂ , the invention proposes to solve the generalized eigenvalue problem

B _p v = λA _pp n (1) associated to the p-th subdomain W ₂ , _p of the coarse partition P ₂ . To solve problem (1 ), it is assumed that a good solution can be found within the subspace generated by the columns of Z _1[p] . The following change of variable may thus be performed: v = Z _1[p] w.

According to this change of variable, a solution of problem (1 ) may be found by resolving the following generalized eigenvalue problem: The interest of the above change of problem is that problem (2) is computationally much cheaper to solve than problem (1 ), because the dimensions of the matrices and are much smaller

than the dimensions of the matrices B _p and A _pp , as represented in Figure 5. The eigenvalues l and eigenvectors w of the general eigenvalue problem (2): may be computed for all p e I ₂ (i.e. for

all the subdomains of the coarse partition P ₂ ). It has to be noticed that

has the same SPD (or approximate SPD) properties than A _pp .

In addition, a second threshold ₂ may be chosen for determining the “relevant” eigenvectors w. For instance, it may be decided to keep only the eigenvectors w associated with eigenvalues l below the predefined second threshold ₂ (i.e. such that λ < ₂ ), with ₂ [[, 1 ]. In one embodiment, ₁ and ₂ may be fixed such that q ₁ ³ ₂ . Furthermore, and ₂ may be chosen as being close where e is a predefined real value small enough, for instance e = 0.1 or e = 0.05).

Each relevant eigenvector w may be prolonged into a vector x of the same height than the matrix A _c by completing with zero values for the indexes that do not correspond to a received line, as in the method of PCT/IB2017/001566 (see vector x in Figure 2b). For each of subdomain W ₂ , _p (with p Î I ₂ ) of the coarse partition P ₂ , the general eigenvalues and the corresponding prolonged vectors

may thus be determined. It is then possible to order the vectors

according to the order of the corresponding (for instance, an ascending order: The eigenvectors having respective eigenvalues

greater than the predetermined threshold ₂ may be set to zero.

The part z\ of the projector Z ₂ associated to the subdomain W _{2 ,p} may then be determined based on the ordered prolonged vectors For instance, may be obtained by concatenating the relevant eigenvectors according to the respective eigenvalue of the relevant eigenvector.

The projector Z ₂ may then be obtained by concatenating the for all p Î I ₂ (i.e. for all the subdomains W ₂ , _p of the coarse partition P ₂ ), according to the respective order value of the subdomain W ₂ , _p .

This is equivalent to determining vectors corresponding to

eigenvectors having respective eigenvalues below the predetermined threshold (q ₂ ) for all p Î I ₂ and concatenating said vectors according to:

- firstly, the respective order value of the subdomain W ₂ , _p for which the vector c is determined; and

- secondly, the respective eigenvalue of the relevant eigenvector.

Once Z ₂ is determined, the projector Z may be computed as follows: Z= Z ₁ Z ₂ .

Equivalently, for each relevant eigenvector w, a vector v may be computed according to Then, each selected vector v may be

prolonged into a vector V of the same height than the matrix A _c by completing with zero values for the indexes that do not correspond to a received line. The projector Z may then be obtained by concatenating the vectors determined for all p E I ₂ (i.e. for all the subdomains W ₂ , _p of the

coarse partition P ₂ ), according to:

- firstly, the respective order value of the subdomain W ₂ , _p for which the vector is determined; and

- secondly, the respective eigenvalue of the relevant eigenvector. Of course, the present invention may be applied for any number of levels: the projector Z _l+1 of a higher partition level (P _i+1 ) applied to the coarse system can be determined from the projector Z _t of a lower partition level {P _l P ₁ being a subpartition of P _l+1 ) according to the above method, with two respective thresholds (for instance, on can choose

Figure 6 is a flow chart of a method for determining a projector Z in one embodiment of the present invention.

As described above, the projector Z ₁ associated to the fine partition may first be determined 601 based on the method of PCT/IB2017/001566 recalled above, with a first threshold 9 ₁ .

For each subdomain W ₂ , _p of P ₂ , a corresponding submatrice Z _1[p] may be extracted 602 from the determined projector Z ₁ . Each Z _1[p] is the part of Z ₁ corresponding to the subdomains W _{1 ,i} of P ₁ that constitute the subdomain W ₂ , _p of P ₂ (a submatrice Z _1[p] thus defined is represented in Figure 4).

Then, for each p Î I ₂ , the eigenvalues l and eigenvectors w of the generalized eigenvalue problem:

may be found 603. It is possible to keep only the“relevant” eigenvectors, i.e. the eigenvectors associated with eigenvalues l below a second predefined threshold ₂ (i.e. eigenvectors w associated to For instance, the

second predefined threshold ₂ may be chosen such that

In one embodiment, each relevant eigenvector solution of the above

problem (2) may be prolonged into a vector cz of the same height than the matrix A _c by completing with zero values for the indexes that do not correspond to a received line. Z ₂ may then be determined by concatenating said vectors according to:

- firstly, the respective order value of the subdomain W ₂ , _p for which the vector is determined; and

- secondly, the respective eigenvalue of the relevant eigenvector.

