PATENT ABSTRACT
An apparatus and method to determine fractions of various phases in a multiphase fluid. The apparatus includes main body including an interior configured receive a multiphase fluid and an exterior. The apparatus senses fluid pressure of multiphase fluid received in the interior and senses a fluid temperature of the multiphase fluid. The apparatus transmits an ultrasonic wave into the fluid and detects the transmitted wave to determine its velocity and attenuation. The apparatus may adjust the determined velocity and attenuation based on the temperature and pressure of the fluid to compensate for a difference between the sensed temperature and pressure and a standard temperature and pressure. The apparatus determines a gas fraction, water fraction, and a non-water fluid fraction of the multiphase fluid based on the sensed fluid pressure, the sensed fluid temperature, and the velocity and attenuation of the ultrasonic wave in the multiphase fluid.

PATENT DESCRIPTION
CROSS REFERENCES TO RELATED APPLICATIONS 
     This application is a divisional of U.S. application Ser. No. 11/829,361, filed Jul. 27, 2007, the entire content of which is incorporated herein by reference. 
    
    
     BACKGROUND 
     1. Field of Invention 
     This invention is directed to an ultrasound multiphase meter and method to detect and quantify the phase fractions and flow rates in a multiphase flow stream. 
     2. Description of the Related Art 
     Multiphase meters have attracted the attention of the oil production industry because of their accuracy and cost savings as opposed to analyzing discrete samples of multiphase fluid to determine fractions of oil, water, and gas. Development of accurate and compact multiphase metering devices that can be installed at well heads in remote onshore fields and unmanned offshore platforms continues to be a technological challenge. Data acquired by such devices may be used in reservoir management and production allocation inasmuch as the particular volumetric fractions of oil, water, and gas can be determined. Therefore this data is highly valuable. However, conventional devices have had difficulty in producing an accurate measurement of various properties of the monitored multiphase while withstanding the harsh environments in which such devices are typically installed. 
     Multiple array ultrasound devices that may acquire real time spatial data from volumetric specimens have been developed for medical applications. Advanced data and signal processing systems and display technologies have been developed for aerospace and the defense industries. These technologies are unsuitable to quantify phase fractions and flow rates of oil, gas, and water in a multiphase flow stream in an oil pipeline. In particular, the field conditions of the oil production environment are extremely harsh because of high pressure and temperatures and because of abrasive particles such as sand. Furthermore, the presence of gas bubbles in the flow streams as well as effects due to high temperature and pressure in the pipeline require specialized models to obtain accurate data collection and analysis. 
     SUMMARY OF THE INVENTION 
     According to one aspect of the invention, there is provided an apparatus configured to determine fractions of various phases in a multiphase fluid. The apparatus typically includes a main body including an interior configured to receive a multiphase fluid and an exterior, a pressure sensor configured to sense fluid pressure of multiphase fluid received in the interior, and a temperature sensor configured to sense a fluid temperature of the multiphase fluid. The apparatus also includes an ultrasonic transducer configured to transmit an ultrasonic wave in the multiphase fluid and an ultrasonic sensor configured to detect the ultrasonic wave transmitted by the ultrasonic transducer. A computer determines a gas fraction, water fraction, and a non-water fluid fraction of the multiphase fluid based on the sensed fluid pressure, the sensed fluid temperature, and at least one characteristic of the detected ultrasonic wave in the multiphase fluid. The at least one characteristic can be at least one of the velocity of the ultrasonic wave and the attenuation of the ultrasonic wave experienced as the wave travels from the ultrasonic transducer to the ultrasonic sensor. 
     According to another aspect of the invention, there is provided a method of determining fractions of various phases in a multiphase fluid. The method includes sensing a pressure of the multiphase fluid and sensing a temperature of the multiphase fluid. The method also includes transmitting an ultrasonic wave in the multiphase fluid, detecting the ultrasonic wave transmitted in the multiphase fluid, and determining a gas fraction, water fraction, and non-water fluid fraction of the multiphase fluid based on the sensed pressure, sensed temperature, and at least one characteristic of the detected ultrasonic wave. The at least one characteristic can be at least one of the velocity of the transmitted ultrasonic wave when it is detected and the attenuation of the ultrasonic wave when it is detected. 
     The apparatus can include a flow meter configured to measure an amount of fluid flowing in the apparatus. Based on this amount and the data determined regarding the water and gas fractions, the fraction of remaining fluid in the multiphase fluid can be determined via simple subtraction of the gas and water fractions from the total. 
     A multiphase meter according to one aspect of the invention includes a positive displacement volume flow meter, an ultrasonic phase fraction detector and a Doppler ultrasound detector. This multiphase meter includes two compact multi-transducer flow sensors designed to determine the water content and gas fraction in the oil mixture as well as to measure the liquid and gas phase velocities. Due to the harsh well head environment the ultrasound phase fraction detector is preferably made with appropriate thermal and mechanical strength. 
     According to another aspect of the invention, there is provided a computer readable medium containing program instructions for execution on a processor, which when executed by the processor, results in performance of certain steps including obtaining data relating to a pressure of a multiphase fluid and obtaining data relating to a temperature of the multiphase fluid. The steps also include receiving data regarding an ultrasonic wave transmitted in the multiphase fluid and determining a gas fraction, water fraction, and non-water fluid fraction of the multiphase fluid based on the data relating the pressure of the multiphase fluid, the data relating to the temperature of the multiphase fluid, and the data regarding the ultrasonic wave transmitted in the multiphase fluid. 
     According to another aspect of the invention, there is provided an apparatus configured to determine fractions of various phases in a multiphase fluid. The apparatus includes a main body including an interior configured receive a multiphase fluid and an exterior, a pressure sensor configured to sense fluid pressure of multiphase fluid received in the interior, and a temperature sensor configured to sense a fluid temperature of the multiphase fluid. The apparatus also includes an ultrasonic transducer configured to transmit an ultrasonic wave in the multiphase fluid and an ultrasonic sensor configured to detect the ultrasonic wave transmitted by the ultrasonic transducer. This aspect of the invention also provides means for determining a gas fraction, water fraction, and a non-water fluid fraction of the multiphase fluid based on the sensed fluid pressure, the sensed fluid temperature, and at least one characteristic of the detected ultrasonic wave in the multiphase fluid. The at least one characteristic can be at least one of the velocity of the ultrasonic wave and the attenuation of the ultrasonic wave experienced as the wave travels from the ultrasonic transducer to the ultrasonic sensor. 
     One application of the invention is to detect phase fractions of oil (non-water fluid), gas, and water in an oil pipeline, where attenuation of an ultrasound signal is used to determine the fraction, by volume, of gas bubbles in an oil/water/gas mixture, and velocity of a transmitted ultrasonic wave is used to determine the water fraction of the mixture. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A more complete appreciation of the invention and many of the attendant advantages thereof will be readily obtained as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings, in which like reference numerals refer to identical or corresponding parts throughout the several views, and in which: 
         FIG. 1A  is a block diagram schematically showing one aspect of the present invention; 
         FIG. 1B  is a schematic including cross sections of a pipeline with arrays, transmitters, pressure, and temperature sensors of an exemplary embodiment of the invention installed; 
         FIG. 2  is a block diagram of a computer system and its main components; 
         FIGS. 3A and 3B  are a graphical user interface for data analysis including a user interface for data analysis according to an example embodiment of the invention; 
         FIG. 4A  is a detail of the graphical user interface of  FIGS. 3A and 3B  including data acquisition parameters and signal generation and monitoring according to an example embodiment of the invention; 
         FIG. 4B  is a detail of the graphical user interface of  FIGS. 3A and 3B  including temperature pressure, time and transmitter amplitude monitoring and adjustment according to an example embodiment of the invention; 
         FIG. 5A  is a detail of the graphical user interface of  FIGS. 3A and 3B  including peak detection and amplitude and velocity calculation according to an example embodiment of the invention; 
         FIG. 5B  is a detail of the graphical user interface of  FIGS. 3A and 3B  including a data save interface according to an example embodiment of the invention; 
         FIG. 6  is a graph showing the dependence of the ultrasound pulse velocity versus temperature in water; 
         FIG. 7  is a graph showing the pressure dependence of the ultrasound wave velocity; 
         FIG. 8  is a graph showing the temperature and pressure dependence of ultrasound wave velocity; 
         FIG. 9  is a graph showing experimental data for ultrasound velocity versus water fraction measurements; 
         FIG. 10  is a graph showing experimental data for water fractions measurements corrected for pressure and temperature effects; 
         FIG. 11  is a graph showing experimental data and theoretical curve for water cut versus ultrasound wave velocity; 
         FIG. 12  is a graph showing peak detection for calculation of the ultrasound signal amplitude; 
         FIG. 13  is a graph showing measured ultrasound wave amplitudes versus gas volumetric fraction; 
         FIG. 14  is a graph showing the ultrasound transmission factor versus gas volumetric fraction; 
         FIG. 15  is a graph showing the ultrasound transmission factor versus gas volumetric fraction in the 10 to 50% GVF range; 
         FIG. 16  is a graph showing pressure compensated experimental data for the ultrasound transmission factor versus gas volumetric fraction in the 10 to 50% range; 
         FIG. 17  is a graph showing pressure and temperature compensated experimental data for the ultrasound transmission factor versus gas volumetric fraction in the 10 to 50% range; and 
         FIG. 18  is a graph showing gas volumetric fraction in the 10 to 50% range versus ultrasound transmission factor. 
     
