Scope
This document describes the Sucker Rod Pump Analysis Tool in detail. It focuses on the solution output, technical background, and calculation methods. The Datagration SRP solution uses both modified Everitt-Jennings wave equations and the Gibbs equation. The wave equation estimates the downhole calculated card from the measured surface downhole card. SCADA systems typically provide the data and the downhole card. The Gibbs equation is used to estimate the viscous damping forces.
This SRP Article has the following categories:
Abstract
The Sucker Rod Pump Analysis Tool provides a number of KPIs which serve as a diagnostic tool for the rods and pump conditions. Pump design parameters as well as field operational data are used. In its essence, the solution uses dynamometer card data to help clients identify rod pump problems, estimate production volumes, and perform stress analysis along the rod string.
In general, the solution performs the following tasks:
- Calculate the downhole pump card from the surface dynamometer card.
- Predict the health status of the pump and reports the confidence in the predicted result.
- Calculate several downhole pump parameters: maximum plunger stroke, effective plunger stroke, pump fillage, and pump deliverability (production rate).
- Calculate the potential production rates, depending on the inflow performance of the pay-zone, and compare these values to the actual well test and calculated values.
- Perform stress analysis: calculate the minimum stress, maximum stress, and maximum allowable stress along the rod’s profile.
- Calculate the polished rod power and the pump card power and estimate the different efficiencies of the artificial lift system.
Method
Even though the surface dynamometer card is easily obtained due to the ease of installation of sensors for measuring the load and position of the polished rod, the analysis of the operation of the pump requires the monitoring of the load and position at the downhole pump.
As mentioned in the previous section, the analysis tool calculates a downhole pump card from the surface dynamometer card. These calculations are applied in PetroVisor through an external activity that is based on the modified Everitt-Jennings algorithm for solving the one-dimensional wave equation in the rod string.
The wave equation takes the polished rod load and position measurements at the surface as boundary conditions for evaluating the plunger load and position, or in fact, the load and position of any point along the rod string.
Note: a wave equation solution for deviated wells is currently (October 2021) under development, which takes into account the well trajectory and the resulting friction between the rod and the tubing.
The production engineer can visually examine the calculated cards in order to understand the actual performance of the pump. Both cards are plotted on the same graph, in addition to other parameters that are calculated separately in PetroVisor using the rod data, produced fluid properties, and the fluid level in the well, such as:
- Fluid load (𝑭𝒐) on the plunger. This is the differential pressure across the plunger during the upstroke multiplied by the plunger’s area.
Where
is the differential pressure across the plunger, and 
is the cross sectional area of the plunger.
Where
is the well head pressure, 
is the annulus pressure, 
is the specific gravity of the produced liquids, and 
is the fluid level in the annulus (from wellhead to top of fluid). - Maximum fluid load (𝑭𝒐 𝑴𝒂𝒙). This is the fluid load assuming no upthrust on the plunger during the upstroke. That’s a zero fluid level in the annulus above the pump setting depth and a zero annulus pressure.
Where
is the pump setting depth. - Rod weight in fluid (𝑾𝒓𝒇). This is the rod weight in air minus the buoyancy force on the rod when it’s submerged in the produced fluids.
Where
is the rod weight in air, 
is the buoyant force on the rod when it’s submerged in the produced fluids, 
is the rod length, 
is the rod’s cross sectional area, and 
is the density of the steel rods. -
𝑾𝒓𝒇+𝑭𝒐 𝑴𝒂𝒙
-
Required counterweight (𝑪𝑩𝑬).
Reported also on the dynamometer cards plot is the prediction of the pump health status and the confidence in the prediction result. A machine learning algorithm was used to predict the downhole pumping condition of the pump by giving the most likely diagnosis. The calculated downhole card is fed into a convolutional neural network (CNN) that predicts the pump’s downhole condition using image recognition.
