FIELD OF THE INVENTION

This disclosure relates to evaluating fluid flow in an oil or gas well, and more particularly relates to a system and method of evaluating multiphase fluid flow in a wellbore using temperature and pressure measurements.
BACKGROUND OF THE INVENTION

Reliable and accurate downhole temperature and pressure measurements have been available in the petroleum industry for the past several years. Permanent downhole pressure monitoring equipment has now been installed in a number of producing basins around the world, with successful measurement operations exceeding five or more years at this time. Downhole permanent temperature measurements have also become more common, with both conventional or fiber optic thermal measurements currently available for most reservoir conditions. While continuous pressure and temperature readings provide an important part of understanding oil and gas production, quantitative information must typically be obtained using other types of data.

For example, the quantitative evaluation of the production or injection profile in an oil and/or gas well has traditionally involved the use of production log measurements of flow rate, pressure, density, and fluid holdup to derive estimates of the wellbore fluid mixture phase velocities, densities, pressure distributions, and completed interval inflow or outflow contributions. Modern production logs can be used in many situations to obtain the necessary measurements that are required to perform these quantitative computations. The measurements made in these cases however are periodic and reflect the wellbore fluid inflows/outflows at the time that the production log was run. Unfortunately, the known art does not provide a solution to obtain continuous or realtime quantitative measurements and evaluations using downhole pressure and temperature readings obtained from a plurality of sensors in the wellbore.
SUMMARY OF THE INVENTION

The present invention relates to a system, method and program product that provides a computational model and evaluation technique for using array pressure and temperature measurements obtained in a flow conduit to evaluate the phase flow rates and velocities, fluid phase holdup, slip velocities between fluid phases, and mixture density and viscosity. These values can then be used, for instance, to quantify the inflow and outflow contributions of completed zones in a wellbore.

In one embodiment, there is a system for analyzing multiphase flow in a wellbore, comprising: an input system for receiving pressure and temperature readings from a pair of sensors located in the wellbore; a computation system that utilizes a flow analysis model to generate a set of wellbore fluid properties from the pressure and temperature readings, wherein the set of wellbore fluid properties includes at least one of: a fluid mixture value, a phase velocity value, a flow rate, a mixture density, a mixture viscosity, a fluid holdup, and a slip velocity; and a system for outputting the wellbore fluid properties.

In a second embodiment, there is a method for analyzing multiphase flow in a wellbore, comprising: obtaining pressure and temperature readings from a pair of sensors located in the wellbore; utilizing a flow analysis model to generate a set of wellbore fluid properties from the pressure and temperature readings, wherein the set of wellbore fluid properties includes at least one of: a fluid mixture value, a phase velocity value, a flow rate, a mixture density, a mixture viscosity, a fluid holdup, and a slip velocity; and outputting the wellbore fluid properties.

In a third embodiment, there is a computer readable medium for storing a computer program product, which when executed by a computer system analyzes multiphase flow in a wellbore, comprising: program code for inputting pressure and temperature readings from a pair of sensors located in the wellbore; program code for implementing a flow analysis model to generate a set of wellbore fluid properties from the pressure and temperature readings, wherein the set of wellbore fluid properties includes at least one of: a fluid mixture value, a phase velocity value, a flow rate, a mixture density, a mixture viscosity, a fluid holdup, and a slip velocity; and program code for outputting the wellbore fluid properties.

An advantage of this invention is the implementation of a quantitative evaluation methodology for characterizing the temperature, pressure, wellbore fluid mixture density and viscosity, and fluid holdup distributions in a wellbore using the temperature and pressure distributions in the well. This is achieved by the development and use of a comprehensive multiphase capillary flow analysis model. The results provide a reliable, accurate, and continuous characterization of the wellbore fluid flow properties such as pressure, temperature, mixture density, mixture viscosity, fluid phase holdup distributions, and completed zone inflow/outflow contributions.

This invention is directly applicable in wellbore environments and conditions in which modern production logging techniques may not be readily accessible or may not be deployable as a result of the wellbore geometry, well depth, water depth, or other operational and economical considerations.

The illustrative aspects of the present invention are designed to solve the problems herein described and other problems not discussed.
BRIEF DESCRIPTION OF THE DRAWINGS

These and other features of this invention will be more readily understood from the following detailed description of the various aspects of the invention taken in conjunction with the accompanying drawings.

FIG. 1 depicts a computer system having a multiphase flow analysis system in accordance with an embodiment of the present invention.

FIG. 2 depicts an embodiment of a multiphase flow analysis system that provides a contribution analysis in accordance with an embodiment of the present invention.

FIG. 3 depicts a graph showing a Fanning friction factor correlated with the Reynolds number and relative roughness for singlephase flow systems.

FIG. 4 depicts a graph showing a friction factor showing the transition between the laminar and turbulent flow regimes in multiphase flow systems.

The drawings are merely schematic representations, not intended to portray specific parameters of the invention. The drawings are intended to depict only typical embodiments of the invention, and therefore should not be considered as limiting the scope of the invention. In the drawings, like numbering represents like elements.
DETAILED DESCRIPTION OF THE INVENTION

Referring to the drawings, FIG. 1 depicts an overview of an illustrative system 11 for implementing aspects of the present invention. As shown, a computer system 10 is provided that includes a multiphase flow analysis system 18 for analyzing fluid characteristics flowing through a wellbore 34. Also provided are at least two sensors 30, 32 placed within the wellbore to provide multipoint pressure and temperature readings.

