Calculating downhole cards in deviated wells

Abstract

Diagnosing a pump apparatus having a downhole pump disposed in a deviated wellbore characterizes axial and transverse displacement of a rod string with two coupled non-linear differential equations of fourth order, which include axial and transverse equations of motion. To solve the equations, derivatives are replaced with finite difference analogs. Initial axial displacement of the rod string is calculated by assuming there is no transverse displacement and solving the axial equation. Initial axial force is calculated using the initial axial displacement and assuming there is no transverse displacement. Initial transverse displacement is calculated using the initial axial force and the initial axial displacement. Axial force and friction force are calculated using the initial displacements, and the axial displacement at the downhole pump is calculated by solving the axial equation with the axial force and the friction force. Load at the downhole pump is calculated so a downhole card can be generated.

Claims

1. A method of diagnosing a pump apparatus having a downhole pump disposed in a deviated wellbore, having a controller of limited processing capability, and having a motor at a surface of the deviated wellbore, the downhole pump reciprocated in the deviated wellbore by a rod string operatively moved by the motor, the method comprising: obtaining surface measurements indicative of surface load and surface position of the rod string at the surface; characterizing axial and transverse displacement of the rod string with two coupled non-linear differential equations of fourth order including an axial equation of motion and a transverse equation of motion by replacing derivatives of the two coupled non-linear differential equations with finite difference analogs; solving the finite difference analogs of the two coupled non-linear differential equations by performing calculating steps with the controller of limited processing capability comprising: initially calculating initial axial displacement of the rod string by assuming there is no transverse displacement and by solving the axial equation of motion; initially calculating initial axial force using the initial axial displacement and assuming there is no transverse displacement; first calculating initial transverse displacement of the rod string by using the initial axial force as initially calculated and using the initial axial displacement as initially calculated; second calculating axial force and friction force by using the initial axial displacement as initially calculated and using the initial transverse displacement as first calculated; and obtaining an axial displacement determination at the downhole pump by third solving the axial equation of motion using the axial force and the friction force as second calculated; calculating with the controller of limited processing capability load at the downhole pump; generating with the controller of limited processing capability a downhole card representative of the load relative to the axial displacement determination of the downhole pump obtained by the calculating steps; and modifying at least one parameter of the pump apparatus based on the generated downhole card by changing operation of the motor.

2. The method of claim 1, wherein obtaining the surface measurements comprises measuring the surface load and the surface position of the rod string at the surface.

3. The method of claim 1, wherein obtaining the surface measurements comprises obtaining the surface measurement from a memory storing the surface measurement.

4. The method of claim 1, wherein the transverse equation of motion is defined by: $\mathrm{EI}\frac{{?}^2}{?s^2}\left[\frac{{?}^{2}v}{?s^2}+\frac{1}{R_{?}}\right]+?A\frac{{?}^{2}v}{?t^2}+n_t+n_p+D_t\frac{?v}{?t}+\frac{F}{R}-?\mathrm{gA}\sin ?=0,$ wherein EI is a bending stiffness of a rod element, E is Young's modulus of elasticity, I is a bending moment, R.sub.? is a radius of curvature of tubing in the deviated wellbore, ? is a density of the rod element, A is a cross-sectional area of the rod element, n.sub.t is a transverse normal force from tubing in the deviated wellbore, an n.sub.p is a transverse normal force from liquid under pressure p, D.sub.t is a viscous damping factor in a transverse direction, F is an axial force on the rod element, $\frac{1}{R}$ is an actual radius of curvature given by $\frac{1}{R}=\frac{1}{R_{?}}+\frac{{?}^{2}v}{?s^2},$ g is acceleration due to gravity, ? is an angle of inclination, and t is time.

5. The method of claim 1, wherein the axial equation of motion is defined by: $\frac{{?}^{2}u}{?s^2}+\frac{?v}{?s}.Math.\frac{{?}^{2}v}{?s^2}-\frac{1}{a^2}\frac{{?}^{2}u}{?t^2}-\frac{D}{\mathrm{AE}}\frac{?u}{?t}+\frac{?g}{E}\cos ?-\frac{F_t}{\mathrm{AE}}=0,$ wherein u(t) is axial displacement of a rod element, a is acoustic velocity of the rod element, A is a cross sectional area of the rod element, E is Young's modulus of elasticity of the rod element, ? is a density of the rod element, g is acceleration due to gravity, ? is an angle of inclination, D is a viscous damping coefficient, F.sub.t is a friction force from tubing, s is a length measured along the rod element, and t is time.

