Systems and methods for determining manufacturing or operating parameters for a deviated downhole well component, including a method that includes representing a tubular string as nodes separated by segments, determining transfer matrices for determining an ith node's state vector from an ith−1 node's state vector, and defining initial state vector values for the reference node. The nodes are numerable from 1 to N with an initial, mechanically constrained reference node representable with i=0, and each is associated with a state vector describing a corresponding node position and one or more forces present at said node. The method further includes applying the transfer matrices to obtain each of the state vectors' values, deriving from at least one of the state vectors a parameter value for said component, and specifying a component having said parameter value. The parameter value can include a centralizer or stabilizer composition, manufacturing dimensions, or position.
Legal claims defining the scope of protection, as filed with the USPTO.
1. A method for determining manufacturing or operating parameters for a deviated downhole well component, the method comprising: representing a tubular string prior to cementing in a borehole as a sequence of nodes separated by tapered segments, said nodes being numerable from i=1 to N with an initial, mechanically constrained reference node located at a casing hanger representable with i=0, a final, mechanically constrained N th node located at a shoe representable with i=N, and each node being associated with a state vector describing a position of the corresponding node in the borehole and one or more forces present on the tubular string at said corresponding node; determining a sequence of transfer matrices enabling the determination of an i th node's state vector from an i th −1 node's state vector; defining values of an initial state vector for the reference node; applying the transfer matrices to obtain values for each of the state vectors; deriving from at least one of the state vectors a parameter value for said component, the parameter value being in a set consisting of a composition, manufacturing dimensions, and a position for a centralizer or stabilizer of the tubular string; and specifying a component having said parameter value, wherein said parameter value facilitates manufacture of the component or positioning of the component for cementing in the borehole.
2. The method of claim 1 , wherein said specifying includes providing the composition or a dimensional specification to a manufacturer of said component.
3. The method of claim 1 , wherein said specifying includes providing the position for the centralizer or stabilizer to an installer of said component.
4. The method of claim 1 , wherein each state vector comprises a vertical position u i , a horizontal position v i , and an inclination angle α i , associated with node i; and further comprises a vertical force F xi , a horizontal force F hi and a bending moment M i present at node i; and wherein the state vector is representable as [u i , v i , α i , F xi , F hi , M i , 1] T .
5. The method of claim 4 , wherein the transfer matrix for the i th node, said i th node associated with a tubular segment, is representable as: [ 1 0 - l i h ( ( l i h ) 3 6 × ( EI ) i + l i x ( EA ) i ) 0 - ( l i h ) 2 2 × ( EI ) i l i x + Δ α i - 1 0 × l i h 0 1 l i x 0 ( - ( l i x ) 3 6 × ( EI ) i + l i h ( EA ) i ) ( l i x ) 2 2 × ( EI ) i l i h - Δ α i - 1 0 × l i x 0 0 1 - ( l i h ) 2 2 × ( EI ) i ( l i x ) 2 2 × ( EI ) i ( l i x ) 2 + ( l i h ) 2 ( EI ) i Δ α i 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 l i h - l i x 1 0 0 0 0 0 0 0 1 ] i where, l i h is a horizontal distance defined as (v i −v i−1 ); l i x is a vertical distance defined as (u i −u i−1 ); Δα i−1 0 is a change in inclination angle at the i th −1 node defined as (α i−1 −α 0 ); Δα i 0 is a change in inclination angle at the i th node defined as (α i −α 0 ); (EI) i is a product of Young's modulus and a moment of inertia of the component at the i th node; and (EA) i is a product of Young's modulus and a cross-sectional area of the component at the i th node.
6. The method of claim 4 , wherein said deriving includes deriving an axial force F αi present at the i th node.
7. The method of claim 4 , wherein the transfer matrix for the i th node, said i th node associated with a flow restriction, is representable as: [ 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 F p x 0 0 0 0 0 1 F p h 0 0 0 0 0 0 1 ] i where, F p x is a vertical force present on a plug located at the i th node; and F p h is a horizontal force present on the plug.
8. The method of claim 1 , wherein the N th node is also mechanically constrained.
9. The method of claim 1 , wherein the component comprises a running string selected from the group consisting of a well casing string, a drillstring, a production string and a coil tubing.
10. The method of claim 9 , wherein the running string comprises a tapered segment, a cross-section size change, a packer or a plug.
