Workflow for Pipe Prover Waterdraw Calibration Metrics Computation: Case Studies with Python Programming

Precise computation of pipe prover calibration metrics is a challenge for metrology technicians and petroleum inspectors. This study has proposed a comprehensive workflow and a Python code for the computation of pipe prover waterdraw calibration metrics. The methodology has been applied to unidirectional and bidirectional pipe prover waterdraw calibration case studies. The computed base prover volume of the pipe prover calibrated through a unidirectional calibration is 4509.4 dm³ at the base temperature of 20°C when the calibrated round-trip base prover volume for the second one is 6019.178 dm³ at 20°C.

Introduction

In the oil and gas industry, the counting of the amount of hydrocarbons produced, refined (or processed), stored, transferred (or offloaded), and transported through pipelines or by tankers and trucks, the volume of petroleum products stored, transferred, and transported is a daily task. This is done with the metering kids, installations that contain a liquid/gas flow meter to measure the actual volume, volume at reference conditions of the amount of transferred liquid [1]. The flow meters need to be calibrated before first commercial use and recalibrated at frequencies defined by the host countries’ regulations [1], [2] to ensure they have or keep having the required performances.

Calibration is the means of assessing the accuracy of a measurement tool in conditions and through the procedures set up by standardization agencies and regulations. According to British Measurement Standards [3], calibration is the operation that, under specified conditions, establishes a relation between the quantity values (with measurement uncertainties) provided by measurement standards and corresponding indications (with associated measurement uncertainties). A calibration is carried out through prover-by-prover approach, that is, with another calibrated measurement tool considered as a reference, the prover [4].

Pipe provers are most used for large-volume measurement flowmeter calibration. Pipe provers are often installed as an integral part of high-value custody transfer and fiscal metering stations, which are dedicated to a particular set of flow meters and duty [3]. Their configuration is shown in Fig. 1. Before being installed and operated, the prover must be calibrated, ideally by an independent third party, to establish the base volume(s) and to demonstrate that the prover can meet the repeatability requirements stated in the standards [5].

Fig. 1. Configuration of pipe prover kid [2].

The calibration fluid can be a clean reference fluid or the actual product. The waterdraw method is the preferred calibration technique for conventional provers. It involves displacing or “drawing off” a volume of water between the prover detectors into field standard test measures, which are calibrated and traceable to a national standard [2]. Fig. 2 illustrates the configuration of a waterdraw pipe prover system.

Fig. 2. Waterdraw pipe prover calibration system [2].

Regarding the direction of calibration fluid (water in waterdraw calibration), two types of calibrations are encountered: the unidirectional calibration and the bidirectional calibration. In a uni-directional prover, the displacer makes one pass [3], traveling in a single direction to complete a prove run, while in a bi-directional prover, the displacer makes two passes, traveling in the forward and then reverse direction to complete a prove run [2].

This study aims to propose a comprehensive workflow for the computation of pipe prover waterdraw calibration metrics. The workflow will be of great usefulness for the national metrology agencies employees and other inspectors that are responsible for calibration operations inspection and to the technicians of companies that provide pipe prover calibration services.

Two case studies will be carried out on real-world calibration data with Microsoft Excel and Python Notebook through Python programming. One will be on a unidirectional pipe prover waterdraw calibration, and the second one over a bidirectional pipe prover waterdraw calibration. Python code will be shared in this paper.

Materials and Methods

Materials

As highlighted in the introduction, two case studies will be carried out. One on a unidirectional pipe prover waterdraw calibration and the second one over a bidirectional pipe prover waterdraw calibration.

The main materials used for the case studies are technical data of the pipe prover and test measures (prover tanks) including the measures base volumes indicated by their previous calibration. Moreover, the calibration operating temperatures of the prover and measures, as well as the pressures of the provers, will be used. Microsoft Excel and Python Notebook are the software used. Microsoft Excel has been used for calibration data manipulation while Python Notebook has served for calibration metrics computation.

For both calibrations three runs have been performed. The characteristics of the provers and test measures are summarized Tables I–IV while the calibration data are those of Tables V–VIII.

Table I. Pipe Prover Characteristics for Unidirectional Calibration

Characteristics	Values
Wall thickness, WT (mm)	15.35
Inner diameter, ID (mm)	620.88
Elasticity, E (1/kPa)	183000000
Coefficient of thermal expansion, Gp (1/°F)	0.0000019082
Base temperature, Tb (°C)	20

Table II. Test Measures Characteristics for Unidirectional Calibration

Characteristics	Values
	M1	M2
Base measure volume, BMVj (dm3)	1000	500
Origine level on the scale, H0j (mm)	158.322	149.7
Unit volume, Vj (m1/mm)	46.93	48.866
Coefficient of thermal expansion, Gj (1/°F)	0.000061	0.00005
Base temperature, Tb (°C)	20	20