6. The method of claim 1, wherein modifying the at least one parameter of the pump apparatus based on the generated downhole card comprises stopping the motor or adjusting a speed of the motor.

7. A non-transitory program storage device having program instructions stored thereon for causing a programmable control device to perform a method of diagnosing a pump apparatus having a downhole pump disposed in a deviated wellbore, having a controller of limited processing capability, and having a motor at a surface of the deviated wellbore, the downhole pump reciprocated in the deviated wellbore by a rod string operatively moved by the motor, the method comprising: obtaining surface measurements indicative of surface load and surface position of the rod string at the surface; characterizing axial and transverse displacement of the rod string with two coupled non-linear differential equations of fourth order including an axial equation of motion and a transverse equation of motion by replacing derivatives of the two coupled non-linear differential equations with finite difference analogs; solving the finite difference analogs of the two coupled non-linear differential equations by performing calculating steps with the controller of limited processing capability comprising: initially calculating initial axial displacement of the rod string by assuming there is no transverse displacement and by solving the axial equation of motion; initially calculating initial axial force using the initial axial displacement and assuming there is no transverse displacement; first calculating initial transverse displacement of the rod string by using the initial axial force as initially calculated and using the initial axial displacement as initially calculated; second calculating axial force and friction force by using the initial axial displacement as initially calculated and using the initial transverse displacement as first calculated; and obtaining an axial displacement determination at the downhole pump by third solving the axial equation of motion using the axial force and the friction force as second calculated; calculating with the controller of limited processing capability load at the downhole pump; generating with the controller of limited processing capability a downhole card representative of the load relative to the axial displacement determination of the downhole pump obtained by the calculating steps; and modifying at least one parameter of the pump apparatus based on the generated downhole card by changing operation of the motor.

8. The program storage device of claim 7, wherein obtaining the surface measurements comprises measuring the surface load and the surface position of the rod string at the surface.

9. The program storage device of claim 7, wherein obtaining the surface measurements comprises obtaining the surface measurement from a memory storing the surface measurement.

10. The program storage device of claim 7, wherein the transverse equation of motion is defined by: $\mathrm{EI}\frac{{?}^2}{?s^2}\left[\frac{{?}^{2}v}{?s^2}+\frac{1}{R_{?}}\right]+?A\frac{{?}^{2}v}{?t^2}+n_t+n_p+D_t\frac{?v}{?t}+\frac{F}{R}-?\mathrm{gA}\sin ?=0,$ wherein EI is a bending stiffness of a rod element, E is Young's modulus of elasticity, I is a bending moment, R.sub.? is a radius of curvature of tubing in the deviated wellbore, ? is a density of the rod element, A is a cross-sectional area of the rod element, n.sub.t is a transverse normal force from tubing in the deviated wellbore, an n.sub.p is a transverse normal force from liquid under pressure p, D.sub.t is a viscous damping factor in a transverse direction, F is an axial force on the rod element, $\frac{1}{R}$ is an actual radius of curvature given by $\frac{1}{R}=\frac{1}{R_{?}}+\frac{{?}^{2}v}{?s^2},$ g is acceleration due to gravity, ? is an angle of inclination, and t is time.

11. The program storage device of claim 7, wherein the axial equation of motion is defined by: $\frac{{?}^{2}u}{?s^2}+\frac{?v}{?s}.Math.\frac{{?}^{2}v}{?s^2}-\frac{1}{a^2}\frac{{?}^{2}u}{?t^2}-\frac{D}{\mathrm{AE}}\frac{?u}{?t}+\frac{?g}{E}\cos ?-\frac{F_t}{\mathrm{AE}}=0,$ wherein u(t) is axial displacement of a rod element, a is acoustic velocity of the rod element, A is a cross sectional area of the rod element, E is Young's modulus of elasticity of the rod element, ? is a density of the rod element, g is acceleration due to gravity, ? is an angle of inclination, D is a viscous damping coefficient, F.sub.t is a friction force from tubing, s is a length measured along the rod element, and t is time.

