CA2913203C - Methods and systems for determining manufacturing and operating parameters for a deviated downhole well component - Google Patents

Abstract

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 i th node's state vector from an i th -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.

Description

METHODS AND SYSTEMS FOR DETERMINING MANUFACTURING AND
OPERATING PARAMETERS FOR A DEVIATED DOWNHOLF WELL COMPONENT
CROSS-REFERENCE TO RELATED APPLICATION
This application claims priority to Provisional U.S. Application Ser. No.
61/837,986, titled "Methods and Systems for Modeling a Deviated Downhole Well Component"
and filed June 21, 2013 by Robello Samuel and Zhenying Wang.
BACKGROUND
io As world demand for petrochemical products has continued to increase, oil and gas companies have had to expand their exploration and production efforts into developing increasingly deep wells. As a result, the structures constructed to form a well must be capable of operating under larger loads and stresses than ever before. Because failures can have costly consequences, it is important to design all well structures with appropriate safety margins.
15 One example of' such a structure is the well casing. A well casing is a tubular structure generally made of a steel pipe surrounded by a concrete layer that secures the steel pipe to the surrounding formation, thus defining the outside wall of the well. The concrete provides support to the steel pipe, as well as additional isolation layer between the formation and fluids flowing within the casing. In order to determine the correct materials and dimensions for the 20 various casing components, engineers frequently perform computer simulations to model various casing configurations under simulated downhole conditions. The simulations provide the engineer with information regarding the various loads and stresses to which the casing might be subjected, and enable potential designs to be evaluated.
But casing designs arc only as good as the underlying simulation model. While 25 simulations of single section casings in vertical wells arc generally well understood and produce accurate results, tapered casings, deviated casings and casings with fluid flow restrictions represent indeterminate complex mechanical systems that can be very difficult or impractical to model using existing techniques. While methods do exist wherein these more complex systems are modeled as simpler single-section vertical wells with the results being 30 adjusted to include additional safety margins, such methods can incur a significant risk, given the lack of quantifiable data to support the selected margins.

= CA 02913203 2015-11-20 BRIEF DESCRIPTION OF THE DRAWINGS
A better understanding of the various disclosed embodiments can be obtained when the following detailed description is considered in conjunction with the attached drawings, in which:
FIG. 1 shows an illustrative downhole well with a well casing modeled using the disclosed systems and methods.
FIG. 2 shows the various parameters describing the forces operating on an illustrative well casing string segment.
FIG. 3 shows an illustrative computer system suitable for performing the disclosed methods.
FIG. 4 shows an illustrative example of the disclosed methods.
It should be understood that the drawings and corresponding detailed description do not limit the disclosure, but on the contrary, they provide the foundation for understanding all modifications, equivalents, and alternatives falling within the scope of the appended claims.
DETAILED DESCRIPTION
The paragraphs that follow describe illustrative systems and methods for determining manufacturing or operating parameters for a deviated downhole well component.
Examples zo are provided within the context of a tapered and deviated well casing that is mechanically constrained at opposite ends. The mechanics of such a casing are explained and illustrated, and matrices are presented that mathematically describe the known and unknown forces acting at various points along the casing. Finally, methods and systems are described that combine the various matrices to compute the forces present along the casing.
The disclosed systems and methods are best understood when described in an illustrative usage context. Accordingly, FIG. 1 shows an illustrative drilling environment. A
drilling platform 2 supports a derrick 4 having a traveling block 6 for raising and lowering a drillstring 8 into borehole 30. A top drive 10 supports and rotates the drillstring 8 as it is lowered through the wellhead 12. A pump 20 circulates drilling fluid through a feed pipe 22 to top drive 10, downhole through the interior of drill string 8, through orifices in a downhole

