Flexible and adaptive formulations for complex reservoir simulations

• US 9,058,446 B2
• Filed: 06/29/2011
• Issued: 06/16/2015
• Est. Priority Date: 09/20/2010
• Status: Active Grant

• Current Assignee: Exxonmobil Upstream Research Company (Exxon Mobil Corporation)
• Original Assignee: Exxonmobil Upstream Research Company (Exxon Mobil Corporation)
• Inventors: Lu, Pengbo, Beckner, Bret L.
• Primary Examiner(s): Chad, Aniss
• Application Number:
  US13/812,828
• Publication Number:
  US 20130124173A1
• Time in Patent Office: 1,448 Days
• Field of Search: 703/2, 703/10
• US Class Current: 1/1
• CPC Class Codes: E21B 49/00 Testing the nature of borehole walls; Formation testing; Methods or apparatus for obtaining samples of soil or well fluids, specially adapted to earth drilling or wells
• CPC Class Codes: G01V 11/00 Prospecting or detecting by methods combining techniques covered by two or more of main groups G01V1/00 - G01V9/00
• CPC Class Codes: G06F 17/11 for solving equations , e.g. nonlinear equations, general mathematical optimization problems optimization specially adapted for a specific administrative, business or logistic context G06Q10/04
• CPC Class Codes: G06F 30/20 Design optimisation, verification or simulation optimisation, verification or simulation of circuit designs G06F30/30

Abstract

A method for performing a simulation of a subsurface hydrocarbon reservoir is disclosed. Each cell in a reservoir model has an equation set representing a reservoir property. A stability limit is determined for each cell. Each cell is assigned to an explicit or implicit formulation. A solution to the system of equations is solved for using an initial guess and an explicit or implicit formulation. A stability limit is calculated for the converged cells. When the number of unconverged cells is greater than a predetermined amount, a reduced nonlinear system is constructed with a list of unconverged cells. The reduced nonlinear system is solved with the implicit formulation, and other cells are solved with the explicit formulation. Parts of the method are repeated until all equation sets satisfy a convergence criterion and a stability criterion, and the solved solution is output.

Claims

1. A method of performing a simulation of a subsurface hydrocarbon reservoir, the reservoir being approximated by a reservoir model having a plurality of cells, each cell having associated therewith an equation set representing a reservoir property, the method comprising:

(a) determining a stability limit for each of the plurality of cells;

(b) assigning each cell to one of an explicit formulation or an implicit formulation;

(c) providing an initial guess to a solution for a system of equations formed using the equation set for each cell in the plurality of cells;

(d) using the initial guess to solve for a solution to the system of equations using an explicit formulation for cells assigned thereto and an implicit formulation for cells assigned thereto;

(e) establishing a list of unconverged cells, the unconverged cells having equation sets that have not satisfied a convergence criterion;

(f) calculating a stability limit for each of the converged cells, the converged cells having equation sets that have satisfied the convergence criterion;

(g) when the number of unconverged cells is greater than a predetermined amount, constructing a reduced nonlinear system with the list of unconverged cells, the reduced nonlinear system being assigned to be solved with the implicit formulation and other cells being assigned to be solved with the explicit formulation;

(h) repeating parts (d), (e), (f) and (g), substituting the solved solution for the initial guess or the most recent solved solution and substituting the equation sets corresponding to the cells in the list of unconverged cells for the system of equations or equation sets from the most recent iteration, until all equation sets satisfy the convergence criterion and a stability criterion; and

(i) when all equation sets satisfy the convergence criterion and the stability criterion, outputting the solved solution as a result for a timestep of a simulation of the subsurface reservoir.

2. The method of claim 1, further comprising using an iterative root-finding method with the initial guess to solve for the solution to the system of equations.

3. The method of claim 1 wherein the iterative root-finding method comprises Newton'"'"'s Method.

4. The method of claim 1, further comprising adding, to the list of unconverged cells, converged cells with one or more reservoir properties exhibiting changes greater than a preset amount.