Table III. Pipe Prover Characteristics for Bidirectional Calibration

Characteristics	Values
Wall thickness, WT (mm)	15.07
Inner diameter, ID (mm)	580.15
Elasticity, E (1/kPa)	178200000
Coefficient of thermal expansion, Gp (1/°F)	0.0000017975
Base temperature, Tb (°C)	20

Table IV. Test Measures Characteristics for Bidirectional Calibration

Characteristics	Values
	M1	M2
Base measure volume, BMVj (dm3)	500	500
Origine level on the scale, H0j (mm)	161.47	149.7
Unit volume, Vj (m1/mm)	49.174	48.866
Coefficient of thermal expansion, Gj (1/°F)	0.00005	0.00005
Base temperature, Tb (°C)	20	20

Table V. Unidirectional Pipe Prover Calibration Data

Run	Measure	Fill	Reading (mm)	TP (°C)	TM (°C)
1	M1	F1	165.8	30.29	30
1	M1	F2	167.7	30.15	30.05
1	M1	F3	172.1	30.08	30.95
1	M2	F1	170	30.72	30.92
1	M2	F2	168	30.06	30
1	M2	F3	167	30.12	30
2	M1	F1	172	29.22	28.68
2	M1	F2	178.35	30.65	30
2	M1	F3	175.45	29.42	29.98
2	M2	F1	160.2	30.45	29.12
2	M2	F2	160.4	29.12	28.12
2	M2	F3	159.4	29.45	28.96
3	M1	F1	174.9	31.84	30.71
3	M1	F2	177.8	31.93	30.84
3	M1	F3	177.5	31.9	30.83
3	M2	F1	163.8	31.92	30.81
3	M2	F2	167.5	31.91	30.86
3	M2	F3	160.7	31.89	30.85

Table VI. Unidirectional Pipe Prover Calibration Pressure Data

Run	PP (kPa)
1	100.95
2	112
3	105.87

Table VII. Bidirectional Pipe Prover Calibration Data

Pass	Measure	Fill	Reading (mm)	TP (°C)	TM (°C)
Forward 1	M1	F1	175.8	31.29	31
Forward 1	M1	F2	177.7	31.15	31.05
Forward 1	M1	F3	182.1	31.08	30.95
Forward 1	M2	F1	180	30.72	30.92
Forward 1	M2	F2	178	31.06	31
Forward 1	M2	F3	177	31.12	31
Reverse 1	M1	F1	172.8	29.35	29
Reverse 1	M1	F2	180.28	30.15	30
Reverse 1	M1	F3	183.7	29.12	28.88
Reverse 1	M2	F1	178.85	29.15	30.2
Reverse 1	M2	F2	181	30.24	29.45
Reverse 1	M2	F3	188.25	29.75	30.08
Forward 2	M1	F1	182	30.22	29.68
Forward 2	M1	F2	188.25	30.65	30
Forward 2	M1	F3	175	30.42	28.98
Forward 2	M2	F1	169.2	30.45	30.12
Forward 2	M2	F2	169.4	30.12	30.12
Forward 2	M2	F3	169.4	30.45	28.96
Reverse 2	M1	F1	175.9	30.67	30.69
Reverse 2	M1	F2	172.23	30.75	30.65
Reverse 2	M1	F3	182	30.72	30.64
Reverse 2	M2	F1	182	30.73	29
Reverse 2	M2	F2	178.5	30.73	29
Reverse 2	M2	F3	176	30.74	30.66
Forward 3	M1	F1	174.9	31.84	30.71
Forward 3	M1	F2	177.8	31.93	30.84
Forward 3	M1	F3	177.5	31.9	30.83
Forward 3	M2	F1	173.8	31.92	30.81
Forward 3	M2	F2	171.5	31.91	30.86
Forward 3	M2	F3	180.7	31.89	30.85
Reverse 3	M1	F1	178.9	30.85	30.81
Reverse 3	M1	F2	175.8	30.83	30.77
Reverse 3	M1	F3	179.5	30.8	30.74
Reverse 3	M2	F1	173.8	30.87	30.81
Reverse 3	M2	F2	180.5	30.8	30.76
Reverse 3	M2	F3	180.5	30.81	30.75

Table VIII. Bidirectional Pipe Prover Calibration Pressure Data

Run	PP (kPa)
Forward 1	102.71
Reverse 1	95.7
Forward 2	100.75
Reverse 2	115.31
Forward 3	117.3
Reverse 3	115.41

Methods

In this section a summary on the pipe prover waterdraw calibration procedure will be underlined first. It will help to understand the usefulness and the essence of the metrics to be computed. Thereafter, the workflow for Pipe Prover Waterdraw Calibration Metrics Computation will be will be presented in details.