12. The program storage device of claim 7, wherein modifying the at least one parameter of the pump apparatus based on the generated downhole card comprises stopping the motor or adjusting a speed of the motor.

13. A controller for a pump apparatus having a surface motor and having a downhole pump, the downhole pump disposed in a deviated wellbore and reciprocated by a rod string disposed in the deviated wellbore, the controller comprising: one or more interfaces obtaining surface measurements indicative of surface load and surface position of the rod string at the surface; memory in communication with the one or more interfaces and storing first characteristics of the deviated wellbore and second characteristics of the rod string, the memory storing a model characterizing axial and transverse displacement of the rod string with two coupled non-linear differential equations of fourth order including an axial equation of motion and a transverse equation of motion having derivatives replaced with finite difference analogs; and a processing unit in communication with the one or more interfaces and the memory, the processing unit having limited processing capability and configured to: solve the finite difference analogs of the two coupled non-linear differential equations by performing calculating steps comprising: initially calculate initial axial displacement of the rod string by assuming there is no transverse displacement and by solving the axial equation of motion, initially calculate initial axial force using the initial axial displacement and assuming there is no transverse displacement, first calculate initial transverse displacement of the rod string by using the initial axial force as initially calculated and using the initial axial displacement as initially calculated, second calculate axial force and friction force using the initial axial displacement as initially calculated and using the initial transverse displacement as initially calculated, and solve the axial equation of motion using the axial force and the friction force as second calculated to obtain an axial displacement determination at the downhole pump, calculate load at the downhole pump, generate a downhole card representative of the load relative to the axial displacement determination of the downhole pump obtained by the calculating steps, and change operation of the motor to modify at least one parameter of the pump apparatus based on the generated downhole card.

14. The controller of claim 13, wherein the transverse equation of motion is defined by: $\mathrm{EI}\frac{{?}^2}{?s^2}\left[\frac{{?}^{2}v}{?s^2}+\frac{1}{R_{?}}\right]+?A\frac{{?}^{2}v}{?t^2}+n_t+n_p+D_t\frac{?v}{?t}+\frac{F}{R}-?\mathrm{gA}\sin ?=0,$ wherein EI is a bending stiffness of a rod element, E is Young's modulus of elasticity, I is a bending moment, R.sub.? is a radius of curvature of tubing in the deviated wellbore, ? is a density of the rod element, A is a cross-sectional area of the rod element, n.sub.t is a transverse normal force from tubing in the deviated wellbore, an n.sub.p is a transverse normal force from liquid under pressure p, D.sub.t is a viscous damping factor in a transverse direction, F is an axial force on the rod element, $\frac{1}{R}$ is an actual radius or curvature given by $\frac{1}{R}=\frac{1}{R_{?}}+\frac{{?}^{2}v}{?s^2},$ g is acceleration due to gravity, ? is an angle of inclination, and t is time.

15. The controller of claim 13, wherein the axial equation of motion is defined by: $\frac{{?}^{2}u}{?s^2}+\frac{?v}{?s}.Math.\frac{{?}^{2}v}{?s^2}-\frac{1}{a^2}\frac{{?}^{2}u}{?t^2}-\frac{D}{\mathrm{AE}}\frac{?u}{?t}+\frac{?g}{E}\cos ?-\frac{F_t}{\mathrm{AE}}=0,$ wherein u(t) is axial displacement of a rod element, a is acoustic velocity of the rod element, A is a cross sectional area of the rod element, E is Young's modulus of elasticity of the rod element, ? is a density of the rod element, g is acceleration due to gravity, ? is an angle of inclination, D is a viscous damping coefficient, F.sub.t is a friction force from tubing, s is a length measured along the rod element, and t is time.

16. The controller of claim 13, wherein to change operation of the motor to modify the at least one parameter of the pump apparatus based on the generated downhole card, the processing unit is configured to stop the motor or adjust a speed of the motor.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 illustrates a sucker rod pump system with a controller for controlling the system's pump.

(2) FIG. 2 illustrates iteration on netstroke and damping factor for the modified Everitt-Jennings algorithm to compute a pump card according to the prior art.

(3) FIG. 3A diagrams a vertical well model.

(4) FIG. 3B diagrams a deviated well model.

(5) FIG. 4 diagrams dynamic behavior of a rod element of a sucker rod pump system for a deviated well.

