Estimating flow rate at a pump

Abstract

A method for determining an estimated flow rate of fluid flow in a pump comprises: obtaining measurements of the pressure and temperature of fluid at the intake to the pump, the pressure and temperature of the fluid at the discharge from the pump, and the electrical power supplied to the pump; determining values representing either the density of the fluid and the specific heat capacity of the fluid, or the specific fluid enthalpy based on measurements and/or historical data; and calculating an estimated efficiency of the pump and an estimated flow rate of the fluid based on the measured electrical power, the measured temperatures, the measured pressures, the determined value for density and the determined value for specific heat capacity or the determined value for specific fluid enthalpy.

Claims

1. A method for determining an estimated mass flow rate of multiphase fluids in the oil and gas industry in a pump system including a pump, the method comprising: obtaining measurements of a pressure and a temperature of a multiphase fluid at an intake to the pump, a pressure and a temperature of the multiphase fluid at a discharge from the pump, and power supplied to the pump system; determining values representing a specific fluid enthalpy based on measurements and fluid models and/or historical data; calculating an estimated efficiency of the pump and an estimated mass flow rate of the multiphase fluid based on the supplied power, the measured temperatures, the measured pressures, and the determined value for specific fluid enthalpy; at least one of using pump specific efficiency vs mass flow rate curves plotted for varying viscosities of multiphase fluids, determined using calibration data for the pump, to determine a viscosity for the multiphase fluid by determining a viscosity curve with which the estimated flow rate and the estimated pump efficiency correspond, using pump power vs mass flow rate curves plotted for varying viscosities of multiphase fluids, determined using calibration data for the pump, to determine a viscosity for the multiphase fluid by determining a viscosity curve with which the estimated flow rate and the estimated pump power correspond, and obtaining a differential pressure of the pump, and using pump specific differential pressure vs mass flow rate curves plotted for varying viscosities of multiphase fluids, determined using calibration data for the pump, to determine a viscosity for the multiphase fluid by determining a viscosity curve with which the estimated flow rate and the estimated pump differential pressure correspond; and comparing the calculated estimated efficiency of the pump to a manufacturer's prescribed efficiency values of the pump, and when the calculated estimated efficiency of the pump is outside of the manufacturer's prescribed efficiency values, issuing an alert which indicates at least one of an error in measured input data, wear on the pump, damage to the pump, or failure of the pump.

2. The method as claimed in claim 1, wherein the multiphase fluid passing through the pump is sampled or collected downstream of the pump, with measurements being taken to determine values for the specific enthalpy.

3. The method as claimed in claim 1, wherein estimated values for the specific fluid enthalpy are used, with the estimated values being derived from fluid property models and from the pressures and temperatures that are measured at the pump intake and discharge.

4. The method as claimed in claim 1, further comprising modifying a pump device in order that it includes temperature and pressure sensors for the intake and the discharge.

5. The method as claimed in claim 1, further comprising determining if a stable flow condition exists before using measured temperature and pressure values in the calculating of the estimated efficiency and mass flow rate.

6. The method as claimed in claim 1, further comprising allowing a predetermined time period to elapse after the system is initiated or after an unstable flow condition is known to be present or has been detected.

7. The method as claimed in claim 6, further comprising allowing a predetermined time period to elapse after a change is made to a control of the pump or to other flow control devices affecting the flow at the pump.

8. The method as claimed in claim 6, wherein the predetermined time period is at least 5 minutes.

9. The method as claimed in claim 1, further comprising checking for changes above a certain threshold in one or more of the measured pressures, the measured temperatures, the power supplied to the pump, or the calculated pump efficiency during a predetermined time period before using measured temperature and pressure values in the calculating of the estimated efficiency and mass flow rate.

10. The method as claimed in claim 1, further comprising comparing the discharge and intake temperatures and checking that the discharge temperature is higher than or at least the same as the intake temperature.

11. The method as claimed in claim 1, wherein the calculating of the estimated efficiency for the pump and the estimated mass flow rate is carried out based on an assumption that the power supplied to the pump is converted to mechanical energy and to heat in the pumped fluid, and that there is no loss of either mass or heat.

12. The method as claimed in claim 1, wherein mass and energy balances are used to determine the mass flow rate based on the measured pressure and temperature values, a known cross-sectional area for the pump, and the determined values for specific fluid enthalpy.

13. The method as claimed in claim 1, further comprising using calibration data for the pump to determine a gas fraction in the pump for compressible fluids based on the estimated mass flow rate and the efficiency.

14. The method as claimed in claim 1, wherein the pump is a centrifugal pump.

15. The method as claimed in claim 1, wherein the pump is an electrical submersible pump (ESP).

16. A non-transient computer programme product comprising instructions that, when executed, will configure a data processing device to carry out the method as claimed in claim 1.

