triplex mud pump output formula price

Rig pump output, normally in volume per stroke, of mud pumps on the rig is  one of important figures that we really need to know because we will use pump out put figures to calculate many parameters such as bottom up strokes,  wash out depth, tracking drilling fluid, etc. In this post, you will learn how to calculate pump out put for triplex pump and duplex pump in bothOilfield and Metric Unit.

Pump Output per Stroke (PO): The calculator returns the pump output per stroke in barrels (bbl).  However this can be automatically converted to other volume units (e.g. gallons or liters) via the pull-down menu.

A triplex mud (or slush) pump has three horizontal plungers (cylinders) driven off of one crankshaft. Triplex mud pumps are often used for oil drilling.

Pump OutputDuplex Pump OutputLitres/Stroke @ 90% Efficiency (2” Rod Diameter)Liner Diamerter (mm)StrokeLength(mm)101 108 114 121 127 133 140 146 152 159 165 170 178 184 190 197 203 209 216203 5.40 6.19 6.99 7.78 8.73 6.69 10.6 11.5 12.7 13.8 15 16.2 17.4 18.9254 6.67 7.62 8.58 6.69 10.8 12.0 13.3 14.6 15.9 17.3 18.7 20.0 21.9 23.6305 7.78 9.90 10.10 11.40 12.9 14.3 15.9 17.3 19.1 20.7 22.6 24.3 26.2 28.3 30.4356 14.6 16.4 18.0 19.9 21.8 23.8 25.9 28.0 30.2 32.4 35.0 37.4 39.9381 15.6 17.3 19.2 21.1 23.2 25.3 27.5 29.7 32.3 34.7 37.4 39.9 42.8406 16.7 18.6 20.5 22.6 24.8 27.0 29.4 32.3 34.5 37.0 39.7 42.8 45.6 48.6457 18.4 20.7 22.7 25.3 27.8 30.2 32.7 35.6 38.5 41.3 44.5 47.7 51.1 54.4508 20.3 22.7 25.1 28.0 30.5 33.4 36.4 39.4 46.2 45.9 49.4 53.1 56.8 60.4559 49.8 53.5 57.3 61.1 65.1 69.2 73.5610 71.1 75.6 80.2Note: For pump output in m 3 /stroke, move the decimal point 3 places to the left.Duplex Mud PumpsThe pistons on a duplex mud pump work in both directions, so that the rear cylinder has thepump rod moving through its swept volume and occupying some volume. The difference incalculations for a duplex vs. a triplex pump is that the displacement volume of this pump rodmust be subtracted from the volume in one of the cylinders, plus the difference in number ofpumping cylinders; 4 for a duplex and 3 for a triplex. Duplex pumps generally have longerstrokes (in the 10 to 18 in. range) and operate at lower rate; in the 40 to 80 stroke/minrange.The general equation to calculate output of a duplex pump is:Pump output (litres/stroke) = ,Where:ID = ID of the linerOD = OD of the rodL = Length of the pump strokeEff = Pump efficiency (decimal)1800, 505 – 3 rd Street SW Calgary, Alberta, Canada T2P 3E6 Telephone: 403.547.2906 Fax: 403.547.3129Email: info@hitechfluid.com Web: www.hitechfluid.com

GDEP is the original creator of the drilling pump and continues to set the standard for durable, high-quality drilling pumps that can withstand the world’s toughest drilling environments. Starting with our PZ7 and rounding out with the market"s most popular pump, the PZ1600, our PZ Series of pumps are the perfect choice for today"s high-pressure drilling applications.

To install Triplex Pump Output bbl/stk on your Android device, just click the green Continue To App button above to start the installation process. The app is listed on our website since 2013-09-06 and was downloaded 22 times. We have already checked if the download link is safe, however for your own protection we recommend that you scan the downloaded app with your antivirus. Your antivirus may detect the Triplex Pump Output bbl/stk as malware as malware if the download link to com.afislite.triplexpump is broken.

Once the Triplex Pump Output bbl/stk is shown in the Google Play listing of your Android device, you can start its download and installation. Tap on the Install button located below the search bar and to the right of the app icon.

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Multiplex piston pumps are positive-displacement reciprocating pumps that are configured with two or more plungers, and are often used in both drilling and well service operations. The most common multiplex pump may be equipped with three pistons (triplex pumps), and are discussed more herein. However, pumps with more or less than three pistons may be used for different applications. For example, quintuplex pumps are available and may generate less flow noise. In some low-cost applications, duplex pumps are also used. In a typical drilling rig configuration, multiplex piston pumps may be installed and operated simultaneously.

Multiplex pumps used in well service activities generally are capable of handling a wide range of fluid types, including corrosive fluids, abrasive fluids and slurries containing relatively large particulates.

When multiplex pumps are used, it is common practice to count the number of strokes to determine the volume of the fluid being pumped. The number of strokes a piston or plunger in a pump completes in a unit of time may be referred to as the stroke speed (typically measured in “strokes per minute” (SPM)). Generally, as the stroke speed increases, the flow rate of fluid being pumped by a triplex pump is also increased.

Rig operators may refer to the size (pump capacity) and the number of strokes to determine the pumped volume, represented in Equation A, below. The pumped volume may be estimated by multiplying the number of strokes by the fluid displaced during one stroke. The number of strokes may be obtained by the number of turns performed by the pump crankshaft multiplied by the number of pistons (plungers) of the pump.

The capacity is the theoretical displacement of one piston during its full stroke. The capacity may be calculated as 2 times the radius (R) of the rotational path of a crankpin pivot point around the crankshaft times the area (A) of the piston cross section. The flow rate is the pumped volume per unit of time, represented by Equation B, below.

As an example, a triplex pump having three pistons of 5-gallon capacity, and rotation speed of 40 RPM (revolution per minute) may have the following flow rate:

Known methods of estimating pumped volume and flow rate are estimates and/or theoretical calculations. However, the fluid volume discharged by each stroke of a multiplex pump is commonly lower that the theoretical capacity due to multiples effect such leakage, valve closing delay, and fluid compressibility.

In one aspect, embodiments of the present disclosure relate to methods that include determining a rotational position of a crankshaft in a multiplex pump from one or more sensors disposed on the crankshaft, determining a position of each of a plurality of pistons along a corresponding pump bore in relation to a total stroke length of each piston and a connecting rod length, calculating an individual theoretical displaced volume of fluid for each of a plurality of chambers in the multiplex pump based on the rotational position of the crankshaft, and summing the individual theoretical displaced volumes to determine a total theoretical pumped volume by the multiplex pump.