    
    
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
       FIG. 1A  is a block diagram showing the functional relationship between a computer system  201  controlling and monitoring an Ultrasound Phase Fraction Detector  1  (UPFD  1 ) and the components within the UPFD  1 . As shown in  FIG. 1A , the computer system is connected to a sensor array  6 , transducer(s)  4 , pressure sensor P 1 , temperature sensor T 1 , and flow meter F 1  (optional), all of which are connected to the main body  8  or configured to sense information from a main body  8 . The main body  8  receives a multiphase fluid, and the computer system  201  receives information sent by the sensor array  6 , pressure sensor P 1 , temperature sensor T 1 , and flow meter F 1  (optional) relating to the multiphase fluid. The computer system  201  then analyzes the information to determine fraction of the multiphase fluid that is water, the fraction that is gas, and the fraction that is a non-water fluid, such as diesel fuel or oil. 
       FIG. 1B  shows one example of a UPFD  1  according to the present invention. In order to achieve the desired thermal and mechanical strength for industrial use, the body of the UPFD  1  of the embodiment shown in  FIG. 1B  is typically manufactured from materials that have similar mechanical and thermal characteristics to standard oil pipes. The main body  8  of the UPFD  1  may be formed from a segment of 316 stainless steel 4″ pipe configured to withstand 300 psi and temperatures in excess of 500° F. Fittings  7  are attached, for example, by welding to the pipe in order to position and fix the ultrasound elements. The fittings are preferably made from 316 stainless steel, which allows welding to the pipe, or from brass, but other materials may be used. This construction enhances thermal and mechanical strength while providing good electrical conductivity for corresponding electrical grounding. Other materials such as 304 stainless steel, black pipe, or polymer pipes may be used, depending on the conditions and costs of a particular installation. 
     As further shown in  FIG. 1B , sensitive elements are placed aligned with the inside wall of the pipe. In this configuration the sensitive elements are protected by a relatively thin metallic layer  9 , a layer of TEFLON (FRP OR PTFE)  5 , or other suitable material that does not excessively reduce the ability to capture the ultrasound signal. The thickness of the TEFLON layer may be, for example, 1.6 mm. Appropriate gaskets and other sealing compounds can also be used in the construction to facilitate sealing of the sensitive elements. 
     A plurality of piezoelectric (PZT) elements are embedded in the TEFLON layer as an ultrasonic sensor array  6 . The sensor array  6  is typically mechanically fixed in a fitting  7 , which may be made from stainless steel or brass. Use of these materials in the fitting  7  allows for an operating temperature range from −350° to 500° F. and pressures in excess of 300 psi. The fitting  7  can be a threaded connection or other such mechanical coupling. Alternatively, the sensor array  6  can be bonded to the wall of the main body  8  with an adhesive. 
     Ultrasound transducers  4  attached to the main body  8  are used as pulse transmitters. The ultrasound transducers  4  are preferably suitable for use in rugged industrial environments. The ultrasound transducers also preferably have an acoustic impedance matching most metals. If desired, an ultrasound pulse can be transmitted through a small thickness of the pipe, thus protecting the ultrasound sensor from the fluid inside the pipe. Transmitting through the wall of the pipe facilitates sealing. The ultrasound transducer  4  may be one such as those available from VALPEYFISHER, for example, model no. CS0.508HP-TC, which have a 1″ active element with a 500 KHz frequency. 
     One example of the invention uses two pairs of transmitter and receiver arrays, one pair orthogonal to the other, so that the signal spans the entire cross sectional area of the pipe. This arrangement can also be used for flow velocity measurement by cross correlation of the signals. 
     As will be discussed, pressure and temperature are important parameters in determining the phase fractions and flow rates of oil, gas, and water in a multiphase flow stream. The propagation velocity of an ultrasound pulse depends on the temperature and pressure of the oil-water mixture  3  in which the pulse travels. Thus, UPFD  1  is provided with a pressure sensor P 1  and temperature sensor T 1  in order to allow these properties to be taken into account in the determination of the phase fractions. 
     A STELLAR TECHNOLOGY INC. pressure sensor model no. GT1800-500G-000 with the following characteristics was found suitable as a pressure sensor P 1  in the above installation. 
     
       
         
               
               
             
           
               
                   
               
             
             
               
                 Standard pressure range: 
                 0-500 PSIg 
               
               
                 Proof pressure: 
                 750 PSIg 
               
               
                 Burst pressure: 
                 1000 PSIg 
               
               
                 Operating media: 
                 fluids and gases compatible with 17-4PH 
               
               
                   
                 stainless steel 
               
               
                 Resolution: 
                 infinite (analog) 
               
               
                 Compensated temp. range: 
                 +30° F. to +130° F. 
               
               
                 Operating temp. range: 
                 −40° F. to +185° F. 
               
               
                 Enclosure: 
                 Body and pressure cavity of stainless 
               
               
                   
                 steel, environmentally sealed 
               
               
                   
               
             
          
         
       
     
     As a temperature sensor T 1 , a series T750-250 temperature sensor, also produced by STELLAR TECHNOLOGY INC., with the following characteristics was found suitable. 
     
       
         
               
               
             
           
               
                   
               
             
             
               
                 Operating temp. range: 
                 −65° F. to +250° F. 
               
               
                 Burst pressure: 
                 30,000 PSI minimum 
               
               
                 Operating media: 
                 fluids and gases compatible with 17-4PH 
               
               
                   
                 stainless steel 
               
               
                 Resolution: 
                 infinite (analog) 
               
               
                 Enclosure: 
                 Body and pressure cavity of stainless 
               
               
                   
                 steel, hermetically sealed 
               
               
                   
               
             
          
         
       
     
     A flow meter F 1  shown on the left side of the main body  8  in  FIG. 1B  is typically provided to determine the total flow through the UPFD  1 . Various types of flow meters may be used for this application. For example, paddle wheel flow meters may be used. 
     A controller or computer system  201  (shown in  FIG. 2 ) uses a combination of data from the pressure sensor P 1 , flow meter F 1 , temperature sensor, T 1 , and sensor array  6  to detect and quantify phase fractions and flow rates of oil, gas, and water in a multiphase flow stream inside the main body  8 . 
       FIG. 2  illustrates a typical computer system  201  upon which an embodiment of the present invention may be implemented. All or merely selected processing components of the embodiments discussed herein may by implemented. The computer system  201  includes a bus  202  and a processor  203 . The computer system  201  also includes a main memory  204 , such as a random access memory (RAM). The computer system  201  further includes a read only memory (ROM)  205  or other static storage device (e.g., programmable ROM (PROM), erasable PROM (EPROM), and electrically erasable PROM (EEPROM)) coupled to the bus  202 . 
     The computer system  201  also includes a disk controller  206  to control one or more storage devices such as a magnetic hard disk  207  and a removable media drive  208  (e.g., floppy disk drive). 
     The computer system  201  may also include a display controller  209  coupled to the bus  202  to control a display  210  for displaying information to a computer user. The computer system may include input devices, such as a keyboard  211  and a pointing device  231 . 
     The computer system  201  performs a portion or all of the processing steps of the invention by virtue of the processor  203  executing one or more sequences of one or more instructions contained in a memory, such as the main memory  204 . Such instructions may be read into the main memory  204  from another computer readable medium, such as a hard disk  207  or a removable media drive  208 . 
     Software is stored on any one or on a combination of computer readable media to control the computer system  201 , to drive a device or devices for implementing the invention, and for enabling the computer system  201  to interact with a human user. 
     The term “computer readable medium” as used herein refers to any medium that participates in providing instructions to the processor  203  for execution. A computer readable medium may take many forms, including but not limited to, non-volatile media, volatile media, and transmission media. Non-volatile media includes, for example, optical, magnetic disks, and magneto-optical disks, such as the hard disk  207  or the removable media drive  208 . Volatile media includes dynamic memory, such as the main memory  204 . Transmission media includes coaxial cables, copper wire and fiber optics, including the wires that make up the bus  202 . 
     Various forms of computer readable media may be involved in carrying out one or more sequences of one or more instructions to processor  203  for execution. For example, the instructions may initially be carried on a magnetic disk of a remote computer. 
     The computer system  201  may also include a communication interface  213  coupled to the bus  202 . The communication interface  213  provides a two-way data communication coupling to a network link  214  that is connected to, for example, a local area network (LAN)  215 , or to another communications network  216  such as the Internet. Wireless links may also be implemented for such connection. The communication interface  213  sends and receives electrical, electromagnetic or optical signals that carry digital data streams representing various types of information. 
     The network link  214  typically provides data communication through one or more networks to other data devices. For example, the network link  214  may provide a connection to another computer through a local network  215  (e.g., a LAN) or through equipment operated by a service provider, which provides communication services through a communications network  216 . The local network  214  and the communications network  216  use, for example, electrical, electromagnetic, or optical signals that carry digital data streams, and the associated physical layer. The computer system  201  can transmit and receive data, including program code, through the network(s)  215  and  216 , the network link  214  and the communication interface  213 . Moreover, the network link  214  may provide a connection through a LAN  215  to a mobile device  217  such as a personal digital assistant (PDA) laptop computer, or cellular telephone. 
     The functionality of the UPFD  1  shown in  FIGS. 1A and 1B  may be completely automated. A signal received from the sensor array  6  is monitored and parameters of a pulse applied to the ultrasound transducers  4  are adjusted accordingly. 
     A first set of parameters that can be adjusted are: delay time from the application of the pulse at the transmitter, acquire time, and the number of data sets to be averaged for noise suppression purposes. Also, the user may monitor the signal received at each ultrasound array channel in real time. Systematic background noise can be monitored and automatically subtracted from the analyzed waveform.  FIGS. 3A ,  3 B, and  4 A show typical displays of a user interface  14  displaying the amplitude vs. time, acquire time, and start delay time for one set of parameters. 
     Temperature and pressure measurements from the area around the transducers  4  are digitized and are monitored in real time as shown in the user interface  14  depicted in  FIG. 4B . Also, the current time and the voltage applied to the ultrasound transmitters are recorded. A transmitter voltage can be adjusted by the operator (mostly for setup and maintenance) or, when the feedback button is activated, directly by the system. When large amounts of gas bubbles are present inside a volume monitored by the system, the attenuation of the transmitted signal is large, thus requiring a large pulse voltage at the transmitters. When only a small fraction of gas bubbles is present, the amplitude of the pulse applied to the ultrasound transducers  4  is preferably reduced in order to prevent clipping (truncation of the top part of the wave of the signal). This increase and decrease of the pulse amplitude, when done automatically, effectively increases the dynamic range of the flowmeter without operator intervention. Also, a mathematical model, discussed hereafter, to account for high pressure and temperature effect is typically incorporated in the system. 
     Data received by the pressure sensor P 1 , temperature sensor T 1 , and flow meter F 1  is automatically analyzed and the results are stored on, for example, the computer hard drive at programmable intervals of time. These results can then be retrieved from a remote location by a simple network connection. The arrival of the ultrasound pulse is detected at each element of the sensor array  6  (shown in  FIGS. 1A and 1B ) and then the positive and the negative peaks in the waveform are detected. The time lapse between the positive and the negative peaks is then calculated, and if the time lapse is in the acceptable range (determined by the frequency of the pulse) the pulse amplitude and the time of flight are calculated. 
     A “Good Amplitude” test is performed by checking for signal clipping and for time differential between the positive and the negative peaks. No clipping means that there are no flat hysteresis portions on the top and bottom portions of the signal where it is maximum or minimum. The time differential means that the minimum and maximum are measured during the same cycle. 
     If a “Good Amplitude” test is made, and if the signal was not clipped, see “Clipping” indicator in  FIG. 4B , a state vector is formed and saved. The state vector includes the current time, delay acquisition time, the acquisition time, number of waveforms averaged, pressure, temperature, transmitter amplitude, the two transducer average velocities, total average velocity, the two transducer amplitudes, and total average amplitude. 
     Two save data modes are implemented as shown in  FIG. 5B . The first mode, “Save Data”, saves only the state vector at programmed time intervals. The second mode, labeled “Save Raw Data”, saves not only the state vector but the actual signals received at each transducer. While the first “Save Data” mode is designed for the routine functionality of the flowmeter, the second mode is designed for calibration, testing, and debugging. 
     Experimental work shows that the propagation velocity of the ultrasound pulse depends significantly on the temperature of oil-water mixture. Temperature compensation equations have been established which are effective in the range of 20-45° C. Since the temperature variation on an actual oil field can be even larger, the domain of the equations was extended to span from 0° C. to 100° C. 
     Nonlinear behavior is readily apparent for such a large temperature interval as shown in  FIG. 6 . In general it is accepted that the speed of sound (SOS) in liquids increases with temperature. Nevertheless, for temperatures larger than 75° C. this behavior changes significantly as is shown in  FIG. 6 , and the SOS actually decreases as temperature increases. The ultrasound velocity has a large variation, approx 155 m/s, which is larger than the difference in velocity between the two media under consideration (water and oil). Thus, the effective compensation of the temperature effect is beneficial for precise measurements of water cuts. 
     In order to quantify the temperature effect on the wave velocity, a fifth order polynomial regression was performed on the experimental data, obtaining:
 