17. A non-transient computer programme product comprising instructions that, when executed, will configure a data processing device: to receive measurements of pressure and temperature of a multiphase fluid at an intake to a pump, pressure and temperature of the multiphase fluid at a discharge from the pump, and electrical power supplied to the pump; to receive or to determine values representing a specific fluid enthalpy, based on measurements and/or historical data; to calculate an estimated efficiency for the pump and an estimated mass flow rate of the multiphase fluid based on the measured electrical power, the measured temperatures the measured pressures, a known cross-sectional area, and the value for specific fluid enthalpy; at least one of to use pump specific efficiency vs mass flow rate curves plotted for varying viscosities of multiphase fluids, determined using calibration data for the pump to determine a viscosity for the multiphase fluid by determining a viscosity curve with which the estimated flow rate and the estimated pump efficiency correspond, to use pump power vs mass flow rate curves plotted for varying viscosities of multiphase fluids, determined using calibration data for the pump, to determine a viscosity for the multiphase fluid by determining a viscosity curve with which the estimated flow rate and the estimated pump power correspond, and to obtain a differential pressure of the pump, and to use pump specific differential pressure vs mass flow rate curves plotted for varying viscosities of multiphase fluids, determined using calibration data for the pump, to determine a viscosity for the multiphase fluid by determining a viscosity curve with which the estimated flow rate and the estimated pump differential pressure correspond; and to compare the calculated estimated efficiency of the pump to a manufacturer's prescribed efficiency values of the pump, and when the calculated estimated efficiency of the pump is outside of the manufacturer's prescribed efficiency values, to issue an alert which indicates at least one of an error in measured input data, wear on the pump, damage to the pump, or failure of the pump.

18. An apparatus for estimating flow rate of multiphase fluids in the oil and gas industry at a pump, the apparatus comprising a data processing device arranged: to receive measurements of pressure and temperature of a multiphase fluid at an intake to a pump, pressure and temperature of the multiphase fluid at a discharge from the pump, and electrical power supplied to the pump; to receive or to determine values representing a specific fluid enthalpy based on measurements and/or historical data; to calculate an estimated efficiency for the pump and an estimated mass flow rate of the multiphase fluid based on the measured electrical power, the measured temperatures the measured pressures, a known cross-sectional area, and the value for specific fluid enthalpy; at least one of to use pump specific efficiency vs mass flow rate curves plotted for varying viscosities of multiphase fluids, determined using calibration data for the pump to determine a viscosity for the multiphase fluid by determining a viscosity curve with which the estimated flow rate and the estimated pump efficiency correspond, to use pump power vs mass flow rate curves plotted for varying viscosities of multiphase fluids, determined using calibration data for the pump, to determine a viscosity for the multiphase fluid by determining a viscosity curve with which the estimated flow rate and the estimated pump power correspond, and to obtain a differential pressure of the pump, and to use pump specific differential pressure vs mass flow rate curves plotted for varying viscosities of multiphase fluids, determined using calibration data for the pump, to determine a viscosity for the multiphase fluid by determining a viscosity curve with which the estimated flow rate and the estimated pump differential pressure correspond; and to compare the calculated estimated efficiency of the pump to a manufacturer's prescribed efficiency values of the pump, and when the calculated estimated efficiency of the pump is outside of the manufacturer's prescribed efficiency values, to issue an alert which indicates at least one of an error in measured input data, wear on the pump, damage to the pump, or failure of the pump.

19. The apparatus as claimed in claim 18, wherein the data processing device is arranged to carry out a method for determining an estimated mass flow rate of the multiphase fluid in a pump system including the pump, the method comprising: obtaining measurements of the pressure and temperature of the multiphase fluid at the intake to the pump, the pressure and temperature of the multiphase fluid at the discharge from the pump, and the power supplied to the pump system; determining values representing the specific fluid enthalpy based on measurements and fluid models and/or historical data; and calculating the estimated efficiency of the pump and the estimated mass flow rate of the multiphase fluid based on the supplied power, the measured temperatures, the measured pressures, and the determined value for specific fluid enthalpy.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Certain preferred embodiments of the invention will now be described by way of example only and with reference to the accompanying drawings in which:

(2) FIG. 1 shows an ESP with motor and pump inside the well;

(3) FIG. 2 is a plot showing measured flow rate from an ESP test—solid line—and the flow rate estimated with the model in equation (15)—dashed line;

(4) FIG. 3 is a plot showing measured efficiency from the ESP test—solid line—and efficiency estimated with the model in equation (16)—dashed line;