Multiphase flow analysis system 18 includes a pressure and temperature input system 20 for obtaining pressure and temperature readings from each sensor 30, 32 in a continuous, as needed, or periodic manner. Also included is a computation system 22 that utilizes a flow analysis model 24 for computing wellbore fluid properties, including one or more of: (1) the fluid mixture; (2) phase velocities; (3) flow rates; (4) mixture density; (5) mixture viscosity; (6) fluid holdups; and (7) estimates of the slip velocities between the wellbore liquid and gas phases and between the oil and water phases, if those phases are present in the system. Wellbore fluid properties 28 may be computed and outputted by output system 29 in an “ondemand” manner, i.e., continuously, as needed, periodically, in realtime, etc. It is also possible to output the wellbore fluid properties when preselected system conditions are reached, such as anomalous incidents or trends, conditions exceeding thresholds, etc. A description of the flow analysis model 24 and how the computations may be implemented is provided below.

Also included in multiphase flow analysis system 18 is a contribution analysis system 26 to quantitatively evaluate an oil or gas well with multiple production or injection zones. For example, FIG. 2 depicts a well 50 having multiple production zones that include a main branch 42, a first contribution branch 44, and a second contribution branch 46. In this case, multipoint measurements are obtained with sets of sensors configured into a multipoint measurements array. In particular, the complex multibranched well 50 is fitted with three sets of sensors (A, B, and C). Each set is strategically located to obtain contribution readings from each different zone in the well. Namely, sensor set C is located to obtain readings for main branch 42; sensor set B is located to obtain contribution readings from main branch 42 and first contribution branch 44; and sensor set A is located to obtain contribution readings from main branch 42, first contribution branch 44, and second contribution branch 46.

A contribution analysis 40 may be obtained for each contribution branch 44, 46 by subtracting all the downstream fluid property computations. For instance, by subtracting computation values obtained from sensor set C from computation values obtained from sensor set B, a contribution analysis 40 for the first contribution branch 44 can be obtained. Similarly, by subtracting computation values obtained from sensor sets B and C from computation values obtained from sensor set A, a contribution analysis 40 for the second contribution branch 46 can be obtained. Contribution analysis 40 for main branch 42 is simply obtained from sensor set C, which has no additional downstream contributions.

Note that each sensor set A, B, C may include more than two sensors in order to provide redundancy. In this example, each set is shown including four sensors, e.g., set A includes sensors A1, A2, A3, and A4. This thus allows six different sensor pairs (e.g., A1A2, A1A3, A1A4, A2A3, A2A4, A3A4) to be used as a basis calculating wellbore fluid properties. Any one or more of the sensor pairs may be used for evaluation purposes. While FIG. 2 depicts a well that has a main branch and first and second contribution branches, embodiments of the invention may also be used with a wellbore having only a main branch with different inflow or outflow zones, such as a cased well having separate perforated intervals.

As noted in FIG. 1, computation system 22 provides a flow analysis model 24 for generating wellbore fluid properties 28 for a sensor pair 30, 32. An explanation for how such properties may be obtained from temperature and pressure readings from sensor pair 30, 32 begins with a review of the fundamental governing relationships that pertain to multiphase fluid flow in a tubular conduit (e.g., a wellbore). The following notation is used throughout the discussion.
Variable Description

 D Flow conduit inside diameter
 f Fanning friction factor
 f_{o }Fraction of oil in liquid component of the system
 f_{w }Fraction of water in liquid component of the system
 g Gravitational acceleration
 H_{L }Liquid holdup
 h_{L }Elevation at end of flow conduit segment
 h_{0 }Elevation at start of flow conduit segment
 L Measured length of the flow conduit segment
 N_{Re }Reynolds number
 P_{L }Pressure at outlet end of flow conduit segment
 P_{0 }Pressure at start end of flow conduit segment
 q_{g }Insitu gas volumetric flow rate
 q_{L }Insitu liquid volumetric flow rate
 V Fluid average velocity in circular conduit
 V_{m }Wellbore fluid mixture superficial velocity, ft/s
 V_{sg }Gas superficial velocity, ft/s
 V_{sgL }Gasliquid slip velocity, ft/s
 V_{sL }Liquid superficial velocity, ft/s
 V_{so }Oil superficial velocity, ft/s
 V_{sow }Oilwater slip velocity, ft/s
 V_{sw }Water superficial velocity, ft/s
 Y_{g }Gas holdup
 Y_{L }Liquid holdup
 Y_{o }Oil holdup
 Y_{w }Water holdup
Greek Description

 α Wellbore deviation angle from vertical, deg
 ε Pipe roughness
 λ_{L }Noslip liquid holdup
 μ_{g }Gas dynamic viscosity, cp
 μ_{L }Liquid dynamic viscosity, cp
 μ_{m }Fluid mixture dynamic viscosity, cp
 μ_{o }Oil dynamic viscosity, cp
 μ_{w }Water dynamic viscosity, cp
 υ_{m }Fluid mixture kinematic viscosity, cpcu ft/lbs
 ρ_{g }Gas density, lbs/cu ft
 ρ_{L }Liquid density, lbs/cu ft
 ρ_{m }Fluid mixture density, lbs/cu ft
 ρ_{o }Oil density, lbs/cu ft
 ρ_{osg }Oil density, g/cc
 ρ_{w }Water density, lbs/cu ft
 ρ_{wsg }Water density, g/cc

One of the fundamental parameters that can be used to quantify and correlate the level of inertial to viscous forces in a fluid flowing in a circular conduit is the Reynolds number. This dimensionless parameter is defined in Eq. 1.

$\begin{array}{cc}{N}_{\mathrm{Re}}=\frac{{\mathrm{DV}}_{m}}{{\upsilon}_{m}}& \left(1\right)\end{array}$

The kinematic viscosity of a fluid mixture appearing in Eq. 1 is defined as the ratio of the dynamic fluid viscosity to its density. This relationship is expressed mathematically in Eq. 2.