 V   w ( T )= k   0   +k   1    T+k   2   T   2   +k   3    T   3+    k   4    T   4+    k   5    T   5    Eq. 1
 
     Where V w (T) is the ultrasound wave velocity, T is the temperature in ° C., and
         k 0 =1402.4±0.0161   k 1 =5.0356±0.00334   k 2 =−0.057869±0.000212   k 3 =0.00032952±5.45e−06   k 4 =−1.4563e−06±6.07e−08   k 5 =3.2042e−09±2.44e−10
 
are the calculated regression coefficients. See  FIG. 6 . The chi-squared test, which indicates how close a mathematical expression represents a data set, provides a value of χ 2 =0.079760, and reflects excellent agreement of the model with the experimental data. The corresponding temperature dependence curve was also plotted in  FIG. 6  as a solid line. The approximation error is in general less than 0.25 m/s which corresponds to a percentage error of only 0.015%.
       

     This mathematical expression can be used to correlate the ultrasound wave velocity at any desired temperature with the velocity measured at a known experimental temperature, i.e., the expression adjusts the value of the ultrasound wave velocity to compensate for the change of speed of sound due to temperature differences. Such an adjustment effectively eliminates the undesired temperature effect on wave velocity measurements, thus enhancing the accuracy of the water fraction estimation. In order to obtain an expression useful for temperature correction two different temperatures are considered: T 0  (the standard temperature) and T 1  (the temperature at which the experiments are taking place). The wave velocity for the two temperatures are written as,
 
 V   w ( T   0 )= k   0   +k   1    T   0   +k   2   T   0   2   +k   3    T   0   3   +k   4    T   0   4   +k   5    T   0   5  
 
 V   w ( T   1 )= k   0   +k   1    T   1   +k   2   T   1   2   +k   3    T   1   3   +k   4    T   1   4   +k   5    T   1   5  
 
     By subtracting the two equations the following equation is obtained,
 
 V   w ( T   0 )− V   w ( T   1 )= k   1  ( T   0   −T   1 )+ k   2  ( T   0   2   −T   1   2 )+ k   3  ( T   0   3   −T   1   3 )+ k   4  ( T   0   4   −T   1   4 )+ k   5  ( T   0   5   −T   1   5 )
 
and solving for V w (T 0 ),
 
 V   w ( T   0 )= V   w ( T   1 )+ k   1  ( T   0   −T   1 )+ k   2 ( T   0   2   −T   1   2 )+ k   3  ( T   0   3   −T   1   3 )+ k   4  (T 0   4   −T   1   4 )+k 5  ( T   0   5   −T   1   5 )
 
     Since the standard temperature is a constant, the terms containing T 0  may be grouped as the value K 0  as follows,
 
 V   w ( T   0 )= V   w ( T   1 )− k   1    T   1   −k   2    T   1   2    k   3    T   1   3   −k   4    T   1   4   −k   5    T   1   5   +K   0    Eq. 2
 
with
 
 K   0   =k   1    T   0   +k   2    T   0   2    k   3    T   0   3   +k   4    T   0   4   +k   5    T   0   5 .
 
     If the standard temperature is chosen as T 0 =40° C., then K 0  becomes K 0 =126.508. 
     Equation 2 can now be used to calculate the wave velocity at the standard temperature T 0  when the experimental data were measured at a different temperature T 1 . 
     Another important factor that may influence the ultrasound velocity is the pressure of the liquid.  FIG. 7  shows the ultrasound pulse velocity versus pressure at two different temperatures: 20° C. and 50° C. It is apparent that the velocity dependence on pressure is relatively linear for a large range of pressures (0 to 500 psia), that covers the current practical oilfield working values. Moreover, the slope of the regression line is similar even for large variations in temperatures. Even though the pressure effect is not as strong as that of temperature, it is significant and is preferably taken into account when obtaining precise measurements of the water fraction. 
     In practice, the combined effect of both pressure and temperature are preferably addressed in the determination of the phase fractions and flow rates of oil, gas, and water in a multiphase flow stream. The three curves presented in  FIG. 8  represent the temperature dependence of the wave velocity for pressures varying from 0 to 200 psig. The temperature range presented in the figure was restricted between 40 and 100° C. as being particularly relevant for the working conditions of an oilfield. Nevertheless the range of the theoretical expressions and experimental data extend from 0 to 100° C. 
     In order to compensate for the effect of pressure, an additional factor is added to equation 1. The equation becomes,
 
 V   w ( T )=k 0   +k   1    T+k   2   T   2   +k   3    T   3+    k   4    T   4   +k   5    T   5   +k   p    P    Eq. 3
 
where P is the pressure measured in PSIg and the pressure coefficient k p =0.014 is determined experimentally.
 
     The excellent agreement of Eq. 3 with experimental data is apparent in  FIG. 8  where the dots represent the experimental data and the continuous lines are the approximating mathematical expressions. The errors for all data, from 0° C. to 100° C. are smaller than 0.1%. In order to correct the experimental data for both parameters one can first calculate the pressure compensation factor,
 
 Kp=k   p  ( P   0   −P )
 
based on the actual measured pressure P and the standard pressure P 0 , then apply equation 2 to determine the wave velocity at standard temperature. This procedure was followed in the analysis of all the experimental data.
 
Experimental Testing in a Full-scale Flow Loop
 
     Extensive testing was performed in a full scale flow loop in a Multiphase Flow Laboratory. The loop has the ability to provide a total flow up to 8000 bpd of multiphase product and to inject up to 50% gas fraction. In the following description of testing, the liquid phase was a mixture of Diesel and water, and the gas phase was nitrogen. 
     Water Cut Measurements 
     In order to determine the percentage of water in the oil-water-gas mixture the ultrasound wave propagation velocity is first accurately determined. The temperature and pressure were simultaneously recorded in the following experiments since these values affect the wave velocity. Then, proper pressure and temperature compensation was applied according to the formulation shown above. A set of 23 experiments was conducted involving mixtures of Diesel and water at different concentrations. The time of flight was measured for the four channels of each detector, and the average was used to calculate the ultrasound wave velocity. The flow through the loop was varied from 2500 to 8000 bpd, and five data sets were acquired per flow rate. Table 1 shows the experimental conditions and the measured ultrasound velocities. The values for the water cut and mass flow were provided by the test loop instrumentation. The temperature and pressure values were recorder by the sensors T 1  and P 1 . The measurement for the 100% water was done in the laboratory, prior to the field experiments. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Experimental data for different water fractions. 
               
             
          
           
               
                   
                 Water Cut 
                 Flow 
                 Pressure 
                 Temp 
                 V1 
                 V2 
                 V3 
                 V4 
                 V5 
               
               
                 Exp 
                 [%] 
                 [lb/min] 
                 [PSIg] 
                 [° C.] 
                 [m/s] 
                 [m/s] 
                 [m/s] 
                 [m/s] 
                 [m/s] 
               
               
                   
               
             
          
           
               
                 1 
                 93.3512 
                 610.932 
                 196.522 
                 39.921 
                 1476.9 
                 1476.35 
                 1476.99 
                 1476.79 
                 1477.47 
               
               
                 2 
                 89.3916 
                 944.969 
                 193.408 
                 40.279 
                 1475.42 
                 1475.73 
                 1469.54 
                 1473.2 
                 1474.56 
               
               
                 3 
                 88.2417 
                 1409.21 
                 186.111 
                 42.075 
                 1474.4 
                 1476.95 
                 1475.32 
                 1474.24 
                 1476.68 
               
               
                 4 
                 85.3726 
                 1863.5 
                 170.86 
                 42.272 
                 1467.96 
                 1469.35 
                 1472.34 
                 1470.38 
                 1471.42 
               
               
                 5 
                 78.7691 
                 572.958 
                 182.508 
                 42.043 
                 1454.61 
                 1452.93 
                 1456.25 
                 1453.22 
                 1455.57 
               
               
                 6 
                 75.6754 
                 938.935 
                 191.255 
                 42.545 
                 1452.36 
                 1451.88 
                 1452.64 
                 1453.75 
                 1450.24 
               
               
                 7 
                 76.0026 
                 1380.34 
                 180.264 
                 42.641 
                 1450.49 
                 1449.56 
                 1449.24 
                 1450.38 
                 1449.62 
               
               
                 8 
                 73.1397 
                 1845.29 
                 167.684 
                 42.983 
                 1444.65 
                 1446.93 
                 1444.49 
                 1448.32 
                 1445.95 
               
               
                 9 
                 43.4616 
                 557.824 
                 192.11 
                 44.826 
                 1380.46 
                 1378.98 
                 1381.21 
                 1376.05 
                 1382.67 
               
               
                 10 
                 49.4808 
                 885.374 
                 190.614 
                 45.564 
                 1388.78 
                 1391.26 
                 1392.19 
                 1387.4 
                 1388.91 
               
               
                 11 
                 46.647 
                 1311.74 
                 179.958 
                 48.321 
                 1385.94 
                 1384.3 
                 1388.35 
                 1389.42 
                 1389.2 
               
               
                 12 
                 30.3514 
                 537.335 
                 200.034 
                 44.249 
                 1348.27 
                 1350.73 
                 1351.48 
                 1352.07 
                 1349.14 
               
               
                 13 
                 25.7556 
                 861.36 
                 196.202 
                 44.378 
                 1344.51 
                 1345.88 
                 1342.23 
                 1344.19 
                 1350.41 
               
               
                 14 
                 21.9624 
                 1253.07 
                 183.516 
                 46.654 
                 1339.22 
                 1340.38 
                 1338.94 
                 1339.82 
                 1336.07 
               
               
                 15 
                 21.7098 
                 1599.17 
                 180.676 
                 48.048 
                 1346.37 
                 1346.85 
                 1345.63 
                 1345.16 
                 1346.21 
               