In another aspect, embodiments of the present disclosure relate to methods that include providing a multiplex pumping system having multiple multiplex pumps for pumping fluid downhole in a drilling operation, and calculating a volumetric efficiency of a first multiplex pump while the multiplex pumping system pumps fluid downhole, wherein the volumetric efficiency is calculated from a suction flow rate into the first multiplex pump and a theoretical discharge volume pumped out of the first multiplex pump.

In another aspect, embodiments of the present disclosure relate to systems that include a fluid source, multiple multiplex pumps, each multiplex pump having a crankshaft, at least one position sensor disposed on the crankshaft, multiple chambers, each chamber having an inlet in fluid communication with the fluid source via an inlet flowline and an outlet, multiple pistons, each piston slidingly engaged within the chamber, and multiple connecting rods, each connecting rod extending from one of the pistons to the crankshaft, a motor connected to the crankshaft, and a calibration tank selectively in fluid communication with the inlet of one of the multiplex pumps at a time.

In yet another aspect, embodiments of the present disclosure relate to systems that include multiple triplex pumps fluidly connected to a fluid source via inlet flowlines, a Coriolis meter disposed along a first inlet flowline, and at least one secondary flowline fluidly connecting a portion of the first inlet flowline upstream the Coriolis meter to one or more different inlet flowlines.

FIG. 3 shows a graph of the theoretical discharge rates of individual pistons in a triplex pump and the collective theoretical flow rate of the triplex pump.

FIG. 4 shows a graph of parameters during a calibration process for determining volumetric efficiency of a triplex pump according to embodiments of the present disclosure.

FIG. 5 shows a graph of the potential operating range of a triplex pump during a calibration process according to embodiments of the present disclosure.

FIG. 6A shows a graph of volumetric efficiency determined from the calibration process of a triplex pump according to embodiments of the present disclosure.

FIG. 6C shows a triplex efficiency curve corresponding to a given fluid compressibility Cfl_calin Graph A that can be normalized for an ideal fluid, and after obtaining corrected efficiency for each point i, Graph B may be generated to show the efficiency performance of the multiplex pump for the ideal fluid.

FIG. 7C shows multiplex pump behavior when the valves do not close instantaneously at the end of the suction stoke and the effect of closing delay for the discharge valves.

FIG. 7E graphically shows the relationship between multiplex pump efficiency and the potential operating range of the multiplex pump during calibration.

FIG. 7G includes a graph, Graph “D,” showing a multiplex pump affected by valve closing delay but no leakage, and a graph, Graph “E,” showing a piston pump affected by leak-rate and no effect of valve closing delay.

FIG. 7H shows data from a calibration normalized for an incompressible fluid in Graph B and plotted in a graph, Graph F, versus pump cycle time (period) in place of speed.

FIG. 10 shows a graph of the relationship between fluid level over time for a pumping process to determine apparent viscosity according to embodiments of the present disclosure.

FIG. 11 is a graph showing the relationship between the change in the fluid levels in the fluid source and calibration tank at different pump speeds and different valve positions.

FIG. 12C shows a graph of mechanical efficiency as a function of pump speed and discharge pressure according to embodiments of the present disclosure.

Embodiments of the present disclosure relate generally to accurate flow rate measurements of fluid being pumped downhole based on rotation speed and crankshaft instantaneous position in a triplex pump (or other multiplex pump) taking in account the pump efficiency. Some embodiments relate to methods that include determining the pump efficiency during normal operations, such as drilling a new portion of a well. Some embodiments relate to methods that include determining the contribution of different elements affecting the overall pump efficiency. Some embodiments relate to methods that include continuously verifying if the data from the last accepted calibration is still applicable with adequate results.

Embodiments of the present disclosure relate to multiplex pumps, including pumps having two or more pistons, such as duplex pumps, triplex pumps, quadraplex pumps, quintuplex pumps and others. However, because triplex pumps may be relatively more common in the field, discussion of multiplex pumps used in accordance with embodiments of the present disclosure may be simplified by referring to a triplex pump as an example of a multiplex pump. Thus, embodiments discussed herein referring to a triplex pump may also apply to multiplex pumps having more or less than three pistons.

The volumetric efficiency of a triplex pump may be obtained by calibrating the triplex pump at location and during drilling, such that calibration may be done without incurring non-productive time (“NPT”). The calibration may be performed at different flow rates and discharge pressure.

NPT refers to time when drilling operations do not occur, for example, where pumping drilling fluid downhole is paused for some reason. For example, NPT may include time from when a drill bit is pulled out of a wellbore to when it is run back to same depth to resume drilling, time required to nipple up and nipple down a BOP stack, pressure test of BOP, tripping of drill string, slip and cut time, and casing run times. Operations such as make up or laid down BHA, logging, fishing, jarring, wait on crew and equipment may also be part of NPT.

According to embodiments of the present disclosure, calibration methods for determining a triplex efficiency without incurring non-productive time (NPT) may include determining and comparing tank levels of fluid to be pumped through a triplex pump. Further, calibration methods may include outputting fluid at a discharge pressure similar to or within the range of the pressure of fluid being discharged during a flow rate measurement period.

Embodiments of the present disclosure also include flow measurement systems. A flow measurement system may include a fluid source, such as a mud tank, a calibration tank, and at least one triplex pump connected to the fluid source and the calibration tank. Each triplex pump may include a crankshaft, three chambers, three pistons slidingly engaged within each chamber, and connecting rods extending from each of the pistons to the crankshaft. A motor may be connected to the crankshaft to rotate the crankshaft at a rotational speed.

FIGS. 1A and 1B show examples of different configurations of a system according to embodiments of the present disclosure. The system 100 includes a fluid source 110, which is shown as being a mud tank. However, other fluid sources may be provided in systems of the present disclosure. One or more sensors 111 may be provided in or proximate to the fluid source 110 to detect a fluid level of the fluid source 110. The fluid source may be in fluid communication with one or more triplex pumps 120a-cvia an inlet flowline 130. In the embodiment shown, the system includes three triplex pumps 120a, 120b, 120c, but more or less triplex pumps may be used in other systems. A valve 132a-cmay be positioned along the inlet flowline 130, which may allow fluid to flow from the fluid source 110 to a connected triplex pump 120a-cor block fluid flow from the fluid source 110. In the embodiment shown, a pump 136a-c(e.g., a centrifugal pump) may be provided along the inlet flowline 130a-c, where the pump 136a-cmay pump fluid from the fluid source 110 to a triplex pump 120a-c. Further, the inlet flowline 130a-cmay have a flow meter 134a-cpositioned between the fluid source 110 and a connected triplex pump 120a-cto detect a flow rate of fluid moving from the fluid source 110 to the triplex pump 120a-c. In some embodiments, the flowmeter 134a-cmay be installed in the flowline between the centrifugal pump 136a-cand a triplex pump 120a-c. The inlet flowline 130a-cmay connect to an inlet 121 to the triplex pump 120a-c, such that fluid may flow from the inlet flowline 130 into a triplex pump 120a-c.