Pipe Prover Waterdraw Calibration Procedure

A combination of pipe prover volume calibration shown by British National Measurement System [3], waterdraw calibration studied by Brendan [2] and the volumetric prover calibration of Manual of Petroleum Measurement Standards [6], lends to pipe provers waterdraw calibration procedure summarized as follows: Calibration skid preparation: It consists of checking whether all the required equipment, instruments, and devices are in place and working properly. The main are those shown in Fig. 2. At this stage, the number of test measures required for a fast and continuous calibration is foreseen. Pre-calibration circulation: Pumping and circulating water through the flowline from the water storage tank and back to itself. This helps to clean the flowline and fill it with water before the start of the calibration. Start pumping water to push the sphere till it is detected by the pipe prover entry detector. Once the sphere entry is detected: (1) fill the first test measure to the upper neck and, while preparing for the next test measure, (2) record the prover starting pressure and the prover temperature, the test measure temperature, and the scale reading, (3) drain measure down to the storage tank. (1) Fill another measure to upper neck and while preparing for the next test measure, (2) record the prover temperature, the test measure temperature and the scale reading, and (3) drain measure down to the storage tank. Repeat steps 4 and 5 till the sphere is detected by the pipe prover out detector. The last fill of the run is stopped when the sphere reaches the detector. This is the end of the first test run or one-way pass for a unidirectional calibration and the end of the first forward pass for a bidirectional calibration For a bidirectional calibration, perform steps 3, 4, and 5 circulating in the opposite direction to water direction in step 3. This is the end of the first reverse (or back) pass and the end of the first run (round-trip) for the bidirectional calibration. For a unidirectional calibration, repeat steps 3–6 three or five times. For a unidirectional calibration, repeat steps 3–7 three or five times or even more.

Pipe Prover Waterdraw Calibration Metrics Computation

The main metrics to be computed are the base prover volumes, which are the volumes of the prover at reference pressure and temperature conditions. Indeed, a pipe prover calibration aims to determine the prover actual volumes at given pressures and temperatures. Its calculations require those of the adjusted base measure volume and the volume conversion/correction factors. The way of computing one of the amounts used for a calibration operation success assessment, the repeatability, will be highlighted in the paper.

Adjusted Base Measure Volume Computation

Let M be the number of test measures (measure tanks) used, $n_j$, $1\le j\le M$, the number of fills of the measure $M_j$ for each pass or run, $H_{oj}$, $1\le j\le M$, the level on the scale set at the neck of $M_j$ correspond to its base volume and N the number of runs (or forward passes or reverse/back passes) used for a pipe prover calibration.

The adjusted base measure volume $BMV{a}_{ijk}$ of a fill $F_k$ of the measure $M_j$ during the run (or forward pass) i, $1\le i\le N$, is given by (1) [7]. where $BM{V}_j$ is the original base volume or indicated volume of the measure $M_j$ at the base temperature, $V_j$ is the unit volume of the measure neck scale, $H_{0j}$ is the level on the scale set at the neck of $M_j$ corresponding to its base volume and $H_{ijk}$ is the water level read on the scale.

(1) $$\begin{array}{r}BMV{a}_{ijk}=BM{V}_j+V_j(H_{ijk}-H_{0j})\end{array}$$

The adjusted base measure volume $BM{V}^{\prime }{a}_{ijk}$ of a fill $F_k$ of the measure $M_j$ during the reverse backward pass i, $1\le i\le N$, is given by (2) [7]: where $BM{V}_j$ is the original base volume or indicated volume of the measure $M_j$ at the base temperature, $V_j$ is the unit volume of the measure neck scale, $H_{0j}$ is the level on the scale set at the neck of $M_j$ corresponding to its base volume and $H_{ijk}^{\prime }$ is the water level read on the scale.

(2) $$\begin{array}{r}BM{V}^{\prime }{a}_{ijk}=BM{V}_j+V_j(H_{ijk}^{\prime }-H_{0j})\end{array}$$

Base Prover Volume Computation

The computation of the base prover volume requires the knowledge of the volume correction/conversion factors for the effects of temperature on the water, on the measures steel and on the prover steel for each fill, and the volume correction factors for the effects of pressure on water in the prover and on the steel of the prover for each run (or pass) [8]. Indeed, the base prover volume is the global or full calibration prover volume which computation relies on the adjusted base measures volume and the correction factors. The base prover volume is said to be the global or full calibration prover volume because the prover volume can be computed on the basis of a pass or a run results.

Most of the corrections for the effect of temperature are made to refer everything to a reference condition rather than the difference in conditions [3]. The common reference/base temperatures used are 15°C, 20°C, or 60°C ([3] and [6]).

The main corrections to be performed are [7]:

- The correction for the difference in temperature of water between the measures and the prover,

- The correction for temperature on the steel shell of measures and the prover,

- The correction for pressure on the steel shell of the prover and on water in the prover.