               
                 16 
                 10.7222 
                 521.125 
                 195.072 
                 48.043 
                 1329.24 
                 1328.05 
                 1330.87 
                 1331.04 
                 1330.84 
               
               
                 17 
                 9.43934 
                 824.633 
                 192.446 
                 48.139 
                 1328.84 
                 1327.42 
                 1328.52 
                 1329.84 
                 1328.38 
               
               
                 18 
                 10.2833 
                 1217.12 
                 182.966 
                 48.39 
                 1326.45 
                 1326.82 
                 1325.31 
                 1327.08 
                 1327.16 
               
               
                 19 
                 10.1669 
                 1593.75 
                 170.356 
                 48.503 
                 1326.02 
                 1326.47 
                 1325.94 
                 1326.48 
                 1326.81 
               
               
                 20 
                 0.846378 
                 502.645 
                 196.202 
                 47.642 
                 1320.11 
                 1320.23 
                 1320.46 
                 1320.16 
                 1320.04 
               
               
                 21 
                 0.874467 
                 803.238 
                 194.431 
                 48.059 
                 1317.19 
                 1318.03 
                 1317.29 
                 1316.35 
                 1317.7 
               
               
                 22 
                 0.829387 
                 1198.86 
                 187.21 
                 48.223 
                 1318.02 
                 1318.45 
                 1317.56 
                 1317.65 
                 1317.48 
               
               
                 23 
                 0.830137 
                 1548.35 
                 174.188 
                 48.603 
                 1316.09 
                 1315.47 
                 1315.35 
                 1315.42 
                 1314.84 
               
               
                 24 
                 100 
                 — 
                 105.05 
                 24.7 
                 1480.46 
                 1482.73 
                 1481.92 
                 1480.63 
                 1481.59 
               
               
                   
               
             
          
         
       
     
     As seen in the table the experimental data presents a slight pressure drop at higher flow rates and relatively small variations in temperature for each data set. Since the direction of the wave propagation is perpendicular to the flow we do not expect large variations in wave velocity with the flow rate. Indeed, the differences seen in Table 1 are not large and probably mostly due to the observed changes in pressure and temperature. 
     The five velocity measurements V1-V5 were then averaged to obtain (Vav), and the standard deviation (σ V ) of the data set, and the standard deviation of the mean, or the standard error, (σ Vav ) both in natural units and as percentage were calculated as seen in Table 2. The average Vav was then used as the predictor of the true value of the velocity of the ultrasound signal. The standard deviation of the mean is the accepted measurement of the precision of measurements. Thus, the measured valued of the velocity for each experiment would be,
 
Vav±σ Vav  
 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Error analysis for ultrasound velocity for different water fractions. 
               
             
          
           
               
                   
                 Water Cut 
                 Flow 
                 Pressure 
                 Temp 
                 Vav 
                 σ V   
                   
                 σ Vav   
               
               
                 Exp 
                 [%] 
                 [lb/min] 
                 [PSIg] 
                 [° C.] 
                 [m/s] 
                 [m/s] 
                 σ Vav   
                 [%] 
               
               
                   
               
             
          
           
               
                 1 
                 93.3512 
                 610.932 
                 196.522 
                 39.921 
                 1476.90 
                 0.4024 
                 0.1799 
                 0.0122 
               
               
                 2 
                 89.3916 
                 944.969 
                 193.408 
                 40.279 
                 1473.69 
                 2.5186 
                 1.1264 
                 0.0764 
               
               
                 3 
                 88.2417 
                 1409.21 
                 186.111 
                 42.075 
                 1475.52 
                 1.2573 
                 0.5623 
                 0.0381 
               
               
                 4 
                 85.3726 
                 1863.5 
                 170.86 
                 42.272 
                 1470.29 
                 1.7176 
                 0.7681 
                 0.0522 
               
               
                 5 
                 78.7691 
                 572.958 
                 182.508 
                 42.043 
                 1454.52 
                 1.4424 
                 0.6450 
                 0.0443 
               
               
                 6 
                 75.6754 
                 938.935 
                 191.255 
                 42.545 
                 1452.17 
                 1.2809 
                 0.5728 
                 0.0394 
               
               
                 7 
                 76.0026 
                 1380.34 
                 180.264 
                 42.641 
                 1449.86 
                 0.5476 
                 0.2449 
                 0.0169 
               
               
                 8 
                 73.1397 
                 1845.29 
                 167.684 
                 42.983 
                 1446.07 
                 1.6070 
                 0.7187 
                 0.0497 
               
               
                 9 
                 43.4616 
                 557.824 
                 192.11 
                 44.826 
                 1379.87 
                 2.5183 
                 1.1262 
                 0.0816 
               
               
                 10 
                 49.4808 
                 885.374 
                 190.614 
                 45.564 
                 1389.71 
                 1.9618 
                 0.8773 
                 0.0631 
               
               
                 11 
                 46.647 
                 1311.74 
                 179.958 
                 48.321 
                 1387.44 
                 2.2336 
                 0.9989 
                 0.0720 
               
               
                 12 
                 30.3514 
                 537.335 
                 200.034 
                 44.249 
                 1350.34 
                 1.5945 
                 0.7131 
                 0.0528 
               
               
                 13 
                 25.7556 
                 861.36 
                 196.202 
                 44.378 
                 1345.44 
                 3.0670 
                 1.3716 
                 0.1019 
               
               
                 14 
                 21.9624 
                 1253.07 
                 183.516 
                 46.654 
                 1338.89 
                 1.6695 
                 0.7466 
                 0.0558 
               
               
                 15 
                 21.7098 
                 1599.17 
                 180.676 
                 48.048 
                 1346.04 
                 0.6588 
                 0.2946 
                 0.0219 
               
               
                 16 
                 10.7222 
                 521.125 
                 195.072 
                 48.043 
                 1330.01 
                 1.3157 
                 0.5884 
                 0.0442 
               
               
                 17 
                 9.43934 
                 824.633 
                 192.446 
                 48.139 
                 1328.60 
                 0.8721 
                 0.3900 
                 0.0294 
               
               
                 18 
                 10.2833 
                 1217.12 
                 182.966 
                 48.39 
                 1326.56 
                 0.7537 
                 0.3371 
                 0.0254 
               
               
                 19 
                 10.1669 
                 1593.75 
                 170.356 
                 48.503 
                 1326.34 
                 0.3605 
                 0.1612 
                 0.0122 
               
               
                 20 
                 0.846378 
                 502.645 
                 196.202 
                 47.642 
                 1320.20 
                 0.1611 
                 0.0720 
                 0.0055 
               
               
                 21 
                 0.874467 
                 803.238 
                 194.431 
                 48.059 
                 1317.31 
                 0.6338 
                 0.2835 
                 0.0215 
               
               
                 22 
                 0.829387 
                 1198.86 
                 187.21 
                 48.223 
                 1317.83 
                 0.4026 
                 0.1800 
                 0.0137 
               
               
                 23 
                 0.830137 
                 1548.35 
                 174.188 
                 48.603 
                 1315.43 
                 0.4449 
                 0.1990 
                 0.0151 
               
               
                 24 
                 100 
                 — 
                 105.05 
                 24.7 
                 1481.47 
                 0.9394 
                 0.4201 
                 0.0284 
               
               
                   
               
             
          
         
       
     
     The experimental data and a polynomial regression curve for the averaged velocities are plotted in  FIG. 9 . The χ 2 =740.907 is relatively small considering the absolute value of the experimental data, reflecting a good mathematical dependence. 
     The experimental data was then corrected for the effects of pressure and temperature, as described above. The values of P 0 =200 PSIg and T 0 =40° C. were selected since they are closer to the actual field conditions. The results are presented in Table 3. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Experimental data for different water fractions corrected for pressure and temperature effects. 
               
             
          
           
               
                   
                 Water Cut 
                 Flow 
                 Pressure 
                 Temp 
                 Vc1 
                 Vc2 
                 Vc3 
                 Vc4 
                 Vc5 
               
               
                 Exp 
                 [%] 
                 [lb/min] 
                 [PSIg] 
                 [° C.] 
                 [m/s] 
                 [m/s] 
                 [m/s] 
                 [m/s] 
                 [m/s] 
               
               
                   
               
             
          
           
               
                 1 
                 93.3512 
                 610.932 
                 196.522 
                 39.921 
                 1477.08 
                 1476.53 
                 1477.17 
                 1476.97 
                 1477.65 
               
               
                 2 
                 89.3916 
                 944.969 
                 193.408 
                 40.279 
                 1475.05 
                 1475.36 
                 1469.17 
                 1472.83 
                 1474.19 
               
               
                 3 
                 88.2417 
                 1409.21 
                 186.111 
                 42.075 
                 1471.28 
                 1473.83 
                 1472.2 
                 1471.12 
                 1473.56 
               
               
                 4 
                 85.3726 
                 1863.5 
                 170.86 
                 42.272 
                 1464.76 
                 1466.15 
                 1469.14 
                 1467.18 
                 1468.22 
               
               
                 5 
                 78.7691 
                 572.958 
                 182.508 
                 42.043 
                 1451.59 
                 1449.91 
                 1453.23 
                 1450.2 
                 1452.55 
               
               
                 6 
                 75.6754 
                 938.935 
                 191.255 
                 42.545 
                 1448.46 
                 1447.98 
                 1448.74 
                 1449.85 
                 1446.34 
               
               
                 7 
                 76.0026 
                 1380.34 
                 180.264 
                 42.641 
                 1446.6 
                 1445.67 
                 1445.35 
                 1446.49 
                 1445.73 
               
               
                 8 
                 73.1397 
                 1845.29 
                 167.684 
                 42.983 
                 1440.42 
                 1442.7 
                 1440.26 
                 1444.09 
                 1441.72 
               
               
                 9 
                 43.4616 
                 557.824 
                 192.11 
                 44.826 
                 1373.26 
                 1371.78 
                 1374.01 
                 1368.85 
                 1375.47 
               
               
                 10 
                 49.4808 
                 885.374 
                 190.614 
                 45.564 
                 1380.6 
                 1383.08 
                 1384.01 
                 1379.22 
                 1380.73 
               
               
                 11 
                 46.647 
                 1311.74 
                 179.958 
                 48.321 
                 1374.44 
                 1372.8 
                 1376.85 
                 1377.92 
                 1377.7 
               
               
                 12 
                 30.3514 
                 537.335 
                 200.034 
                 44.249 
                 1341.76 
                 1344.22 
                 1344.97 
                 1345.56 
                 1342.63 
               
               
                 13 
                 25.7556 
                 861.36 
                 196.202 
                 44.378 
                 1337.87 
                 1339.24 
                 1335.59 
                 1337.55 
                 1343.77 
               