$\begin{array}{cc}{\upsilon}_{m}=\frac{{\mu}_{m}}{{\rho}_{m}}& \left(2\right)\end{array}$

The general relationship that describes the pressure loss exhibited due to fluid flow in a circular tubular conduit is given by Fanning's equation. Note that gravitational effects have been included in this expression.

$\begin{array}{cc}\frac{{P}_{0}{P}_{L}+{\rho}_{m}\ue89eg\ue8a0\left({h}_{0}{h}_{L}\right)}{L}=\frac{2\ue89e{\rho}_{m}\ue89e{\mathrm{fV}}_{m}^{2}}{D}& \left(3\right)\end{array}$

Substitution of Eqs. 2 and 3 into Eq. 1 results in expression that can be used to correlate the Reynolds number and friction factor to the conduit dimensions, the pressure loss over a given length of conduit, and the physical properties of the fluid flowing in the conduit. Note that the relationship given in Eq. 4 is explicitly independent of the fluid velocity, except that the effect of this parameter is implicitly manifested in the fluid flow problem in the form of the Reynolds number.

$\begin{array}{cc}\begin{array}{c}{N}_{\mathrm{Re}}\ue89e\sqrt{f}=\ue89e\frac{{D}^{\frac{3}{2}}\ue89e{{\rho}_{m}^{\frac{1}{2}}\ue8a0\left[{P}_{0}{P}_{L}+{\rho}_{m}\ue89eg\ue8a0\left({h}_{0}{h}_{L}\right)\right]}^{\frac{1}{2}}}{\sqrt{2\ue89eL}\ue89e{\mu}_{m}}\\ =\ue89e\frac{{{D}^{\frac{3}{2}}\ue8a0\left[{P}_{0}{P}_{L}+{\rho}_{m}\ue89eg\ue8a0\left({h}_{0}{h}_{L}\right)\right]}^{\frac{1}{2}}}{\sqrt{2\ue89eL\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\rho}_{m}}\ue89e{\upsilon}_{m}}\end{array}& \left(4\right)\end{array}$

The Fanning friction factor encountered in Eqs. 3 and 4 is a function of the Reynolds number and the relative roughness of the conduit in which the flow occurs. The Fanning friction factor is correlated with the Reynolds number and relative roughness as presented in FIG. 3. This friction factor correlation is generally considered to be applicable to singlephase fluid flow. Note that there is an unstable transition regime for Reynolds numbers in the range of 2,000 to 3,000. For Reynolds numbers below about 2,000, laminar flow conditions exist. For Reynolds numbers greater than approximately 3,000, turbulent flow conditions are generally considered to prevail.

Based upon gasliquid experimental data, the friction factor that is applicable for multiphase flow generally tends to have a smooth, continuous transition between the laminar and turbulent flow regimes. This transition regime behavior is depicted in FIG. 4. This transition regime behavior was found to be valid for the typical range of pipe relative roughness values that are commonly found in commercially available oilfield tubular goods

$\left(\frac{\varepsilon}{D}\le 0.004\right).$

Note that in this case, the transition regime is a smooth transition that deviates from that of laminar flow at a Reynolds number of approximately 1,000, characterized by the inertialturbulent friction factor values at higher Reynolds numbers.

The Fanning friction factor that corresponds to the laminar flow regime in FIGS. 3 and 4 (N_{Re}<2000 and N_{Re}<1000, respectively) can be described mathematically with the relationship given in Eq. 5.

$\begin{array}{cc}f=\frac{16}{{N}_{\mathrm{Re}}}& \left(5\right)\end{array}$

The Fanning friction factor that corresponds to the turbulent flow regime (N_{Re}>3,000) can be accurately evaluated using the relationship described in Colebrook, C. F.: “Turbulent Flow in Pipes, with Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws,” J. Inst. Civil Engs., London, (19381939).

The evaluation of this expression requires an iterative numerical solution procedure and is presented in Eq. 6.

$\begin{array}{cc}\frac{1}{\sqrt{f}}=4\ue89e\mathrm{log}\ue8a0\left(0.269\ue89e\frac{\varepsilon}{D}+\frac{1.255}{{N}_{\mathrm{Re}}\ue89e\sqrt{f}}\right)& \left(6\right)\end{array}$

In addition to the governing fluid flow relationships presented thus far, a conservation of mass relationship for the fluids present in the system can also be defined. The fluid mixture density is generally computed in multiphase flow analyses in the manner depicted in Eq. 7. However, an alternate form of this relationship for flow in horizontal circular conduits is described in Dukler, A. E.: “GasLiquid Flow in Pipelines,” AGA, API, Vol. I, Research Results, (May 1969). That expression is given in Eq. 8.

$\begin{array}{cc}{\rho}_{m}={\rho}_{L}\ue89e{Y}_{L}+{\rho}_{g}\ue8a0\left(1{Y}_{L}\right)& \left(7\right)\\ {\rho}_{m}={\rho}_{L}\ue8a0\left(\frac{{\lambda}_{L}^{2}}{{Y}_{L}}\right)+{\rho}_{g}\left(\frac{{\left(1{\lambda}_{L}\right)}^{2}}{1{Y}_{L}}\right)& \left(8\right)\end{array}$

The noslip liquid holdup is utilized in Dukler's alternate fluid mixture density relationship. The noslip liquid holdup is defined in Eq. 9.

$\begin{array}{cc}{\lambda}_{L}=\frac{{q}_{L}}{{q}_{L}+{q}_{g}}& \left(9\right)\end{array}$

In a similar manner, the fluid mixture dynamic viscosity can be evaluated by various means. Hagedorn, A. R. and Brown, K. E.: “The Effect of Liquid Viscosity in Vertical TwoPhase Flow,” JPT, (Feb. 1964), 203, suggested that the fluid mixture viscosity in a multiphase flow system should be evaluated in the manner given by Eq. 10.