The system 100 may further include a calibration tank 150, which may hold fluid for introduction into the triplex pumps 120a-c. Fluid may be selectively flowed from the calibration tank 150 to an inlet flowline 130a-cthrough a calibration flowline 160, where a valve 162a-cdisposed along the calibration flowline 160a-cmay allow or prevent fluid flow from the calibration tank 150 to the inlet flowline 130a-c. In the embodiment shown, the calibration flowline 160a-cmay introduce fluid from the calibration tank 150 into the inlet flowline 130a-cat a location upstream of the flow meter 134a-cand the pump 136a-c. In such embodiments, the pump 136a-cmay pump fluid from the calibration flowline 160a-cthrough the inlet flowline 130a-cto the triplex pump inlet 121.

Each triplex pump 120a-cmay have a crankshaft 122 connected to three pistons 123 by connecting rods 124, where a first axial end of each connecting rod 124 is connected to the crankshaft 122, and the second axial end of each connecting rod 124 is connected to a piston 123. Each piston 123 is positioned within the triplex pump 120a-cto slidingly engage within a liner or bore 125. In operation, the crankshaft 122 may rotate, thereby moving the connecting rods 124, where the connecting rods 124 translate rotational movement from the crankshaft 122 into linear movement to push and pull the pistons 123 through the bores 125. Linear movement of the pistons 123 through the bores 125 may suction fluid into the triplex pump 120a-cthrough a suction valve 126 disposed in the triplex pump inlet 121 and discharge fluid out of the triplex pump 120a-cthrough a discharge valve 127 disposed in an outlet 128 of the triplex pump 120a-c. Discharged fluid from the triplex pump 120a-cmay flow through a discharge flowline 140, to be pumped downhole, for example. In the embodiment shown, the outlets 128 of each of the three triplex pumps 120a-care fluidly connected to the discharge flowline 140. A pressure sensor 142 may be positioned along the discharge flowline 140 to monitor the pressure of fluid being discharged from the triplex pumps 120a-c.

The triplex pumps 120a-care designed so that the pistons 123 may be easily replaced, for example to change the size of the piston 123 and the bore 125. For example, a piston having a relatively larger cross-sectional area may be used in triplex pumps 120a-cto provide a large capacity (as defined in Equations 1 and 2) and to provide high flow and lower discharge pressure (e.g., for drilling shallow wells), and a piston having a relatively smaller cross-sectional area may be used in the triplex pumps 120a-cto provide lower flow and higher discharge pressure (e.g., for drilling deeper wells).

The triplex pump 120a-cis a type of positive displacement pump, and may be considered as a volumetric flow measurement system. For each triplex pump 120a-c, the position of the crankshaft 122 may be continuously determined by an encoder 170 to accurately monitor the crankshaft position, for example, to monitor the rotational position of the crankshaft within 5 degrees or less accuracy. For example, an encoder may be positioned on a pinion shaft in a triplex pump to monitor the crankshaft position. Furthermore, the crankshaft position may be determined from a reference angle from a top dead position of the crankshaft, which may be obtained by a dedicated sensor 171 positioned at a highest point of the crankshaft. Using one or more position sensors 170, 171 on the crankshaft, the crankshaft position may be known at any moment, which may be used to determine the amount of fluid ejected into the discharge flowline 140 due to the movement of the piston 123. Namely, the ejected flow of fluid is related to the instantaneous velocity of the piston 123, where the instantaneous velocity of the piston 123 is the derivative of the piston"s position, and the piston"s position is related to the crankshaft position. In some embodiments, the encoder 170 may be absolute, so that the position measurement may be referred to a defined origin (or reference point), which merges measurements from sensors 171 into 170.

Triplex pumps are referred to throughout this disclosure in methods and systems for pumping fluid downhole. The methods and systems disclosed herein may also be applicable to multiplex pumps having two chambers and corresponding pistons or more than three chambers and corresponding pistons. For example, a system for pumping fluid downhole may include a fluid source, a calibration tank, and at least one multiplex pump, where each multiplex pump has a crankshaft rotatable by a motor 180, at least one position sensor disposed on the crankshaft, multiple chambers (each chamber having an inlet in fluid communication with the fluid source and the calibration tank via an inlet flowline and an outlet), multiple pistons, each piston slidingly engaged within the chamber, and multiple connecting rods, each connecting rod extending from one of the pistons to the crankshaft.

Referring still to FIG. 1A, a motor 180 is connected to each crankshaft 122 to drive and rotate the crankshaft 122 at a rotational speed. The motor 180 may be coupled to the pump crankshaft via a speed reduction system, possibly involving a gear reducer of belt drive system. A torque sensor 182 may be positioned between the motor 180 and crankshaft 122 to measure the amount of torque transferred to the crankshaft 122, or a torque sensor may be positioned on the crankshaft. For each pump, a variable frequency drive (VFD) 190a-cmay be in communication with the motor 180 to change the motor speed of the pump 120a-c.

Further, sensors as described herein (e.g., sensors 170, 171 disposed around the crankshaft 122) may be in wireless communication with or may be wired to a programmable logic controller (“PLC”), depending on, for example, the types of sensors being used and the location of the sensor in the system, where the PLC may receive signals from the sensors and mediate data transmission to a computational device. The PLC may continuously monitor the state of the sensors and transmit data to the computational device. For example, a PLC may provide real-time feedback of pressure, temperature, frequency, position and/or other measurements provided from the sensor signals. The PLC (not shown) of the triplex pump may communicate with the rig central computer system.

As shown in FIGS. 1A and 1B, a rig control system 192 may be used to perform methods disclosed herein. The rig control system 192 may control the VFD 190a-cof each triplex pump 120a-c. The rig control system 192 may receive measurements (such as pump speed and crankshaft potion, discharge pressure . . . ) for calculating pump characteristics, and also, the rig control system 192 can control the valve 132a-cand valves 162a-callowing control of feeding fluid to the triplex pumps.

FIG. 1B describes an alternative embodiment of the system 100, where each triplex pump 120a-cis fed via a buffer tank 112 which can be isolated from a main active tank holding the fluid source 110. One or more sensors 113 may be provided in or proximate to the buffer tanks 112 to detect a fluid level of the fluid in each buffer tank 112. A flow meter 134 may be positioned along the flow line between each buffer tank 112 and triplex pump 120a-c. Such embodiment may provide multiple operating modes with less changing of valve controls.