2 tool (not shown), back to the surface via the annulus around drillstring 8, and into a retention pit 24. The drilling fluid aids in maintaining the borehole integrity.
Because boreholes are routinely drilled to ten thousand feet or more in depth and can be steered horizontally for perhaps twice that distance, a well casing string is inserted into the borehole and is cemented to the borehole wall to provide support to the borehole and isolation between the formation and the fluids flowing within the well casing string. In the example of FIG. 1, the upper end of well casing string 14 is attached to and mechanically constrained by a casing hanger, located at the end of casing header 11. As well casing string 14 extends downhole it may be tapered to provide additional support between the casing string and the borehole wall and to reduce the overall weight of the string.
Well casing string 14 of FIG. 1, for example, includes tapered reductions 13 and 15. Well casing string 14 also curves to conform to the shape of the deviated well shown. A shoe 16 is located at the end of well casing string 14, wherein shoe 16 and stabilizers 18 fix and mechanically constrain the lower end of well casing string 14 within borehole 30.
As can be seen in the illustrative example of FIG. 1, prior to being cemented to the borehole wall, well casing string 14 is supported and mechanically constrained by just two points at either end of the well casing string. The deviation of the well casing string, as well as the reductions in the cross-sectional are of the well casing string at the reductions, produce complex three-dimensional forces that act upon the well. The resulting service load zo distribution on the well casing string causes the string to become a statically complicated indeterminate mechanical system. Piston forces acting on plugs within the casing (e.g., cementing plugs) also produce forces similar to those at the reduction, further complicating the system.
In at least some illustrative embodiments, the forces acting on the above-described well casing string are modeled by dividing the system into a series of N
subsystems that each only interact with adjacent subsystems, and then determining the forces acting on each subsystem in sequence. FIG. 2 shows an illustrative well casing string subsystem identified as segment 202 that is defined by two nodes 204 and 206. Each node is described by a state vector that includes information regarding the location of the node relative to a reference node, and also includes information describing the forces present at the node.
Each illustrative state vector is defined as, [u, v, ai F, Fhi M1 1]7; (1) where, ui is the true vertical depth (TVD) of node i relative to the reference node;
vi is the horizontal distance of node i from the reference node;
ai is the inclination angle of the casing segment at node i;
Fxi is the vertical force present at node i;
F hi is the horizontal force present at node i; and Mi is the bending moment present at node i.
The reference node of the illustrative example is located at the casing hanger and is designated as node 0, and the node at the opposite end of the well casing string and furthest lo away from node 0 is designated as node N.
After dividing the illustrative well casing string into segments, the unknown forces acting on each segment's downhole node are determined by starting at the reference node at one end of the first segment (where all elements of the state vector are known) and determining the state vector for the next node at the opposite end of the segment. The state vector for the downhole node of the first segment is determined by computing a cross product of a transfer matrix and the reference node's state vector. This transfer matrix is defined based upon the known elements of the downhole node's state vector, the previous node's state vector and/or the reference node's state vector. For the first segment the previous node is also the reference node (i.e., 1-1=0). In at least some illustrative embodiments, the transfer matrix T for a well casing string segment defined between nodes i and i-1 is described using known state vector elements as, f.h)a t,h) 1 0 _______________ + __ ) J Atx X ih 2 6X(Fai CEA) X (El), # 1-1 Oirfi 1,,h Off 0 1 iT Ace(1 x F
t 6x(Et) i kE)j 2)3(En I
(IP2 i ___ (2) no N i 2x (ui), 2x(Er), (E/):

where, is the horizontal distance defined as (vi-1T is the vertical distance defined as (ui¨ ui_i);
6.4Li is the change in inclination angle at the ith-1 node defined as (a/4 ¨
ao);
Lex? is the change in inclination angle at the ith node defined as (ai¨ ao);
(El); is the product of Young's modulus and a moment of inertia of the component at the ith node; and (EA)1 is the product of Young's modulus and a cross-sectional area of the component at the ith node.
For the first node, i=1 and the state vector V1 is expressed as, = x Vo (3).
Equation (3) expresses the state vector VI as a set of constrained linear equations for the well casing segment that can be solved to determine the unknown forces present at node 1. For each subsequent node i, the cross product of the node's transfer matrix and the reference node's state vector is combined with prior cross products for nodes 1 through i-1 to determine the state vector for node i, thus determining the unknown forces at each node i (i.e., Fx1, Fhi is and/or M1). Once these forces are known, the axial force at a node can be computed as, Fa, = Fsõ. X eos(cri) Fhi x sin(r1) (4).
In at least some illustrative embodiments, a transfer matrix method (TMM) is used to perform the combination of cross products. Using TMM, the combination of cross products producing the ith node's state vector is expressed as, 17, alt x (5).
Because each of the prior products of products for each node 1 through i-1 have already been calculated, the product of products does not need to be recalculated for each node. Instead, each node's transfer matrix can be combined with a cumulative transfer matrix, thus avoiding duplicative computations. For example, for a three node string this would be expressed as, Tacc = ; V1 = T,,,X Vo (6), To cc = Tam X r2; V-, Tacc x 170 = (Ti x 1-2) x (7), and T
= Tacc X r3; 173 = Taõ. x Vo = (7.3. x T2, X T3) X Vo (8). acz The cross product Ti x T2 has already been calculated in equation (7) and saved as cumulative transfer matrix Tace, which is reused without recalculation in equation (8).