$\begin{array}{cc}{\mu}_{m}={\mu}_{L}^{{Y}_{L}}\ue89e{\mu}_{g}^{1{Y}_{L}}& \left(10\right)\end{array}$

The fluid mixture dynamic viscosity has been more commonly estimated in previous investigations of multiphase fluid flow using a holdupweighted combination of the liquid and gas viscosities, given by Eq. 11.

μ_{m}=μ_{L} Y _{L}+μ_{g}(1−Y _{L}) (11)

A relationship for the fluid mixture dynamic viscosity that is identical to that given in Eq. 11 has been proposed, except that the noslip liquid holdup is the weighting parameter used in those analyses rather than the slipadjusted liquid holdup.

μ_{m}=μ_{L}λ_{L}+μ_{g}(1−λ_{L}) (12)

Where required, the kinematic viscosity (Eq. 3) can be evaluated using the fluid mixture density obtained with Eqs. 7 or 8 and dynamic fluid mixture viscosity evaluated with Eqs. 10, 11, or 12. An alternative approach is to evaluate the kinematic viscosity of the fluid mixture in a manner analogous to that used for the holdupweighted mixture density and viscosity. This expression is given in Eq. 13.

$\begin{array}{cc}{\upsilon}_{m}=\frac{{\mu}_{L}}{{\rho}_{L}}\ue89e{Y}_{L}+\frac{{\mu}_{g}}{{\rho}_{g}}\ue89e\left(1{Y}_{L}\right)& \left(13\right)\end{array}$

Regardless of the particular fluid mixture density and viscosity relationship used in the analysis, most of the previous investigations of multiphase flow in pipe have evaluated the liquid density and dynamic viscosity of oil and water mixtures using the relationships given in Eqs. 14 and 15. Note that other mixture viscosity models may be used in the analysis such a medium emulsion model for oilwater vertical flow systems.

ρ_{L}=ρ_{o}f_{o}+ρ_{w}f_{w } (14)

μ_{L}=μ_{o}f_{o}+μ_{w}f_{w } (15)

Typically when these relationships for computing the liquid density and dynamic viscosity of oilwater systems are used, the fraction of oil and water are often evaluated assuming noslip conditions. However, a similar analysis could also be performed using an appropriate slip relationship between the water and the less dense oil phase in the system without any loss in generality. In addition, the slip velocity relationship between the oil and water phases in a twophase liquid flow system can be reliably determined using Eq. 16. An illustrative embodiment provided herein utilizes this relationship (Eq. 16) for the oilwater slip velocity for wellbore inclinations up to about 70 degrees, but the invention may also use other applicable oilwater slip velocity correlations. This disclosure includes but is not limited to the use of only a single oilwater slip velocity relationship in the invention.

$\begin{array}{cc}{V}_{\mathrm{sow}}=0.6569\ue89e{\left({\rho}_{\mathrm{wsg}}{\rho}_{\mathrm{osg}}\right)}^{0.25}\ue89e\mathrm{exp}\ue8a0\left[0.788\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{ln}\ue8a0\left(\frac{1.85}{{\rho}_{\mathrm{wsg}}{\rho}_{\mathrm{osg}}}\right)\ue89e\left(1\frac{{Y}_{w}}{{Y}_{L}}\right)\right]\ue89e\left(1+0.04\ue89e\alpha \right)& \left(16\right)\end{array}$

The fundamental definition of the slip velocity between the oil and water phases in a twophase oilwater system is given by Eq. 17. It is noted that the slip velocity between the oil and water phases is simply the difference between the average velocities of the oil and water phases. Note that when the definition of the slip velocity between the oil and water phases (twophase relationship) is applied to a threephase analysis (as is considered in this invention), the holdups of the oil and water phases must be normalized by the liquid holdup in the threephase system. This normalization of the oil and water phase holdups to the liquid holdup (oil+water) in a threephase system is presented in Eq. 17.

$\begin{array}{cc}{V}_{\mathrm{sow}}=\frac{{V}_{\mathrm{so}}\ue89e{Y}_{L}}{{Y}_{o}}\frac{{V}_{\mathrm{sw}}\ue89e{Y}_{L}}{{Y}_{w}}& \left(17\right)\end{array}$

A similar slip velocity relationship exists between the gas and liquid phases in a multiphase system. An accurate and reliable correlation for estimating the slip velocity between the gas and liquid phases in a multiphase system is given in Eq. 18. Other gasliquid slip velocity relationships may also be used in the computational analysis described in this invention disclosure. While the gasliquid slip velocity relationship given in Eq. 18 provides an illustrative embodiment, the use of other applicable gasliquid slip velocity correlations may also be utilized and fall within the scope of this invention.

V _{sgL}=[(0.95−Y _{g} ^{2})^{0.5}+0.025](1+0.04α) (18)

The fundamental definition of the slip velocity relationship between the gas and liquid phases in a multiphase system is given by Eq. 19.

$\begin{array}{cc}{V}_{\mathrm{sgL}}=\frac{{V}_{\mathrm{sg}}}{{Y}_{g}}\frac{{V}_{\mathrm{sL}}}{{Y}_{L}}& \left(19\right)\end{array}$

The liquid mixture superficial velocity in a multiphase system is the sum of the oil and water superficial velocities.

V _{sL} =V _{so} +V _{sw } (20)

The sum of the holdups of each of the fluid phases must total to 1, the sum of all of the fluids in the system.

1=Y _{o} +Y _{w} +Y _{g} =Y _{L} +Y _{g } (21)

The wellbore mixture fluid superficial velocity is the sum of the superficial velocities of each of the fluid phases present in the system.