A computing device may be any type of server, desktop, embedded, or other computer hardware. The computing device may include at least one or more computer processor(s), and a memory module (e.g., random access memory (RAM), flash memory, etc.), interfacing with the computer processor(s). The processor may be, for example, a central processing unit (CPU), a graphics processing unit (GPU), and application-specific integrated circuit (ASIC), a field-programmable gate array (FPGA), etc. The processor may include one or more processor cores, i.e., circuits that read and execute program instructions. The processor may further be equipped with a memory controller that provides an interface to the memory module. The memory controller may include the logic necessary to write and to read to/from the memory module and to refresh the memory of the memory module, e.g., if the memory is dynamic random-access memory (DRAM). The memory controller may be a component of the processor, or it may be a separate component, interfacing with the processor. The processor may further include an input/output (I/O) interface that may allow connection of various communication buses, including, for example, a peripheral component interconnect express (PCIe) bus, but also a conventional peripheral component interconnect (PCI) bus, a universal serial bus (USB), etc. to the processor. In some embodiments, the I/O interface may alternatively not be integrated in the processor, but may instead be implemented in one or more separate chips, interfacing with the processor. Further, parts of the I/O interface may be integrated in the processor, whereas other parts may be implemented elsewhere.

In addition, a computing device may also include one or more storage device(s) (e.g., a hard disk, an optical drive such as a compact disk (CD) drive or digital versatile disk (DVD) drive, a flash memory stick, etc.), and numerous other elements and functionalities. The computing device may further include one or more output device(s) such as a screen (e.g., a liquid crystal display (LCD), a plasma display, touchscreen, cathode ray tube (CRT) monitor, projector, or other display device), and input device(s) (e.g., a keyboard and a mouse), thus enabling a user to interact with the computing device. The computing device may be connected to a network (e.g., a local area network (LAN), a wide area network (WAN) such as the Internet, mobile network, or any other type of network) via a network interface connection (not shown).

Memory may provide permanent storage to the operating system for data used by the processor. Read-only memory (ROM) may store data permanently for the operating system, and random access memory (RAM) may store status information for input and output devices, along with values for timers, counters, and internal devices. Data from PLCs may be uploaded onto a processor using a programming device, for example, a computer. Data from PLCs may be used, for example, in calibration processes according to embodiments of the present disclosure.

According to embodiments of the present disclosure, in-time measurements indicating performance of different components in triplex pumping systems may be used to calibrate one or more triplex pump efficiencies, thereby allowing for more accurate determination of the amount of fluid being pumped downhole from the triplex pumping system. The validity of triplex efficiency estimation may be tracked and used to improve triplex efficiency estimation. Triplex inefficiency may be caused, for example, by leaks past a seal between a piston and chamber, delay from the time the piston takes to switch directions, which may suction or discharge unintended fluid into or out of the chamber, and/or from leaks through suction and/or discharge portions of a piston cycle such as from leaks in inlet and/or outlet valves.

FIG. 2A shows a diagram of an example of a triplex pump according to embodiments of the present disclosure, where external leakage in the triplex pump 200 may be measured. The triplex pump 200 includes three pistons 210 (one is shown) connected to a crankshaft 220 by connecting rods 230 (one is shown) and a pushing rod 231 attached to each-other at a knuckle 232. Each piston 210 is disposed in a liner or a bore 240, where the piston 210 may slide back and forth through the chamber bore 240 as the crankshaft 220 rotates and moves the connecting rods 230 as well as the pushing rod 231 back and forth. When the piston 210 slides in a direction away from the crankshaft 220, fluid may be pushed through a discharge valve 242, and when the piston 210 slides in the opposite direction, fluid may be suctioned into the chamber 224 through a suction valve 244. A motor 270 may rotate the crankshaft 220 at a rotational speed, possibly via a speed reduction system involving belt transmission, chain transmission and gears. The pistons 210 and bores 240 are arranged in a side-by-side manner along a plane extending radially outward from the crankshaft 220. Further, the components of the triplex pump 300 may be held within a pump frame, or housing 202 (i.e., the crankcase).

A collection box 250 may be positioned under the pushing rod 231 and under the liners or bores 240. In some embodiments, a single collection box 250 may have a size/area large enough to collect fluid that leaks from each of the three pistons 210 in the triplex pump 200. In some embodiments, three collection boxes may each be sized and positioned under each piston in the triplex pump, such that individual collection boxes collect fluid that leaks from individual pistons. Further, in some embodiments, a pump 260 (e.g., a progressive cavity pump) and a pump drive motor 265, configured to drive the pump 260, may be provided in the triplex pump 200 to pump fluid collected in the collection box 250 out of the collection box 250. The pump 260 may pump fluid collected in the collection box 250 through a cleaning line 261 extending therefrom to jet fluid onto a piston 210 to clean the piston 210.

A control valve 262 may be provided along a flowline 264 in fluid communication with the collection box 250 to allow or prevent fluid flow through the flowline 264 towards a fluid source (e.g. a mud tank). For example, when the fluid level in the collection box 250 is too high or reaches a preselected level, the control valve 262 may be opened so that the extra fluid can be transferred out of the collection box 250.

Referring now to FIG. 2B, a schematic of triplex pump operation is provided to show more detail on a relationship between fluid discharge volume to piston movement and crankshaft rotation in the triplex pump 200. The piston 210 slides back and forth inside the liner or bore 240. The triplex pump 200 includes three pistons 210 and three corresponding chambers 224, where a connecting rod 230 connects each piston to a crankshaft 220, via the crankpin 233. The three crankpins may be offset by 120 degrees. In the case of quintuplex pump, the quintuplex pump would be equipped with five pistons and the crankshaft would have five crankpins offset by 72 degrees. One of the piston 210, liner or bore 240, chamber 224 and connecting rod 230 assembly is shown in FIG. 2B. The chamber 224 has a suction valve 244 positioned in an inlet 221 and a discharge valve 242 positioned in an outlet 228. As the piston 210 moves through the chamber 240, the piston 210 movement may suction fluid through the inlet 221 as the piston 210 moves toward the crankshaft 220, and the piston 210 movement may discharge fluid through the outlet 228 as the piston 210 moves away from the crankshaft 220. For a given position of the piston 210, the corresponding chamber volume 224 in the chamber 240 may be determined based on the piston axial position.

The instantaneous theoretical flow rate of fluid being moved by one piston through a triplex pump may be calculated by measuring an angular position of a crankshaft in the triplex pump versus time and applying Equations 2 and 3. The instantaneous flow rate of the other two pistons in the triplex pump may also be determined by the same logic, where an angular offset of 120 and 240 degree are added to the angle θ of Equation 1. Application of Equation 2 to each of the pistons in a triplex pump are shown below in Equations 2a-2c.