(5) FIG. 4 shows the recorded temperature increase from the ESP test—solid line—and the same values estimated with the model in equation (15)—dashed line—when the measured flow rate was used as an input for the mode;

(6) FIG. 5 shows a plot of flow rate from a flow loop for an ESP test—blue line—and the flow rate estimated with the model in equation (23)—red line;

(7) FIG. 6 is a plot of efficiency from the flow loop—solid line—and the efficiency estimated with equation (26)—dashed line and equation (27)—bold line:

(8) FIG. 7 shows, in the top plot, the flow rate obtained from a flow loop for an ESP test—solid line—and the flow rate estimated with the multiphase model as shown in equation (23)—dashed line, and, in the bottom plot, the gas volume fraction;

(9) FIG. 8 shows viscosity obtained from the flow loop—solid line—and viscosity estimated with the multiphase model—dashed line; and

(10) FIG. 9 is an example of ESP efficiency curves for various fluid viscosities.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

(11) The estimation of flow rate through a pump will be described below with reference to an Electrical Submersible Pump (ESP) as an example. The ESP is a centrifugal pump which is installed inside the well with the electrical motor as shown in FIG. 1. The motor and the pump convert electric power to heat and mechanical energy for the pumped fluid. The example calculations relate firstly to an incompressible fluid, for example an oil and water mixture, and secondly to a compressible fluid, for example an oil, water and gas mixture.

(12) The pump supplied power, P.sub.Pump, will be converted to mechanical energy, P.sub.Fluid, or heat, Q.sub.Fluid, in the pumped fluid. It is assumed that the fluid mass inside the pump is constant, and there is no loss of either mass (no leak out of the well) or heat from the system (fluid is heated). ζ is specific enthalpy. Applying mass and energy balances will give:
w=w.sub.In=w.sub.Out≈q.sub.Inβ.sub.MixIn≈q.sub.Outβ.sub.MixOut  (1)
P.sub.Pump=w(ζ.sub.Out.sup.0−ζ.sub.In.sup.0)+ΔQ.sub.Loss=ΔP.sub.Fluid,Mecanical+ΔQ.sub.Fluid,Heat+ΔQ.sub.Loss  (2)

(13) The stagnation enthalpy, ζ.sup.0, and the specific enthalpy, ζ, as expressed as set out below.
ζ.sup.0=ζ+½v.sup.2+gh
ζ=u+pv

(14) where v is the fluid velocity, g is specific gravitation, h is the height, u is internal energy, and p is pressure.

(15) For downhole pumps both the motor and the pump is placed in the flowing pipe, thus the heat loss from the pumping system can be neglected since this heat is transferred to the fluid and the temperature on the discharge is measured. Then equation (2) can be written:
P.sub.Pump=W(ζ.sub.Out−ζ.sub.In)=ΔP.sub.Fluid,Mechanical+ΔQ.sub.Fluid,Heat  (2a)

(16) Based on the equations (1) and (2a) the mass flow rate can be determined using specific fluid enthalpy obtain by estimation or measurement of the fluid in accordance with equations (44) to (47) below. Alternatively, using density and specific heat capacity then following model may be derived to determine the mass flow rate, w:

(17) $\begin{array}{cc}{w}_{\mathrm{total}}=f\left(P_{\mathrm{Pump}},p_{\mathrm{In}},p_{\mathrm{Out}},T_{\mathrm{In}},T_{\mathrm{Out}},{\rho }_{\mathrm{MixIn}},{\rho }_{\mathrm{MixOut}},c_{P_{—}\mathrm{MixIn}},c_{P_{—}\mathrm{MixOut}},A\right)& \left(3\right)\\ P_{\mathrm{Pump}}=\stackrel{.}{m}{\mathrm{gh}}_{\mathrm{Out}}-\stackrel{.}{m}{\mathrm{gh}}_{\mathrm{In}}+\frac{1}{2}\left(\stackrel{.}{m}{v}_{\mathrm{Out}}^2-\stackrel{.}{m}{v}_{\mathrm{In}}^2\right)+c_{P_{—}\mathrm{MixOut}}\stackrel{.}{m}{T}_{\mathrm{Out}}-c_{P_{—}\mathrm{MixIn}}\stackrel{.}{m}{T}_{\mathrm{In}}& \left(4\right)\\ P_{\mathrm{Pump}}=w\left({\mathrm{gh}}_{\mathrm{Out}}-{\mathrm{gh}}_{\mathrm{In}}+\frac{1}{2}\left(v_{\mathrm{Out}}^2-v_{\mathrm{In}}^2\right)+c_{P_{—}\mathrm{MixOut}}{T}_{\mathrm{Out}}+c_{P_{—}\mathrm{MixIn}}{T}_{\mathrm{In}}\right)& \left(5\right)\\ p_2=p_1+\rho g\left(h_2-h_1\right)& \left(6\right)\\ {\mathrm{gh}}_2-{\mathrm{gh}}_1=\frac{p_2-p_1}{\rho }& \left(7\right)\\ w=\stackrel{.}{m}=\rho q=\rho \mathrm{Av}& \left(8\right)\\ P_{\mathrm{Pump}}=w\left(\frac{p_{\mathrm{Out}}}{{\rho }_{\mathrm{MixOut}}}-\frac{p_{\mathrm{In}}}{{\rho }_{\mathrm{MixIn}}}+c_{P_{—}\mathrm{MixOut}}{T}_{\mathrm{Out}}-c_{P_{—}\mathrm{MixIn}}{T}_{\mathrm{In}}\right)+\frac{w^3}{2}\left(\frac{1}{{\rho }_{\mathrm{MixOut}}^2{A}_{\mathrm{Out}}^2}-\frac{1}{{\rho }_{\mathrm{MixIn}}^2{A}_{\mathrm{In}}^2}\right)& \left(9\right)\end{array}$