5. The method of claim 1, further comprising adding, to the list of unconverged cells, converged cells that do not satisfy the stability criterion.

6. The method of claim 1, further comprising adding, to the list of unconverged cells, any cell that is neighbor to a cell in the list of unconverged cells and that does not satisfy the stability criterion.

7. The method of claim 1, wherein the number of neighbor cells is between 1 and N-W, where N is the number of the plurality of cells in the reservoir model and W is the number of cells having equation sets that satisfy the convergence criterion.

8. The method of claim 1, wherein any cell that is neighbor to a cell in the list of unconverged cells and that does satisfy the stability criterion is assigned to be solved using the explicit formulation.

9. The method of claim 1, wherein the reservoir property is at least one of fluid pressure, fluid saturation, and fluid flow.

10. The method of claim 1, wherein some or all of the method is performed using a computer.

11. The method of claim 1, wherein the timestep of the simulation of the subsurface reservoir is a first timestep, the method further comprising:

(j) repeating parts (a) through (i) at additional timesteps; and

(k) outputting the solved solutions for the first timestep and the additional timesteps, the solved solutions simulating the subsurface reservoir over time.

12. The method of claim 1, wherein outputting the solved solution includes displaying the solved solution.

13. The method of claim 1, wherein the predetermined amount is zero.

14. The method of claim 1, further including employing a post-Newton material balance corrector when all equation sets satisfy the convergence criterion and the stability criterion.

15. The method of claim 14, wherein the post-Newton material balance corrector employs an explicit molar update or total volumetric flux conservation.

16. A method of performing a simulation of a subsurface hydrocarbon reservoir, the reservoir being approximated by a reservoir model having a plurality of cells, each cell having associated therewith an equation set representing a reservoir property including at least one of fluid pressure, saturation, and fluid flow, the method comprising:

(a) determining a stability limit for each of the plurality of cells;

(b) assigning each cell to one of an explicit formulation or an implicit formulation;

(c) providing an initial guess to a solution for a system of equations formed using the equation set for each cell in the plurality of cells;

(d) using the initial guess to solve for a solution to the system of equations using an explicit formulation for cells assigned thereto and an implicit formulation for cells assigned thereto;

(e) establishing a list of unconverged cells, the unconverged cells having equation sets that have not satisfied a convergence criterion;

(f) calculating a stability limit for each of the converged cells, the converged cells having equation sets that have satisfied the convergence criterion;

(g) when the number of unconverged cells is greater than a predetermined amount, adding, to the list of unconverged cells,

converged cells with one or more reservoir properties exhibiting changes greater than a preset amount,

converged cells that do not satisfy the stability criterion, and

any cell that is neighbor to a cell in the list of unconverged cells and that does not satisfy the stability criterion,

and constructing a reduced nonlinear system with the list of unconverged cells, the reduced nonlinear system being assigned to be solved with the implicit formulation and other cells being assigned to be solved with the explicit formulation;

(h) repeating parts (d), (e), (f) and (g), substituting the solved solution for the initial guess or the most recent solved solution and substituting the equation sets corresponding to the cells in the list of unconverged cells for the system of equations or equation sets from the most recent iteration, until all equation sets satisfy the convergence criterion and a stability criterion; and

(i) when all equation sets satisfy the convergence criterion and the stability criterion, outputting the solved solution as a result for a timestep of a simulation of the subsurface reservoir.

17. The method of claim 16, wherein the timestep of the simulation of the subsurface reservoir is a first timestep, the method further comprising:

(j) repeating parts (a) through (i) at additional timesteps; and

(k) outputting the solved solutions for the first timestep and the additional timesteps, the solved solutions simulating the subsurface reservoir over time.

18. The method of claim 17, further including employing a post-Newton material balance corrector when all equation sets satisfy the convergence criterion and the stability criterion.