For each crankshaft angle θ, the instantaneous positive individual discharged theoretical flow rates of fluid being moved through each of the three bores in the chamber by each of the three pistons in the triplex pump may be summed together to determine a total theoretical flow rate of the triplex pump. For each angle θ, in addition to summing the instantaneous theoretical flow rates of each piston, verification that a piston is in the discharge phase may be performed. Otherwise, the corresponding discharged flow-rate is zero. In practical condition, all pistons of a multiplex pump do not discharge simultaneously, as at least one piston is in suction mode while another piston is in discharge mode.

Equations 3a-c are related to the instantaneous discharge flow rate of each piston in a multiplex pump. Similar equations may be written for the instantaneous suction flow rate.

Such calculated total theoretical flow rate (sum of the individual discharged theoretical flow rates of each piston in a multiplex pump) may be used to analyze the overall behavior of the multiplex pump.

For example, FIG. 3 shows a graph 10 of theoretical discharged flow rate of individual pistons in a triplex pump as the angular position of the crankshaft changes. When the flow rate ratio of a piston is zero (or below), the piston is in suction mode (the piston is moving in a direction toward the crankshaft, displayed as dashed line in FIG. 3), and when the flow rate ratio of the piston is positive, the piston is in discharge mode (where the piston is moving in an opposite direction away from the crankshaft, displayed as solid line in FIG. 3). The theoretical instantaneous discharge (or suction) flow rate of each individual piston (calculated along changes in crankshaft angular position) may be summed together. For each given position of the crankshaft, less than all the pistons in a multiplex pump (e.g., one or two pistons) can be in discharge mode (or suction mode). The instantaneous summed flow rate is plotted on a graph 20 of the total theoretical flow rate of the triplex as a whole at the different angular positions of the crankshaft. As shown in graph 20, the theoretical flow rate varies from the average theoretical flow rate, and thus, it may be advantageous to use the theoretical flow rate given it is more precise depending on the crankshaft annular position at a moment in time. The shape of the theoretical flow curves shown in FIG. 3 is calculated from a triplex pump having an R/L ratio of the rotational path radius R (offset of the crankpin to the crankshaft axis) to the connecting rod length L equal to 0.25. However, the shape of the theoretical flow curves may vary with different R/L ratios (e.g., a theoretical flow curve may be flatter or steeper).

FIG. 3 describes the flow output for a theoretical pump with “ideal” operation with the pistons alternatively generating suction or discharge. However, the flow effect between suction and discharge is only a shift by 180 degrees for each piston. FIG. 3 also displays the instantaneous flow in discharge phase and in suction phase.

In multiplex pumps having more than three chambers and corresponding pistons, the theoretical flow rate may have a higher frequency of peaks. For example, in a multiplex pump having five chambers, five pistons slidingly engaged in the liners or bores, and five connecting rods connecting the pistons to a crankshaft, five overlapping theoretical flow rate paths for each corresponding piston and chamber may be plotted and summed together to graph the total theoretical flow rate of the multiplex as a whole at the different angular positions of the crankshaft, as shown in FIG. 3 for a triplex pump.

Methods of the present disclosure may include determining a rotational position of a crankshaft in a multiplex pump (e.g., triplex pump) from one or more sensors disposed near (or on) the crankshaft, calculating individual discharged theoretical flow rates of fluid through each of the chambers in the multiplex pump based on the rotational position of the crankshaft, and summing the individual theoretical flow rates (which includes the effect of the angular shift between pistons) to determine a total theoretical flow rate of the multiplex pump.

A multiplex pump has a limited volumetric efficiency, where fluid ejected from a single chamber is lower than the theoretical displaced volume from a single piston stroke through the chamber. Inefficiencies in fluid ejection from a multiplex chamber may be caused, for example, by delay in valve closing, where part of the fluid returns across the valve, leaks through the seal of the valve when closed, and leaks through seals around the piston (between the piston and chamber). Cavitation may be ignored/negligible in system designs where a “charge” pump (e.g., pump 136a-cin FIGS. 1A-B) may insure that the pump chamber is properly filled when the piston reaches the end of the suction portion of a piston stroke.

According to embodiments of the present disclosure, a volumetric efficiency of a triplex pump may be calculated by calibrating the triplex pump. A triplex pump may be calibrated by comparing a known value of fluid volume pumped by the triplex pump to a calculated “theoretical discharge volume” of the triplex pump. The known value of fluid volume pumped by a triplex pump may be obtained from the variation of fluid volume in one or more calibration tanks (e.g., calibration tank 150 in FIG. 1A). The change of fluid volume in the tank is typically obtained via the usage of fluid level measurement device (e.g., sensor 151 in FIG. 1A) in the tank associated with the geometrical description of the tank: for example (but not limited to), the tank may have a rectangular cross-section, or trapezoidal cross-section, or part of a horizontal cylinder, vertical cylinder, etc. In some embodiments, a known volume of fluid pumped into a triplex pump may be measured by a Coriolis meter.

The pump volumetric efficiency may be calculated as the ratio of a calibration tank(s) volume variation and the “theoretical discharged volume” during the duration of the pumping calibration period.

Once the known value of fluid volume pumped into the triplex pump during a calibration period is measured and the theoretical discharge volume of the triplex pump during the calibration period is calculated, the volumetric efficiency of the triplex pump may be determined. The calibration is preferably performed for one selected triplex pump in a given time (e.g., in FIGS. 1A-B, a first calibration for triplex pump 120a, a second calibration for triplex pump 120b, and a third calibration for triplex pump 120c). With installation involving multiple triplex pumps, each triplex may be calibrated in a succession of calibration period.

For example, referring again to FIG. 1A, a method may include performing calibration of one selected pump while the normal rig process is not (or at minimum) affected. To achieve such process, each triplex pump 120a-cmay be connected to the calibration tank 150 separately, while the other triplex pumps 120a-cstay connected to the main tank 110. The discharge of the triplex pumps 120a-cmay be interconnected to a common discharge flowline 140 which may deliver the pumped fluid into the well so that normal rig activities may be performed (i.e., drilling). In the following description, triplex pump 120ais considered in the calibration mode, while the two other triplex pumps (120band 120c) are operating normally. However, the three multiplex pumps maintain delivering fluid to the well for the drilling process.