(18) The electric power, P.sub.Pump, to the pump (or the motor) can be measured or calculated from the voltage and current:
P.sub.Pump=ΦU.sub.MotorI.sub.Motor=Φ(U.sub.Motor−R.sub.CableI.sub.Motor)  (10)
Φ=PF√{square root over (3)}={right arrow over (3)} sin(Q.sub.U∠1)  (11)

(19) If the pumped fluid is incompressible, the fluid density at the intake and the discharge will be equal. If it is assumed that the flowing areas are equal as well, the equation can be simplified as:

(20) $\begin{array}{cc}{P}_{\mathrm{Pump}}=w\left(\frac{p_{\mathrm{Out}}-p_{\mathrm{In}}}{\rho }+c_{P_{—}\mathrm{MixOut}}{T}_{\mathrm{Out}}-c_{P_{—}\mathrm{MixIn}}{T}_{\mathrm{In}}\right)& \left(12\right)\end{array}$
If the specific heat capacity is the same at the inlet and outlet as well, the equation can be expressed:

(21) $\begin{array}{cc}{P}_{\mathrm{Pump}}=w\left(\frac{p_{\mathrm{Out}}-p_{\mathrm{In}}}{\rho }+c_P\left(T_{\mathrm{Out}}-T_{\mathrm{In}}\right)\right)& \left(13\right)\end{array}$

(22) Now the mass flow rate, w, or the volume flow rate, q, can be determined:

(23) $\begin{array}{cc}w=\frac{\rho P_{\mathrm{Pump}}}{p_{\mathrm{Out}}-p_{\mathrm{In}}+c_P\rho \left(T_{\mathrm{Out}}-T_{\mathrm{In}}\right)}=\frac{\eta .Math.\rho P_{\mathrm{Pump}}}{p_{\mathrm{Out}}-p_{\mathrm{In}}}& \left(14\right)\\ q=\frac{P_{\mathrm{Pump}}}{p_{\mathrm{Out}}-p_{\mathrm{In}}+c_P\rho \left(T_{\mathrm{Out}}-T_{\mathrm{In}}\right)}=\frac{\eta P_{\mathrm{Pump}}}{p_{\mathrm{Out}}-p_{\mathrm{In}}}& \left(15\right)\end{array}$

(24) FIG. 2 shows the results of an example calculation using equation (15). A Centrilift™ ESP was tested with all relevant input and output parameters being measured, and the flow rates also being measured. The specific heat capacity was c.sub.P=1800, and the remaining variables were measured in the flow loop of the test rig. FIG. 2 illustrates the measured flow rate as compared to the flow rate estimated using equation (15). It will be appreciated that there is a close correspondence between the two flow rates, especially during steady state/stable flow conditions.

(25) The pump efficiency, η, may be expressed as the pump power turned into mechanical fluid energy:

(26) $\begin{array}{cc}\eta =\frac{w\left(p_{\mathrm{Out}}-p_{\mathrm{In}}\right)}{\rho P_{\mathrm{Motor}}}=\frac{q\left(p_{\mathrm{Out}}-p_{\mathrm{In}}\right)}{P_{\mathrm{Motor}}}& \left(16\right)\end{array}$

(27) FIG. 3 shows the results of an example calculation using equation (16). The same Centrilift™ ESP test was used as for FIG. 2. FIG. 3 illustrates the measured efficiency as compared to the efficiency estimated using equation (16). It will be appreciated that there is a close correspondence between the two efficiency values, especially during steady state/stable flow conditions.