11. The method of claim 1 , further comprising cementing the component having said parameter value in the borehole.
12. The method of claim 1 , further comprising manufacturing the component having said parameter value or positioning the component having said parameter value in the borehole.
13. A system that determines manufacturing and operating parameters for a deviated downhole well component, the system comprising: a memory having deviated downhole well component modeling software; and one or more processors coupled to the memory, the software causing the one or more processors to: represent a tubular string prior to cementing in a borehole as a sequence of nodes separated by tapered segments, said nodes being numerable from i=1 to N with an initial, mechanically constrained reference node located at a casing hanger representable with i=0, a final, mechanically constrained N th node located at a shoe representable with i=N, and each node being associated with a state vector describing a position of the corresponding node in the borehole and one or more forces present on the tubular string at said corresponding node; determine a sequence of transfer matrices enabling the determination of an i th node's state vector from an i th −1 node's state vector; define values of an initial state vector for the reference node; apply the transfer matrices to obtain values for each of the state vectors; derive from at least one of the state vectors a parameter value for said component, the parameter value being in a set consisting of a composition, manufacturing dimensions, and a position for a centralizer or stabilizer of the tubular string; and specify a component having said parameter value, wherein said parameter value facilitates manufacture of the component or positioning of the component for cementing in the borehole.
14. The system of claim 13 , wherein the one or more processors specify said component at least in part by providing the composition or a dimensional specification to a manufacturer of said component.
15. The system of claim 14 , wherein the transfer matrix for the i th node, said i th node associated with a tubular segment, is representable as: [ 1 0 - l i h ( ( l i h ) 3 6 × ( EI ) i + l i x ( EA ) i ) 0 - ( l i h ) 2 2 × ( EI ) i l i x + Δ α i - 1 0 × l i h 0 1 l i x 0 ( - ( l i x ) 3 6 × ( EI ) i + l i h ( EA ) i ) ( l i x ) 2 2 × ( EI ) i l i h - Δ α i - 1 0 × l i x 0 0 1 - ( l i h ) 2 2 × ( EI ) i ( l i x ) 2 2 × ( EI ) i ( l i x ) 2 + ( l i h ) 2 ( EI ) i Δ α i 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 l i h - l i x 1 0 0 0 0 0 0 0 1 ] i where, l i h is a horizontal distance defined as (v i −v i−1 ); l i x is a vertical distance defined as (u i −u i−1 ); Δα i−1 0 is a change in inclination angle at the i th −1 node defined as (α i−1 −α 0 ); Δα i 0 is a change in inclination angle at the i th node defined as (α i −α 0 ); (EI) i is a product of Young's modulus and a moment of inertia of the component at the i th node; and (EA) i is a product of Young's modulus and a cross-sectional area of the component at the i th node.
16. The system of claim 13 , wherein the one or more processors specify said component at least in part by providing the position for the centralizer or stabilizer to an installer of said component.
17. The system of claim 13 , wherein each state vector comprises a vertical position u i , a horizontal position v i , and an inclination angle α i , associated with node i; and further comprises a vertical force F xi , a horizontal force F hi and a bending moment M i present at node i; and wherein the state vector is representable as [u i , v i , α i , F xi , F hi , M i , 1] T .
18. The system of claim 17 , wherein the one or more processors derive said parameter value at least in part by deriving an axial force F αi present at the i th node.
19. The system of claim 17 , wherein the transfer matrix for the i th node, said i th node associated with a flow restriction, is representable as: [ 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 F p x 0 0 0 0 0 1 F p h 0 0 0 0 0 0 1 ] i where, F p x is a vertical force present on a plug located at the i th node; and F p h is a horizontal force present on the plug.
20. The system of claim 13 , wherein the N th node is also mechanically constrain ed.
21. The system of claim 13 , wherein the component comprises a running string selected from the group consisting of a well casing string, a drillstring, a production string and a coil tubing.
22. The system of claim 21 , wherein the running string comprises a tapered segment, a cross-section size change, a packer or a plug.
23. The system of claim 13 , further comprising cementing the component having said parameter value in the borehole.
24. The system of claim 13 , further comprising manufacturing the component having said parameter value or positioning the component having said parameter value in the borehole.
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January 17, 2014
August 20, 2019
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