A supervised machine learning technique was used, and the algorithm was trained using thousands of labelled dynamometer cards and their respective health status. Based on our database, over 12,000 dynamometer cards with their relevant downhole conditions were used in the training of a CNN that can predict pumping downhole condition with an accuracy that reaches 89% on a black box testing set.
The probability of the given classification is also calculated and reported to reflect the confidence in the algorithm’s prediction. For example, if the output of the algorithm is “Gas Interference” with a “probability of 0.85,” it means that the classifier is 85% certain that the input card looks like those cards that were classified as gas interreference in the training stage.
Depending on availability and quality of dynamometer cards, the following problems can be identified:
- Gas interference.
- Inoperative pump (gas lock, complete downhole blockage).
- Standing valve leak.
- Slight rotation in cards due to polished rod bent.
- Rod parted.
- Pump tagging down.
- High viscous fluid friction.
- Insufficient inflow (fluid pound or downhole blockage).
- Traveling valve leak.
- Pump tagging up.
- Viscous fluid friction is slightly high.
- Normal.
- No card is available (in case of missing input data).
The difference between output 11 and 7 is that 11 is a slight viscous effect that is close to normal.
For convenience, the most important loads from the dynamometer cards graph are reported in a table format next to the dynamometer cards graph. These are:
- Fluid load (𝑭𝒐).
- Maximum Fluid load (𝑭𝒐 𝑴𝒂𝒙).
- Peak polished rod load (𝑷𝑷𝑹𝑳). That’s the maximum load value on the surface dynamometer card.
- Minimum polished rod load (𝑴𝑷𝑹𝑳). That’s the minimum load value on the surface dynamometer card.
- Peak pump load (𝑷𝑷𝑼𝑴𝑷𝑳). That’s the maximum load value on the downhole dynamometer card.
- Minimum pump load (𝑴𝑷𝑼𝑴𝑷𝑳). That’s the minimum load value on the surface dynamometer card.
- Required counterweight (𝑪𝑩𝑬).
Rod travel related parameters are also reported next to the dynamometer cards graph:
- Stroke per minute (𝑺𝑷𝑴).
- Polished rod stroke.
- Maximum plunger travel (𝑴𝑷𝑻). That’s the maximum stroke length on the downhole dynamometer card.
- Effective plunger travel (𝑬𝑷𝑻). This is the stroke length on the downhole dynamometer card at the transfer point, which is the point at which the load is transferred from the standing valve to the travelling valve. Comparing the effective plunger travel to the maximum plunger travel reflects how “full” the downhole pump is.
The maximum and effective plunger travels (strokes) are calculated in the external activity as follows:
-
- The load over time and position over time information is taken from the downhole card.
- The top of stroke (TOS) is indicated by the maximum of the position over time. This corresponds to the maximum plunger travel.
- The position of the transfer point (TP) is indicated by the absolute minimum of the first derivative of the load as a function of time. As mentioned earlier, this corresponds to the effective plunger travel.
Figure 1: Top Of Stroke (TOS) and Transfer Point (TP)
The analysis tool calculates the polished rod power and the pump card power and estimate the different efficiencies of the artificial lift system. The input and calculated powers, in addition to the estimated efficiencies are reported:
- Electric power to the motor (
). This is the measured electric power input at the motor’s terminals.
-
- Polished rod power (
). This is the polished rod power required at the surface, determined from the surface dynamometer card.
Where
is the area of the surface dynamometer card (polished rod load in kN versus polished rod stroke in m), 
is the pumping speed in strokes per minute. - Pump power (
). This is the downhole pump power determined from the downhole dynamometer card in the same manner as with the polished rod power.
Where
is the area of the downhole dynamometer card. - Lifting efficiency (
). This is the ration of the pump’s hydraulic power used for fluid lifting 
to the polished rod power 
.
- Surface efficiency (
). This is the ratio of the polished rod power to the electrical power input at the prime mover’s terminals 
.