               
                 14 
                 21.9624 
                 1253.07 
                 183.516 
                 46.654 
                 1329.72 
                 1330.88 
                 1329.44 
                 1330.32 
                 1326.57 
               
               
                 15 
                 21.7098 
                 1599.17 
                 180.676 
                 48.048 
                 1335.19 
                 1335.67 
                 1334.45 
                 1333.98 
                 1335.03 
               
               
                 16 
                 10.7222 
                 521.125 
                 195.072 
                 48.043 
                 1317.86 
                 1316.67 
                 1319.49 
                 1319.66 
                 1319.46 
               
               
                 17 
                 9.43934 
                 824.633 
                 192.446 
                 48.139 
                 1317.38 
                 1315.96 
                 1317.06 
                 1318.38 
                 1316.92 
               
               
                 18 
                 10.2833 
                 1217.12 
                 182.966 
                 48.39 
                 1314.83 
                 1315.2 
                 1313.69 
                 1315.46 
                 1315.54 
               
               
                 19 
                 10.1669 
                 1593.75 
                 170.356 
                 48.503 
                 1314.44 
                 1314.89 
                 1314.36 
                 1314.9 
                 1315.23 
               
               
                 20 
                 0.846378 
                 502.645 
                 196.202 
                 47.642 
                 1309.2 
                 1309.32 
                 1309.55 
                 1309.25 
                 1309.13 
               
               
                 21 
                 0.874467 
                 803.238 
                 194.431 
                 48.059 
                 1305.8 
                 1306.64 
                 1305.9 
                 1304.96 
                 1306.31 
               
               
                 22 
                 0.829387 
                 1198.86 
                 187.21 
                 48.223 
                 1306.54 
                 1306.97 
                 1306.08 
                 1306.17 
                 1306 
               
               
                 23 
                 0.830137 
                 1548.35 
                 174.188 
                 48.603 
                 1304.34 
                 1303.72 
                 1303.6 
                 1303.67 
                 1303.09 
               
               
                 24 
                 100 
                 — 
                 105.05 
                 24.7 
                 1514.78 
                 1517.05 
                 1516.24 
                 1514.95 
                 1515.91 
               
               
                   
               
             
          
         
       
     
     Table 4 presents the pressure and temperature compensated averaged velocities and the error analysis. As expected the change in the standard error is very small since the compensation procedure implies mathematical operations with exact numbers. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 Error analysis for ultrasound velocity for different water fractions. 
               
             
          
           
               
                   
                 Water Cut 
                 Flow 
                 Pressure 
                 Temp 
                 Vav 
                 σ V   
                   
                 σ Vav   
               
               
                 Exp 
                 [%] 
                 [lb/min] 
                 [PSIg] 
                 [° C.] 
                 [m/s] 
                 [m/s] 
                 σ Vav   
                 [%] 
               
               
                   
               
             
          
           
               
                 1 
                 93.3512 
                 610.932 
                 196.522 
                 39.921 
                 1477.08 
                 0.4024 
                 0.1799 
                 0.0122 
               
               
                 2 
                 89.3916 
                 944.969 
                 193.408 
                 40.279 
                 1473.32 
                 2.5186 
                 1.1264 
                 0.0765 
               
               
                 3 
                 88.2417 
                 1409.21 
                 186.111 
                 42.075 
                 1472.40 
                 1.2573 
                 0.5623 
                 0.0382 
               
               
                 4 
                 85.3726 
                 1863.5 
                 170.86 
                 42.272 
                 1467.09 
                 1.7176 
                 0.7681 
                 0.0524 
               
               
                 5 
                 78.7691 
                 572.958 
                 182.508 
                 42.043 
                 1451.50 
                 1.4424 
                 0.6450 
                 0.0444 
               
               
                 6 
                 75.6754 
                 938.935 
                 191.255 
                 42.545 
                 1448.27 
                 1.2809 
                 0.5728 
                 0.0396 
               
               
                 7 
                 76.0026 
                 1380.34 
                 180.264 
                 42.641 
                 1445.97 
                 0.5476 
                 0.2449 
                 0.0169 
               
               
                 8 
                 73.1397 
                 1845.29 
                 167.684 
                 42.983 
                 1441.84 
                 1.6070 
                 0.7187 
                 0.0498 
               
               
                 9 
                 43.4616 
                 557.824 
                 192.11 
                 44.826 
                 1372.67 
                 2.5183 
                 1.1262 
                 0.0820 
               
               
                 10 
                 49.4808 
                 885.374 
                 190.614 
                 45.564 
                 1381.53 
                 1.9618 
                 0.8773 
                 0.0635 
               
               
                 11 
                 46.647 
                 1311.74 
                 179.958 
                 48.321 
                 1375.94 
                 2.2336 
                 0.9989 
                 0.0726 
               
               
                 12 
                 30.3514 
                 537.335 
                 200.034 
                 44.249 
                 1343.83 
                 1.5945 
                 0.7131 
                 0.0531 
               
               
                 13 
                 25.7556 
                 861.36 
                 196.202 
                 44.378 
                 1338.80 
                 3.0670 
                 1.3716 
                 0.1024 
               
               
                 14 
                 21.9624 
                 1253.07 
                 183.516 
                 46.654 
                 1329.39 
                 1.6695 
                 0.7466 
                 0.0562 
               
               
                 15 
                 21.7098 
                 1599.17 
                 180.676 
                 48.048 
                 1334.86 
                 0.6588 
                 0.2946 
                 0.0221 
               
               
                 16 
                 10.7222 
                 521.125 
                 195.072 
                 48.043 
                 1318.63 
                 1.3157 
                 0.5884 
                 0.0446 
               
               
                 17 
                 9.43934 
                 824.633 
                 192.446 
                 48.139 
                 1317.14 
                 0.8721 
                 0.3900 
                 0.0296 
               
               
                 18 
                 10.2833 
                 1217.12 
                 182.966 
                 48.39 
                 1314.94 
                 0.7537 
                 0.3371 
                 0.0256 
               
               
                 19 
                 10.1669 
                 1593.75 
                 170.356 
                 48.503 
                 1314.76 
                 0.3605 
                 0.1612 
                 0.0123 
               
               
                 20 
                 0.846378 
                 502.645 
                 196.202 
                 47.642 
                 1309.29 
                 0.1611 
                 0.0720 
                 0.0055 
               
               
                 21 
                 0.874467 
                 803.238 
                 194.431 
                 48.059 
                 1305.92 
                 0.6338 
                 0.2835 
                 0.0217 
               
               
                 22 
                 0.829387 
                 1198.86 
                 187.21 
                 48.223 
                 1306.35 
                 0.4026 
                 0.1800 
                 0.0138 
               
               
                 23 
                 0.830137 
                 1548.35 
                 174.188 
                 48.603 
                 1303.68 
                 0.4449 
                 0.1990 
                 0.0153 
               
               
                 24 
                 100 
                 — 
                 105.05 
                 24.7 
                 1515.79 
                 0.9394 
                 0.4201 
                 0.0277 
               
               
                   
               
             
          
         
       
     
     As can be seen in Tables 3, 4, and  FIG. 10 , after the pressure and temperature corrections, the differences between the velocities measured at different flow rates are reduced in magnitude. A quadratic regression was performed on the averaged values of the wave velocities, expressed as,
 
 V   w ( wc )= k   0   +k   1    wc+k   2    wc   2  
 
     Where V w (wc) is the wave velocity, we is the water fraction in %. The regression coefficients were obtained as,
 
 k   0 =1304.2±1.81
 
 k   1 =1.1453±0.109
 
 k   2 =0.008942±0.00112
 
     The value of χ 2 =387.449 is almost twice as small as the value obtained for the same type of regression for the uncorrected data. It reflects excellent agreement of the model with the experimentally measured data. This expression can be used to calculate the ultrasound wave velocity that we expect to measure for a certain water fraction at a certain temperature and pressure. 
     Since it is desirable to determine the amount of water in the mixture (for example, an oil, water, and gas mixture), one calculates the water cut from a measured wave velocity. Solving the quadratic equation for water cut wc we obtain the following water cut determination equation: 
             wc   =         -     k   1       +         k   1   2     -     4   ⁢           ⁢       k   2     ⁡     (       k   0     +     V   w       )                 2   ⁢           ⁢     k   2               
Substituting the numerical values for k 0 , k 1 , and k 2  we arrived at the following expression,
 
 wc= 55.9159√{square root over (0.035768 V   w −45.3369)}−64.0405   Eq. 4
 
Equation 4 can be used to determine the water cut in the water-oil liquid mixture, based on the measured wave velocity. The experimental data, the theoretical curve obtained using Equation 4, and ±5% error curves are presented in  FIG. 11 . All experimental data are included in the ±5% error range, with most data points presenting less then 3% error from the theoretical curve. Thus, the UPFD  1  proved to have good precision and accuracy in experimental conditions that simulate the actual oilfield environment.
 
Gas Fraction Measurements
 
     Research has shown that the gas volumetric fraction in an oil-water-gas mixture is exponentially correlated with the attenuation of the ultrasound wave. In order to determine the wave attenuation, one measures the amplitude of the wave that propagates from the ultrasound transmitter to the sensor array  6  positioned on an opposing wall of the UPFD  1 . 
     A set of 10 experiments was conducted involving two mixtures of diesel and water at 25% and a set of 5 experiments at 50% water cut. The flow through the loop was varied from 4000 to 6000 bpd, and the gas volumetric fraction (GVF) was set at 0%, 10%, 20%, 30%, 40%, and 50%. The actual values for the GVF, water cut, and the flow rates were measured by the test loop equipment. The experimental data represents 3 minute averages of the measured values. Pressure and temperatures were measured by dedicated sensors. The amplitude of an average of 50 ultrasound signals was determined for each channel of the detector, and then averaged for the eight channels of the flowmeter. 
     For this purpose the amplitude of the ultrasound signal is determined by detecting the first positive and first negative peaks on the recorded signal and calculates the amplitude.  FIG. 12 . A “Good Amplitude” test is performed by checking for signal clipping and for a time differential between positive and the negative peaks. The transmitter voltage was adjusted such that no clipping occurred. For calibration purposes, only a predetermined set of values for the transmitter voltages were allowed: 3, 30, 50, 70, 90, and 100V. 
     Five data sets were acquired per each flow rate, water cut, and gas volumetric fraction combination. The temperature and pressure were simultaneously recorded, since they could affect the measurements. Table 5 shows the experimental conditions, and the measured amplitudes for the five data points A1-A5. The values of the measured amplitudes are shown as digitized by the data acquisition cards, with no physical units. 
     The five amplitude measurements A1-A5 were then averaged (Aav), and the standard deviation (σ A ) of the data set, and the standard deviation of the mean, or the standard error, (σ Aav ) both in natural units and as percentage were calculated as seen in Table 6. The average Aav was then used as the predictor of the true value of the amplitude of the transmitted ultrasound signal. The standard deviation of the mean is the accepted measurement of the precision of measurements. Thus, the measured valued of the amplitude for each experiment would be,
 
Aav±σ Aav  
 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 5 
               
             
             
               
                   
               
               
                 Experimental data for different gas volumetric fractions and 25% water cut. 
               