(28) Alternatively, from the fluid temperature increase:

(29) $\begin{array}{cc}\eta =\frac{P_{\mathrm{Motor}}-{\mathrm{wc}}_P\left(T_{\mathrm{Out}}-T_{\mathrm{In}}\right)}{P_{\mathrm{Motor}}}& \left(17\right)\end{array}$

(30) Combining the equations (16) and (17) enables direct calculation of the pump efficiency independent of the applied break horse power, P.sub.Pump via equation (18) The pump efficiency should be verified to be in the pump range (as specified by the manufacturer) and only valid if T.sub.Out>T.sub.In+ΔT.sub.min and p.sub.Out>P.sub.In+Δp.sub.Min before applied for flow rate estimation according to equation (14) or (15). If the pump efficiency from the calculation is outside of the range from the manufacturer then an alert is made regarding potentially bad measurements or some other outside influence leading to bad results, such as an unstable flow during the measurement period.

(31) $\begin{array}{cc}\eta =\frac{1}{1+c_P\rho \frac{T_{\mathrm{Out}}-T_{\mathrm{In}}}{p_{\mathrm{Out}}-p_{\mathrm{In}}}}& \left(18\right)\end{array}$

(32) Both the estimated flow rate (FIG. 2) pump efficiency (FIG. 3) deviate from the flow loop measurements. The main uncertainty of the input data is the measured temperature change. To evaluate this input variable, the model has been used to calculate the temperature increase and compare with the flow loop measurement. The measured and estimated temperature changes (to achieve the flow loop measured flow rate) are shown in FIG. 4.

(33) If the pumped fluid is compressible, the calculations are more complex as indicated in equation (19). Since this model is a 3.sup.rd order equation, a non-linear method (e.g. Newton Raphson) may be used to solve the equation. If we assume that the cross sectional areas at intake and discharge are equal, the equation can be written:

(34) $\begin{array}{cc}{P}_{\mathrm{Pump}}=w\left(\frac{p_{\mathrm{Out}}}{{\rho }_{\mathrm{MixOut}}}-\frac{p_{\mathrm{In}}}{{\rho }_{\mathrm{MixIn}}}+c_{P_{—}\mathrm{MixOut}}{T}_{\mathrm{Out}}-c_{P_{—}\mathrm{MixIn}}{T}_{\mathrm{In}}\right)+\frac{w^3}{2A^2}\left(\frac{1}{{\rho }_{\mathrm{MixOut}}^2}-\frac{1}{{\rho }_{\mathrm{MixIn}}^2}\right)& \left(19\right)\end{array}$

(35) The next assumptions are that the phase mass fractions, x.sub.i, and fluid (phase) properties are known and that there is no slip between the phases. The following relations for mixed heat capacity and density may be established:

(36) $\begin{array}{cc}{c}_{P_{—}\mathrm{Mix}}=\underset{i=1}{\overset{N}{.Math.}}\left[x_i\left(p,T\right).Math.c_{P_{—}i}\left(p,T\right)\right]& \left(20\right)\\ {\rho }_{\mathrm{Mix}}={\left[\underset{i=1}{\overset{N}{.Math.}}\frac{x_i\left(p,T\right)}{{\rho }_i\left(p,T\right)}\right]}^{-1}& \left(21\right)\end{array}$

(37) The flow rate may be formulated as 3.sup.rd order equation:

(38) 0$\begin{array}{cc}{w}^3+w\frac{2A^2}{\frac{1}{{\rho }_{\mathrm{MixOut}}^2}-\frac{1}{{\rho }_{\mathrm{MixIn}}^2}}\left(\frac{p_{\mathrm{Out}}}{{\rho }_{\mathrm{MixOut}}}-\frac{p_{\mathrm{In}}}{{\rho }_{\mathrm{MixIn}}}+c_{P_{—}\mathrm{MixOut}}{T}_{\mathrm{Out}}-c_{P_{—}\mathrm{MixIn}}{T}_{\mathrm{In}}\right)=\frac{2A^2}{\frac{1}{{\rho }_{\mathrm{MixOut}}^2}-\frac{1}{{\rho }_{\mathrm{MixIn}}^2}}{P}_{\mathrm{Pump}}& \left(22\right)\\ w^3+w.Math.c_1\left(A,{\rho }_{\mathrm{MixIn}},{\rho }_{\mathrm{MixOut}},c_{P_{—}\mathrm{MixIn}},c_{P_{—}\mathrm{MixOut}},p_{\mathrm{In}},p_{\mathrm{Out}},T_{\mathrm{In}},T_{\mathrm{Out}}\right)+c_0\left(A,{\rho }_{\mathrm{MixIn}},{\rho }_{\mathrm{MixOut}},P_{\mathrm{Pump}}\right)=0& \left(23\right)\end{array}$