- System efficiency (
). This is the ratio of the pump’s hydraulic power 
to the electrical power input at the prime mover’s terminals . 
- Pump volumetric efficiency (
). This is the ratio of the actual production rate from well test at the surface 
to the downhole pump displacement 
. 
The convention we are using here is that the pump displacement is calculated using the maximum plunger travel, and that it accounts for the fluid shrinkage between the downhole pump and the wellhead. This way, a pump volumetric efficiency lower than 1.0 would be attributed to partial fillage of the downhole pump and/or fluid slippage past the plunger..webp?width=622&height=75&name=image-png-Sep-17-2024-07-50-24-3272-PM%20(1).webp)
Where
is the formation volume factor of the produced fluid. If no well test data are available on a certain day, the pump volumetric efficiency can be also estimated using calculated flowrates.
Patterson’s equation is used to calculate the slippage through the clearance between the plunger and the barrel, where
is the plunger’s diameter, 
is the clearance between the plunger and the barrel, 
is the length of the plunger, and 
is the produced fluid viscosity. A user-defined (given) value for the volumetric efficiency is also reported that reflects the operator’s experience of their field.
On the other hand, if frequent well test data are available, and consequently volumetric pump efficiencies, the flowrate can be calculated in the periods between the tests using the last available volumetric efficiency, if it’s believed to not have changed significantly.
- Polished rod power (
Different pressure and pressure-related values are also reported:
- Pump intake pressure (
), calculated from fluid level in the annulus. The calculation ignores the gas gradient in the annulus and assumes a constant density for the whole liquid column in the annulus. 
- Pump intake pressure (
), calculated from the fluid load on the pump card. 
- Well head pressure (𝑾𝑯𝑷).
- Annulus pressure (
).
One of the main outputs of the sucker rod pump solution is the production rate calculations. Production rates are normally back-allocated, and well-tests are needed to be performed from time to time to measure the actual flowrate and help calibrate the back-allocation process. However, with knowing the effective plunger stroke length, which is an output of the wave equation external activity, the flowrate can be calculated continuously given the rest of the available data such as the downhole pump dimensions and the uptime.
Calculated flow rates are reported next to the values from well tests. These are:
- Liquid rate. This is the calculated flowrate as described in the pump volumetric efficiency.
- Oil rate calculated using the last available water-cut value from well tests.
- Water rate calculated using the last available water-cut value from well tests.
- Gas rate calculated using the last available gas-oil-ratio value from well tests.
Potential flowrates are also estimated and reported. The potential flow rate refers to how much a well "can" produce at a certain time. If the pump's SPM is increased, more fluid volumes are produced, the bottomhole pressure drops and consequently the fluid level above reservoir. However, care must be taken not to decrease the fluid level below the pump level otherwise gas will go into the pump and might lead to fluid pound. This means that there is a limit to which flowrate can be increased: that's an increase in flowrate that will result in a decrease in fluid level from a certain depth to pump setting depth.
This new flowrate is the well's production potential, and it can be estimated by calculating the bottomhole pressure that corresponds to a fluid level at the pump depth, and using this pressure in an IPR equation to calculate the new flow rates.
Since the loads along the entire rod string can be calculated by the external activity, it’s possible to calculate the stresses along the string. The maximum (
) and minimum stress (
) values in the rod string are calculated, in addition to the maximum allowable stress (
) as per the Modified Goodman diagram/equation for the rod steel:
Where 
is the minimum tensile strength of the steel, 
is the slope of the 
curve in the Modified Goodman diagram, and 
is a service factor that accounts for the corrosivity of the rod environment.
These stresses are calculated along the rod’s profile and plotted on the same graph. This helps the operator to keep the stress value under the maximum allowable stress to avoid rod failure.
A plot of the peak and minimum polished rod load over time is provided to observe this KPI. For example, a gradual increase in the PPRL and a gradual decrease in the MPRL would indicate an added source of friction inside the well over time, such as wax deposition.
