CN105247166A - 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

Method and system for determining manufacturing parameters and operating parameters of a deviated downhole well assembly

Cross Reference to Related Applications

This application claims priority to provisional U.S. application serial No. 61/837,986 entitled "methods and systems for modeling and device down well component" filed 2013 on 21.6.s..

Background

As the world's demand for petrochemical products continues to grow, oil and gas companies have had to extend their exploration and production efforts to develop deeper and deeper wells. Thus, these structures configured to form wells must be capable of operating under greater loads and stresses than previously. Because failure can have costly consequences, it is important that all wells be designed with appropriate safety margins.

One example of such a structure is a well casing. Well casing is a tubular structure typically made of steel pipes surrounded by a layer of concrete which secures the steel pipes to the surrounding ground, thus defining the outer wall of the well. The concrete provides support for the steel pipe and additional insulation between the formation and the fluid flowing within the casing. To determine the appropriate materials and dimensions for the various casing assemblies, engineers typically perform computer simulations to model the various casing configurations under simulated downhole conditions. The simulation provides the engineer with information about the various loads and stresses to which the casing may be subjected and enables evaluation of the potential design.

The casing design is only comparable to the underlying simulation model. While simulation of a single casing in a vertical well is generally well understood and produces accurate results, tapered casings, deviated casings, and casings with fluid flow restrictions represent uncertain complex mechanical systems that can be very difficult or impractical to model using prior art techniques. While there do exist methods in which these more complex systems are modeled as simpler single vertical wells, with these results adjusted to include additional safety margins, such methods may pose significant risks given the lack of quantifiable data to support the selected margins.

Drawings

The various disclosed embodiments can be better understood when the following detailed description is considered in conjunction with the following drawings, in which:

FIG. 1 shows an illustrative downhole well having a well casing modeled using the disclosed systems and methods.

FIG. 2 shows various parameters describing forces acting on an illustrative well casing string.

FIG. 3 shows an illustrative computer system suitable for performing the disclosed method.

Fig. 4 shows an illustrative example of the disclosed method.

It should be understood that the drawings and corresponding detailed description do not limit the disclosure, but rather they provide the basis for understanding all modifications, equivalents, and alternatives falling within the scope of the appended claims.

Detailed Description

The following paragraphs describe illustrative systems and methods for determining manufacturing or operating parameters of a deviated downhole well assembly. Examples are provided within the context of a conically deviated well casing that is mechanically restrained at opposite ends. The mechanics of such a casing is illustrated and described, and a matrix is presented that mathematically describes the known and unknown forces acting at various points along the casing. Finally, methods and systems are described that combine various matrices to calculate the force exhibited along the casing.

The disclosed systems and methods are best understood when described in the context of illustrative use. Thus, fig. 1 shows an illustrative drilling environment. The drilling platform 2 supports a derrick 4 having a traveling block 6 for raising and lowering a drill string 8 into a borehole 30. The top drive 10 supports and rotates the drill string 8 as the drill string 8 is lowered through the wellhead 12. The pump 20 circulates drilling fluid through a feed pipe 22 to the top drive 10, downhole through the interior of the drill string 8, through an orifice in a downhole tool (not shown), back to the surface through a annulus around the drill string 8, and into a holding pit 24. The drilling fluid helps maintain the integrity of the wellbore.

Because wellbores are typically drilled to depths of ten thousand feet or more and can be controlled in the horizontal direction to perhaps twice that distance, a casing string is inserted into the wellbore and cemented to the wellbore wall to provide support for the wellbore and isolation between the formation and fluids flowing within the well casing string. In the example of fig. 1, the upper end of the well casing string 14 is attached to a casing hanger 15 and is mechanically restrained by the casing hanger 15, which is located at the end of the casing joint 11. As the well casing string 14 is run downhole, it may be tapered to provide additional support between the casing string and the wellbore wall and reduce the overall weight of the string. For example, the well casing string 14 of FIG. 1 includes tapered reductions 13 and 15. The well casing string 14 is also curved to conform to the shape of the deviated well as shown. The carrier plate 16 is positioned at the end of the well casing string 14, with the carrier plate 16 and stabilizer 18 securing and mechanically restraining the lower end of the well casing string 14 within the wellbore 30.

