Stable method and apparatus for solving S-shaped non-linear functions utilizing modified Newton-Raphson algorithms
원문보기
IPC분류정보
국가/구분
United States(US) Patent
등록
국제특허분류(IPC7판)
G06F-017/10
G06G-007/48
G06G-007/00
출원번호
US-0377760
(2006-03-15)
등록번호
US-7505882
(2009-03-17)
발명자
/ 주소
Jenny,Patrick
Tchelepi,Hamdi A.
Lee,Seong H.
출원인 / 주소
Chevron U.S.A. Inc.
대리인 / 주소
Northcutt,Christopher D.
인용정보
피인용 횟수 :
5인용 특허 :
0
초록▼
An apparatus and method are provided for solving a non-linear S-shaped function F=ƒ(S) which is representative of a property S in a physical system, such saturation in a reservoir simulation. A Newton iteration (T) is performed on the function ƒ(S) at Sv to determine a next iterative value
An apparatus and method are provided for solving a non-linear S-shaped function F=ƒ(S) which is representative of a property S in a physical system, such saturation in a reservoir simulation. A Newton iteration (T) is performed on the function ƒ(S) at Sv to determine a next iterative value Sv+1. It is then determined whether Sv+1 is located on the opposite side of the inflection point Sc from Sv. If Sv+1 is located on the opposite side of the inflection point from Sv, then Sv+1 is set to Sl, a modified new estimate. The modified new estimate, Sl, is preferably set to either the inflection point, Sc, or to an average value between Sv and Sv+1, i.e., Sl=0.5(Sv+Sv+1). The above steps are repeated until Sv+1 is within the predetermined convergence criteria. Also, solution algorithms are described for two-phase and three-phase flow with gravity and capillary pressure.
대표청구항▼
What is claimed is: 1. A method for modeling with a reservoir simulator by solving a non-linear S-shaped function F=ƒ(S) which is representative of a property S in a physical subterranean reservoir system having fluid flow therein, the S-shaped function having an inflection point Sc and a pred
What is claimed is: 1. A method for modeling with a reservoir simulator by solving a non-linear S-shaped function F=ƒ(S) which is representative of a property S in a physical subterranean reservoir system having fluid flow therein, the S-shaped function having an inflection point Sc and a predetermined value of F to which ƒ(S) converges, the method comprising the steps of: a) by a computer, selecting an initial value Sv+1 as a potential solution to the function ƒ(S); b) performing a Newton iteration (T) on the function ƒ(S) at Sv to determine a next iterative value Sv+1; c) determining whether Sv+1 is located on the opposite side of the inflection point Sc from Sv; d) setting Sv+1=Sl, a modified new estimate, if Sv+1 is located on the opposite side of the inflection point of ƒ(S) from Sv; e) determining whether Sv+1 is within a predetermined convergence criteria; f) setting Sv=Sv+1 if Sv+1 is not within a predetermined convergence criteria; g) repeating steps (b)-(f) until Sv+1 is within the predetermined convergence criteria; h) selecting Sv+1 as a satisfactory solution S to the S-shaped function F=ƒ(S); and i) utilizing the satisfactory solution S to the S-shaped function ƒ(S) for modeling the fluid flow within the subterranean reservoir with the reservoir simulator. 2. The method of claim 1 wherein the modified new estimate Sl is set to Sl=αSv+(1-α)Sv+1, where 0l is set to Sc, the inflection point of the function ƒ(S). 6. The method of claim 1 wherein the S-shaped function ƒ(S) is a fractional flow function utilized in the reservoir simulator to represent characteristics of the fluid flow within the subterranean reservoir. 7. The method of claim 1 wherein the S-shaped function ƒ(S) mathematically comports with the following mathematical expression: where S=saturation of a first phase in a two fluid flow problem; ƒ(S)=a function of saturation S; μ1=viscosity of fluid in a first phase; and μ2=viscosity of fluid in a second phase. 8. The method of claim 1 wherein Sv+1 is within the predetermined convergence criteria if ε>|Sv+1-Sv| where ε is predetermined tolerance limit. 