The commonly used methods for constructing p( A) b are:īoth methods minimize the residual norm over all vectors in span at iteration m. Basically, these techniques approximate A -1 b by p( A) b. These Krylov subspaces are spaces spanned by vectors of the form p( A) v, where p is a polynomial. Iterative solution based on projection onto Krylov subspaces is typically used. If A is large, solution of the matrix equations is impractical using direct methods such as Gaussian elimination because of computer storage or CPU time requirements. The well variables are then obtained by back substitution as In this case, the well variables may be directly eliminated, and the iterative solution is on the implicitly defined matrix system Where x w is the well variable-solution vector and x R is the reservoir variable-solution vector. The matrix A typically has associated well constraint equations and well variables and may be partitioned in block 2 × 2 form as In most formulations, pressure is an unknown for each cell. In the adaptive implicit formulation, there is a variable number of unknowns per cell. In the fully implicit formulation, there is a fixed number n of unknowns per cell where n ≥ 2. In the Implicit Pressure Explicit Saturation (IMPES) formulation, there is a single unknown per cell pressure. The matrix problem involves solving Ax=b, where A is typically a large sparse matrix, b is the right-side vector, and x is the vector of unknowns. For nonisothermal problems, an energy balance is added to the system.
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