Multi-stage linear solution for implicit reservoir simulation

A reservoir simulation system including:

• memory to store equations obtained from three sequential stages that represent fluid flow properties of a reservoir, the reservoir associated with a plurality of grid blocks and a plurality of physical components;
• one or more processors to:
• in a first stage generate a first matrix, based on the equations, that includes mass conservation information for the plurality of physical components and volume balance information for the plurality of grid blocks;
• in a second stage transform the mass conservation information and the volume balance information for the same plurality of grid blocks used in the first matrix to generate a second matrix by:
• generating a component phase density matrix to represent densities of the plurality of physical components in each of at least two physical phases;
• generating a phase volume derivative matrix to represent volume changes of the at least two physical phases;
• and
• generating a restriction operator and a prolongation operator, based on the component phase density matrix and the phase volume derivative matrix, for use in transforming the first matrix;
• in a third stage using a constrained pressure residual method, reduce the second matrix to remove saturation information to generate a third matrix that includes pressure information for the same plurality of grid blocks used in the second matrix, wherein the pressure information is the only unknown in the third matrix;
• solve the third matrix to generate a first pressure solution for the plurality of grid blocks;
• solve the second matrix based on the first pressure solution to generate a first saturation solution and a second pressure solution for the plurality of grid blocks;
• and
• use the first saturation solution and the second pressure solution to generate a solution of the first matrix in the first stage;
• generate a simulation that predicts behavior of the reservoir based on the solution of the first matrix;
• and
• recover resources from the reservoir based upon the simulation.

A non-transitory machine-readable storage device having instructions stored thereon which, when performed by a machine, cause the machine to perform operations, the operations comprising:

• accessing equations in three sequential stages that represent fluid flow properties of a reservoir, the reservoir associated with a plurality of grid blocks and a plurality of physical components;
• in a first stage generating a first matrix, based on the equations, that includes mass conservation information for the plurality of physical components and volume balance information for the plurality of grid blocks;
• in a second stage transforming the mass conservation information and the volume balance information in the first matrix into saturation information and pressure information included in a second matrix for the same plurality of grid blocks used in the first stage;
• in a third stage using a constrained pressure residual method, reducing the second matrix to remove the saturation information to generate a third matrix that includes the pressure information for the same plurality of grid blocks used in the second stage, wherein the pressure information is the only unknown in the third matrix;
• solving the third matrix to generate a first pressure solution for the plurality of grid blocks;
• solving the second matrix based on the first pressure solution to generate a first saturation solution and a second pressure solution for the plurality of grid blocks;
• and
• using the first saturation solution and the second pressure solution to generate a solution of the first matrix in the first stage;
• generating a simulation that predicts behavior of the reservoir based on the solution of the first matrix;
• and
• recover resources from the reservoir based upon the simulation.