(6) FIG. 5 illustrates a flowchart of a process for calculating downhole data for a sucker rod pump system in a deviated well.

(7) FIG. 6A illustrates a pump controller according the present disclosure for a sucker-rod pump system.

(8) FIG. 6B illustrates a schematic of the pump controller for controlling/diagnosing the sucker-rod pump system according to the present disclosure.

DETAILED DESCRIPTION OF THE DISCLOSURE

(9) According to the present disclosure, the modified Everitt-Jennings algorithm is used to compute downhole data from surface data by solving the one dimensional damped wave equation with finite differences. The one-dimensional damped wave equation, however, only takes into consideration friction of a viscous nature and ignores any type of mechanical friction. If the well is substantially vertical, mechanical friction is negligible, and the obtained downhole data may be accurate. However, in deviated or horizontal wells, mechanical friction between the rods, couplings, and tubing needs to be considered. According to this disclosure, the modified Everitt-Jennings method is adapted to utilize finite differences to incorporate mechanical friction factors in the calculation of downhole data in deviated or horizontal wells.

(10) To do this, the teachings of the present disclosure use a finite difference approach to treat a system of two coupled non-linear differential equations, which encompass the forces acting on a rod element in a deviated well. The axial displacement and the transverse displacement of the rod element are considered, providing a complete model for analyzing the downhole conditions. As such, the teachings of the present disclosure utilize the equations as derived by Lukasiewicz, which has been described previously and incorporated herein by reference. The axial and transverse equations of motion for the rod element have been noted in the background section of the present disclosure.

(11) Referring now to FIG. 5, a process 100 is illustrated in flowchart form for solving of the system of coupled differential equations for axial and transverse displacement of a rod element in a deviated well. For the derivatives that appear in the previous equations (2), (3), and (4), Taylor series approximations are used to generate finite difference analogs (Block 102). For the first and second derivatives, a first-order-correct central difference and a second-order-correct central difference are used, respectively. For more details on the derivation of the second derivative analog with respect to displacement. See Everitt, T. A. and Jennings, J. W.: An Improved Finite-Difference Calculation of Downhole Dynamometer Cards for Sucker-Rod Pumps, SPE 18189, 1988, which is incorporated herein by reference.

(12) In particular, the finite difference analogs are as follows (the subscript i represents the node at an axial distance of the rod string and the subscript j represents the timestep). For the space discretization, the finite difference analogs are:

(13) 0$\frac{?u}{?s_{i,j}}=\frac{\left(u_{i+1,j}-u_{i,j}\right)}{?s}-\frac{?s}{2}\frac{{?}^{2}u}{?s^2},\frac{?v}{?s_{i,j}}=\frac{\left(v_{i+1,j}-v_{i,j}\right)}{?s}-\frac{?s}{2}\frac{{?}^{2}v}{?s^2},\frac{{?}^{2}u}{?s_{i,j}^2}=\frac{\left(u_{i+1,j}-2u_{i,j}+u_{i-1,j}\right)}{?s^2}-\frac{?s^2}{12}\frac{{?}^{4}u}{?s^4},\frac{{?}^{2}v}{?s_{i,j}^2}=\frac{\left(v_{i+1,j}-2v_{i,j}+u_{i-1,j}\right)}{?s^2}-\frac{?s^2}{12}\frac{{?}^{4}t}{?s^4}.$

(14) For the discretization in time, the finite difference analogs are:

(15) $\frac{?u}{?t_{i,j}}=\frac{\left(u_{i,j+1}-u_{i,j}\right)}{?t}-\frac{?t}{2}\frac{{?}^{2}u}{?t^2},\frac{?v}{?t_{i,j}}=\frac{\left(v_{i,j+1}-v_{i,j}\right)}{?t}-\frac{?t}{2}\frac{{?}^{2}v}{?t^2},\frac{{?}^{2}u}{?t_{i,j}^2}=\frac{\left(u_{i,j+1}-2u_{i,j}+u_{i,j-1}\right)}{?t^2}-\frac{?t^2}{12}\frac{{?}^{4}u}{?t^4},\frac{{?}^{2}v}{?t_{i,j}^2}=\frac{\left(v_{i,j+1}-2v_{i,j}+v_{i,j-1}\right)}{?t^2}-\frac{?t^2}{12}\frac{{?}^{4}v}{?t^4}.$

(16) The analogs for the derivatives with respect to time are straight forward. However, the derivatives with respect to space of a degree greater than one preferably have the finite difference analogs split into several equations to accommodate different taper properties of the rod string 28. Splitting the finite difference analogs into several equations primarily allows one to pick a change in length ?s of the curved rod so that values for the position, load, and stress can be calculated at chosen steps down the wellbore as opposed to having to interpolate between fixed points. This option allows a user more freedom to refine the discretization to optimize stress analysis.