19. A computer program product comprising a non-transitory computer readable storage medium and computer executable logic recorded on said non-transitory computer readable storage medium, the computer program product further comprising:

(a) code for determining a stability limit for each of a plurality of cells in a reservoir model that approximates a subsurface hydrocarbon reservoir, each cell having associated therewith an equation set representing a reservoir property;

(b) code for assigning each cell to one of an explicit formulation or an implicit formulation;

(c) code for providing an initial guess to a solution for a system of equations formed using the equation set for each cell in the plurality of cells;

(d) code for using the initial guess to solve for a solution to the system of equations using an explicit formulation for cells assigned thereto and an implicit formulation for cells assigned thereto;

(e) code for establishing a list of unconverged cells, the unconverged cells having equation sets that have not satisfied a convergence criterion;

(f) code for calculating a stability limit for each of the converged cells, the converged cells having equation sets that have satisfied the convergence criterion;

(g) code for constructing a reduced nonlinear system with the list of unconverged cells when the number of unconverged cells is greater than a predetermined amount, the reduced nonlinear system being assigned to be solved with the implicit formulation and other cells being assigned to be solved with the explicit formulation;

(h) code for repeating parts (d), (e), (f) and (g), substituting the solved solution for the initial guess or the most recent solved solution and substituting the equation sets corresponding to the cells in the list of unconverged cells for the system of equations or equation sets from the most recent iteration, until all equation sets satisfy the convergence criterion and a stability criterion; and

(i) code for outputting the solved solution as a result for a timestep of a simulation of the subsurface reservoir when all equation sets satisfy the convergence criterion and the stability criterion.

20. The computer program product of claim 19, further comprising code for adding, to the list of unconverged cells,

converged cells with one or more reservoir properties exhibiting changes greater than a preset amount,

converged cells that do not satisfy the stability criterion, and

any cell that is neighbor to a cell in the list of unconverged cells and that does not satisfy the stability criterion,

when the number of unconverged cells is greater than a predetermined amount.

21. The computer program product of claim 19, wherein the timestep is a first timestep, and further comprising:

(j) code for repeating parts (a) through (i) at additional timesteps; and

(l) outputting the solved solutions for the first timestep and the additional timesteps, the solved solutions simulating the subsurface reservoir over time.

22. A method of managing hydrocarbon resources, comprising:

(a) approximating a subsurface hydrocarbon reservoir with a reservoir model having a plurality of cells, each cell having associated therewith an equation set representing a reservoir property;

(b) determining a stability limit for each of the plurality of cells;

(c) assigning each cell to one of an explicit formulation or an implicit formulation;

(d) providing an initial guess to a solution for a system of equations formed using the equation set for each cell in the plurality of cells;

(e) using the initial guess to solve for a solution to the system of equations using an explicit formulation for cells assigned thereto and an implicit formulation for cells assigned thereto;

(f) establishing a list of unconverged cells, the unconverged cells having equation sets that have not satisfied a convergence criterion;

(g) calculating a stability limit for each of the converged cells, the converged cells having equation sets that have satisfied the convergence criterion;

(h) when the number of unconverged cells is greater than a predetermined amount, constructing a reduced nonlinear system with the list of unconverged cells, the reduced nonlinear system being assigned to be solved with the implicit formulation and other cells being assigned to be solved with the explicit formulation;

(i) repeating parts (e), (f), (g) and (h), substituting the solved solution for the initial guess or the most recent solved solution and substituting the equation sets corresponding to the cells in the list of unconverged cells for the system of equations or equation sets from the most recent iteration, until all equation sets satisfy the convergence criterion and a stability criterion;

(j) when all equation sets satisfy the convergence criterion and the stability criterion, outputting the solved solution as a result of a timestep of a simulation of the subsurface reservoir; and

(k) managing hydrocarbon resources using the simulation of the subsurface reservoir.

23. The method of claim 22, wherein the simulated characteristic is fluid flow in the subsurface reservoir.

24. The method of claim 22, wherein managing hydrocarbons comprises extracting hydrocarbons from the subsurface reservoir.