As can be seen in the illustrative example of fig. 1, the well casing string 14 is supported and mechanically restrained by only two points at either end of the well casing string prior to bonding to the wellbore wall. The deflection of the well casing string and the reduction of the cross-sectional area of the well casing string at the reduced section creates complex three-dimensional forces acting on the well. The resulting distribution of the service load on the well casing string results in the string being a statically complex indeterminate mechanical system. The piston force acting on the plunger within the casing (e.g., the cementing plunger) also generates a force similar to the force at the reduction, further complicating the system.

In at least some illustrative embodiments, the forces acting on the well casing string described above are modeled by dividing the system into a series of N subsystems that each interact only with adjacent subsystems, and then determining the forces acting on each subsystem in turn. Fig. 2 shows an illustrative well casing string system, identified as a segment 202 bounded by two nodes 204 and 206. Each node is described by a state vector that includes information about the position of the node relative to a reference node, and also includes information describing the forces present at the node. Each of the declarative state vectors is defined as,

$V_i={\left[\begin{array}{ccccccc}{u}_i& v_i& {\mathrm{\α}}_i& F_{xi}& F_{hi}& M_i& 1\end{array}\right]}_i^T---\left(1\right)$

wherein,

u_itrue Vertical Depth (TVD) for node i relative to the reference node;

v_iis the horizontal distance of node i from the reference node;

α_iis the inclination of the casing section at node i;

F_xiis the vertical force presented at node i;

F_hiis the horizontal force present at node i; and

M_ithe bending moment present at node i.

The reference node of the illustrative example is positioned at the casing hanger and designated node 0, and the node at the opposite end of the well casing string and furthest from node 0 is designated node N.

After the illustrative well casing string is divided into segments, the unknown forces acting on the downhole nodes of each segment are determined by starting at a reference node located at one end of the first segment (where all elements of the state vector are known) and determining the state vector of the next node located at the opposite end of the segment. The state vector of the downhole node of the first segment is determined by calculating the cross product of the transition matrix and the state vector of the reference node. Based on the state vector of the downhole node, the state vector of the previous node and/or the known elements of the state vector of the reference nodeThis transition matrix is defined. For the first paragraph, the previous node is also the reference node (i.e., i-1 ═ 0). In at least some illustrative embodiments, the transfer matrix T for a casing string segment defined between nodes i and i-1 is determined using known state vector elements_iIt is described that in the description above,

wherein,

is a horizontal distance, defined as (v)_i–v_i-1)；

Is a vertical distance, defined as (u)_i–u_i-1)；

Is i^th-change in inclination at node 1, defined as (α)_i-1–α₀)；

Is i^thThe change in inclination at the node point is defined as (α)_i–α₀)；

(EI)_iIs i^thThe product of the Young's modulus and moment of inertia of the component at the node; and

(EA)_iis i^thThe product of the Young's modulus and the cross-sectional area of the component at the node.

For the first node, 1 equals i and the state vector V₁In the expression of,

V₁＝T₁×V₀(3)。

equation (3) puts the state vector V₁Expressed as a system of constrained linear equations for the well casing section, which can be solved to determine the unknown force present at node 1. For each subsequent node i, the cross product of the transition matrix of the node and the state vector of the reference node is combined with the previous cross products of nodes 1 through i-1 to determine the state vector of node i, and thus the unknown force (i.e., F) at each node i_xi、F_hiAnd/or M_i). Once these forces are known, the axial force at the node can be calculated as,

F_αi＝F_xi×cos(α_i)+F_hi×sin(α_i)(4)。

in at least some illustrative embodiments, the combination of cross products is performed using a Transition Matrix Method (TMM). By using TMM, i will be generated^thThe cross-product combination of the state vectors of the nodes is expressed as,

$V_i=\left({\Π}_1^i{T}_i\right)\×V_0---\left(5\right).$

since each of the previous products of the products of each node 1 to i-1 has been calculated, steps are required to recalculate the product of each node. Instead, the transition matrix for each node may be combined with the cumulative transition matrix, thus avoiding duplicate computations. For a three-node column, for example, this can be expressed as,

T_acc＝T₁；V₁＝T_acc×V₀(6)，

T_acc＝T_acc×T₂；V₂＝T_acc×V₀＝(T₁×T₂)×V₀(7) and an

T_acc＝T_acc×T₃；V₃＝T_acc×V₀＝(T₁×T₂×T₃)×V₀(8)。

Cross product T₁×T₂Has been calculated in equation (7) and saved as the cumulative transition matrix T_accThe cumulative transition matrix is reused without recalculation in equation (8).