Volume Correction Factor for Temperature Difference of Water Between the Measure and the Prover

The correction for the difference in temperature of water between the measure and the prover is the correction for the temperature difference between the water when in the individual test measures and the water when in the prover [7]. According to [9], the volume correction factor for temperature difference between the measure and the prover $CTD{W}_{ijk}$ for a fill $F_k$ of the measure $M_j$ during the run (or forward pass) i, is the ratio of the water density at its temperature in the measure to its density at the starting temperature of the prover over the concerned fill (3). where $RHOW$ is the water density, $T{M}_{ijk}$ and $T{P}_{ijk}$ are the temperature of water in the measure $M_j$ and the prover during the fill $F_k$ of the run (or forward pass) i.

(3) $$\begin{array}{r}CTD{W}_{ijk}=\frac{RHOW\left(T{M}_{ijk}\right)}{RHOW\left(T{P}_{ijk}\right)}\end{array}$$

The volume correction factor for temperature difference between the measure and the prover $CTD{W}_{ijk}^{\prime }$ for a fill $F_k$ of the measure $M_j$ during the backward pass i, is the ratio of the water density at its temperature in the measure to its density at the starting temperature of the prover over the concerned fill (4). where $RHO{W}^{\prime }$ is the water density, $T{M}_{ijk}^{\prime }$ and $T{P}_{ijk}^{\prime }$ are the temperature of water in the measure $M_j$ and the prover during the fill $F_k$ of the backward pass i.

(4) $$\begin{array}{r}CTD{W}_{ijk}^{\prime }=\frac{RHOW\left(T{M}_{ijk}^{\prime }\right)}{RHOW\left(T{P}_{ijk}^{\prime }\right)}\end{array}$$

Manual of Petroleum Measurement Standards [7] proposes that the mathematical model of Wagenbreth can be used for water density estimation when the water temperature is between 1.66°C and 40.56°C (35°F and 105°F) in the prover and between 0.055°C and 40.56°C (32.1°F and 105°F) in the measure. It adds that Kell’s model can be used for water temperatures above 40.56°C (105°F) in the prover or the measure.

Equation (5) is Wagenbreth’s mathematical model for water density estimation: where $RHOW\left(T\right)$ is the water density in kg per m³ and T is the temperature in°C.

(5) $$\begin{array}{rl}RHOW\left(T\right)& ==999.8395639+0.06798299989T\\ & -0.009106025564T^2\\ & +0.0001005272999T^3\\ & -0.000001126713526T^4\\ & +0.000000006591795606T^5\end{array}$$

Volume Correction Factor for Temperature on the Steel of Measures and the Prover

For Manual of Petroleum Measurement Standards [9], the correction factor for temperature on the steel of the measure and the prover $CCT{S}_{ijk}$ for a fill $F_k$ of the measure $M_j$ during the run (or forward pass) i, is the ratio of the correction factor for temperature on the steel of the measure, $CTS{M}_{ijk}$, to the correction factor for temperature on the steel of the prover, $CTS{P}_{ijk}$, over the concerned fill (6):

(6) $$\begin{array}{r}CCT{S}_{ijk}=\frac{CTS{M}_{ijk}}{CTS{P}_{ijk}}\end{array}$$

Similarly, the correction factor for temperature on the steel of the measure and the prover $CCT{S}_{ijk}^{\prime }$ for a fill $F_k$ of the measure $M_j$ during the backward pass i, is the ratio of the correction factor for temperature on the steel of the measure, $CTS{M}_{ijk}^{\prime }$, to the correction factor for temperature on the steel of the prover, $CTS{P}_{ijk}^{\prime }$, over the concerned fill (7).

(7) $$\begin{array}{r}CCT{S}_{ijk}^{\prime }=\frac{CTS{M}_{ijk}^{\prime }}{CTS{P}_{ijk}^{\prime }}\end{array}$$

Equations (8) to (11) give the mathematical relationships for $CTS{M}_{ijk}$, $CTS{M}_{ijk}^{\prime }$, $CTS{P}_{ijk}$ and $CTS{P}_{ijk}^{\prime }$ computations [9]: where $T_b$ the base temperature, $T{M}_{ijk}$ and $T{P}_{ijk}$ are the temperature of water in the measure $M_j$ and the prover during the fill $F_k$ of the run (or forward pass) i, $T{M}_{ijk}^{\prime }$ and $T{P}_{ijk}^{\prime }$ are the temperature of water in the measure $M_j$ and the prover during the fill $F_k$ of the backward pass i, $G_j$ is the cubical of the coefficient of thermal expansion of the measure $M_j$ and $G_p$ are the cubical of the coefficient of thermal expansion of the prover. $G_j$ and $G_p$ are in 1/°C of 1/°F.

(8) $$\begin{array}{r}CTS{M}_{ijk}=1+G_j(T{M}_{ijk}-T_b)\end{array}$$

(9) $$\begin{array}{r}CTS{M}_{ijk}^{\prime }=1+G_j(T{M}_{ijk}^{\prime }-T_b)\end{array}$$

(10) $$\begin{array}{r}CTS{P}_{ijk}=1+G_p(T{P}_{ijk}-T_b)\end{array}$$

(11) $$\begin{array}{r}CTS{P}_{ijk}^{\prime }=1+G_p(T{P}_{ijk}^{\prime }-T_b)\end{array}$$

Volume Correction Factor for Pressure on the Prover Steel and Water in the Prover

Since measures are taken to maintain the pressure as constant as possible over a run (or pass), the pressure on the steel of the prover, which is also the pressure on the water in the prover, is unique for a run (or pass). Therefore, the correction factor for pressure is characteristic of runs (or passes).