(17) To handle the fourth order derivative with respect to displacement, a central finite difference scheme of second order is used:

(18) $\frac{{?}^{4}v}{?s_{i,j}^4}=\frac{v_{i+2,j}-4v_{i+1,j}+6v_{i,j}-4v_{i-1,j}+v_{i-2,j}}{?s^4}-\frac{?s^2}{6}\frac{{?}^{6}v}{?s^6}.$

(19) To run a diagnostic model of a deviated well based on surface measurements and calculate a downhole pump card, the transverse and axial equations of motion (2) and (3) must be solved simultaneously. The teachings of the present disclosure provide a solution to the model for a deviated well, as discussed in detail below.

(20) Without loss of generality, as an initialization step (Block 104), the rod string 28 can be assumed to lie on the tubing 18 to solve for an initial value for the axial displacement u. In other words, it is assumed that there is not transverse displacement, i.e., v=0. In this case, a value of the coefficient of friction (?) (for the friction force acting on the rod string 28 from the tubing 18) is selected as 0.05, which can be based on empirical evidence or other information (Block 106), and the simplified version of the axial equation of motion (4) assuming no transverse motion is solved first (Block 108).

(21) In particular, introducing the finite difference analogs into the simplified version of the axial equation of motion (4) yields:

(22) $u_{i+1,j}=u_{i,j}\left(\frac{\frac{\mathrm{??}{t}^2?s}{R}+2?t^2+2\frac{{\mathrm{?s}}^2}{a^2}-\frac{D?s^2?t}{\mathrm{AE}}}{?t^2+\frac{\mathrm{??}{t}^2?s^2}{R}}\right)+u_{i,j+1}\left(\frac{\frac{?s^2}{a^2}-\frac{D?s^2?t}{\mathrm{AE}}}{?t^2+\frac{\mathrm{??}{t}^2?s^2}{R}}\right)+u_{i-1,j}\left(\frac{-?t^2}{?t^2+\frac{\mathrm{??}{t}^2?s^2}{R}}\right)+u_{i,j-1}\left(\frac{\frac{?s^2}{a^2}}{?t^2+\frac{\mathrm{??}{t}^2?s^2}{R}}\right)-\cos ?\left(\frac{-\frac{?g?s^2?t^2}{E}}{?t^2+\frac{\mathrm{??}{t}^2?s^2}{R}}\right)+?g\sin ?\left(\frac{\frac{??s^2?t^2}{E}}{?t^2+\frac{\mathrm{??}{t}^2?s^2}{R}}\right).$

(23) Next, still assuming that there is not transverse displacement, the axial force F is calculated (Block 110), and the transverse equation of motion (2) is solved accordingly (Block 112). In particular, introducing the finite difference analogs into the axial force and the transverse equation of motion (2) yields:

(24) $F=\mathrm{AE}\left[\frac{?u}{?s}\right]=\mathrm{AE}\left(\frac{\left(u_{i+1,j}-u_{i,j}\right)}{?s}\right),v_{i+2,j}=4v_{i+1,j}-4v_{i-1,j}-v_{i-2,j}-v_{i,j}\left(6-\frac{\frac{2?A}{?t^2}-\frac{D_t}{?t}}{\frac{\mathrm{EI}}{?s^4}}\right)-v_{i,j+1}\left(\frac{\frac{?A}{?t^2}-\frac{D_t}{?t}}{\frac{\mathrm{EI}}{?s^4}}\right)-v_{i,j-1}\left(\frac{?A?s^4}{?t^2\mathrm{EI}}\right)-u_{i+1,j}\left(\frac{A?s^3}{\mathrm{RI}}\right)-u_{i,j}\left(-\frac{A?s^3}{\mathrm{RI}}\right)-\left(\frac{?s^4}{\mathrm{EI}}\right)\left(\mathrm{NT}+\frac{?{\mathrm{pr}}^2}{R_{?}}-?\mathrm{gA}\sin ?\right).$