In addition to the previously calculated and reported parameters and KPIs, several design data are reported for the SRP unit, the rods, and the well completion.
Basic SRP unit and rod data:
- Unit class.
- Unit API classification.
- Rod grade.
- Rod service factor.
Rod data for each of the taper sizes:
- Rod length.
- Rod relative length.
- Rod weight.
Completion data:
- Tubing outside diameter.
- Tubing nominal weight per unit length.
- Casing outside diameter.
- Casing nominal weight per unit length.
- Tubing anchor depth.
- Pay-zone depth.
Wave Equation Details
Vertical Wells
Introduction
The behavior of the sucker rod string is modeled by using the 1D damped wave equation. The model and its solution is described by Everitt, T.A. and Jennings, J.W. (1992).
The wave equation describes the propagation of waves in a continuous medium and is given by
u is the displacement of a rod-element, compared to its resting position. v is the speed of sound in the material (wave propagation velocity). c is the damping factor.
Numerical Solution
The wave equation is solved numerically by using the finite difference method (FDM): The derivatives of displacement are substituted by finite differences or Taylor series approximations.
The rod string is sectioned into nodes that are the distance dx apart from each other. The displacement of the i-th node at time step j is described by u[i][j]. These u[i][j] can be arranged in a grid (see picture).
The wave equation is solved for the displacement u[i+1][j]. This solution is dependent of the displacement of four other nodes u[i-1][j], u[i][j-], u[i][j] and u[i][j+1].
The displacement of the first two rows of the grid are filled by using the information of the surface card (2 formulas for u[0][j] and u[1][j]). From here the whole grid can be solved by using the formula for u[i+1][j].
The movement of the plunger is described by the last row of the grid: the series of displacements in time of the last node 
. The force acting on the plunger is calculated by using Hooks law and the displacements of the node above the plunger 
. Force and movement of the plunger are combined to get the downhole card.
Viscous Damping Coefficient Calculation
A viscous damping force is acting upon each node of the rod string. It is described by the term c*du/dt in the wave equation (Gibbs’ damping). In this model this damping force is the only way to dissipate energy. Choosing the correct value for the damping coefficient c is important to receive an accurate shape of the downhole card.
Everitt provides an iterative method for selecting a damping coefficient. He defines the damping coefficient c by:
This equation compares the mechanical power 
put into the system at the polished rod to the hydraulic power 
of the fluid flowing out of the well. The power difference is to be dissipated by the damping term of the wave equation.
The mechanical power at the polished rod is taken from the area of the surface card and the pump frequency. The hydraulic power of the fluid is given by the flowrate, fluid level, fluid density and well head pressure.
A direct calculation of the damping coefficient is not possible. The damping coefficient, flowrate and hydraulic power are dependent on the plunger stroke length. The plunger stroke length is taken from the downhole card. The shape of the downhole card is only correct if the wave equation was solved using an accurate damping coefficient.
To get out of this predicament Everitt provides an iterative method to obtain the damping coefficient:
- Get the polished rod power and polished rod stroke length from the surface card.
- Estimate the fluid flowrate by using 80% of the polished rod stroke as the plunger stroke length.
- Calculate the hydraulic power using the roughly estimated plunger stroke length.
- Calculate the value for the damping coefficient c with estimated values.
- Solve the grid using the estimated value for the damping coefficient. Generate the downhole card.
- Determine the plunger stroke length from the downhole card.
- Use the new plunger stroke length to calculate the flowrate and hydraulic power.
- Calculate a new damping coefficient with the updated values for hydraulic power and plunger stroke length.
- Solve the grid using the new value for the damping coefficient. Generate the downhole card.
Repeat step 6 to step 9 two times to reach a stable solution for the damping coefficient and the plunger stroke length.
The hydraulic power is now compared to the mechanical power of the plunger: - Calculate the mechanical power of the plunger from the area of the downhole card and the pump frequency.