It should be noted that the transfer matrix is not limited to the particular implementation of equation (2). For example, the casing string and the reduced portion of the cross-sectional area of the casing, such as a plunger, such as a cementing plunger, present within the reduced portion 15 of FIG. 1, may be represented by a much simpler transfer matrix. In at least some illustrative embodiments, the plunger and the reducer positioned at node i are represented as,

${\left[\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& F_p^x\\ 0& 0& 0& 0& 0& 1& F_p^h\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right]}_i---\left(8\right)$

wherein,

to be positioned at i^thA plunger or reduced portion at a node point exhibiting a vertical force; and

is the horizontal force present on the plunger or reduced portion.

Other transfer matrices may include, for example, parameters describing the load imposed on the casing string by the salt formation (i.e., "salt load"). Those skilled in the art will appreciate the wide variety of transfer matrices suitable for use with the methods described herein, and all such variations of transfer matrices are within the scope of the present disclosure.

The arithmetic method of equations (6) through (8) used to calculate equation (5) is suitably implemented by software executing on a computer system, such as the illustrative system shown in fig. 3. Hardware and software components of computer system 300 are shown that, in at least some illustrative embodiments, implement at least a portion of the matrix-based well casing string modeling shown as method 400 in fig. 4 (described in more detail below). The computer system 300 operates in accordance with software (which may be stored on a non-transitory information storage medium 340) and enables a user to interact with the system through a keyboard 334, a pointing device 335 (e.g., a mouse), and a display 336 in order to configure, control, and monitor the performance of matrix-based well casing string modeling.

Display interface 352, processor 356, peripheral interface 358, information storage 360, network interface 362, and memory 370 are located within processing subsystem 330 of computer system 300. A bus 364 couples each of these elements to each other and transports their communications. The network interface 362 enables communication with other systems (e.g., via the internet with a central database server that houses additional modeling parameters and is adapted to store modeling results). In accordance with user input received through the peripheral interface 358 and program instructions from the memory 370 and/or information storage device 360, the processor 356 processes and applies input from the user to the well casing string data to perform the disclosed methods and present the results to the user. Storage 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 are respectively retrieved by processor 356 for execution. These modules include: a user interface module that processes user input provided through keyboard 334 and pointing device 335 via peripheral interface 358; a vector definition module 374 that defines a state vector for each node; a matrix definition module 376 that defines a transition matrix for the segment; a cross-product module 378 that computes a cross-product that updates the cumulative transition matrix; a transition matrix method module 380 that determines unknown state vector elements using the accumulated transition matrix from the cross-product module 378; a parameter derivation module 382 that derives well component manufacturing parameters or well component centralizer positions; and a presentation module 384 that provides the derived manufacturing parameters or centralizer positions to the manufacturer or operator (e.g., by graphically presenting the size of the casing segment or centralizer positions along the length of the casing prior to cementing in place).

FIG. 4 shows an illustrative method of implementing the matrix-based modeling described above, at least a portion of which may be implemented by software executing on computer system 300. It should be noted that while the embodiment of FIG. 3 shows various software modules executing on computer system 300, in other illustrative embodiments, some or all of these modules may execute on two or more computers within a networked system and/or a distributed system. Referring to both fig. 3 and 4, the state vector of the reference node (node 0) at the beginning of the casing string is defined by user input (user interface module 372) or using previously stored data (e.g., data on information storage device 360) (block 402; vector definition module 374). The node index i is incremented from 0 to 1 and the total product is initialized to zero (block 402; TMM module 382). Will present i^thThe state vector for a node, here node 1, is defined (block 404; vector definition module 374) in a manner similar to that used for the reference node, but with at least one unknown state vector element therein.

