Methods for performing reservoir simulation

A method for performing reservoir simulation by solving a mixed implicit-IMPES matrix (MIIM) equation, wherein the MIIM equation arises from a Newton iteration of a variable implicit reservoir model, wherein the variable implicit reservoir model comprises a plurality of cells including implicit cells and IMPES cells, wherein the MIIM equation includes a first scalar IMPES equation for each of the IMPES cells and a set of implicit equations for each of the implicit cells, the method comprising:

• a) constructing a global IMPES pressure equation from the MIIM equation, wherein said constructing the global IMPES pressure equation comprises:
• constructing a second scalar IMPES pressure equation for each of the implicit cells from the set of implicit equations corresponding to the implicit cell;
• and
• concatenating the first scalar IMPES pressure equation for each of the IMPES cells and the second scalar IMPES pressure equation for each of the implicit cells;
• b) solving the global IMPES pressure equation for pressure changes;
• c) computing first residuals at the implicit cells in response to the pressure changes;
• d) determining improved saturations and improved pressures by performing one or more iterations with a selected preconditioner at the implicit cells;
• e) computing second residuals at the implicit cells and at a subset of the IMES cells that are in flow communication with any of the implicit cells in response to the improved saturations and improved pressures;
• f) determining if a convergence condition based on the second residuals is satisfied;
• g) repeating b) through f) until the convergence condition is satisfied;
• h) computing a final solution estimate for the MIIM equation from the pressure changes, improved saturations and improved pressures after the convergence condition is satisfied;
• i) applying the final solution estimate to determine behavior of the reservoir model at a future discrete time value.

A method for performing reservoir simulation by solving a mixed implicit-IMPES matrix (MIIM) equation, wherein the MIIM equation arises from a Newton iteration of a variable implicit reservoir model, wherein the variable implicit reservoir model comprises a plurality of cells including implicit cells and IMPES cells, wherein the MIIM equation includes a first scalar IMPES equation for each of the IMPES cells and a set of implicit equations for each of the implicit cells, the method comprising:

• a) constructing a global IMPES pressure equation from the MIIM equation, wherein said constructing the global IMPES pressure equation comprises:
• constructing a second scalar IMPES pressure equation for each of the implicit cells from the set of implicit equations corresponding to the implicit cell;
• and
• concatenating the first scalar IMPES pressure equation for each of the IMPES cells and the second scalar IMPES pressure equation for each of the implicit cells;
• b) solving the global IMPES pressure equation for first pressures;
• c) computing improved saturations at the implicit cells;
• d) determining if a convergence condition is satisfied;
• e) repeatedly performing b) through d) until the convergence condition is satisfied;
• f) computing a final solution estimate for the MIIM equation using the improved saturations and first pressures after the convergence condition is satisfied;
• i) applying the final solution estimate to determine behavior of the reservoir model at a future discrete time value.

A method for performing reservoir simulation by solving an implicit linear equation arising in a Newton iteration of an implicit reservoir model, wherein the reservoir model comprises a plurality of cells, the method comprising:

• a) constructing a global IMPES pressure equation from the implicit linear equation, wherein the global IMPES pressure equation comprises one scalar IMPES pressure equation for each of the plurality of cells;
• b) solving the global IMPES pressure equation to determine first pressure values, wherein one of the first pressure values is associated with each of the plurality of cells;
• c) constructing a complementary matrix equation in terms of unknowns other than pressure, wherein the complementary matrix equation is constructed using a constraint of conserving total velocity between cells;
• d) solving the complementary matrix equation to determine improved estimates of the unknowns other than pressure at each of the plurality of cells;
• e) constructing a composite solution change which combines a first solution change associated with the first pressure values and a second solution change associated with the improved estimates of unknowns other than pressure;
• f) providing the composite solution change to a solution accelerator;
• g) the solution accelerator generating an accelerated solution change;
• h) determining if a convergence condition is satisfied;
• i) repeating (b) through (h) until the convergence condition is satisfied;
• j) computing a final solution estimate based on the accelerated solution change after the convergence condition is satisfied;
• k) applying the final solution estimate to predict properties of reservoir fluids at a future time value.

A method for performing reservoir simulation using total velocity sequential preconditioning, wherein the reservoir is sub-divided into a plurality of cells, the method comprising:

• formulating finite difference equations which describe a behavior of reservoir fluids over a timestep;
• solving the finite difference equations by performing one or more Newton iterations, where each of said one or more Newton iteration comprises:
• a) constructing a linear approximation for each non-linear term in the finite difference equations;
• b) constructing an implicit matrix equation based on the finite difference equations and the linear approximations;
• c) solving the implicit matrix equation, wherein said solving the implicit matrix equation comprises:
• (c1) constructing a complementary matrix equation in terms of unknowns other than pressure using a constraint of conserving total velocity between cells;
• (c2) solving the complementary matrix equation for improved estimates of unknowns other than pressure;
• repeatedly performing said solving the finite difference equations in order to predict behavior of the reservoir fluids over time.

A method for performing reservoir simulation by solving an implicit matrix equation arising from a Newton iteration of an implicit reservoir model, wherein the reservoir model comprises a plurality of cells, wherein the implicit matrix equation includes unknown variables, the method comprising:

• a) constructing a global IMPES pressure equation using the implicit matrix equation, wherein the global IMPES pressure equation comprises one scalar IMPES pressure equation for each of the plurality of cells;
• b) solving the global IMPES pressure equation to determine first pressure values, wherein one of the first pressure values is associated with each of the plurality of cells;
• c) computing improved estimates of the unknown variables other than pressure by performing one or more iterations of a preconditioner;
• d) constructing a composite solution change by combining a first solution change associated with the first pressure values and a second solution change associated with the improved estimates of the unknown variables other than pressure;
• e) providing the composite solution change to a solution accelerator;
• f) the solution accelerator generating an accelerated solution change in response to the composite solution change;
• g) repeatedly performing b) through f) until a convergence criteria is satisfied;
• i) computing a final solution estimate based on the accelerated solution change after the convergence criteria is satisfied;
• wherein the final solution change is utilized to predict a behavior of reservoir fluids at a future discrete time value.

A method for performing reservoir simulation by solving an implicit matrix equation arising from a Newton iteration of an implicit reservoir model, wherein the implicit reservoir model comprises a plurality of cells, wherein the implicit matrix equation is expressed in terms of unknown variables including pressures, the method comprising:

• a) constructing a global IMPES pressure equation using the implicit matrix equation, wherein the global IMPES pressure equation comprises one scalar IMPES pressure equation for each of the plurality of cells;
• b) solving the global IMPES pressure equation to determine first improved estimates of the pressures, wherein one of the first improved estimates is associated with each of the plurality of cells;
• c) computing second improved estimates of all the unknowns variables by performing one or more iterations of a preconditioner;
• d) constructing a composite solution change in the unknown variables by combining a first solution change associated with the first improved estimates and a second solution change associated with the second improved estimates;
• e) providing the composite solution change to a solution accelerator;
• f) the solution accelerator generating an accelerated solution change;
• g) determining if a convergence condition is satisfied;
• h) repeatedly performing b) through g) until the convergence condition is satisfied;
• i) computing a final solution estimate based on the accelerated solution change after the convergence condition is satisfied, and applying the final solution estimate to predict properties of reservoir fluids at a future time value.