9. The method of claim 1 wherein Sv+1 is within the predetermined convergence criteria if ε>|F-ƒ(Sv+1)| where ε is predetermined tolerance limit. 10. The method of claim 1 further comprising setting Sv+1 to an upper or lower bound Sb if Sv+1 exceeds those bounds in step (b). 11. The method of claim 1 wherein the S-shaped function F=ƒ(S) is characterized by one of a look-up table or a general analytical function. 12. A program storage device readable by a machine, tangibly embodying a program of instructions executable by the machine to perform method steps for solving a non-linear S-shaped function, F=ƒ(S), which is representative of a property S in a physical subterranean reservoir system having fluid flow therein, the S-shaped function having an inflection point Sc and a predetermined value of F to which ƒ(S) converges, the method comprising the steps of: a) by a computer, selecting an initial value Sv as a potential solution to the function ƒ(S); b) performing a Newton iteration (T) on the function ƒ(S) at Sv to determine a next iterative value Sv+1; c) determining whether Sv+1 is located on the opposite side of the inflection point Sc from Sv; d) setting Sv+1=Sl, a modified new estimate, if Sv+1 is located on the opposite side of the inflection point of ƒ(S) from Sv; e) determining whether Sv+1 is within a predetermined convergence criteria; f) setting Sv=Sv+1 if Sv+1 is not within a predetermined convergence criteria; g) repeating steps (b)-(f) until Sv+1 is within the predetermined convergence criteria; h) selecting Sv+1 as a satisfactory solution to the S-shaped function F=ƒ(S); and i) utilizing the satisfactory solution S to the S-shaped function ƒ(S) for modeling the fluid flow within the subterranean reservoir with the machine. 13. A method for updating a Newton's iteration for a fractional flow function ƒ(S) associated with two-phase flow of a physical subterranean reservoir system with gravity in a grid cell of a reservoir simulator, the method comprising: (a) by a computer, checking the directions of phase velocities and gravity force in a grid cell; (b) if the phase velocities are co-current, applying a regular Newton stability analysis; (i) calculating the directional fractional flows Fh and Fv and its second-order derivatives (F"h and F"v); (ii) calculating Cg for vertical flow (for horizontal flow, Cg=0); (iii) applying a saturation limit 0≦Sv+1≦1; (iv) if F"h(Sv+1)F"h(Sv)v(Sv+1)F"v(Sv)r; (ii) applying a saturation limit: if the cell is not adjacent to the boundary or the neighboring cell does not belong to the different saturation domain, 0≦Sv+1≦S*r or S*rv+1v+1≦1; (iii) if F"h(Sv+1)F"h(Sv)v(Sv+1)F"v(Sv)g1 and Cg2; (c) calculating boundaries of zero fractional flow for phase 1, (L1(S1,S2)) and L2(S1,S2); (d) calculating initial points for saturation, (S1β1,S2β1) and (S1β2,S2β2) on L1(S1,S2) and L2(S1,S2), respectively; (e) checking the sign of F*1 and F*2; (f) from an initial estimate of saturations (S10 and S20), calculating a fractional flow estimate (F10 and F20) and a determinant of a Jacobian matrix; (g) if F*1≧0 and F*2≧0; (i) if the initial estimate of the saturations does not yield a positive condition, take the point at L1(S1β1,S2β1) as the initial estimate of the saturations; (ii) iterate the saturations based on where 0≦Sv+1≦1 is enforced after each iteration; (iii) if F1v2v1(S1β1,S2β1); (iv) if S1+S2>1, normalize the saturation to ensure S1≧0; (v) if under-relax the saturation iteration; (h) if F*12≧0, and Det≧0, set the initial estimate of the saturations at point at L1(S1β1,S2β1); (i) if F*12≧0, and Det121(S1β2,S2β2); (j) if F*120 set the initial estimate of the saturations at point (S1β3,S2β3); and (k) utilizing the updated Newton's iteration for modeling the three-phase flow of the physical subterranean reservoir system with the reservoir simulator.
※ AI-Helper는 부적절한 답변을 할 수 있습니다.