(25) At this point, initial values for the transverse displacement v and the axial displacement u are available. The axial force F and the friction force F.sub.t are solved (Block 114), and the axial equation of motion (3) is solved (Block 116). In particular, introducing the finite difference analogs into the axial force F, the friction force F.sub.t, and the axial equation of motion (3) yields:

(26) $F=\mathrm{AE}\left[\frac{?u}{?s}+\frac{1}{2}{\left(\frac{?v}{?s}\right)}^2\right],F=\mathrm{AE}\left[\frac{\left(u_{i+1,j}-u_{i,j}\right)}{?s}+\frac{v_{i+1,j}^2-2v_{i+1,j}{v}_{i,j}+v_{i,j}^2}{2?s^2}\right].F_t=\frac{S}{R}\mathrm{??}s,F_t=\frac{\mathrm{AE}?}{2R?s}\left[2?s\left(u_{i+1,j}-u_{i,j}\right)+v_{i+1,j}^2-2v_{i+1,j}{v}_{i,j}+v_{i,j}^2\right].u_{i+1,j}=\frac{1}{?s^2-\frac{?}{R}}\left[\begin{array}{c}{u}_{i,j}\left(\frac{2}{?s^2}+\frac{2}{?t^2{?}^2}-\frac{D}{\mathrm{AE}?t^2}-\frac{?}{R}\right)+\\ u_{i-1,j}\left(-\frac{1}{?s^2}\right)+u_{i,j+1}\left(\frac{1}{?t^2{a}^2}+\frac{D}{\mathrm{AE}?t^2}\right)+\\ u_{i,j-1}\left(\frac{1}{?t^2{a}^2}\right)+v_{i+1,j}^2\left(\frac{?}{2R?s}-\frac{1}{?s^3}\right)+\\ v_{i,j}^2\left(\frac{?}{2R?s}-\frac{2}{?s^3}\right)+v_{i+1,j}.Math.v_{i,j}\left(\frac{3}{?s^3}-\frac{1}{?R?s}\right)+\\ v_{i-1,j}.Math.v_{i+1,j}\left(-\frac{1}{?s^3}\right)+v_{i-1,j}.Math.v_{i,j}\left(\frac{1}{?s^3}\right)-\frac{?g}{E}\cos ?\end{array}\right].$

(27) Finally, solving for the displacement u.sub.i+1,j in the above system yields the downhole position at the downhole pump used to calculate the downhole pump card (Block 116). Load at the downhole pump is then computed using Hooke's law (i.e.,

(28) $\mathrm{Load}=\mathrm{EA}\left(\frac{?u}{?s}\right))$
(Block 118). Thus, at this point, the solution can follow the form used in the Everitt-Jennings method.

(29) In particular, solving for the displacement u.sub.i+1,j in the above system requires knowing displacement two nodes behind in space, u.sub.i,j and u.sub.i?1,j relative to the node being calculated u.sub.i+1,j. To start the solution, the displacements u.sub.0,j and u.sub.1,j need to be known for all of the timesteps j. The initial displacement u.sub.0,j is know from the surface measurements of the sucker rod pump system, but the next node's displacement u.sub.1,j is calculated with Hooke's law when the polished rod load, Load.sub.PR (the surface load minus the buoyed weight of the rods), is substituted for the Load and a first-order-correct forward-difference analog is substituted for

(30) $\frac{?u}{?s},$
which yields:

(31) $u_{i,j}=\left({\mathrm{Load}}_{\mathrm{PR},j}?\frac{?s}{\mathrm{EA}}\right)+u_{0,j}$