It should be noted that the transfer matrix is not limited to the specific embodiment of equation (2). For example, plugs such as cementing plugs present within the casing string and reductions in the cross-sectional area of the casing such as reduction 15 of FIG. 1 may be represented by much simpler transfer matrices. In at least some illustrative embodiments, s such plugs and reductions located at a node i are represented as, -1 0 0 0 0 0 0=

0 0 0 1 0 0 0 (8) 0 0 0 0 1 0 Fp':
0 0 0 0 0 1 F;11 where, F; is a vertical force present on a plug or reduction located at the ith node;
and F1P is a horizontal force present on the plug or reduction.
Other transfer matrices may include, for example, parameters that describe the load imposed on a casing string by a salt formation (i.e., "salt loading"). A wide variety of transfer matrices suitable for use with the methods described herein will become apparent to those of ordinary skill in the art, and all such variations of transfer matrices are within the scope of the present disclosure.
The algorithmic approach of equations (6) through (8) for computing equation (5) is suitable for implementation by software executing on a computer system such as the illustrative system shown in FIG. 3. Both hardware and software components of computer system 300 are shown, which in at least some illustrative embodiments implement at least part of the matrix-based well casing string modeling shown as method 400 in FIG. 4 (described in more detail below). Computer system 300 operates in accordance with software (which may be stored on non-transitory information storage media 340) and enables a user to interact with the system via keyboard 334, pointing device 335 (e.g., a mouse) and display 336 to configure, control and monitor the execution of the matrix-based well casing string modeling.
Located within processing subsystem 330 of computer system 300 is a display interface 352, a processor 356, a peripheral interface 358, an information storage device 360, a network interface 362 and a memory 370. Bus 364 couples each of these elements to each other and transports their communications. Network interface 362 enables communications with other systems (e.g., via the Internet with a central database server housing additional modeling parameters and suitable for saving the results of the modeling). In accordance with user input received via peripheral interface 358 and program instructions from memory 370 and/or information storage device 360, processor 356 processes input from the user and applies it to the well casing string data to perform the disclosed methods and present the results to the user. Storage device 360 may be implemented using any number of known non-transitory information storage media, including but not limited to magnetic disks, solid-state storage devices and optical storage disks.
Various software modules are shown loaded into memory 370 of FIG. 3, where they io are each accessed by processor 356 for execution. These modules include:
user interface module, which processes user inputs provided with keyboard 334 and pointing device 335 via peripheral interface 358; vector definition module 374, which defines the state vector for each node; matrix definition module 376, which defines the transfer matrix for a segment; cross product module 378, which computes the cross product that updates the cumulative transfer is matrix; transfer matrix method module 380, which uses the cumulative transfer matrix from cross product module 378 to determine unknown state vector elements; parameter derivation module 382, which derives well component manufacturing parameters or well component centralizer positions; and presentation module 384 which provides the derived manufacturing parameters or centralizer positions to manufacturing or operations personnel (e.g., by 20 graphically presenting the dimensions of casing segments or of centralizer positions along the length of the casing prior to cementing into place).
FIG. 4 shows an illustrative method that implements the above-described matrix-based modeling, at least part of which may be implemented by software executing on computer system 300. It should be noted that although the embodiment of FIG. 3 shows 25 various software modules executing on computer system 300, in other illustrative embodiments some or all of the modules may execute on two or more computers within a networked and/or distributed system. Referring to both FIGS. 3 and 4, the state vector for the reference node (node 0) at the start of a casing string is defined (block 402;
vector definition module 374), either via user input (user interface module 372) or using previously stored data 30 (e.g., data stored on information storage device 360). Node index i is incremented from 0 to 1 and the total product is initialized to zero (block 402; TMM module 382). A
state vector for the current ith node (here node 1) is defined in a manner similar to that used for the reference node (block 404; vector definition module 374), but with at least one unknown state vector element.