The projector Z may then be determined 605 based on the formula

The determination 604 of Z ₂ is optional. In one alternative embodiment, for each relevant eigenvector a respective vector may be computed

according to Then, each selected vector may be prolonged

into a vector of the same height than the matrix A _c by completing with

zero values. The projector Z may then be obtained 605 by concatenating the vectors according to:

- firstly, the respective order value of the subdomain W ₂ , _p for which the vector is determined; and

- secondly, the respective eigenvalue of the relevant eigenvector.

The preconditioner M may then be calculated based on the projector Z, for instance according to one of the following formulas:

or

Of course, other formula linking M and Z may be used. Part of the steps described above can represent steps of an example of a computer program which may be executed by the device of Figure 7.

Figure 7 is a possible embodiment for a device that enables the present invention.

In this embodiment, the device 700 comprise a computer, this computer comprising a memory 705 to store program instructions loadable into a circuit and adapted to cause circuit 704 to carry out the steps of the present invention when the program instructions are run by the circuit 704.

The memory 705 may also store data and useful information for carrying the steps of the present invention as described above.

The circuit 704 may be for instance:

- a processor or a processing unit adapted to interpret instructions in a computer language, the processor or the processing unit may comprise, may be associated with or be attached to a memory comprising the instructions, or

- the association of a processor / processing unit and a memory, the processor or the processing unit adapted to interpret instructions in a computer language, the memory comprising said instructions, or

- an electronic card wherein the steps of the invention are described within silicon, or

- a programmable electronic chip such as a FPGA chip (for « Field- Programmable Gate Array »). This computer comprises an input interface 703 for the reception of data used for the above method (e.g. the reservoir model) according to the invention and an output interface 706 for providing the projector Z or the preconditioner M to an external device 707.

To ease the interaction with the computer, a screen 701 and a keyboard 702 may be provided and connected to the computer circuit 704.

It has to be noted that the projector Z associated with the coarse system obtained with the method of the present invention is different from the projector Z that is obtained by applying the method of PCT/IB2017/001566 to the coarse matrix A _c . The details of this difference are provided hereinafter.

Figure 8 is a representation of submatrices corresponding to the p-th subdomain of the coarse partition.

As represented in Figure 8, let us consider the following notations:

• Z _1[p] is the bloc-diagonal part of the projector Z ₁ corresponding to the subset of subdomains of the fine partition that compose the p-th subdomain W ₂ , _p of the coarse partition P ₂ \

• Z _1[p* ] is the bloc-column part of the projector Z ₁ corresponding to the subset of subdomains of the fine partition P ₁ that compose the p-th subdomain W ₂ , _p of the coarse partition

• is the diagonal part of the coarse matrix z[ _[p] AZ _1[p]

corresponding to the p-th subdomain W ₂ , _p ;

• is the block-row part of the coarse matrix zj; _[p]i 4Z _1[p]

corresponding to the p-th subdomain W ₂ , _p .

The generalized eigenvalue problem corresponding to the p-th subdomain W ₂ , _p in the coarse system is:

We now detail to what correspond the matrices B _p and A _pp in the case where the method of the prior art (i.e. the method of PCT/IB2017/001566) is used on the one hand, and in the case where the present invention is used on the other hand. In both cases, A _pp is the same matrix: A _pp = zl _[p] A _pp Z _1[p] .

In PCT/IB2017/001566, it was proposed applying the same method to the coarse system. This leads to construct:

where the operator Rowsum(M) is the multiplication of M by a vector which all entries are 1 (dimension of the vector is the number of columns in M), and the operator Diag(v) transforms a vector v into a diagonal matrix D which entries are Du = v,.

In other words, B _p is obtained by adding row by row to the diagonal of A _pp the entries of A _{p *} that are not in A _pp .

Now, let us consider the eigenvalue problem induced by the“coarse” partition P ₂ on the matrix A (i.e. we are using the method of the prior art on the matrix A with the coarse partition P ₂ ). This eigenvalue problem is, for the p-th subdomain W ₂ , _p of the partition P _{2 \}

In the method presented here, the idea is that an optimal manner to construct the part of the projector corresponding to the p-th subdomain

is to project this problem on the coarse level using the projector Z _1[p] .

Therefore, we obtain the eigenvalue problem (2) associated with the p-th subdomain W ₂ , _p and the matrix A _c :

Since we obtain the local eigenvalue problem for p-th

subdomain n the coarse matrix A _c :

Hence, with the method presented here,

Since finally obtain:

It is easy to see that this formula is different from the formula obtained for the method of the prior art, at least because the matrix

Diag Rowsum(A _{p *} )

is a (scalar) diagonal matrix whereas the matrix a block

diagonal matrix.

Expressions such as "comprise", "include", "incorporate", "contain", "is" and "have" are to be construed in a non-exclusive manner when interpreting the description and its associated claims, namely construed to allow for other items or components which are not explicitly defined also to be present. Reference to the singular is also to be construed in be a reference to the plural and vice versa.

A person skilled in the art will readily appreciate that various parameters disclosed in the description may be modified and that various embodiments disclosed may be combined without departing from the scope of the invention.

It is noted that the previous examples are 2D examples for the sake of simplicity, but 3D examples may be derived from these 2D examples without difficulties. It is also noted that all the above formula can be rewritten by using the transposition operator. Therefore, even if in the above description, an “index” corresponds to a “row index”, the skilled person would easily understand that an“index” could also correspond to a“column index”.