If the segment associated with the current node is a segment that restricts fluid flow, such as a plunger or cannula cross-sectional area reduction (block 406; matrix definition module 376), a transfer matrix is defined in terms of one or more forces exhibited by the restriction (block 408; matrix definition module 376). If the segment associated with the current node is a well casing string, then the transition matrix is defined in terms of the state vector of the reference node, the state vector of the previous node, and/or the known elements of the state vector of the current node(block 410; matrix definition module 376). Once the transition matrix for node i is defined, the transition matrix T will be accumulated for i ═ 1_accInitialisation to T₁Or for i>1, updating the accumulated transition matrix to the transition matrix T of the current node_iAnd T_accThe cross product of (block 412; matrix definition module 380). The cross product is then used to determine a full state vector for the node, which is used to determine the unknown element values of the state vector for the current node (block 414; matrix definition module 380). In at least some illustrative implementations, if all state vector values are known (e.g., if these values are easily measured at a starting point on the ground), then block 414 is skipped for the first node (i ═ 1).

If additional well casing string segments remain (block 418; TMM module 380), the node index i is incremented and the state vector of the current node becomes the state vector of the previous node (block 416; TMM module 380). The process is then repeated for the next segment and node along the well casing string (blocks 404 through 418). If no additional casing string segments are present (block 418; TMM module 380), then the previously unknown and now calculated elements of each state vector are used as a basis for determining casing manufacturing parameters such as size and composition, or for positioning a centralizer to position the casing within the wellbore (block 420; parameter derivation module 382). The resulting manufacturing parameters or centralizer positions are provided to the manufacturing personnel as a composition and/or dimensional specification or to the well operator/component installer as a centralizer or stabilizer position, respectively (block 422, presentation block 384), concluding the method (block 424).

In at least some illustrative embodiments, the forces calculated at each node along the casing string are indicated on a graphical representation of the well casing string. Once these forces are determined, the axial load (e.g., tension) present at a given node may be calculated, for example, by using equation (4). Casing string parameters such as section length, wall thickness, and material composition may then be determined from the calculated axial loads. These parameters provide the required casing safety margin at a reduced cost as compared to prior methods that overestimate the required casing string parameters. In other illustrative embodiments, the position of one or more centralizers (determined based on the calculated and displayed forces) is located on the graphical representation of the casing string. The force calculated at each node is used to determine the lateral force experienced by casing segment i (positioned between the two nodes i and i + 1), for example by using the following equation,

can be based on a state vector V_iAnd well trajectory to determine the point of contact between the casing and the wellbore wall. The calculated lateral forces and contact points are then used to determine the appropriate centralizer and optimal centralizer positions, for example, at the contact points between the casing section and the wellbore wall.

Numerous other modifications, equivalents, and alternatives will occur to those skilled in the art once the above disclosure is fully appreciated. For example, although the described embodiments use TMM to determine the transition matrix, other analytical methods are suitable for determining the transition matrix for determining the unknown elements of the node of interest. Additionally, while the examples provided are applied in the context of static modeling, static snapshots that are repeated over time may be combined to provide dynamic modeling of the well casing string as well as near real-time or real-time modeling (e.g., for predicting the load on the well casing string as the cementing plunger progresses down the string). Further, while the disclosed embodiments describe modeling a well casing string, any of a wide variety of well components may be modeled and manufacturing and/or operating parameters of the components determined. These well components include, but are not limited to, drill strings, work strings, production strings, and coiled tubing strings. Other well assemblies (e.g., packers) that restrict fluid flow within a running string are also within the scope of the present disclosure. It is intended that the following claims be interpreted to embrace all such modifications, equivalents, and alternatives as applicable.