Equations (12) and (13) gives the relationship for correction factor for pressure on the steel of the prover and on the water in the prover. The correction factor for pressure on the steel of the prover and on the water in the prover $CC{P}_i$ (respectively $CC{P}_i^{\prime }$) for the run or forward pass (respectively backward pass) i, is the product of the correction factor for pressure on the steel of the prover, $CP{S}_i$ (respectively $CP{S}_i^{\prime }$), and the correction factor for temperature on the water in the prover, $CP{W}_i$ (respectively $CP{W}_i^{\prime }$) [9], [10]:

(12) $$\begin{array}{r}CC{P}_i=CP{S}_i\times CP{W}_i\end{array}$$

(13) $$\begin{array}{r}CC{P}_i^{\prime }=CP{S}_i^{\prime }\times CP{W}_i^{\prime }\end{array}$$

$CP{S}_i$, $CP{S}_i^{\prime }$, $CP{W}_i$ and $CP{W}_i^{\prime }$ are calculated with (14) to (17): where $P{P}_i$ is the prover pressure during the run (or forward pass) i, $P{P}_i^{\prime }$ is the prover pressure during backward pass i, $ID$ is the inner diameter of the pipe prover, $E$ is the modulus of elasticity in $ps{i}^{-1}$ or $P{a}^{-1}$, $WT$ is the wall thickness of the prover, $F$ is the compressibility factor of water in the prover in in $ps{i}^{-1}$ or $P{a}^{-1}$.

(14) $$\begin{array}{r}CP{S}_i=1+\frac{P{P}_i\times ID}{E\times WT}\end{array}$$

(15) $$\begin{array}{r}CP{S}_i^{\prime }=1+\frac{P{P}_i^{\prime }\times ID}{E\times WT}\end{array}$$

(16) $$\begin{array}{r}CP{W}_i=\frac{1}{1-F\times P{P}_i}\end{array}$$

(17) $$\begin{array}{r}CP{W}_i^{\prime }=\frac{1}{1-F\times P{P}_i^{\prime }}\end{array}$$

Total Water Fill for Runs or Passes

The total water fill $W{D}_i$ of the pipe prover during the run (or forward pass) i is the sum of the volume of water in the prover corrected for temperature effects at the end of the run (or forward pass). $W{D}_i$ is given by (18): where $BMV{a}_{ijk}$ is the adjusted base measure volume of the fill $F_k$ of the measure $M_j$ during the backward pass i, $CTD{W}_{ijk}$ is the volume correction factor for temperature difference between the measure and the prover for the fill $F_k$ of $M_j$ during the backward pass i, $CCT{S}_{ijk}$ is the the correction factor for temperature on the steel of the measure and the prover for the fill $F_k$ of $M_j$ during the backward pass i, M be the number of test measures (measure tanks) used, $n_j$, $1\le j\le M$, the number of fills of the measure $M_j$.

(18) $$\begin{array}{r}W{D}_i=\sum _{j=1}^M\sum _{k=1}^{n_j}BMV{a}_{ijk}\times CTD{W}_{ijk}\times CCT{S}_{ijk}\end{array}$$

For a bidirectional calibration, the total water fill $W{D}_i^{\prime }$ of the pipe prover during the backward pass i is the sum of the volume of water in the prover corrected for temperature effects at the end of pass. $W{D}_i^{\prime }$ is given by (19): where $BM{V}^{\prime }{a}_{ijk}$ is the adjusted base measure volume of the fill $F_k$ of the measure $M_j$ during the backward pass i, $CTD{W}_{ijk}^{\prime }$ is the volume correction factor for temperature difference between the measure and the prover for the fill $F_k$ of $M_j$ during the backward pass i and $CCT{S}_{ijk}^{\prime }$ the correction factor for temperature on the steel of the measure and the prover for the fill $F_k$ of $M_j$ during the run backward pass i, M be the number of test measures (measure tanks) used, $n_j$, $1\le j\le M$, the number of fills of the measure $M_j$.

(19) $$\begin{array}{r}W{D}_i^{\prime }=\sum _{j=1}^M\sum _{k=1}^{n_j}BM{V}^{\prime }{a}_{ijk}\times CTD{W}_{ijk}^{\prime }\times CCT{S}_{ijk}^{\prime }\end{array}$$

Base Prover Volume for Runs or Passes

The base prover volume $BP{V}_i$ for the run (or forward pass) i, is the total water fill $W{D}_i$ of the pipe prover corrected for pressure effects. $BP{V}_i$ is given by (20): where $W{D}_i$ is total water fill of the pipe prover for the (or forward pass) i and $CC{P}_i$ the correction factor for pressure on the steel of the prover and on the water in the prover.