A method of calibrating triplex pump 120amay include closing a valve 132ato an inlet flowline 130 to the calibrated triplex pump 120a(where the valves 132b-cto the other of the three triplex pumps 120band 120cmay be open or closed during calibration) and opening a valve 162ato a calibration flowline 160 in fluid communication with the inlet 121 of the calibrated triplex pump 120a. The valves 162b-care closed. When the valve 162ato the calibration flowline 160 is open, the number of rotations of the crankshaft 122 of the calibrated triplex pump 120aduring the calibration period may be determined using one or more sensors 170, 171 around the crankshaft 122, while a sensor 151 may be used to measure a true volume of fluid pumped from the calibration tank 150 for the calibration period. The sensor 151 may be a level sensor: with the proper definition of the shape of tank 151, it is possible to determine the volume of fluid in the tank 151 in relation with the measurement of the level sensor 151. The number of rotations (including partial rotations) of the crankshaft 122 during the calibration period may be used to calculate a theoretical transferred volume from the calibrated triplex pump 120aduring the calibration period, which may be compared to the measured volume of fluid pumped from the calibration tank 150 to determine a volumetric efficiency of the first triplex pump 120a.

The “calibration tank pumped volume” may be determined by a level sensor in the calibration tank, for example, by sensor 151 in calibration tank 150 shown in FIG. 1A. This process includes providing a proper description of the calibration tank (shape and size) in order to relate fluid level to the volume of fluid pumped from the calibration tank.

The “theoretical transferred volume” is the volume transferred by the triplex pump during the calibration period. Using the calculated theoretical flow rate through a triplex pump, a theoretical discharge volume of fluid from the triplex pump may be calculated over a period of time. In some embodiments, the “theoretical transfer volume” by the triplex pump corresponding to a given angular rotation of the crankshaft may be calculated by determining the difference of linear positions of the piston corresponding to the final and initial crankshaft angular position (e.g., using position sensors as discussed above). The difference of linear position of the pistons multiplied by piston cross section determines the change in fluid volume in the chamber.

As the multiplex pump has multiple pistons, the “theoretical discharge volume” pumped volume is the sum of the volume displaced by each piston in the multiplex pump for a certain rotational displacement of the crankshaft, including an integer number N of full turns (360 degrees), which is incremented when the crankshaft passes at a certain reference point (e.g., zero degree).

The volume of Npumpingcapacity of a piston may be calculated, with Npumpingbeing the number of completed strokes between the initial and final strokes (but not including the initial and final strokes). The pumped capacity of a piston is the cross section of the piston, A, multiplied by the length of the stroke performed by the piston (twice the distance R).

The same calculation may be done for the other pistons in the multiplex pump by using Equations 5.4 and 5.5 to provide the pumped volume of a second piston, Vol_Piston2, and the pumped volume of a third piston, Vol_Piston3, in a triplex pump. The total pumped volume (VolPump) may be calculated using Equation 6,

Also, by determining the instantaneous position of the crankshaft in a triplex pump, the instantaneous flow rate and the discharge volume for the triplex pump can be determined. The rotational path radius of the connecting rod around the crankshaft and the connecting rod length are used for such determinations. The diameter of the piston is also a parameter to define the flow-rate (via the surface area of the piston cross section).

During a calibration period of a first multiplex pump in a system, one or more other available multiplex pumps may simultaneously discharge fluid to a common discharge line to provide a desired discharge pressure of pumped output fluid. For example, referring again to the system 100 in FIG. 1A, while first triplex pump 120ais being calibrated, the other available triplex pumps 120band 120care connected to the main tank 110 to simultaneously discharge fluid in the common discharge flowline 140 and insure the desired discharged pressure. Triplex pumps 120band 120cmay be calibrated sequentially or at different times from the first triplex pump 120a. While an individual triplex pump (e.g., 120a, 120bor 120c) is being calibrated, one or more of the remaining triplex pumps in the system 100 may continue normal operation.

Because the volumetric efficiency is affected by discharge pressure and triplex pump speed, the calibration of the selected triplex pump may be determined in view of these two parameters. For example, FIG. 4 shows an example of volumetric efficiency calibration cycles graphed over time. When operating on a drilling rig equipped with multiple triplex pumps, the discharge pressure (P1, P2, P3) may be controlled by the total flow rate (Qt1, Qt2, Qt3) through the well while the flow rate shared between the pump in calibration and the other pump(s) may be adjusted. In some embodiments, when using a back pressure system or other managed pressure drilling techniques, the discharge pressure of the triplex pumps may also be determined by the setting of this pressure control system. The theoretical flow rate from a triplex pump being calibrated, Qi1, Qi2, Qi3, Qi4, may be calculated from the crankshaft angular position in the calibrated triplex pump, as discussed above. The theoretical transferred volume by the triplex during the calibration period may also be obtained from the change in crankshaft angular position (including the number of crankshaft revolutions) in the calibrated triplex pump, as discussed above. The discharge pressure P and total discharged flow rate Qt are from the total flow of fluid from each triplex pump in the system (e.g., three triplex pumps 120a-care shown in the system in FIG. 1), whereas the theoretical flow rate Qi is from a single calibrated triplex pump.

The level of a calibration tank, Lcal, used for calibrating the calibrated triplex pump may further be graphed as a function of time during the process of volumetric efficiency calibration to compare the true volume of fluid pumped through the calibrated triplex pump with the theoretical pumped volume during calibration periods. Multiple calibration periods may be performed for a single calibrated triplex pump in a system (where a triplex pump may be referred to as a calibrated triplex pump while it is being calibrated). The downward slopes of the calibration tank level line, Lcal, represent different calibration periods, where the steepness of the downward slope is related to the speed of the calibrated triplex pump (relatively steeper downward slopes are calibration periods having the calibrated triplex pump moving at relatively faster speeds, and relatively shallow downward slopes are from calibration periods having the calibrated triplex pump operating at relatively slower speeds). The upward sloping portions of the calibration tank level line, Lcal, represent periods of refilling the calibration tank (e.g., which may be done in the system 100 of FIG. 1 by opening a refill valve 152 on a refill flowline 154 between a fluid source 110 and the calibration tank 150).

As shown in FIG. 4, multiple calibration cycles (e.g., four calibration cycles, as shown, or more or less than four calibration cycles) may be performed while the discharged pressure and total discharged flow rate are held at P1 and Qt1, respectively. In the embodiment shown, each of the calibration cycles may be performed at different calibrated triplex pump speeds. Multiple calibration cycles may further be performed at different discharge pressure and total discharge flow rates. For example, multiple calibration cycles may be performed while the discharged pressure and total discharged flow rate are held at P2 and Qt2 and for the flow rates Qi1′, Q21′, Q31′, Q41′ of the calibrated triplex, respectively, and also while the discharged pressure and total discharged flow rate are held at P3 and Qt3, respectively. For each calibration condition, the remaining triplex pumps in the system are set to provide a specific flow rate. Such calibration method may be conducted using an overall rig control (e.g., rig control 192 in FIGS. 1A and 1B).