Claims (20)

1. A method for determining a manufacturing parameter or an operating parameter of a deviated downhole well assembly, the method comprising:

representing the pipe string as a sequence of nodes separated by segments, the nodes being calculable from i-1 to N, wherein an initial mechanically limited reference node may be represented by i-0, and each node being associated with a state vector describing the position of the corresponding node and one or more forces present at the corresponding node;

determining can be according to i^th-1 State vector determination of node i^thNode pointA transition matrix sequence of state vectors of (a);

defining values of an initial state vector of the reference node;

applying the transition matrix to obtain values for each of the state vectors;

deriving parameter values for the component from at least one of the state vectors, the parameter values being in the set consisting of composition, fabrication size, and position of a centralizer or stabilizer; and

specifying the component having the parameter value.

2. The method of claim 1, wherein the specifying comprises providing the composition or dimensional specification to a manufacturer of the component.

3. The method of claim 1, wherein the specifying comprises providing the location of the centralizer or stabilizer to an installer of the component.

4. The method of claim 1, wherein each state vector includes a vertical position u associated with a node i_iHorizontal position v_iAnd an angle of inclination α_i(ii) a And further comprising a vertical force F exhibited at node i_xiHorizontal force F_hiAnd bending moment M_i(ii) a And wherein the state vector may be represented as [ u ]_i，v_i，α_i，F_xi，F_hi，M_i，1]^T。

5. The method of claim 4, wherein said i^thThe transition matrix of nodes, the i^thThe nodes are associated with the tubular segments and can be represented as:

${\left[\begin{array}{ccccccc}1& 0& -l_i^h& (\frac{{\left(l_i^h\right)}^3}{6\×{\left(EI\right)}_i}+\frac{l_i^x}{{\left(EA\right)}_i})& 0& -\frac{{\left(l_i^h\right)}^2}{2\×{\left(EI\right)}_i}& l_i^x+{\mathrm{\Δ\α}}_{i-1}^0\×l_i^h\\ 0& 1& l_i^x& 0& (-\frac{{\left(l_i^x\right)}^3}{6\×{\left(EI\right)}_i}+\frac{l_i^h}{{\left(EA\right)}_i})& \frac{{\left(l_i^x\right)}^2}{2\×{\left(EI\right)}_i}& l_i^h-{\mathrm{\Δ\α}}_{i-1}^0\×l_i^x\\ 0& 0& 1& -\frac{{\left(l_i^h\right)}^2}{2\×{\left(EI\right)}_i}& \frac{{\left(l_i^x\right)}^2}{2\×{\left(EI\right)}_i}& \frac{\sqrt{{\left(l_i^x\right)}^2+{\left(l_i^h\right)}^2}}{{\left(EI\right)}_i}& {\mathrm{\Δ\α}}_i^0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& l_i^h& -l_i^x& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right]}_i$

wherein,

is a horizontal distance, defined as (v)_i–v_i-1)；

Is a vertical distance, defined as (u)_i–u_i-1)；

Is i^th-change in inclination at node 1, defined as (α)_i-1–α₀)；

Is i^thThe change in inclination at the node point is defined as (α)_i–α₀)；

(EI)_iIs i^thThe product of the Young's modulus and moment of inertia of the component at the node; and

(EA)_iis i^thThe product of the Young's modulus and the cross-sectional area of the component at the node.

6. The method of claim 4, wherein the deriving comprises deriving the i^thAxial force F exhibited at the node_αi。

7. The method of claim 4, wherein said i^thThe transition matrix of nodes, the i^thThe node is related to current limiting and can be represented as:

${\left[\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& F_p^x\\ 0& 0& 0& 0& 0& 1& F_p^h\\ 0& 0& 0& 0& o& 0& 1\end{array}\right]}_i$

wherein,

to be positioned at said i^thA vertical force exhibited on the plunger at the node; and

is the horizontal force present on the plunger.

8. The method of claim 1, wherein said N^thThe nodes are also mechanically constrained.

9. The method of claim 1, wherein the assembly comprises a running string selected from the group consisting of: a well casing string, a drill string, a production string, and coiled tubing.

10. The method of claim 9, wherein the running string comprises a tapered section, a cross-sectional dimension change, a packer, or a plug.