V _{m} =V _{so} +V _{sw} +V _{sg} =V _{sL} +V _{sg } (22)

The wellbore fluid mixture kinematic viscosity can be evaluated as the sum of the kinematic viscosities of each of the fluid phases and their associated fluid holdups.

$\begin{array}{cc}{\upsilon}_{m}=\frac{{\mu}_{o}\ue89e{Y}_{o}+{\mu}_{w}\ue89e{Y}_{w}+{\mu}_{g}\ue89e{Y}_{g}}{{\rho}_{o}\ue89e{Y}_{o}+{\rho}_{w}\ue89e{Y}_{w}+{\rho}_{g}\ue89e{Y}_{g}}& \left(23\right)\end{array}$

A final governing relationship that may be utilized to resolve the unknowns in the fluid flow problem is an expression relating the insitu mixture density directly to the measured pressure and temperature, and the composition of the fluids in the system. This relationship can be an equationofstate, such as the model proposed by Peng and Robinson. Other equationsof state can also be used to evaluate the mixture density and fluid mixture viscosity at the insitu conditions of temperature and pressure, for a given composition of wellbore fluid.
Evaluation of SinglePhase Flow Metering Parameters

In singlephase flow metering cases, the evaluation of the fluid flow parameters involves the solution of three equations for three unknown parameter values in the problem. In singlephase flow, the unknown parameters that must be determined in the analysis are the fluid superficial velocity (V_{si}), the Reynolds number (N_{Re}), and the Fanning friction factor (f). The i subscript appearing on the phase superficial velocity and fluid properties in Eq. A1 represents the individual fluid phase (oil, gas, or water: i.e. o, g, or w). The definition of the singlephase Reynolds number in conventional oilfield units is given in Eq. A1.

$\begin{array}{cc}{N}_{\mathrm{Re}}=\frac{124\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89eD\ue89e\uf603{V}_{\mathrm{si}}\uf604}{{\gamma}_{i}}=\frac{124\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89eD\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\rho}_{i}\ue89e\uf603\mathrm{Vsi}\uf604}{{\mu}_{i}}& \left(A\ue89e\text{}\ue89e1\right)\end{array}$

The Fanning friction factor for singlephase flow conditions is determined from FIG. 3. The Fanning friction is a function of the Reynolds number and the relative pipe roughness.

For laminar flow conditions (N_{Re}<2,000), the Fanning friction factor given in FIG. 3 is defined by the relationship given in Eq. A2.

$\begin{array}{cc}f=\frac{16}{{N}_{\mathrm{Re}}}& \left(A\ue89e\text{}\ue89e2\right)\end{array}$

Under turbulent flow conditions (N_{R}>3,000), the Fanning friction factor can be determined using the nonlinear ColebrookWhite relationship given in Eq. A3.

$\begin{array}{cc}\frac{1}{\sqrt{f}}=4\ue89e\mathrm{log}\ue8a0\left(0.269\ue89e\frac{\varepsilon}{D}+\frac{1.255}{{N}_{\mathrm{Re}}\ue89e\sqrt{f}}\right)& \left(A\ue89e\text{}\ue89e3\right)\end{array}$

The final relationship that is used to resolve the unknowns in the singlephase flow metering problem is the capillary flow relationship that relates the pressure loss due to frictional and gravitational effects of flow in the conduit to the Reynolds number, fluid properties, and relative pipe roughness is given in Eq. A4 using conventional oilfield units.

$\begin{array}{cc}{N}_{\mathrm{Re}}\ue89e{f}^{1/2}=\frac{1722.9\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e{D}^{3/2}\ue89e{{\rho}_{i}^{1/2}\ue8a0\left[{P}_{0}{P}_{L}+0.006945\ue89e{\rho}_{i}\ue89eL\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta \right]}^{1/2}}{{L}^{1/2}\ue89e{\mu}_{i}}& \left(A\ue89e\text{}\ue89e4\right)\end{array}$

The solution of these relationships for the three unknowns in the problem may for example be accomplished using a nonlinear rootsolving procedure such as SecantNewton. The parameter of variation in the rootsolving procedure is the superficial velocity (V_{si}). With the singlephase fluid physical properties (μ_{i}, ρ_{i}) known as a function of the pressure and temperature, the superficial velocity is used to determine the Reynolds number as defined in Eq. A1, the Fanning friction factor from FIG. 3 (or Eqs. A2 or A3), and the basis function constructed by rearranging the capillary flow relationship given in Eq. A4.

Note that for a singlephase system, the oilwater and gasliquid slip velocities are of course equal to zero. The same is true of the superficial velocities and holdups of the fluid phases not present in the singlephase system.
Evaluation of OilWater TwoPhase Flow Metering Parameters

The solution of twophase flow metering computations using temperature and pressure measurements described in this invention involve the resolution of a nonlinear system of 10 independent relationships for the 10 unknown parameters in the problem. This is true in oilwater, oilgas, and watergas twophase flow metering analyses using distributed temperature and pressure measurements.

For an oilwater system, the unknowns that must be resolved in the analysis are the oil and water holdups, the oil, water, and mixture superficial velocities, mixture density and dynamic viscosity, the slip velocity between the oil and water phases, the Reynolds number and Fanning friction factor, and pressure loss that occurs over the metering length of the flow conduit. Note that the gas holdup and superficial velocity are equal to zero for an oilwater system, as is the gasliquid slip velocity.

The first relationship that is used to construct the multiphase flow metering analysis in oilwater systems is the holdup relationship given in Eq. B1.

1=Y _{o} +Y _{w } (B1)

The mixture density in oilwater twophase flow can be defined by the expression given in Eq. B2.