(32) Since the numerical methodology for solving the system of coupled nonlinear differential equations is similar to the numerical implementation of the modified Everitt-Jennings method, a similar iterative method can be used to calculate the net stroke and damping factor for the deviated well model disclosed herein. See, e.g., Pons-Ehimeakhe, V., Modified Everitt-Jennings Algorithm With Dual Iteration on the Damping Factors, 2012 South Western Petroleum Short Course, Lubbock, Tex., April 18-19. Moreover, to optimize the resolution of the viscous damping in the deviated model disclosed herein, the current algorithm can further include an iteration on single or dual damping factors as disclosed in co-pending application Ser. No. 13/633,167 entitled Calculating Downhole Pump Card With Iterations on Single Damping Factor and Ser. No. 13/633,174 entitled Calculating Downhole Pump Card With Iterations on Dual Damping Factors, which are incorporated herein by reference. Thus, a single damping factor D that covers viscous damping or dual damping factors D.sub.up and D.sub.down in the above equations for the upstroke and downstroke can be iterated on in conjunction with fluid load line calculations and concavity testing to better converge on the appropriate damping for the downhole pump card generated.

(33) Using finite differences to solve the system of coupled differential equations is a useful method for analyzing stress in the sucker rod pump system. Splitting the finite difference analogs for the space discretization allows the model to be valid for a tapered rod string, including steel rods and fiberglass rods with sinker bars. Finally, including Coulombs friction in the analysis of the deviated well model gives a better approximation of the downhole conditions than using a vertical-hole model.

(34) The process 100 disclosed herein, when applied as a diagnostic tool, generates a downhole card without the excess downhole friction caused by deviation and with optimal viscous damping. This process 100 is particularly useful for controlling wells based on the downhole data. As will be appreciated, teachings of the present disclosure can be implemented in digital electronic circuitry, computer hardware, computer firmware, computer software, or any combination thereof. Teachings of the present disclosure can be implemented in a computer program product tangibly embodied in a machine-readable storage device for execution by a programmable processor so that the programmable processor executing program instructions can perform functions of the present disclosure.

(35) To that end, the teachings of the present disclosure can be implemented in a remote processing device or a pump controller. For example, FIG. 6A shows an embodiment of a pump controller 200 installed on a sucker-rod pump system 10, such as a pump jack commonly used to produce fluid from a well. The pump system 10 includes a walking beam 11 connected to a frame 15. The walking beam 11 operatively connects to a polished rod 12 connected via a rod string (not shown) to a downhole pump (not shown), which can be any downhole reciprocating pump as discussed herein. A motor control panel 19 controls a motor 17 to move the walking beam 11 and reciprocate the polished rod 12, which in turn operates the downhole pump. Although a pump jack is shown, other sucker-rod pump systems can be used, such as a strap jack, or any other system that reciprocates a rod string using cables, belts, chains, and hydraulic and pneumatic power systems.

(36) In general, sensors 202 and 204 measure load and position data of the pump system 10 at the surface, and the measured data from the sensors 202 and 204 is relayed to the controller 200. After processing the information, the controller 200 sends signals to the motor control panel 19 to operate the pump system 10. A particular arrangement of controller 200 and sensors 202 and 204 is disclosed in U.S. Pat. No. 7,032,659, which is incorporated herein by reference.

(37) As shown, the controller 200 uses a load sensor 202 to detect the weight of the fluid in the production tubing during operation of the pump system 10 and uses a position sensor 204 to measure the position of the pump system 10 over each cycle of stroke. The position sensor 204 can be any position measurement device used for measuring position relative to the top or bottom of the stroke. For example, the position sensor 204 can be a dual position sensor that produces a continuous position measurement and a discrete switch output that closes and opens at preset positions of the polished rod 12.

(38) Alternatively, the degree of rotation of the pump system's crank arm can provide displacement data. For example, a sensor can determine when the system's crank arm passes a specific location, and a pattern of simulated polished rod displacement versus time can be adjusted to provide an estimate of polished rod positions at times between these crank arm indications. In another alternative, a degree of inclination of the walking beam 11 can provide displacement data. For example, a device can be attached to the walking beam 11 to measure the degree of inclination of the pumping unit.

(39) Load data of the system 10 can be directly measured using a load cell inserted between a polished rod clamp and carrier bar. Alternatively, the strain on the walking beam 11 can provide the load data. Using a load sensor 202, for example, the controller 200 can measure the strain on the polished rod 12 and can then control the pump system 10 based on the strain measured. The load sensor 202 may use any of a variety of strain-measuring devices known to a person of ordinary skill in the art. For example, the load sensor 202 can be a load measurement device used on the pump system 10 that includes a load cell installed on the pumping rod 12 or mounted on the walking beam 11. The load sensor 202 can measure strain in the polished rod 12 and can use a strain-gage transducer welded to the top flange of the walking beam 11.