             
          
           
               
                   
                   
                   
                   
                   
                   
                 Trans. 
                   
                   
                   
                   
                   
               
               
                   
                 GVF 
                 Water Cut 
                 Flow 
                 Pressure 
                 Temp 
                 Volt. 
               
               
                 Exp 
                 [%] 
                 [%] 
                 [lb/min] 
                 [PSIg] 
                 [° C.] 
                 [V] 
                 A1 
                 A2 
                 A3 
                 A4 
                 A5 
               
               
                   
               
             
          
           
               
                 1 
                 0 
                 25.76 
                 861.36 
                 196.9 
                 43.3 
                 3 
                 152.726 
                 150.183 
                 155.424 
                 153.518 
                 150.983 
               
               
                 2 
                 10.32 
                 26.68 
                 842.245 
                 193.92 
                 43.26 
                 30 
                 239.386 
                 235.372 
                 232.539 
                 240.11 
                 236.729 
               
               
                 3 
                 19.89 
                 27.13 
                 839.787 
                 193.98 
                 43.39 
                 50 
                 192.17 
                 196.114 
                 194.729 
                 190.43 
                 188.732 
               
               
                 4 
                 29.60 
                 27.31 
                 856.554 
                 194.05 
                 43.46 
                 70 
                 141.827 
                 143.844 
                 154.223 
                 139.631 
                 135.583 
               
               
                 5 
                 39.66 
                 27.04 
                 873.406 
                 189.67 
                 43.94 
                 90 
                 82.592 
                 85.693 
                 81.928 
                 79.381 
                 80.227 
               
               
                 6 
                 48.86 
                 27.05 
                 834.26 
                 178.51 
                 44.65 
                 100 
                 48.391 
                 52.625 
                 56.582 
                 53.259 
                 50.271 
               
               
                 7 
                 9.495 
                 22.58 
                 1260.48 
                 184.19 
                 45.97 
                 30 
                 224.461 
                 220.539 
                 229.428 
                 227.628 
                 221.527 
               
               
                 8 
                 20.25 
                 23.6 
                 1261.54 
                 184.84 
                 46.01 
                 50 
                 172.428 
                 170.893 
                 168.318 
                 176.253 
                 178.432 
               
               
                 9 
                 29.45 
                 24.15 
                 1264.67 
                 185.01 
                 46.08 
                 70 
                 120.311 
                 118.726 
                 119.422 
                 124.521 
                 115.329 
               
               
                 10 
                 39.17 
                 25.19 
                 1256.81 
                 175.37 
                 46.28 
                 90 
                 67.752 
                 71.317 
                 73.672 
                 68.149 
                 72.483 
               
               
                 11 
                 48.26 
                 26.26 
                 1212.3 
                 157.7 
                 47.27 
                 100 
                 36.249 
                 32.527 
                 34.150 
                 31.372 
                 28.441 
               
               
                   
               
             
          
         
       
     
     The standard error was less then 4% with most values being close to 1%. The larger errors correspond to the smaller recorded amplitudes, since the peak detection is more difficult and less precise in such cases. This was due to the fact that the transmitter voltage for this experiment was limited to 100V. Since the electronics may provide an excitation signal of up to 500V, and in real life measurements the automatic adjustment of the transmitter voltage would keep the detected amplitude larger than 100 digital units, the error would be smaller in a typical measurement environment. Moreover, the 50 ultrasound signals averaged to obtain each amplitude measurement were recorded and processed in less than 1 second. Since such a speed is not necessary for typical gas fraction measurements a larger number of signals could be used in calculation. This would allow enhancing the precision of the measured results. 
       FIG. 13  shows the measured ultrasound wave amplitudes versus gas volumetric fraction. As is evident from  FIG. 13 , the amplitude is strongly dependent on the gas fraction. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 6 
               
             
             
               
                   
               
               
                 Error analysis for amplitude measurement for different 
               
               
                 gas volumetric fractions and 25% water cut. 
               
             
          
           
               
                   
                   
                   
                   
                   
                   
                 Trans. 
                   
                   
                   
                   
               
               
                   
                 GVF 
                 Water Cut 
                 Flow 
                 Pressure 
                 Temp 
                 Volt. 
                   
                   
                   
                 σ Aav   
               
               
                 Exp 
                 [%] 
                 [%] 
                 [lb/min] 
                 [PSIg] 
                 [° C.] 
                 [V] 
                 Aav 
                 σ A   
                 σ Aav   
                 % 
               
               
                   
               
             
          
           
               
                 1 
                 0 
                 25.76 
                 861.36 
                 196.9 
                 43.3 
                 3 
                 152.567 
                 2.08 
                 0.93 
                 0.61 
               
               
                 2 
                 10.32 
                 26.68 
                 842.245 
                 193.92 
                 43.26 
                 30 
                 236.827 
                 3.08 
                 1.38 
                 0.58 
               
               
                 3 
                 19.89 
                 27.13 
                 839.787 
                 193.98 
                 43.39 
                 50 
                 192.435 
                 3.02 
                 1.35 
                 0.70 
               
               
                 4 
                 29.60 
                 27.31 
                 856.554 
                 194.05 
                 43.46 
                 70 
                 143.022 
                 6.97 
                 3.12 
                 2.18 
               
               
                 5 
                 39.66 
                 27.04 
                 873.406 
                 189.67 
                 43.94 
                 90 
                 81.964 
                 2.45 
                 1.10 
                 1.34 
               
               
                 6 
                 48.86 
                 27.05 
                 834.26 
                 178.51 
                 44.65 
                 100 
                 52.225 
                 3.11 
                 1.39 
                 2.66 
               
               
                 7 
                 9.495 
                 22.58 
                 1260.48 
                 184.19 
                 45.97 
                 30 
                 224.717 
                 3.82 
                 1.71 
                 0.76 
               
               
                 8 
                 20.25 
                 23.60 
                 1261.54 
                 184.84 
                 46.01 
                 50 
                 173.265 
                 4.08 
                 1.82 
                 1.05 
               
               
                 9 
                 29.45 
                 24.15 
                 1264.67 
                 185.01 
                 46.08 
                 70 
                 119.662 
                 3.31 
                 1.48 
                 1.24 
               
               
                 10 
                 39.17 
                 25.19 
                 1256.81 
                 175.37 
                 46.28 
                 90 
                 70.674 
                 2.63 
                 1.17 
                 1.66 
               
               
                 11 
                 48.26 
                 26.26 
                 1212.3 
                 157.7 
                 47.27 
                 100 
                 32.547 
                 2.94 
                 1.31 
                 4.04 
               
               
                   
               
             
          
         
       
     
     Since the transmitter voltage changed during the measurements, the amplitude of the transmitted signal does not measure the actual attenuation of the ultrasound wave. Thus, the transmission factor T f  is defined as, 
               T   f     =       A     GT   v       ×   1000           
where A is the wave amplitude, G is the system gain including the preamplifier and the A/D card gains, and T v  is the transmitter voltage. A proportionality factor of 1,000 was used to obtain more readable values.
 
     Table 7 presents the data for the calculated ultrasound signal transmission factor for all the experiments, and Table 8 presents the corresponding statistics. Because the transmission factor is obtained by multiplication and divisions with exact numbers, changes in the percentage standard error are not expected, even though the actual standard deviation of the mean looks smaller. As before, the measured value of the transmission factor is,
 
T f ±σ Tfav 
 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 7 
               
             
             
               
                   
               
               
                 Experimental data for ultrasound signal transmission factor. 
               
             
          
           
               
                   
                 GVF 
                 Water Cut 
                 Flow 
                 Pressure 
                 Temp 
                   
                   
                   
                   
                   
               
               
                 Exp 
                 [%] 
                 [%] 
                 [lb/min] 
                 [PSIg] 
                 [° C.] 
                 Tf1 
                 Tf2 
                 Tf3 
                 Tf4 
                 Tf5 
               
               
                   
               
             
          
           
               
                 1 
                 0 
                 25.76 
                 861.36 
                 196.9 
                 43.3 
                 19.886 
                 19.555 
                 20.238 
                 19.989 
                 19.659 
               
               
                 2 
                 10.32 
                 26.68 
                 842.245 
                 193.92 
                 43.26 
                 3.117 
                 3.065 
                 3.028 
                 3.126 
                 3.082 
               
               
                 3 
                 19.89 
                 27.13 
                 839.787 
                 193.98 
                 43.39 
                 1.501 
                 1.532 
                 1.521 
                 1.488 
                 1.475 
               
               
                 4 
                 29.60 
                 27.31 
                 856.554 
                 194.05 
                 43.46 
                 0.791 
                 0.803 
                 0.861 
                 0.779 
                 0.757 
               
               
                 5 
                 39.66 
                 27.04 
                 873.406 
                 189.67 
                 43.94 
                 0.358 
                 0.372 
                 0.356 
                 0.345 
                 0.348 
               
               
                 6 
                 48.86 
                 27.05 
                 834.26 
                 178.51 
                 44.65 
                 0.189 
                 0.206 
                 0.221 
                 0.208 
                 0.196 
               
               
                 7 
                 9.495 
                 22.58 
                 1260.48 
                 184.19 
                 45.97 
                 2.923 
                 2.872 
                 2.987 
                 2.964 
                 2.885 
               
               
                 8 
                 20.25 
                 23.6 
                 1261.54 
                 184.84 
                 46.01 
                 1.347 
                 1.335 
                 1.315 
                 1.377 
                 1.394 
               
               
                 9 
                 29.45 
                 24.15 
                 1264.67 
                 185.01 
                 46.08 
                 0.671 
                 0.663 
                 0.666 
                 0.695 
                 0.644 
               
               
                 10 
                 39.17 
                 25.19 
                 1256.81 
                 175.37 
                 46.28 
                 0.294 
                 0.310 
                 0.320 
                 0.296 
                 0.315 
               
               
                 11 
                 48.26 
                 26.26 
                 1212.3 
                 157.7 
                 47.27 
                 0.142 
                 0.127 
                 0.133 
                 0.123 
                 0.111 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 8 
               
             
             
               
                   
               
               
                 Error analysis for ultrasound signal transmission factor. 
               