(20) $$\begin{array}{r}BP{V}_i=W{D}_i\times CC{P}_i\end{array}$$

For a bidirectional calibration, the base prover volume $BP{V}_i^{\prime }$ for the backward pass i, is the total water fill $W{D}_i^{\prime }$ of the pipe prover corrected for pressure effects. $BP{V}_i^{\prime }$ is given by (21): where $W{D}_i^{\prime }$ is total water fill of the pipe prover for the backward pass i and $CC{P}_i^{\prime }$ the correction factor for pressure on the steel of the prover and on the water in the prover.

(21) $$\begin{array}{r}BP{V}_i^{\prime }=W{D}_i^{\prime }\times CC{P}_i^{\prime }\end{array}$$

For a bidirectional calibration, the base prover volume $BP{V}_{2,i}$ is for the run i, is the sum of its forward and backward base prover volumes (22): where $BP{V}_i$ and $BP{V}_i^{\prime }$ are the base prover volumes for the forward and backward passes for the run i, respectively.

(22) $$\begin{array}{r}BP{V}_{2,i}=BP{V}_i+BP{V}_i^{\prime }\end{array}$$

Base Prover Volume

The base prover volume $BPV$ is the average of the base prover volumes of the different runs (23) and (24): where N is the number of runs of the calibration, $BP{V}_i$ is the base prover volume of run i for a unidirectional calibration and $BP{V}_{2,i}$ is the base prover volume of run i for a bidirectional calibration.

(23) $$BPV=\frac{1}{N}\sum _{i=1}^{N}BP{V}_i$$

(24) $$\begin{array}{r}BPV=\frac{1}{N}\sum _{i=1}^{N}BP{V}_{2,i}\end{array}$$

Based on the above, the flowcharts of Figs. 3 and 4 present the workflow proposed by the current study for pipe prover waterdraw calibration metrics computation.

Fig. 3. Workflow for unidirectional pipe prover waterdraw calibration metrics computation.

Fig. 4. Workflow for bidirectional pipe prover waterdraw calibration metrics computation.

Pipe Prover Waterdraw Calibration Assessment Metric

The main metric used for calibration assessment is the range percentage. The range percentage of a run $R_i$, $R_{2,i}$ and $R_{2,i}^{\prime }$ of a pipe prover waterdraw calibration is defined as the ratios of the ranges of the run base prover volume datasets to their minimums, that is (25)–(27):

(25) $$\begin{array}{r}{R}_i=100\times \frac{Max{\left\{BP{V}_i\right\}}_{1\le i\le N}-Min{\left\{BP{V}_i\right\}}_{1\le i\le N}}{Min{\left\{BP{V}_i\right\}}_{1\le i\le N}}\end{array}$$

(26) $$\begin{array}{r}{R}_{2,i}=100\times \frac{Max{\left\{BP{V}_i\right\}}_{1\le i\le N}-Min{\left\{BP{V}_i\right\}}_{1\le i\le N}}{Min{\left\{BP{V}_i\right\}}_{1\le i\le N}}\end{array}$$

where N is the number of runs of the calibration, $BP{V}_i$ is the base prover volume of run (or forward pass) i and $BP{V}_i^{\prime }$ are the base prover volume of the backward pass of the run i for a bidirectional calibration.

(27) $$\begin{array}{r}{R}_{2,i}^{\prime }=100\times \frac{Max{\left\{BP{V}_i^{\prime }\right\}}_{1\le i\le N}-Min{\left\{BP{V}_i^{\prime }\right\}}_{1\le i\le N}}{Min{\left\{BP{V}_i^{\prime }\right\}}_{1\le i\le N}}\end{array}$$

Results and Discussion

Python Codes

The case studies have been carried out with Python progamming on Jupiter Notebook through Anaconda. The codes help to compute the different metrics required for pipe prover base volume determination. Figs. 5–8 show the codes written for that purpose. For each case study, the code has been split in two figures to be able to be displayed in the paper. They must be run one after one another.

Case Study Provers Calibration Results

Unidirectional Pipe Prover Calibration Metrics Computation

As highlighted in materials section, the pipe prover has been calibrated through three runs with two test measure. Table IX shows the results for adjusted measure volumes and different corrections for effects of temperature for all measure fills. The results for pressure corrections computation and base prover volume calculation for the three runs are underlined in Table X.

It can be noted through the results of Table X that the base prover volume ranges between 4508.514 dm³ and 4510.719 dm³. The average of the base prover volumes is 4509.4 dm³.

The repeatability of the calibration is 0.00049. Therefore, one can conclude that calibration operations have succeeded, and the calibrated base prover volume is 4509.4 dm³ at the base temperature of 20°C.