If the segment associated with the current node is one that restricts fluid flow, such as a plug or a casing cross-sectional area reduction (block 406; matrix definition module 376) the transfer matrix is defined in terms of one or more forces present at the restriction (block 408; matrix definition module 376). If the segment associated with the current node is a well casing string, the transfer matrix is defined in terms of the reference node's state vector, the previous node's state vector and/or the known elements of the current node's state vector (block 410; matrix definition module 376). Once the transfer matrix is defined for node i, the io cumulative transfer matrix Taõ is either initialized as T1 for i = 1, or updated as the cross product of the current node's transfer matrix Ti and Taõ for i> 1 (block 412;
cross product module 380). The cross product is then utilized to determine the node's full state vector, which is used to determine the unknown element values of the current node's state vector (block 414; TMM module 380). In at least some illustrative embodiments, block 414 is skipped for the first node (i=1) if all the state vector values are already known (e.g., if the values are readily measurable at a starting point at the surface).
If additional well easing string segments remain (block 418; TMM module 380) node index i is incremented and the current node's state vector becomes the prior node's state vector (block 416; TMM module 380). The above-described process is then repeated for the next segment and node along the well casing string (blocks 404 through 418).
If there are no additional well casing string segments (block 418; TMM module 380), the previously unknown and now calculated elements of each state vector are subsequently used as a basis for determining casing manufacturing parameters such as dimensions and composition, or as a basis for locating centralizers to position the casing within a borehole (block 420, parameter derivation module 382). The resulting manufacturing parameters or centralizer position arc respectively provided as composition ancUor dimensional specifications to manufacturing personnel, or as a position for a centralizer or stabilizer to well operators/component installers (block 422, presentation module 384), ending the method (block 424).
In at least some illustrative embodiments, the calculated forces at each node along a casing string are indicated on a graphical representation of the well casing string. Once the forces have been determined, the axial load (e.g., tension) present at a given node can be computed, for example by using equation (4). Casing string parameters such as segment lengths, wall thickness and material compositions may then be determined from the computed axial load. These parameters provide the required casing safety margins at a reduced cost when compared to existing methods that overestimate the required casing string parameters. In other illustrative embodiments, the positions of one or more centralizers (determined based upon the computed and displayed forces) are on the casing string's graphical representation. The calculated forces at each node are used to determine the side force to which a casing segment i (located between two nodes i and 1+1) is subjected, for example by using the equation, SixieForce.: = 4(Fx0+1) ¨ Fx1) + ¨ Fh (9).
The contact points between the casing and borehole wall can be determined from the state vector V, and the well trajectory. The computed side force and the contact points arc then used to determine a suitable centralizer(s) and the best centralizer position(s), for example, at the contact point(s) between the casing segment and the borehole wall.
Numerous other modifications, equivalents, and alternatives, will become apparent to those skilled in the art once the above disclosure is fully appreciated. For example, although the embodiments described use the TMM to determine the transfer matrices, other analytical methods are also suitable to determine transfer matrices used to determine the unknown elements of a node of interest. Further, although the examples provided are applied within the context of static modeling, static snapshots repeatedly performed over time may be combined to provide dynamic as well as near-real-time or real-time modeling of the well casing string (e.g., to predict the loading on a well casing string as a cementing plug progresses down the string). Additionally, although the disclosed embodiments describe modeling well casing strings, any of a wide variety of well components may be modeled to determine the manufacturing and/or operating parameters of said components. These well components include but are not limited to drillstrings, workstrings, production strings and coiled tubing stings. Other well components that restrict fluid flow within a running tubular string (e.g., packers) are also within the scope of the disclosure. It is intended that the following claims be interpreted to embrace all such modifications, equivalents, and alternatives where applicable.