(39) The efficiency of the pump may be expressed as the fraction of the applied power that is transformed into mechanical energy, similar to equation (16), but the fluid compressibility should be accounted for:

(40) $\begin{array}{cc}{P}_{\mathrm{Pump}}\eta =w\left(\frac{p_{\mathrm{Out}}}{{\rho }_{\mathrm{MixOut}}}-\frac{p_{\mathrm{In}}}{{\rho }_{\mathrm{MixIn}}}\right)+\frac{w^3}{2A^2}\left(\frac{1}{{\rho }_{\mathrm{MixOut}}^2}-\frac{1}{{\rho }_{\mathrm{MixIn}}^2}\right)& \left(24\right)\\ P_{\mathrm{Pump}}\left(1-\eta \right)=w\left(c_{P_{—}\mathrm{MixOut}}{T}_{\mathrm{Out}}-c_{P_{—}\mathrm{MixIn}}{T}_{\mathrm{In}}\right)& \left(25\right)\\ \eta =\frac{1-\eta }{c_{P_{—}\mathrm{MixOut}}{T}_{\mathrm{Out}}-c_{P_{—}\mathrm{MixIn}}{T}_{\mathrm{In}}}\left(\frac{p_{\mathrm{Out}}}{{\rho }_{\mathrm{MixOut}}}-\frac{p_{\mathrm{In}}}{{\rho }_{\mathrm{MixIn}}}\right)+\frac{{P_{\mathrm{Pump}}^2\left(1-\eta \right)}^3}{2{A^2\left(c_{P_{—}\mathrm{MixOut}}{T}_{\mathrm{Out}}-c_{P_{—}\mathrm{MixIn}}{T}_{\mathrm{In}}\right)}^3}\left(\frac{1}{{\rho }_{\mathrm{MixOut}}^2}-\frac{1}{{\rho }_{\mathrm{MixIn}}^2}\right)& \left(26\right)\end{array}$

(41) Equation (26) can be reformulated in the efficiency, η, as a 3.sup.rd order formula:

(42) $\begin{array}{cc}{\eta }^3-3{\eta }^2+\left(3+b_1+b_2\right)\eta +\left(b_1-1\right)=0& \left(27\right)\\ b_1=\frac{2{A^2\left(c_{P_{—}\mathrm{MixOut}}{T}_{\mathrm{Out}}-c_{P_{—}\mathrm{MixIn}}{T}_{\mathrm{In}}\right)}^2}{P_{\mathrm{Pump}}^2}\left(\frac{p_{\mathrm{Out}}}{{\rho }_{\mathrm{MixOut}}}-\frac{p_{\mathrm{In}}}{{\rho }_{\mathrm{MixIn}}}\right){\left(\frac{1}{{\rho }_{\mathrm{MixOut}}^2}-\frac{1}{{\rho }_{\mathrm{MixIn}}^2}\right)}^{-1}& \left(28\right)\\ b_2=\frac{2{A^2\left(c_{P_{—}\mathrm{MixOut}}{T}_{\mathrm{Out}}-c_{P_{—}\mathrm{MixIn}}{T}_{\mathrm{In}}\right)}^2}{P_{\mathrm{Pump}}^2}{\left(\frac{1}{{\rho }_{\mathrm{MixOut}}^2}-\frac{1}{{\rho }_{\mathrm{MixIn}}^2}\right)}^{-1}& \left(29\right)\end{array}$

(43) The models in equations (23) and (27) are non-linear and the solution may be found using the Newton-Raphson method. Then the derivate of the function will be used:
f.sub.w.sup.1(w)=3w.sup.2+c.sub.1(A,ρ.sub.MixIn,ρ.sub.MixOut,c.sub.P . . . MixIn,c.sub.P . . . MixOut,p.sub.In,p.sub.Out,T.sub.In,T.sub.Out)  (30)
f.sub.η.sup.1(η)=3η.sup.2−6η+(3+b.sub.1+b.sub.2)  (31)

(44) The solution of the variable is now found by iterations on x (which is w or η in the equations above):

(45) $\begin{array}{cc}{x}_{i+1}=x_i-\frac{f_x\left(x_i\right)}{f_x^1\left(x_i\right)}& \left(32\right)\end{array}$

(46) FIG. 5 shows an example calculation using equation (23). A Centrilift™ ESP was tested with all relevant input and output parameters being measured, and the flow rates also being measured. The specific heat capacity was c.sub.P=1750, and the remaining variables were measured in the flow loop. FIG. 5 illustrates the measured flow rate as compared to the flow rate estimated using equation (23). It will be appreciated that there is a close correspondence between the two flow rates, especially during steady state/stable flow conditions.