Bidirectional Pipe Prover Calibration Metrics Computation

As far as the bidirectional pipe calibration is concerned, the prover has been calibrated through three runs with two test measures, each run composed of two passes, one forward and one backward. The results for adjusted measure volumes and corrections for the effects of temperature for all measure fills are summarized in Table XII. Table XI shows the pressure corrections gotten for the three runs.

The maximum and minimum base prover volume for the forward passes are 3009.23 dm³ and 3009.707 dm³, respectively, while the maximum base prover volume for the backward passes ranges between 3009.329 dm³ and 3010.079 dm³. The average of the base prover volumes is 4509.4 dm³. The round-trip base prover volumes are 6019.542 dm³, 6018.955 dm³, and 6019.036 dm³ with an average of 6019.178 dm³.

The repeatabilities of the calibration are 0.00016 and 0.00025 for forward and backward passes. As a result, the calibrated round-trip base prover volume is 6019.178 dm³ at the base temperature of 20°C.

Conclusions and Recommendation

Pipe prover is one of the key components of volume metering station in oil and gas industry. Precise computation of its calibration metrics is a challenge for metrology technicians and petroleum inspectors. This study has proposed a comprehensive workflow for the computation of pipe prover waterdraw calibration metrics which will help the national metrology agencies employees, calibration operations technicians and inspectors in the performance of their activities.

The results of the two real world case studies prove that the objectives have been achieved since the application of the methodology has served to validate the pipe prover waterdraw calibrations. Indeed, the computed repeatabilities of the pipe prover unidirectional and bidirectional calibration operations gotten are in the acceptance range. The computed base prover volume of the pipe prover calibrated through a unidirectional calibration is 4509.4 dm³ at the base temperature of 20°C when the calibrated round-trip base prover volume for the second is 6019.178 dm³ at 20°C.

We recommend proposing a similar technique for flow meter calibrations for easy and quick validation of such operations.

Appendix

Table IX. Unidirectional Pipe Prover Calibration Metric Computation Results

Run	Measure	Fill	Reading (mm)	TP (°C)	TM (°C)	BMVa (dm3)	CTDW	CTSP	CTSM	CCTS
1	M1	F1	165.8	30.29	30	1000.351	1.000088	1.000035	1.001098	1.001063
1	M1	F2	167.7	30.15	30.05	1000.440	1.000030	1.000035	1.001103	1.001068
1	M1	F3	172.1	30.08	30.95	1000.647	0.999732	1.000035	1.001202	1.001167
1	M2	F1	170	30.72	30.92	500.992	0.999938	1.000037	1.000983	1.000946
1	M2	F2	168	30.06	30	500.894	1.000018	1.000035	1.000900	1.000865
1	M2	F3	167	30.12	30	500.845	1.000036	1.000035	1.000900	1.000865
2	M1	F1	172	29.22	28.68	1000.642	1.000159	1.000032	1.000953	1.000921
2	M1	F2	178.35	30.65	30	1000.940	1.000199	1.000037	1.001098	1.001061
2	M1	F3	175.45	29.42	29.98	1000.804	0.999832	1.000032	1.001096	1.001064
2	M2	F1	160.2	30.45	29.12	500.513	1.000401	1.000036	1.000821	1.000785
2	M2	F2	160.4	29.12	28.12	500.523	1.000291	1.000031	1.000731	1.000700
2	M2	F3	159.4	29.45	28.96	500.474	1.000145	1.000032	1.000806	1.000774
3	M1	F1	174.9	31.84	30.71	1000.778	1.000355	1.000041	1.001176	1.001135
3	M1	F2	177.8	31.93	30.84	1000.914	1.000344	1.000041	1.001190	1.001149
3	M1	F3	177.5	31.9	30.83	1000.900	1.000337	1.000041	1.001189	1.001148
3	M2	F1	163.8	31.92	30.81	500.689	1.000350	1.000041	1.000973	1.000932
3	M2	F2	167.5	31.91	30.86	500.870	1.000331	1.000041	1.000977	1.000936
3	M2	F3	160.7	31.89	30.85	500.538	1.000328	1.000041	1.000976	1.000935

Table X. Unidirectional Pipe Prover Calibration Metric Computation Results (Ctd)

Run	PP (kPa)	WD	CPS	CPW	CCP	BPV (dm3)
1	100.95	4508.825	1.000022	1.000047	1.000069	4508.514
2	112	4509.315	1.000025	1.000052	1.000077	4508.968
3	105.87	4511.044	1.000023	1.000049	1.000072	4510.719

Table XI. Bidirectional Pipe Prover Calibration Metric Computation Results

Pass	PP (kPa)	WD	CPS	CPW	CCP	BPV (dm3)
Forward 1	102.71	3009.674	1.000022	1.000048	1.000070	3009.463
Reverse 1	95.7	3009.426	1.000021	1.000044	1.000065	3009.230
Forward 2	100.75	3009.915	1.000022	1.000047	1.000069	3009.707
Reverse 2	115.31	3010.317	1.000025	1.000054	1.000079	3010.079
Forward 3	117.3	3009.963	1.000025	1.000054	1.000079	3009.725
Reverse 3	115.41	3009.567	1.000025	1.000054	1.000079	3009.329