Claims (20)

WHAT IS CLAIMED IS:

1. A computer-implemented method comprising:
representing a tubular string prior to cementing in the 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 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 a deviated downhole well component, the parameter value being in a set consisting of a composition, manufacturing dimensions, and a position for a centralizer or stabilizer of the tubing string; and at least one of:
cementing said component having said parameter value in the borehole;
manufacturing said component having said parameter value; and positioning said component having said parameter value in the borehole.

2. The method of claim 1 further comprising providing the composition or a dimensional specification to a manufacturer of said component.

3. The method of claim 1 further comprising 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, , and an inclination angle a, , associated with node i;
and the one or more forces comprise a vertical force F xi and a horizontal force F hi present at node i; and the state vector further comprises a bending moment M i present at node i; and wherein the state vector is representable as [u i, v i, a i, F xi, F hi, M i, 1]T.

5. The method of claim 4, wherein the transfer matrix for the ith node, said ith node associated with a tubular segment, is representable as:
where, l~ is a horizontal distance defined as (vi¨ vi-1);
l~ is a vertical distance defined as (ui ¨ ui-1);
.DELTA..alpha.~-1 is a change in inclination angle at the ith-1 node defined as (.alpha.i-1- .alpha.0);
.DELTA..alpha.~-1 is a change in inclination angle at the ith node defined as (.alpha.i¨ .alpha.0);
(El)i is a product of Young's modulus and a moment of inertia of the component at the ith node; and (EA)i is a product of Young's modulus and a cross-sectional area of the component at the ith node.

6. The method of claim 4, wherein said deriving includes deriving an axial force Fai present at the ith node.

7. The method of claim 4, wherein the transfer matrix for the ith node, said ith node associated with a flow restriction, is representable as:
where, F~ is a vertical force present on a plug located at the ith node; and F~ is a horizontal force present on the plug.

8. The method of claim 1, wherein the Nth node is also mechanically constraint.

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. A 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 Nth 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 ith node's state vector from an ith-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 a deviated downhole well 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 at least one of:
cement the component having said parameter value in the borehole manufacture the component having said parameter value; and position the component having said parameter value in the borehole.

12. The system of claim 11, wherein the one or more processors provide the composition or a dimensional specification to a manufacturer of said component.

13. The system of claim 11, wherein the one or more processors provide the position for the centralizer or stabilizer to an installer of said component.

14. The system of claim 11, wherein each state vector comprises a vertical position µ i, a horizontal position v i , and an inclination angle .alpha. i , associated with node i; and the one or more forces comprise a vertical force F xi, and a horizontal force F hi present at node i;
and the state vector further comprises a bending moment m, present at node i;
and wherein the state vector is representable as [µi, v i, .alpha. i, F xi, F
hi, M i, 1]T.

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:
where, ~ is a horizontal distance defined as (v i¨ vi-I);
~ is a vertical distance defined as (µ i¨ µ i-I);
.DELTA..alpha.~ is a change in inclination angle at the i th-1 node defined as (.alpha. i-1¨ .alpha.0);
.DELTA..alpha.~ is a change in inclination angle at the i th node defined as (.alpha. i¨ .alpha. 0);
(El)i is a product of Young's modulus and a moment of inertia of the component at the 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 14, wherein the one or more processors derive said parameter value at least in part by deriving an axial force Fai present at the ith node.

17. The system of claim 14, wherein the transfer matrix for the ith node, said ith node associated with a flow restriction, is representable as:
where.
F~ is a vertical force present on a plug located at the ith node; and F~ is a horizontal force present on the plug.

18. The system of claim 11, wherein the Nth node is also mechanically constrained.

19. The system of claim 11, 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.

20. The system of claim 19, wherein the running string comprises a tapered segment, a cross-section size change, a packer or a plug.