As can be seen in the zoomed in view 40 of the Lcal2 cycle of calibration under discharge pressure P1 and total discharge flow rate Qt1, the theoretical flow rate Qi2 has a pattern of fluctuation from the overlapping periods of suction and discharge between the three pistons in the calibrated triplex pump (as described previously in relation to FIG. 3). Further in the calibration tank (e.g., 151 in FIG. 1A), the fluid volume may also display a fluctuating volume variation in accordance with the flow-rate variation, as the tank volume is related the integral of the pumped flow-rate. Level Lcal 2 may have a pattern of fluctuating rates of level change versus time as the change of volume in the tank corresponds to the pumped flow-rate:

To detect the fluctuation effect on the level measurement performed by the level sensor 151, a pulse dampener may not be present in the suction line of the multiplex pump 120a-c, or if such pulse dampeners are present, they may be de-activated. Furthermore, the sampling rate of the level sensor 151 may be fast enough to be able to acquire multiple level information per revolution of the crankshaft. To be able to determine the valve closing delay with sufficient accuracy, the sampling rate may correspond to an equivalent time of a crankshaft rotation of less than 5 degrees, or even 2 degrees or even 0.5 degree. As a sampling rate, the sampling rate may be faster than 5 millisecond or even 2 millisecond or even 1 millisecond.

When considering Equation 7, it is useful to performed the calibration over a time corresponding to a integer number of revolution of the crankshaft. For each crankshaft revolution, a triplex pump theoretically takes three times the volume of one piston displacement from the suction supply. When the calibration time does not match an entire number of crankshaft revolution, then Equation 6 applies. The calibration procedure described in FIG. 4 may be performed for each triplex pump of the drilling rig system. Such procedures may have no (or limited) influence on the drilling process and may create limited loss of rig productive time.

Calibration of a triplex pump may be limited by a set of parameters of the overall drilling system. For example, FIG. 5 shows a potential operating range of a calibrated triplex pump. The discharge pressure depends mainly on the total flow rate through the well. As the majority of pressure loss occurs inside the drill string, the flow is mostly turbulent and the pressure increases nearly as a square function versus the total discharged flow rate. Several desired discharge pressures Pc from a calibrated triplex pump may be achieved for calibration cycles by imposing preselected total discharge flow rates Qt using all available triplex pumps in the system (i.e., the triplex pumps in the system not being calibrated). When an operating point (e.g., Pcl and Qil) has been selected, the calibration of one triplex pump (the calibrated triplex pump) may be performed, for example, where its flow rate is set for an adequate time to empty the calibration tank. The other available triplex pumps (i.e., the non-calibrated triplex pumps) in the system may provide the rest of the flow rate, such that a total discharge flow rate Qt is delivered from the triplex pump system.

This process may be repeated for the number of calibration points desired for the calibrated triplex pump at the selected discharge pressure Pc. For example, when the calibration of a calibrated triplex pump is performed at a discharge pressure Pc4 corresponding to a total discharge flow rate Qt4 through the well, the operating points for calibration may include Qi1, Qi2, Qi3, Qi4, and Qi5. However, the calibrated triplex pump would not be able to be calibrated with a flow rate higher than Qi5, as this would require higher power than the calibrated triplex pump can deliver under the operating ranges shown in FIG. 4. Further, the calibrated triplex pump would not be able to be calibrated with a flow rate lower than Qi1, as the non-calibrated triplex pumps in the system would have to produce too large hydraulic power to maintain Qt4 and Pc4 in the well under the operating ranges shown in FIG. 4.

The dark shaded portion in the graph represents the zone limiting the lower flow rate for calibration, and the lightly shaded zone represents the suitable operating ranges of the calibrated triplex pump. Additional limiting factors limit the range for calibration operating parameters, such as the maximum discharge pressure allowed by a given pump liner (or bore) size, and the max flow rate that the well and bottom hole assembly (BHA) may tolerate. Different systems (e.g., systems having different multiplex pumps, a different number of multiplex pumps in the system, and/or different well conditions) may have different ranges of operating parameters. For example, the dark shaded zone may be reduced or not present when three or more non-calibrated triplex pumps are available in the system.

For each calibration point (Pc, Qi), the triplex efficiency may be determined, as explained above, and plotted in a graph showing triplex pump efficiency as a function of crank shaft speed, such as shown in FIG. 6A. FIG. 6A is the display of the efficiencies obtained by the calibration period such as described in FIG. 4, while respecting the calibration range as described in FIG. 5.

As seen in FIG. 6A, for a given discharge pressure Pc, the volumetric efficiencies may be dropped slightly at relatively lower crankshaft speeds, which may be due to leakage in the calibrated triplex pump, and relatively lower volumetric efficiencies may be seen at relatively higher crankshaft speeds, which may be due to relatively higher effect of valve delay, such that the volumetric efficiency curves may have a bell shape as crankshaft speed increases. Further, as the given discharge pressure Pc increases (Pc1 being the lowest graphed discharge pressure and Pc7 being the greatest discharge pressure), the likelihood of leakage increases, and thus volumetric efficiency may be lower. Such estimated efficiencies of FIG. 6A from the calibration process may be related to a given fluid compressibility Cfl_cal.

According to embodiments of the present disclosure, a system for pumping drilling fluid downhole using multiple triplex pumps may have one triplex pump being calibrated while all triplex pumps may provide fluid to the drilling process. With such procedure the total flow rate through the well may optionally be modified if the dependence of triplex efficiency on discharge pressure is characterized over a given range of discharge pressure. In some embodiments of a calibration method, one triplex pump may be used for calibration while the remaining triplex pump(s) in the system continue to pump drilling fluid downhole. After one of the triplex pumps has been calibrated, a second triplex pump may be calibrated while the other triplex pump(s) continue to pump drilling fluid downhole.

The external and internal leakages may depend on the discharge pressure, as the discharge pressure may force fluid through the leak areas (e.g., leaks through seals in the multiplex pump). The fluid properties may also affect the leak rate. For example, an increase of density or viscosity may reduce the leak rate.

After the calibration period of the multiple triplex pumps, valves (e.g., valves 132a-cand 162a-cshown in FIG. 1A) are set so that the multiple triplex pumps may take the fluid from the main fluid tank 110, while the calibration tank 150 may not be used. In the case of FIG. 1B, the three valves 132 of the main active tank may be opened. The “rig control system” 192 may perform the proper setting of these valves and also select the optimum operating conditions for the three triplex pumps 120a, 120b, 120c. Further, the rig control system 192 may select which triplex pump should be active (e.g., two versus three pumps) as well as the pump speed (strokes per minute, “SPM”) per triplex pump.