Table XII. Bidirectional Pipe Prover Calibration Metric Computation Results (Ctd)

Pass	Measure	Fill	Reading (mm)	TP (°C)	TM (°C)	BMVa (dm3)	CTDW	CTSP	CTSM	CCTS
Forward 1	M1	F1	175.8	31.29	31	500.705	1.000091	1.000037	1.000990	1.000953
Forward 1	M1	F2	177.7	31.15	31.05	500.798	1.000031	1.000036	1.000994	1.000958
Forward 1	M1	F3	182.1	31.08	30.95	501.014	1.000041	1.000036	1.000986	1.000950
Forward 1	M2	F1	180	30.72	30.92	501.481	0.999938	1.000035	1.000983	1.000948
Forward 1	M2	F2	178	31.06	31	501.383	1.000019	1.000036	1.000990	1.000954
Forward 1	M2	F3	177	31.12	31	501.334	1.000038	1.000036	1.000990	1.000954
Reverse 1	M1	F1	172.8	29.35	29	500.557	1.000104	1.000030	1.000810	1.000780
Reverse 1	M1	F2	180.28	30.15	30	500.925	1.000046	1.000033	1.000900	1.000867
Reverse 1	M1	F3	183.7	29.12	28.88	501.093	1.000071	1.000030	1.000799	1.000769
Reverse 1	M2	F1	178.85	29.15	30.2	501.424	0.999685	1.000030	1.000918	1.000888
Reverse 1	M2	F2	181	30.24	29.45	501.530	1.000239	1.000033	1.000851	1.000818
Reverse 1	M2	F3	188.25	29.75	30.08	501.884	0.999900	1.000032	1.000907	1.000875
Forward 2	M1	F1	182	30.22	29.68	501.010	1.000164	1.000033	1.000871	1.000838
Forward 2	M1	F2	188.25	30.65	30	501.317	1.000199	1.000034	1.000900	1.000866
Forward 2	M1	F3	175	30.42	28.98	500.665	1.000433	1.000034	1.000808	1.000774
Forward 2	M2	F1	169.2	30.45	30.12	500.953	1.000101	1.000034	1.000911	1.000877
Forward 2	M2	F2	169.4	30.12	30.12	500.963	1.000000	1.000033	1.000911	1.000878
Forward 2	M2	F3	169.4	30.45	28.96	500.963	1.000448	1.000034	1.000806	1.000772
Reverse 2	M1	F1	175.9	30.67	30.69	500.710	0.999994	1.000035	1.000962	1.000927
Reverse 2	M1	F2	172.23	30.75	30.65	500.529	1.000031	1.000035	1.000959	1.000924
Reverse 2	M1	F3	182	30.72	30.64	501.010	1.000025	1.000035	1.000958	1.000923
Reverse 2	M2	F1	182	30.73	29	501.578	1.000523	1.000035	1.000810	1.000775
Reverse 2	M2	F2	178.5	30.73	29	501.407	1.000523	1.000035	1.000810	1.000775
Reverse 2	M2	F3	176	30.74	30.66	501.285	1.000025	1.000035	1.000959	1.000924
Forward 3	M1	F1	174.9	31.84	30.71	500.660	1.000355	1.000038	1.000964	1.000926
Forward 3	M1	F2	177.8	31.93	30.84	500.803	1.000344	1.000039	1.000976	1.000937
Forward 3	M1	F3	177.5	31.9	30.83	500.788	1.000337	1.000039	1.000975	1.000936
Forward 3	M2	F1	173.8	31.92	30.81	501.178	1.000350	1.000039	1.000973	1.000934
Forward 3	M2	F3	180.7	31.89	30.85	501.515	1.000328	1.000038	1.000976	1.000938
Reverse 3	M1	F1	178.9	30.85	30.81	500.857	1.000012	1.000035	1.000973	1.000938
Reverse 3	M1	F2	175.8	30.83	30.77	500.705	1.000019	1.000035	1.000969	1.000934
Reverse 3	M1	F3	179.5	30.8	30.74	500.887	1.000019	1.000035	1.000967	1.000932
Reverse 3	M2	F1	173.8	30.87	30.81	501.178	1.000019	1.000035	1.000973	1.000938
Reverse 3	M2	F2	180.5	30.8	30.76	501.505	1.000012	1.000035	1.000968	1.000933
Reverse 3	M2	F3	180.5	30.81	30.75	501.505	1.000019	1.000035	1.000968	1.000933

Fig. 5. Python code for unidirectional pipe prover calibration metric computation.

Fig. 6. Python code for unidirectional pipe prover calibration metric computation (Ctd).

Fig. 7. Python code for bidirectional pipe prover calibration metric computation.

Fig. 8. Python code for bidirectional pipe prover calibration metric computation (Ctd).