             
          
           
               
                   
                 GVF 
                 Water Cut 
                 Flow 
                 Pressure 
                 Temp 
                   
                   
                   
                 σ Tfav   
               
               
                 Exp 
                 [%] 
                 [%] 
                 [lb/min] 
                 [PSIg] 
                 [° C.] 
                 Tfav 
                 σ Tf   
                 σ Tfav   
                 % 
               
               
                   
               
             
          
           
               
                 1 
                 0 
                 25.76 
                 861.36 
                 196.9 
                 43.3 
                 19.865 
                 0.2709 
                 0.1211 
                 0.6098 
               
               
                 2 
                 10.32 
                 26.68 
                 842.245 
                 193.92 
                 43.26 
                 3.084 
                 0.0400 
                 0.0179 
                 0.5805 
               
               
                 3 
                 19.89 
                 27.13 
                 839.787 
                 193.98 
                 43.39 
                 1.503 
                 0.0236 
                 0.0106 
                 0.7024 
               
               
                 4 
                 29.60 
                 27.31 
                 856.554 
                 194.05 
                 43.46 
                 0.798 
                 0.0389 
                 0.0174 
                 2.1799 
               
               
                 5 
                 39.66 
                 27.04 
                 873.406 
                 189.67 
                 43.94 
                 0.356 
                 0.0106 
                 0.0048 
                 1.3359 
               
               
                 6 
                 48.86 
                 27.05 
                 834.26 
                 178.51 
                 44.65 
                 0.204 
                 0.0122 
                 0.0054 
                 2.6641 
               
               
                 7 
                 9.495 
                 22.58 
                 1260.48 
                 184.19 
                 45.97 
                 2.926 
                 0.0497 
                 0.0222 
                 0.7599 
               
               
                 8 
                 20.25 
                 23.6 
                 1261.54 
                 184.84 
                 46.01 
                 1.354 
                 0.0318 
                 0.0142 
                 1.0518 
               
               
                 9 
                 29.45 
                 24.15 
                 1264.67 
                 185.01 
                 46.08 
                 0.668 
                 0.0185 
                 0.0083 
                 1.2358 
               
               
                 10 
                 39.17 
                 25.19 
                 1256.81 
                 175.37 
                 46.28 
                 0.307 
                 0.0114 
                 0.0051 
                 1.6620 
               
               
                 11 
                 48.26 
                 26.26 
                 1212.3 
                 157.7 
                 47.27 
                 0.127 
                 0.0115 
                 0.0051 
                 4.0360 
               
               
                   
               
             
          
         
       
     
     The experimental data for the transmission factor are shown in  FIG. 14  versus the gas volumetric fraction GVF. All experimental data are shown, including the ones corresponding to 50% water cut. 
     The dependence of the ultrasound attenuation to the GVF is exponential. Moreover, there are two different types of interaction between the ultrasound wave and the gas bubbles inside the flow stream. When the GVF is relatively low, and the gas bubbles are small compared to the wavelength, the ultrasound wave is scattered uniformly in all directions. When the GVF is large the attenuation occurs mostly by back reflection. The consequence is that there are two different measurement regimes with two corresponding mathematical models. 
     Thus, the average transmission factor measured for 25% water cut was used to obtain two exponential regressions, one for the 0 to just below 10% GVF and the other for 10 to 50% GVF. 
     For the 0 to just below 10% GVF range the theoretical curve is described by,
 
 T   f   =k   L0   +k   L1   e   −k     L2     GVF    Eq. 5
 
where the regression coefficients were:
 
     k L0 =1.2152±0.0881 
     k L1 =18.65±0.128 
     k L2 =0.22296±0.00727 
     The regression was in good agreement with the data as shown by the small chi-squared test result of χ 2 =0.00881634. 
     For the 10 to 50% GVF range the regression was,
 
 T   f   =k   H0   +k   H1   e   −k     H2     GVF    Eq. 6
 
with
 
     k H0 =0.018501±0.0618 
     k H1 =6.8297±0.334 
     k H2 =0.077899±0.00495 
     and a chi-squared test result of χ 2 =0.0238425. The χ 2  factor is in this case slightly larger because more experimental data were recorded in this interval.  FIG. 15  presents the transmission factor for the 10 to 50% GVF range. This range is considered more relevant for the industrial applications of the meter. 
     While there is a good agreement between the experimental data and the theoretical regression curves, the variations in the pressure and temperature between experiments could have a negative effect on the precision of the measurements. Since the pressure inside the pipe will influence the volume of the gas bubbles, but the back reflection of the ultrasound signal is dependent on the area of the bubbles, the dependence between the transmission factor and the pressure will be of the form, 
                     T     fp   ⁢           ⁢   0       =         T   fp     ⁡     (       P   0       P   p       )         2   3               Eq   .           ⁢   7               
where T fp0  is the transmission factor at a chosen standard pressure P 0 , and T fp  is the transmission factor obtained at the measured pressure P p . Similarly, the temperature influence can be compensated by,
 
                     T     ft   ⁢           ⁢   0       =         T   ft     ⁡     (       T   t       T   0       )         2   3               Eq   .           ⁢   8               
where T ft0  is the transmission factor at a chosen standard temperature T 0 , and T ft  is the transmission factor obtained at the measured temperature T t . Table 9 shows the ultrasound transmission factors corrected for a standard pressure P 0 =200 psi and T 0 =40° C. The pressure compensated experimental data for the ultrasound transmission factor versus gas volumetric fraction in the 10 to 50% range are plotted in  FIG. 16 .  FIG. 17  shows the experimental data compensated for both pressure and temperature. The statistical properties of the data set will be similar with those obtained for uncompensated data since the transformations involved only exact numbers.
 
                                                                                                                         TABLE 9                   Pressure and temperature compensated experimental data for the ultrasound       transmission factor versus gas volumetric fraction in the 10-50% range.                GVF   Water Cut   Flow   Pressure   Temp                               Exp   [%]   [%]   [lb/min]   [PSIg]   [° C.]   Tft1   Tft2   Tft3   Tft4   Tft5   Tftav                    1   0   25.76   861.36   196.9   43.3   21.184   20.831   21.558   21.293   20.942   21.162       2   10.32   26.68   842.245   193.92   43.26   3.353   3.296   3.257   3.363   3.315   3.317       3   19.89   27.13   839.787   193.98   43.39   1.618   1.651   1.639   1.603   1.589   1.620       4   29.60   27.31   856.554   194.05   43.46   0.854   0.866   0.928   0.840   0.816   0.861       5   39.66   27.04   873.406   189.67   43.94   0.395   0.410   0.392   0.380   0.384   0.392       6   48.86   27.05   834.26   178.51   44.65   0.219   0.239   0.257   0.241   0.228   0.237       7   9.495   22.58   1260.48   184.19   45.97   3.388   3.329   3.463   3.436   3.343   3.392       8   20.25   23.6   1261.54   184.84   46.01   1.559   1.545   1.521   1.593   1.613   1.566       9   29.45   24.15   1264.67   185.01   46.08   0.777   0.767   0.771   0.804   0.745   0.773       10   39.17   25.19   1256.81   175.37   46.28   0.354   0.372   0.385   0.356   0.378   0.369       11   48.26   26.26   1212.3   157.7   47.27   0.185   0.166   0.175   0.161   0.146   0.167                    
For the range 0 to 10% GVF the exponential regression:
 
 T   ft   =k   L3   +k   L4   e   −k     L5     GVF    Eq. 9
 
where the regression coefficients values±one standard deviation were:
     k L3 =1.3765±0.0238   k L4 =19.785±0.0347   k L5 =0.22507±0.0019   

     The regression was in excellent agreement with the data as shown by the extremely small χ 2 =0.000653. 
     For the 10 to 50% GVF range the regression equation,
 
 T   ft   =k   H3   +k   H4   e   −k     H5     GVF    Eq. 10
 
with k H3 =0.014777±0.0415
 
     k H4 =7.1369±0.2 
     k H5 =0.074839±0.00288 
     and χ 2 =0.00961833. 
     The standard deviation of the regression coefficients improved by a factor of two and the value of χ 2  reduced by an order of magnitude when compared with the values obtained for the uncompensated experimental data. 
     In order to obtain the gas volumetric fraction from the ultrasound transmission factor measurements, the expressions for the exponential regressions for GVF are solved. Thus, 
     
       
         
           
             
               
                 
                   
                     
                       GVF 
                       = 
                       
                         
                           - 
                           
                             1 
                             
                               k 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 5 
                               
                             
                           
                         
                         ⁢ 
                         ln 
                         ⁢ 
                         
                           
                             
                               
                                 T 
                                 ft 
                               
                               - 
                               
                                 k 
                                 
                                   L 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   3 
                                 
                               
                             
                             
                               k 
                               
                                 L 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 4 
                               
                             
                           
                           ⁡ 
                           
                             [ 
                             % 
                             ] 
                           
                         
                       
                     
                     , 
                     
                         
                     
                     ⁢ 
                     
                       
                         if 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           T 
                           ft 
                         
                       
                       &gt; 
                       3.3 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     and 
                     , 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   11 
                 
               
             
             
               
                 
                   
                     GVF 
                     = 
                     
                       
                         - 
                         
                           1 
                           
                             k 
                             
                               H 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               5 
                             
                           
                         
                       
                       ⁢ 
                       ln 
                       ⁢ 
                       
                         
                           
                             
                               T 
                               ft 
                             
                             - 
                             
                               k 
                               
                                 H 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 3 
                               
                             
                           
                           
                             k 
                             
                               H 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               4 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           % 
                           ] 
                         
                       
                     
                   
                   , 
                   
                       
                   
                   ⁢ 
                   
                     
                       if 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         ft 
                       
                     
                     ≤ 
                     3.3 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   12 
                 
               
             
           
         
       
     
     The results obtained using the multiphase flowmeter data were then compared with the corresponding values of the GVF obtained from the test loop instruments, and the percentage errors were calculated. As shown in Table 10 and  FIG. 18 , all errors are less than 7% with most of them around 2%. It should be noted that the value for the GVF provided by the test loop instruments represents a 3 minutes average. During this time the actual value of the GVF varied to a certain degree. The five measurements recorded with the research multiphase flowmeter represent 1 second snapshots of the GVF that captured the variability of the flow. Thus, the actual precision of the flowmeter is higher than the errors in Table 10 suggest. In actual use, averaging of the signals between measurements may produce much more precise results. 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 10 
               
             
             
               
                   
               
               
                 Measured and experimental GVF for 25% water cut experiments. 
               