- Compare the mechanical power of the plunger to the hydraulic power of the fluid.
- Adjust the damping coefficient accordingly:
- Calculate new downhole card.
Repeat step 10 to step 13 until the difference of
and 
are sufficiently small.
Everitt provides the following flowchart for the iterative method for selecting the damping coefficient:
Deviated Wells
Under-development
SRP Problem Detection
Health Status Detection Using ML Model
Dynamometer cards are transformed from x, y (displacements vs loads) into pixel-wise (32*32*1) (pixels x, pixels y, color channels 1) using image-encoding. Then using a convolutional neural network, a supervised machine learning algorithm was built and trained on more than 12,000 labeled cards (healthy and unhealthy cards) to be able to identify various downhole production conditions on the sucker rod pump (as in the well-known face detection problem).
Figure 2: ML model creation steps
In model improvements, comments/feedback and inaccurate diagnosis are put into a validation set and the model starts to change hyperparameters tuning and the network’s weights in order to get better results on these testing data. Then our constructed model (upper left corner), in parallel with the comments (lower right corner), are combined in order to enhance model weights to capture more accurate card status from downhole cards (reinforced learning to meet certain output).
Figure 3: Model optimization
Required Input Parameters
The following data sets are required:
- Surface Dynamometer card output files.
- Pump specifications.
- Completion/well schematic.
- Operational data: Pump operational settings, PVT data, fluid densities, well head pressure, well test data, etc.
|
Dynamic signals |
Unit |
|
Daily produced oil per time increment |
m3 |
|
Daily produced gas per time increment |
m3 |
|
Daily produced water per time increment |
m3 |
|
Daily well head pressure |
Pa |
|
Time on production |
d |
|
Dynamic fluid level |
m |
|
Liquid above pump |
m |
|
Fluid densities |
kg/m3 |
|
Static signals |
Unit |
|
Tubing length 2 7/8 |
m |
|
Tubing length 2 3/8 |
m |
|
Tubing length 3 1/2 |
m |
|
Tubing length 4 1/2 |
m |
|
Tubing length 3 1/2 |
m |
|
Pump setting depth |
m |
|
Constants from literature |
Unit |
Value |
|
specific weight or density of steel |
lbm/ft3 |
495.30 |
|
gravitational acceleration |
ft/s2 |
32.17 |
|
young's modulus |
psi |
30500000 |
References
Boyun Guo, X. ". (2017). Petroloeum Production Engineering. Gulf Professional Publishing.
Brown, K. (1980). The Technology of Artificial lift Methods (Vol. 2a). Tulsa, OK: PennWell Publishing Company.
Ehimeakhe, V. (October 2010). Calculating Pump Fillage for Well Control Using Transfer Point Location. SPE Eastern Regional Meeting. Morgantown, West Virginia, USA. doi:https://doi.org/10.2118/136595-MS
John Patterson, Kyle Chambliss, Lynn Rowlan, Jim Curfew. (2007). Progress Report #4 on “Fluid Slippage in Down-Hole Rod-Drawn Oil Well Pumps”. SWPSC. Lubbock, Texas.
Pons, V. (October 2014). Optimal Stress Calculations for Sucker Rod Pumping Systems. SPE Artificial Lift Conference & Exhibition. Houston, Texas. doi:https://doi.org/10.2118/171346-MS
Takacs, G. (2003). Power Efficiency of Sucker-Rod Pumping.
Victoria Pons, Jeffrey DaCunha. (Sept. 25 - 28, 2012.). Wave Equation: Derivation and Analysis. 8th Annual Sucker Rod Pumping Workshop. Oklahoma City, Oklahoma.
Everitt, T.A., & Jennings, J.W. (1992). An Improved Finite Difference Calculation of Downhole Dynamometer Cards for Sucker Rod Pumps. SPE Production Engineering.