During this pumping period, the flow rate of each triplex pump may be determined by using the SPM of each triplex pump, allowing determination of the nominal flow rate. The nominal flow rate may be corrected by the pump efficiency corresponding to the triplex pump. The pump efficiency may be obtained from the calibration data (such as displayed in FIG. 6). The selection of the pump efficiency may require the knowledge of the measured discharge pressure (e.g., from pressure sensor 142) in the discharge flowline 140. The total flow-rate discharged in the well is:

Further, according to some embodiments, a triplex pump may be calibrated more than once, for example, to maintain validity of the triplex pump"s determined efficiency. For example, in some embodiments, a triplex pump may be calibrated once a selected period, e.g., where a period may be selected from a number of days, weeks or months.

In one embodiment, the flow calibration method for the rig triplex pumps provides the triplex volumetric efficiency (e.g., FIG. 6A) versus pump speed and discharge pressure. This volumetric efficiency is influenced by the compressibility of fluid which limits the effective stroke to discharge fluid. In some embodiments, the multiplex pump efficiency can be corrected for the effect of fluid compressibility. The effect of fluid compressibility may be understood by observing FIG. 6B.

Referring to FIG. 6B, when considering the effect of pressure at the triplex pump, the discharge valve opens at the beginning of the discharge stroke, when the pressure in a multiplex chamber 340 has reached the same value as in the discharge flowline. During the initial displacement, fluid is being compressed inside the chamber 340. This initial compression stroke LCdepends on the compressibility of the fluid, as well as the “dead volume,” Vdead, in the pump chamber 340.

P0: The suction pressure. This pressure is the pressure generated by the centrifugal pump (e.g., in the range of 50 PSI). To simplify the global estimation, it may be considered as the reference pressure (where P0=0), as this suction pressure is typically low in comparison with the discharge pressure.

The volume ΔVcompdirectly affects the triplex pump volumetric efficiency, as it appears as a reduction of volume per stroke. This effect depends on the fluid compressibility and the discharge pressure.

The effect of fluid compressibility may appear as a delay to opening the discharge valve. However, with compressibility effect, compression elastic energy is not stored in the discharge fluid and would be recovered as fluid expansion as the fluid returns to atmospheric pressure out of the well, thereby not affecting the delta-flow. In some embodiments, the fluid compressibility may be provided to a rig control system, which may be used to ensure that this effect is compensated for in the current pumping action. As described above, the rig control system may also have access to the discharge pressure (e.g., as measured by pressure sensor 142 in FIG. 1A). The fluid compressibility may be obtained by one of the following methods:1. Direct measurement by pressure-volume-temperature (“PVT”) cell (piston chamber and displacement+pressure).

2. Measurement of acoustic velocity in the fluid and calculation of compressibility (based on knowledge of the density). This can be obtained from P-wave propagation through the fluid or via pressure-wave travelling along a tubular such as signal or noise travelling along the mud flow pipes.

For such embodiment, the calibration data of FIG. 6A may be linked to fluid compressibility Cfl_calduring the calibration period (by using the methods described above). This provides a graph of triplex efficiency ηv_calrelated to the calibration with fluid of compressibility Cfl_cal. As shown in FIG. 6C, the triplex efficiency curve of FIG. 6A corresponds to a given fluid compressibility Cfl_calthat can be normalized for an ideal fluid which would be not-compressible. In such case, Cfl=“infinity.”

After obtaining this corrected efficiency for each point i, the graph B of FIG. 6C may be obtained and describes the efficiency performance of the multiplex pump for an incompressible fluid. Such graph can be produced immediately after the calibration cycle of each pump.

Then (as seen in FIG. 6D) during a pumping sequence with several pumps, graph “B” corresponding to an incompressible fluid may be adapted to graph “C” showing current fluid compressibility Cfl_2This adaptation may be performed as soon as a new value of compressibility Cfl_2is obtained. The adaptation procedure may include the following:

The volumetric efficiency may also be influenced by elements including leakage (external leakage at the piston seal and internal leakage at the valve seals) and closing delay at the valves, where fluid may return backwards in place of being pushed forward to the discharge. These elements are affected by different parameters of the fluid and pumping operation. In some embodiments, the control process may be characterized independently of these elements (or based on a defined time schedule) to allow continuous correction of the efficiency calibration, such as displayed in FIG. 6D.

Valve leaks may also affect the pumps efficiency, as fluid continuously returns across the valve when a difference in pressure is generated by the multiplex pump. This may be referred to as “internal leakage,” as no fluid from the internal leakage is ejected out of the multiplex pump. Internal leaks may also reduce the discharge rate. Further, internal leakage may create a spike at the beginning of a stroke, as the piston must reach a sufficient speed so that instantaneous pumping action matches the leak rate.

The effect on flow discharge of the leaks (external and internal leakage) is displayed in FIG. 7A. According to some embodiments of the present disclosure, the leak rate (from internal and/or external leakage) may be determined for a given pump while discharging fluid at a given pressure.

The leak rate (internal and external) primarily depends on the discharge pressure of the multiplex pump. The fluid density and viscosity may also influence the leak rate. During operation of a multiplex pump, one valve may be open for each chamber. However, when the multiplex pump is stopped, the valves may simultaneously hold the pressure, and it is unclear if the pressure is applied onto the piston.

In some embodiments, the leak rate may be considered as being steady for the multiplex pump and does not depend on the pump speed (FIG. 7A). In particular, at low speed, the effect of valve closing delay is minimized while the leakage effect is increased. When using the calibration period of FIG. 4 and the effect of valve closing delay is minimized, multiple leak rates may have been determined for a given fluid with the obtained results computed and plotted in a graph (e.g., as shown in FIG. 7A), and the corresponding time information may be plotted in a graph (leak-rate versus pressure), such as shown in FIG. 7B. A straight line and a parabola may then be fitted on the set of data. The fitting with the highest correlation factor may be selected as the best fitting.

When the parabola fitting is the selected solution and corresponds to the flow across a small orifice (the leak passage), a leak formula may be determined as follows.

After determining a leak rate formula for a given fluid, the leak rate can be continuously adjusted for that multiplex pump, based on the fluid density and the discharge pressure.

In such case, after determination of such leak formula for a given fluid, the leak rate can be estimated continuously based on the fluid viscosity and the discharge pressure.

According to embodiments of the present disclosure, a change in the leak rate in a multiplex pump may be managed. After some elapsed pumping time, one or a few calibration points (e.g., from a short calibration process requiring operation of that multiplex pump at low SPM) may be performed at conditions similar to the initial calibration.