             
          
           
               
                   
                 Experimental (loop) 
                   
                 Measured 
                 Error 
               
               
                 Exp 
                 GVF [%] 
                 Transmission factor 
                 GVF [%] 
                 [%] 
               
               
                   
               
             
          
           
               
                 1 
                 0 
                 21.162 
                 −1.72e−06 
                 −0.008 
               
               
                 2 
                 10.32 
                 3.317 
                 10.29 
                 0.17 
               
               
                 3 
                 19.89 
                 1.620 
                 19.93 
                 −0.25 
               
               
                 4 
                 29.60 
                 0.861 
                 28.49 
                 3.72 
               
               
                 5 
                 39.66 
                 0.392 
                 39.27 
                 0.96 
               
               
                 6 
                 48.86 
                 0.237 
                 46.36 
                 5.08 
               
               
                 7 
                 9.495 
                 3.392 
                 9.99 
                 −5.31 
               
               
                 8 
                 20.25 
                 1.566 
                 20.39 
                 −0.71 
               
               
                 9 
                 29.45 
                 0.773 
                 29.96 
                 −1.72 
               
               
                 10 
                 39.17 
                 0.369 
                 40.12 
                 −2.45 
               
               
                 11 
                 48.26 
                 0.167 
                 51.45 
                 −6.61 
               
               
                   
               
             
          
         
       
     
     The robustness of the GVF measurements was tested in an additional set of 5 experiments using 50% water cut and GVF from 10 to 50%. The raw experimental data, and the pressure and temperature compensated ones, as well as the relevant statistical parameters are presented in Tables 11 to 15. 
     The results provided by the multiphase flowmeter were compared with the corresponding values obtained from the test loop instruments, and the percentage errors were calculated. As seen in Table 16 the errors were less than 6% as predicted. 
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 11 
               
             
             
               
                   
               
               
                 Experimental data for different gas volumetric fractions and 50% water cut. 
               
             
          
           
               
                   
                   
                   
                   
                   
                   
                 Trans. 
                   
                   
                   
                   
                   
               
               
                   
                 GVF 
                 Water Cut 
                 Flow 
                 Pressure 
                 Temp 
                 Volt. 
               
               
                 Exp 
                 [%] 
                 [%] 
                 [lb/min] 
                 [PSIg] 
                 [° C.] 
                 [V] 
                 A1 
                 A2 
                 A3 
                 A4 
                 A5 
               
               
                   
               
             
          
           
               
                 12 
                 9.46 
                 49.93 
                 888.457 
                 189.46 
                 45.02 
                 30 
                 236.425 
                 232.479 
                 228.173 
                 233.735 
                 230.481 
               
               
                 13 
                 19.44 
                 50.71 
                 877.162 
                 189.88 
                 45.28 
                 50 
                 188.385 
                 176.928 
                 185.275 
                 182.342 
                 191.417 
               
               
                 14 
                 28.31 
                 51.44 
                 884.183 
                 189.96 
                 45.72 
                 70 
                 125.624 
                 127.751 
                 134.285 
                 129.378 
                 130.723 
               
               
                 15 
                 40.55 
                 51.34 
                 885.573 
                 182.65 
                 46.21 
                 90 
                 74.291 
                 78.541 
                 75.482 
                 76.552 
                 78.499 
               
               
                 16 
                 49.04 
                 53.16 
                 873.349 
                 172.21 
                 46.77 
                 100 
                 46.249 
                 48.527 
                 44.150 
                 41.372 
                 48.441 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 12 
               
             
             
               
                   
               
               
                 Error analysis for amplitude measurement for different 
               
               
                 gas volumetric fractions and 50% water cut. 
               
             
          
           
               
                   
                   
                   
                   
                   
                   
                 Trans. 
                   
                   
                   
                   
               
               
                   
                 GVF 
                 Water Cut 
                 Flow 
                 Pressure 
                 Temp 
                 Volt. 
                   
                   
                   
                 σ Aav   
               
               
                 Exp 
                 [%] 
                 [%] 
                 [lb/min] 
                 [PSIg] 
                 [° C.] 
                 [V] 
                 Aav 
                 σ A   
                 σ Aav   
                 % 
               
               
                   
               
             
          
           
               
                 12 
                 9.46 
                 49.93 
                 888.457 
                 189.46 
                 45.02 
                 30 
                 232.259 
                 3.14 
                 1.40 
                 0.60 
               
               
                 13 
                 19.44 
                 50.71 
                 877.162 
                 189.88 
                 45.28 
                 50 
                 184.869 
                 5.59 
                 2.50 
                 1.35 
               
               
                 14 
                 28.31 
                 51.44 
                 884.183 
                 189.96 
                 45.72 
                 70 
                 129.552 
                 3.26 
                 1.46 
                 1.12 
               
               
                 15 
                 40.55 
                 51.34 
                 885.573 
                 182.65 
                 46.21 
                 90 
                 76.673 
                 1.87 
                 0.83 
                 1.09 
               
               
                 16 
                 49.04 
                 53.16 
                 873.349 
                 172.21 
                 46.77 
                 100 
                 45.747 
                 3.04 
                 1.36 
                 2.97 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 13 
               
             
             
               
                   
               
               
                 Experimental data for ultrasound signal transmission factor at 50% water cut. 
               
             
          
           
               
                   
                 GVF 
                 Water Cut 
                 Flow 
                 Pressure 
                 Temp 
                   
                   
                   
                   
                   
               
               
                 Exp 
                 [%] 
                 [%] 
                 [lb/min] 
                 [PSIg] 
                 [° C.] 
                 Tf1 
                 Tf2 
                 Tf3 
                 Tf4 
                 Tf5 
               
               
                   
               
             
          
           
               
                 12 
                 9.46 
                 49.93 
                 888.457 
                 189.46 
                 45.02 
                 3.079 
                 3.027 
                 2.971 
                 3.043 
                 3.001 
               
               
                 13 
                 19.44 
                 50.71 
                 877.162 
                 189.88 
                 45.28 
                 1.472 
                 1.382 
                 1.448 
                 1.425 
                 1.495 
               
               
                 14 
                 28.31 
                 51.44 
                 884.183 
                 189.96 
                 45.72 
                 0.701 
                 0.713 
                 0.749 
                 0.722 
                 0.729 
               
               
                 15 
                 40.55 
                 51.34 
                 885.573 
                 182.65 
                 46.21 
                 0.322 
                 0.341 
                 0.328 
                 0.332 
                 0.341 
               
               
                 16 
                 49.04 
                 33.16 
                 873.349 
                 172.21 
                 46.77 
                 0.181 
                 0.190 
                 0.172 
                 0.162 
                 0.189 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 14 
               
             
             
               
                   
               
               
                 Error analysis for ultrasound signal transmission factor at 50% water cut. 
               
             
          
           
               
                   
                 GVF 
                 Water Cut 
                 Flow 
                 Pressure 
                 Temp 
                   
                   
                   
                 σ Tfav   
               
               
                 Exp 
                 [%] 
                 [%] 
                 [lb/min] 
                 [PSIg] 
                 [° C.] 
                 Tfav 
                 σ Tf   
                 σ Tfav   
                 % 
               
               
                   
               
             
          
           
               
                 12 
                 9.46 
                 49.93 
                 888.457 
                 189.46 
                 45.02 
                 3.024 
                 0.0409 
                 0.0183 
                 0.6047 
               
               
                 13 
                 19.44 
                 50.71 
                 877.162 
                 189.88 
                 45.28 
                 1.444 
                 0.0436 
                 0.0195 
                 1.3509 
               
               
                 14 
                 28.31 
                 51.44 
                 884.183 
                 189.96 
                 45.72 
                 0.723 
                 0.0182 
                 0.0081 
                 1.1248 
               
               
                 15 
                 40.55 
                 51.34 
                 885.573 
                 182.65 
                 46.21 
                 0.333 
                 0.0081 
                 0.0036 
                 1.0888 
               
               
                 16 
                 49.04 
                 33.16 
                 873.349 
                 172.21 
                 46.77 
                 0.179 
                 0.0119 
                 0.0053 
                 2.9702 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 15 
               
             
             
               
                   
               
               
                 Pressure and temperature compensated experimental data for the ultrasound transmission 
               
               
                 factor versus gas volumetric fraction in the 10-50% range at 50% water cut. 
               
             
          
           
               
                   
                 GVF 
                 Water Cut 
                 Flow 
                 Pressure 
                 Temp 
                   
                   
                   
                   
                   
                   
               
               
                 Exp 
                 [%] 
                 [%] 
                 [lb/min] 
                 [PSIg] 
                 [° C.] 
                 Tft1 
                 Tft2 
                 Tft3 
                 Tft4 
                 Tft5 
                 Tftav 
               
               
                   
               
             
          
           
               
                 12 
                 9.46 
                 49.93 
                 888.457 
                 189.46 
                 45.02 
                 3.454 
                 3.396 
                 3.333 
                 3.414 
                 3.367 
                 3.393 
               
               
                 13 
                 19.44 
                 50.71 
                 877.162 
                 189.88 
                 45.28 
                 1.655 
                 1.554 
                 1.628 
                 1.602 
                 1.682 
                 1.624 
               
               
                 14 
                 28.31 
                 51.44 
                 884.183 
                 189.96 
                 45.72 
                 0.793 
                 0.807 
                 0.848 
                 0.817 
                 0.825 
                 0.818 
               
               
                 15 
                 40.55 
                 51.34 
                 885.573 
                 182.65 
                 46.21 
                 0.377 
                 0.399 
                 0.383 
                 0.389 
                 0.398 
                 0.389 
               
               
                 16 
                 49.04 
                 33.16 
                 873.349 
                 172.21 
                 46.77 
                 0.222 
                 0.232 
                 0.211 
                 0.198 
                 0.232 
                 0.219 
               
               
                   
               
             
          
         
       
     
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 16 
               
             
             
               
                   
               
               
                 Measured and experimental GVF for 50% water cut experiments. 
               
             
          
           
               
                   
                 Experimental 
                   
                 Measured 
                 Error 
               
               
                 Exp 
                 GVF [%] 
                 Transmission factor 
                 GVF [%] 
                 [%] 
               
               
                   
               
             
          
           
               
                 12 
                 9.46 
                 3.39276 
                 9.995 
                 −5.648 
               
               
                 13 
                 19.44 
                 1.62405 
                 19.9 
                 −2.358 
               
               
                 14 
                 28.31 
                 0.817902 
                 29.19 
                 −3.124 
               
               
                 15 
                 40.55 
                 0.389222 
                 39.39 
                 2.881 
               
               
                 16 
                 49.04 
                 0.219136 
                 47.48 
                 3.18 
               
               
                   
               
             
          
         
       
     
     The percentage of oil in the oil/water/gas mixture is determined by subtracting the sum of the determined water and gas fractions from the total. Total flow may be determined in any suitable manner, for example, via a flow meter F 1 . Thus, the flow of oil (or non-water fluid) may be determined by subtraction as follows Flow Rate of Oil=Total Flow−Total Flow (water fraction+gas fraction) 
     Various modifications and variations of the present invention are possible in light of the above teachings. It is therefore to be understood that within the scope of the appended claims, the invention may be practiced otherwise than as specifically described herein. For example, while described in terms of both software and hardware components interactively cooperating, it is contemplated that the system described herein may be practiced entirely in software.