ρ_{m}=ρ_{o} Y _{o}+ρ_{w} Y _{w } (B2)

The simultaneous solution of Eqs. B1 and B2 results in expressions for the oil and water holdups, expressed in terms of the unknown mixture density.

$\begin{array}{cc}{Y}_{o}=\frac{{\rho}_{w}{\rho}_{m}}{{\rho}_{w}{\rho}_{o}}& \left(B\ue89e\text{}\ue89e3\right)\\ {T}_{w}=\frac{{\rho}_{m}{\rho}_{o}}{{\rho}_{w}{\rho}_{o}}& \left(B\ue89e\text{}\ue89e4\right)\end{array}$

The twophase oilwater flow mixture dynamic viscosity for nonemulsion fluid systems may be expressed by the relationship given in Eq. B3.

μ_{m}=μ_{o} Y _{o}+μ_{w} Y _{w } (B5)

In terms of the unknown mixture density, the mixture viscosity is defined as given in Eq. B6.

$\begin{array}{cc}{\mu}_{m}=\frac{{\mu}_{o}\ue8a0\left({\rho}_{w}{\rho}_{m}\right)+{\mu}_{w}\ue8a0\left({\rho}_{m}{\rho}_{o}\right)}{{\rho}_{w}{\rho}_{o}}& \left(B\ue89e\text{}\ue89e6\right)\end{array}$

The mixture superficial velocity of an oilwater twophase system is the sum of the superficial velocities of the oil and water phases.

V _{m} =V _{so} +V _{sw } (B7)

The superficial mass velocity of a twophase oilwater flow stream is best characterized using Eqs. B2, B7, and an equationofstate. An expression that relates Eqs. B2 and B7 to the mass velocity is given by Eq. B8.

ρ_{m} V _{m}=(ρ_{o} Y _{o}+ρ_{w} Y _{w})(V _{so} +V _{sw}) (B8)

Expressions for estimating the oil and water superficial velocities expressed in terms of the mixture superficial velocity and density may be obtained using the definition of the mass velocity given in Eq. B8, in combination with an independent equationofstate for computing the mixture density using the temperature, pressure, and fluid composition. With the two measurements (differential pressure and temperature), two parameters may be resolved in the oilwater twophase system analysis, the mixture density and the velocity.

A slip velocity relationship that is applicable for oil and water multiphase systems is presented in Eq. B9, expressed in terms of conventional oilfield units. This relationship relates the slip between the oil and water phases to the differences in densities of the two fluids and the conduit inclination angle.

$\begin{array}{cc}{V}_{\mathrm{sow}}=0.6569\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\left(\frac{{\rho}_{w}{\rho}_{o}}{62.428}\right)}^{0.25}\ue89e\mathrm{exp}\ue8a0\left[0.788\ue89e\mathrm{ln}\ue8a0\left(\frac{115.5}{{\rho}_{w}{\rho}_{o}}\right)\ue89e\left(\frac{{\rho}_{w}{\rho}_{o}}{{\rho}_{w}{\rho}_{o}}\right)\right]\ue89e\left(1+0.04\ue89e\alpha \right)& \left(B\ue89e\text{}\ue89e9\right)\end{array}$

The Reynolds number of oilwater twophase flow in a circular conduit is given by Eq. B10.

$\begin{array}{cc}{N}_{\mathrm{Re}}=\frac{124\ue89e{\rho}_{m}\ue89eD\ue89e\uf603{V}_{m}\uf604}{{\mu}_{m}}& \left(B\ue89e\text{}\ue89e10\right)\end{array}$

Substitution of the oil and water superficial velocities, holdups (Eqs. B3 and B4), and the oilwater slip relationship into the definition of the Reynolds number (Eq. B9), results in an expression for Reynolds number that represents one component of the rootsolving procedure basis function.
The resulting expression can be used in conjunction with the capillary flow relationship for a twophase oilwater system, given in Eq. B11, to construct a basis function for a nonlinear rootsolving procedure with the mixture density as the variable parameter.

$\begin{array}{cc}{N}_{\mathrm{Re}}=\frac{1722.9\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e{D}^{3/2}\ue89e{{\rho}_{m}^{1/2}\ue8a0\left({\rho}_{w}{\rho}_{o}\right)\ue8a0\left[\begin{array}{c}{P}_{0}{P}_{L}+\\ 0.006945\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e{\rho}_{m}\ue89eL\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta \end{array}\right]}^{1/2}}{\sqrt{\mathrm{Lf}}\ue8a0\left[{\mu}_{o}\ue8a0\left({\rho}_{w}{\rho}_{m}\right)+{\mu}_{w}\ue8a0\left({\rho}_{m}{\rho}_{o}\right)\right]}& \left(B\ue89e\text{}\ue89e11\right)\end{array}$

Once the system mixture density has been determined with the rootsolving procedure, the oilwater system slip velocity is evaluated with Eq. B19, and the mixture velocity is evaluated with Eq. B12.

$\begin{array}{cc}{V}_{m}=\frac{\left[{\rho}_{w}\ue8a0\left({\rho}_{o}{\rho}_{m}\right)+{\rho}_{o}\ue8a0\left({\rho}_{w}{\rho}_{m}\right)\right]\ue89e{V}_{\mathrm{sow}}}{{\rho}_{w}^{2}{\rho}_{o}^{2}}& \left(B\ue89e\text{}\ue89e12\right)\end{array}$

The Reynolds number can then be determined with Eq. B10 and the Fanning friction factor (f) is obtained with FIG. 4, which shows the Fanning friction factor for multiphase flow systems (or with the laminar or ColebrookWhite turbulent flow relationships).