11. A system for determining a manufacturing parameter or an operating parameter of a deviated downhole well assembly, the system comprising:

a memory having deviated downhole assembly modeling software; and

one or more processors coupled to the memory, the software causing the one or more processors to:

representing the pipe string as a sequence of nodes separated by segments, the nodes being calculable from i-1 to N, wherein an initial mechanically limited reference node may be represented by i-0, and each node being associated with a state vector describing the position of the corresponding node and one or more forces present at the corresponding node;

determining can be according to i^th-1 State vector determination of node i^thA transition matrix sequence of state vectors of the nodes;

defining values of an initial state vector of the reference node;

applying the transition matrix to obtain values for each of the state vectors;

deriving parameter values for the component from at least one of the state vectors, the parameter values being in the set consisting of composition, fabrication size, and position of a centralizer or stabilizer; and

specifying the component having the parameter value.

12. The system of claim 11, wherein the one or more processors specify the component at least in part by providing the composition or dimensional specification to a manufacturer of the component.

13. The system of claim 11, wherein the one or more processors specify the component at least in part by providing the location of the centralizer or stabilizer to an installer of the component.

14. The system of claim 11, wherein each state vector includes a vertical position u associated with a node i_iHorizontal position v_iAnd an angle of inclination α_i(ii) a And further comprising a vertical force F exhibited at node i_xiHorizontal force F_hiAnd bending moment M_i(ii) a And wherein the state vector may be represented as [ u ]_i，v_i，α_i，F_xi，F_hi，M_i，1]^T。

15. The system of claim 12, wherein said i^thThe matrix of nodes, the i^thThe nodes are associated with the tubular segments and can be represented as:

${\left[\begin{array}{ccccccc}1& 0& -l_i^h& (\frac{{\left(l_i^h\right)}^3}{6\×{\left(EI\right)}_i}+\frac{l_i^x}{{\left(EA\right)}_i})& 0& -\frac{{\left(l_i^h\right)}^2}{2\×{\left(EI\right)}_i}& l_i^x+{\mathrm{\Δ\α}}_{i-1}^0\×l_i^h\\ 0& 1& l_i^x& 0& (-\frac{{\left(l_i^x\right)}^3}{6\×{\left(EI\right)}_i}+\frac{l_i^h}{{\left(EA\right)}_i})& \frac{{\left(l_i^x\right)}^2}{2\×{\left(EI\right)}_i}& l_i^h-{\mathrm{\Δ\α}}_{i-1}^0\×l_i^x\\ 0& 0& 1& -\frac{{\left(l_i^h\right)}^2}{2\×{\left(EI\right)}_i}& \frac{{\left(l_i^x\right)}^2}{2\×{\left(EI\right)}_i}& \frac{\sqrt{{\left(l_i^x\right)}^2+{\left(l_i^h\right)}^2}}{{\left(EI\right)}_i}& {\mathrm{\Δ\α}}_i^0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& l_i^h& -l_i^x& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right]}_i$

wherein,

is a horizontal distance, defined as (v)_i–v_i-1)；

Is a vertical distance, defined as (u)_i–u_i-1)；

Is i^th-change in inclination at node 1, defined as (α)_i-1–α₀)；

Is i^thThe change in inclination at the node point is defined as (α)_i–α₀)；

(EI)_iIs i^thThe product of the Young's modulus and moment of inertia of the component at the node; and

(EA)_iis i^thThe product of the Young's modulus and the cross-sectional area of the component at the node.

16. The system of claim 14, wherein the one or more processors are at least in part by deriving the i^thAxial force F exhibited at the node_αiTo derive the parameter values.

17. The system of claim 14, wherein said i^thThe transition matrix of nodes, the i^thThe node is related to current limiting and can be represented as:

${\left[\begin{array}{ccccccc}1& 0& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& F_p^x\\ 0& 0& 0& 0& 0& 1& F_p^h\\ 0& 0& 0& 0& o& 0& 1\end{array}\right]}_i$

wherein,

to be positioned at said i^thA vertical force exhibited on the plunger at the node; and

is the horizontal force present on the plunger.

18. The system of claim 11, wherein said N^thThe nodes are also mechanically constrained.

19. The system of claim 11, wherein the assembly comprises a running string selected from the group consisting of: a well casing string, a drill string, a production string, and coiled tubing.

20. The system of claim 19, wherein the running string comprises a tapered section, a cross-sectional dimension change, a packer, or a ram.