(47) FIG. 6 shows data from the same test in relation to efficiency. The measured values for efficiency are plotted along with calculated values from equations (26) and (27) based on pressure and temperature differentials respectively.

(48) FIG. 7 illustrates, in the top plot, a comparison of flow rates from the flow loop for an ESP test and flow rates calculated using equation (23), with the bottom plot showing the gas volume fraction (GVF). It will be appreciated that the large variations in GVF result in substantial variations in compressibility of the fluid through the pump. Nonetheless, the estimated flow rates from equation (23) follow the measured values closely.

(49) The pump curves and the viscosity correction factors may be used to find the fluid viscosity as shown in equation (38).

(50) $\begin{array}{cc}{\mu }_0={\mu }_{\mathrm{Estimate}}& \left(33\right)\\ {\mathrm{VCF}}_Q=f\left({\mu }_{\mathrm{Estimate}},f\right)& \left(34\right)\\ w_{\mathrm{base}}=w_{\mathrm{Estimate}}\left(\frac{f_0}{f}\right)\frac{1}{{\mathrm{VCF}}_Q}\left(\frac{{\rho }_{\mathrm{water}}}{{\rho }_{\mathrm{Mix}}}\right)& \left(35\right)\\ {\eta }_{\mathrm{base}}=f_{\mathrm{LUT}}\left(w_0,{\eta }_0,w_{\mathrm{base}}\right)& \left(36\right)\\ {\mathrm{VCF}}_{\eta }=\frac{{\eta }_{\mathrm{Estimate}}}{{\eta }_{\mathrm{base}}}& \left(37\right)\\ {\mu }_{\mathrm{Estimate}}=\left(f\left({\mathrm{VCF}}_{\eta },f\right)\right)& \left(38\right)\end{array}$

(51) While the absolute difference between the estimated viscosity and the estimated viscosity at the previous iteration is greater than a defined tolerance ε, iterate on equations (33)—(38) to estimate the viscosity. Equation (33) saves the viscosity estimated at the previous step to be compared with the estimated viscosity of the current step. Initially, this can be set to an initial guess value defined by the user. Viscosity correction factor for flow is computed for this viscosity (34) The estimated flow rate w.sub.Estimate from (14) is converted to reference base case (water, 60 Hz) using viscosity correction factor and affinity laws (35). The base efficiency η.sub.base is then computed from the reference base curve implemented in the form of a lookup table (36). Using the base efficiency and the estimated efficiency η.sub.Estimate from (16), (17) or (18), the viscosity correction factor for efficiency is computed (37). The viscosity is then estimated from the VCF function for efficiency (38). This is then iterated until the change in the viscosity is less than a defined tolerance value. FIG. 8 shows the viscosity as measured from the flow loop for the ESP test compared against the calculated viscosity using the model described above.

(52) It is also possible to determine the fluid viscosity from efficiency, flow rate, mass flow rate, pump frequency, and pump specific calibration data/efficiency curves if this is available. An example of such curves is shown in FIG. 9. When the flow rate and efficiency is known then viscosity can be read from the graph. As well as this, other pump specific relationships like Dp vs flow, or power vs flow may be used.

(53) For Multiphase flow through the ESP, the GVF can be estimated using the Gas correction factor GCF and the estimated efficiency from (16), (17) or (18). When gas enters the ESP, the performance of the pump is affected and this can be observed in the differential pressure and the brake horsepower (BHP) of the pump. To account for the effect to gas in the system, the co-called Gas correction factors are modelled for dp and BHP. The GCF for dp and BHP are primarily a function of the gas fraction or the GVF and are modelled using experimental data. Estimation of the GVF using the GCF can be done using the equations below.

(54) $\begin{array}{cc}{\mathrm{VCF}}_Q=f\left({\mu }_{\mathrm{Estimate}},f\right)& \left(39\right)\\ w_{\mathrm{base}}=w_{\mathrm{Estimate}}\left(\frac{f_0}{f}\right)\frac{1}{{\mathrm{VCF}}_Q}\left(\frac{{\rho }_{\mathrm{water}}}{{\rho }_{\mathrm{Mix}}}\right)& \left(40\right)\\ {\eta }_{\mathrm{base}}=f_{\mathrm{LUT}}\left(w_0,{\eta }_0,w_{\mathrm{base}}\right)& \left(41\right)\\ {\mathrm{GCF}}_{\eta }=\frac{{\eta }_{\mathrm{Estimate}}}{{\eta }_{\mathrm{base}}}& \left(42\right)\\ x_G=f\left({\mathrm{GFC}}_{\eta }\right)& \left(43\right)\end{array}$