The oil and water phase superficial velocities are subsequently evaluated using expressions derived from the mixture and mass velocity relationships (Eqs. B7 and B8), and the oil and water holdups are evaluated with Eqs. B3 and B4. The mixture dynamic viscosity can then be readily evaluated using Eq. B5 or B6.
Evaluation of OilGas TwoPhase Flow Metering Parameters

Metering analyses using distributed temperature and pressure measurements in a twophase oilgas system involves the determination of 10 unknown parameter values using 10 independent relationships, some of which are nonlinear and/or piecewise continuous. The unknown parameters that must be resolved in an oilgas twophase system analysis are the following: oil and gas holdups, superficial velocities, the mixture superficial velocity, density and viscosity, and the gasliquid slip velocity, Reynolds number and Fanning friction factor. The water holdup and superficial velocity in this case are equal to zero, as is the oilwater slip velocity. Essentially with the two physical measurements that are being made in this case, the differential pressure and the temperature, the mixture density and superficial velocity can be resolved.

The holdup relationship for a twophase oilgas system is given in Eq. C1.

1=Y _{o} +Y _{g } (C1)

The mixture density is defined as in Eq. C2.

ρ_{m}=ρ_{o} Y _{o}+ρ_{g} Y _{g } (C2)

The solution of these two relationships results in expressions for the oil and gas holdups in terms of the mixture density.

$\begin{array}{cc}{Y}_{o}=\frac{{\rho}_{m}{\rho}_{g}}{{\rho}_{o}{\rho}_{g}}& \left(C\ue89e\text{}\ue89e3\right)\\ {Y}_{g}=\frac{{\rho}_{o}{\rho}_{m}}{{\rho}_{o}{\rho}_{g}}& \left(C\ue89e\text{}\ue89e4\right)\end{array}$

The mixture viscosity in a twophase oilgas system is generally defined in one of two ways, with the more common relationship given in Eq. C5 or with the HagedornBrown model given in Eq. C6.

μ_{m}=μ_{o} Y _{o}+μ_{g} Y _{g } (C5)

μ_{m}=μ_{o} ^{Y} ^{ o }μ_{g} ^{Y} ^{ g } (C6)

These expressions can be readily rewritten in terms of the oil and gas holdups given in Eqs. C3 and C4 as functions of the mixture density.

$\begin{array}{cc}{\mu}_{m}=\frac{{\mu}_{o}\ue8a0\left({\rho}_{m}{\rho}_{g}\right)+{\mu}_{g}\ue8a0\left({\rho}_{o}{\rho}_{m}\right)}{{\rho}_{o}{\rho}_{g}}& \left(C\ue89e\text{}\ue89e7\right)\\ {\mu}_{m}={\mu}_{o}^{\left(\frac{{\rho}_{m}{\rho}_{g}}{{\rho}_{o}{\rho}_{g}}\right)}\ue89e{\mu}_{g}^{\left(\frac{{\rho}_{o}{\rho}_{m}}{{\rho}_{o}{\rho}_{g}}\right)}& \left(C\ue89e\text{}\ue89e8\right)\end{array}$

The mixture superficial velocity of the twophase system is defined in Eq. C9, with the superficial mass velocity being evaluated with Eq. C10.

V _{m} =V _{so} +V _{sg } (C9)

ρ_{m} V _{m}=(ρ_{o} Y _{o}+ρ_{g} Y _{g})(V _{so} +V _{sg}) (C10)

Solution of Eqs. C9 and C10, with substitution of the previously determined relationships for the holdups (Eqs. C3 and C4), the oil and gas superficial velocities can be expressed in terms of the mixture density and superficial velocity. The mixture density in this case is best characterized using an accurate equationofstate to determine the densities of the liquid and vapor phases in the system.
The gasliquid slip velocity relationship is defined for an oilgas twophase system as shown in Eq. C11.

$\begin{array}{cc}\frac{{V}_{\mathrm{sg}}}{{Y}_{g}}\frac{{V}_{\mathrm{so}}}{{Y}_{o}}=\left[{\left(0.95{Y}_{g}^{2}\right)}^{0.5}+0.025\right]\ue89e\left(1+0.04\ue89e\alpha \right)& \left(C\ue89e\text{}\ue89e11\right)\end{array}$

The Reynolds number of a twophase oilgas flow is determined using Eq. C12.

$\begin{array}{cc}{N}_{\mathrm{Re}}=\frac{124\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e{\rho}_{m}\ue89eD\ue89e\uf603{V}_{m}\uf604}{{\mu}_{m}}& \left(C\ue89e\text{}\ue89e12\right)\end{array}$

Substitution of the mixture superficial velocity given by Eq. C9 into the Reynolds number relationship (Eq. C12) yields an expression that can be used to construct a basis function for a rootsolving procedure to evaluate the unknown parameter values in the oilgas twophase flow metering problem.

Another expression for the Reynolds number can be obtained from the capillary flow relationship that describes the pressure differential in the flow conduit due to frictional and gravitational effects. This relationship is given in Eq. C13 and is used to complete the construction of the rootsolving basis function used in the analysis. Note that the Fanning friction factor appearing in Eq. C13 is obtained from FIG. 4 as a function of the Reynolds number and the relative pipe roughness, or by the solution of the laminar or turbulent flow relationships that correspond to the graphical solution of the Fanning friction factor.