(40) Alternatively, the load sensor 202 can be a strain measuring device that clamps on to a load-bearing surface of the walking beam 11 or any convenient location as disclosed in U.S. Pat. No. 5,423,224. In another example, the load sensor 202 can use an assembly similar to what is disclosed in U.S. Pat. No. 7,032,659, which is incorporated herein by reference in its entirety.

(41) Finally, the amplitude and frequency of the electrical power signal applied to the motor 17 can be used to determine motor rotation (i.e. displacement data) and motor torque (i.e. load data). In this way, the motor speed and the displacement of the polished rod can provide a series of motor speed and displacement data pairs at a plurality of displacements along the polished rod. That displacement data which represents a complete stroke of the pump system 10 can then be converted to load on the rod string and displacement of the rod string at a plurality of displacements along the polished rod, as described in U.S. Pat. No. 4,490,094.

(42) Details of the pump controller 200 are schematically shown in FIG. 6B. In general, the controller 200 includes one or more sensor interfaces 212 receiving measurements from the load and position sensors 202 and 204. Additional inputs of the controller 200 can connect to other devices, such as an infrared water-cut meter, an acoustic sounding device (ASD) provide real-time data which can be logged for pressure buildup analysis and real-time calibration for fluid-level control. The controller 200 also include a power system (not shown), as conventionally provided.

(43) The controller 200 can have software 222 and data 224 stored in memory 220. The memory 220 can be a battery-backed volatile memory or a non-volatile memory, such as a one-time programmable memory or a flash memory. Further, the memory 220 may be any combination of suitable external and internal memories.

(44) The software 222 can include motor control software and pump diagnostic software, and the data 224 stored can be the measurements logged from the various load and position sensors 202 and 204 and calculation results. The data 224 in the memory 220 stores characteristics of the well, including the depth, azimuth, and inclination of points along the well, which can be derived from drilling and survey data. Because the rod string may be tapered as is sometimes the case, the data 224 in the memory 220 can also store characteristics of the sucker rods taper, such as depth, diameter, weight, and length of various sections of the rod.

(45) A processing unit 210 having one or more processors then processes the measurements by storing the measurement as data 224 in the memory 220 and by running the software 222 to make various calculations as detailed herein. For example, the processing unit 210 obtains outputs from the surface sensors, such as the load and position measurements from then sensors 202 and 204. In turn, the processing unit 210 correlates the output from the load sensor 202 to the position of the polished rod 12 and determines the load experienced by the polished rod 12 during the stroke cycles. Using the software 212, the processing unit 210 then calculates the downhole card indicative of the load and position of the downhole pump.

(46) To control the pump system 10, the pump controller 200 preferably uses an unabbreviated Everitt-Jennings algorithm with finite differences to solve the wave equation. The controller 200 calculates pump fillage and optimizes production on each stroke. This information is used to minimize fluid pounding by stopping or slowing down the pump system 10 at the assigned pump fillage setting. The pump controller 200 can also analyze the downhole pump card and determine potential problems associated with the pump and its operation. This is so because the shape, pattern, and other features associated with the downhole pump card represents various conditions of the pump and its operation.

(47) After processing the measurements, the controller 200 sends signals to the motor control panel 19 to operate the pump system 10. For example, one or more communication interfaces 214 communicate with the motor control panel 19 to control operation of the pump system 10, such as shutting off the motor 17 to prevent pump-off, etc. The communication interfaces 214 can be capable of suitable forms of communications, and they may also communicate data and calculation results to a remote site using any appropriate communication method.

(48) The foregoing description of preferred and other embodiments is not intended to limit or restrict the scope or applicability of the inventive concepts conceived of by the Applicants. It will be appreciated with the benefit of the present disclosure that features described above in accordance with any embodiment or aspect of the disclosed subject matter can be utilized, either alone or in combination, with any other described feature, in any other embodiment or aspect of the disclosed subject matter.

(49) In exchange for disclosing the inventive concepts contained herein, the Applicants desire all patent rights afforded by the appended claims. Therefore, it is intended that the appended claims include all modifications and alterations to the full extent that they come within the scope of the following claims or the equivalents thereof.