(55) As noted above, specific fluid enthalpy values can be determined as an alternative to using density and specific heat capacity. In this case, starting with equations (1) and (2a) above, then the equations below can be used to find the efficiency and hence the flow rate. The fluid enthalpy can be found from fluid data bases, and thus calculated based on the actual pressure and temperature

(56) $\begin{array}{cc}w=\frac{{ϛ}_{\mathrm{Out}}\left(p_{\mathrm{Out}},T_{\mathrm{Out}}\right)-{ϛ}_{\mathrm{In}}\left(p_{\mathrm{In}},T_{\mathrm{In}}\right)}{P_{\mathrm{Pump}}}& \left(44\right)\\ {ϛ}_{\mathrm{Pos}}=\underset{i=1}{\overset{N}{.Math.}}{x}_i{ϛ}_{i,\mathrm{Pos}}\left(p_{\mathrm{Pos}},T_{\mathrm{Pos}}\right)& \left(45\right)\end{array}$

(57) where x.sub.i is the mass fraction of phase no i.

(58) Further the efficiency η.sub.i may be found from the enthalpy directly as:

(59) $\begin{array}{cc}\eta =\frac{{ϛ}_{\mathrm{Out}}\left(p_{\mathrm{Out}},T_{\mathrm{In}}\right)-{ϛ}_{\mathrm{In}}\left(p_{\mathrm{In}},T_{\mathrm{In}}\right)}{{ϛ}_{\mathrm{Out}}\left(p_{\mathrm{Out}},T_{\mathrm{Out}}\right)-{ϛ}_{\mathrm{In}}\left(p_{\mathrm{In}},T_{\mathrm{In}}\right)}=\frac{\mathrm{\Delta ϛ}\left(\Delta p,T_{\mathrm{In}}\right)}{\mathrm{\Delta ϛ}\left(\Delta p,\Delta T\right)}& \left(46\right)\end{array}$
The enthalpy function may be defined as a polynomial function in p and T:
ζ.sub.i=a.sub.0+a.sub.pp+a.sub.TT+a.sub.pTpT+ . . . +a.sub.MpNTp.sup.MT.sup.n  (47)

(60) By way of a summary a review of an example process is set out below. To ensure robust and accurate flow rate calculations some precautions are necessary as explained above. First, the input measurements P.sub.Pump, p.sub.In, T.sub.In, p.sub.Out, T.sub.Out are obtained. Then these values are checked to ensure that they are suitable, for example by checking that the last value is in line with the measurements during a given preceding period (e.g. not deviate more than a given value from the average value during the last 5 minutes). This ensures that the measurements are stable. It is also possible to adjust for dynamic effects by predicting the measurement(s) based on the input data. An estimator such as a Kalman filter could be used in this case. If the measurements are not suitable then in some cases they might be corrected using a model or the like. Alternatively the process may require a waiting period of the type discussed above to allow for a steady stable state to be achieved.

(61) If the measurements are suitable then the appropriate parts of the above model are utilised to calculate the pump efficiency will be determined. The calculated efficiency should be verified, and if it is in line with manufacturer data then it is accepted. If the efficiency is outside the range, then an alert is made to indicate that something is wrong either with the:

(62) 1. Measured input data (operator should check),

(63) 2. Pump (wear, tear, failure) which should be notified, or

(64) 3. Applied fluid properties (fluid model update may be required)

(65) If desired and the pump specific parameters are available then the fluid viscosity may be directly determined.

(66) In the above analysis since the ESP used as an example has the motor and the pump submerged in the fluid the heat loss can be ignored. The proposed technique can be adapted for any pump or compressor where the applied electric power is measured or can be determined. The same model can be used to find the flow rate provided that the heat loss, Q.sub.Loss, is compensated for. This may be done with a simple model:
Q.sub.Loss≈H.sub.A(T.sub.Out−T.sub.Env)  (48)

(67) The heat transfer coefficient, H.sub.A, may be adapted or determined from the equipment size and location. The environment temperature, T.sub.Env, should also be measured for this situation.

(68) Thus, it will be appreciated that the proposed estimation of flow rate provides accurate results, as evidenced by the comparisons in the Figures, and furthermore that there are various advantages compared to known systems as in US 2013/317762 and US 2015/211906. The fluid viscosity is not required to be measured or estimated, which simplifies the calculations significantly and also allows for a greater range of fluids to be the basis for the calculations, including for example heavy oils. The compressibility (i.e. gas fraction) will not significantly affect the accuracy of the results, which means that the estimation technique provides important advantages in relation to multiphase flow, such as pumping of oil and gas mixtures. In addition, it is not necessary to test the pump with varying fluid viscosity for calibration purposes in order to obtain accurate flow rate estimates, although if suitable calibration data is available then this can advantageously be used to determine the fluid viscosity.