$\begin{array}{cc}{N}_{\mathrm{Re}}=\frac{1722.9\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e{D}^{3/2}\ue89e{{\rho}_{m}^{1/2}\ue8a0\left({\rho}_{o}{\rho}_{g}\right)\ue8a0\left[\begin{array}{c}{P}_{0}{P}_{L}+\\ 0.006945\ue89e{\rho}_{m}\ue89eL\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta \end{array}\right]}^{1/2}}{\sqrt{\mathrm{Lf}}\ue8a0\left[{\mu}_{o}\ue8a0\left({\rho}_{m}{\rho}_{g}\right)+{\mu}_{g}\ue8a0\left({\rho}_{o}{\rho}_{m}\right)\right]}& \left(C\ue89e\text{}\ue89e13\right)\end{array}$

The unknown parameter used as the variable of the rootsolving procedure in this analysis is the mixture density. Once the mixture density has been determined, the Reynolds number can be readily evaluated using either Eq. C12 or C13. The mixture superficial velocity is then evaluated with Eq. C9. The oil and gas phase holdups may be determined with Eqs. C3 and C4, followed by the mixture dynamic viscosity computed with Eq. C5.

A similar solution methodology can be developed for the alternate mixture viscosity relationship given in Eq. C6, as that given when Eq. C5 is used. The solution methodology developed in this invention is applicable in general for all oilgas twophase flow cases. Substitution for an alternate dynamic viscosity or gasliquid slip velocity relationship is permitted in the analysis.
Evaluation of WaterGas TwoPhase Flow Metering Parameters

In a watergas twophase flow metering system developed using distributed temperature and pressure measurements, the evaluation of the 10 unknown parameters require the use of 10 independent functional relationships involving those parameters in order to resolve the multiphase flow metering problem. The unknown parameter values that must be determined from the metering analysis are the water and gas holdups, the water and gas superficial velocities, the mixture superficial velocity, density and dynamic viscosity, the gasliquid slip velocity, Reynolds number and Fanning friction factor. In a manner similar to that described previously for the other twophase flow metering analyses, a nonlinear rootsolving procedure is required to resolve the unknowns of the problem. Note that in a twophase watergas flow metering analysis, the oil holdup and superficial velocity are equal to zero, as well as is the oilwater slip velocity.

The holdup relationship that is applicable for a twophase watergas metering analysis is given by Eq. D1.

1=Y _{w} +Y _{g } (D1)

The mixture density of the watergas twophase flow is defined by Eq. D2.

ρ_{m}=ρ_{w} Y _{w}+ρ_{g} Y _{g } (D2)

Simultaneous solution of Eqs. D1 and D2 results in expressions for the water and gas holdups, expressed in terms of the fluid mixture density of the watergas system.

$\begin{array}{cc}{Y}_{w}=\frac{{\rho}_{m}{\rho}_{g}}{{\rho}_{w}{\rho}_{g}}& \left(D\ue89e\text{}\ue89e3\right)\\ {Y}_{g}=\frac{{\rho}_{w}{\rho}_{m}}{{\rho}_{w}{\rho}_{g}}& \left(D\ue89e\text{}\ue89e4\right)\end{array}$

There are at least two fluid mixture viscosity relationships that can be used for characterizing the dynamic fluid viscosity in a watergas twophase metering analysis. The more commonly used of these is the relationship given in Eq. D5, with an alternate fluid mixture viscosity relationship proposed by Hagedorn and Brown given in Eq. D6.

μ_{m}=μ_{w} Y _{w}+μ_{g} Y _{g } (D5)

μ_{m}=μ_{w} ^{Y} ^{ w }μ_{g} ^{Y} ^{ g } (D6)

Application of the holdup relationships obtained in Eqs. D3 and D4 in the mixture viscosity model given by Eq. D5, results in a fluid mixture viscosity relationship that is only a function of the unknown fluid mixture density.

$\begin{array}{cc}{\mu}_{m}=\frac{{\mu}_{w}\ue8a0\left({\rho}_{m}{\rho}_{g}\right)+{\mu}_{g}\ue8a0\left({\rho}_{w}{\rho}_{m}\right)}{{\rho}_{w}{\rho}_{g}}& \left(D\ue89e\text{}\ue89e7\right)\end{array}$

The mixture superficial velocity in a watergas twophase flow metering analysis is the sum of the water and gas phase superficial velocities.

V _{m} =V _{sw} +V _{sg } (D8)

The superficial mass velocity in the watergas system can be evaluated as defined in Eq. D9.

ρ_{m} V _{m}=(ρ_{w} Y _{w}+ρ_{g} Y _{g})(V _{sw} +V _{sg}) (D9)

Solution of Eqs. D8 and D9, with the definitions of the water and gas holdups previously obtained in Eqs. D3 and D4, the superficial velocity of the water and gas phases can be expressed in terms of the mixture density and superficial velocity.
The gasliquid slip velocity can be evaluated using the slip velocity relationship presented in Eq. D10.

$\begin{array}{cc}\frac{{V}_{\mathrm{sg}}}{{Y}_{g}}\frac{{V}_{\mathrm{sw}}}{{Y}_{w}}=\left[{\left(0.95{Y}_{g}^{2}\right)}^{0.5}+0.025\right]\ue89e\left(1+0.04\ue89e\alpha \right)& \left(D\ue89e\text{}\ue89e10\right)\end{array}$

The fluid mixture superficial velocity is given in Eq. D8 and the Reynolds number for a watergas multiphase flow is defined by the relationship given in Eq. D11.

$\begin{array}{cc}{N}_{\mathrm{Re}}=\frac{124\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e{\rho}_{m}\ue89eD\ue89e\uf603{V}_{m}\uf604}{{\mu}_{m}}& \left(D\ue89e\text{}\ue89e11\right)\end{array}$

Substitution of the results of mixture dynamic viscosity (Eq. D7) and superficial velocity (Eq. D8) in the Reynolds number relationship results in one component of the basis function for evaluating the unknowns in the multiphase metering problem.

The other component of the basis function (alternate Reynolds number relationship) is obtained from the capillary flow relationship that relates the pressure differential observed in flow in a conduit to the frictional and gravitational effects.

$\begin{array}{cc}{N}_{\mathrm{Re}}=\frac{1722.9\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e{D}^{3/2}\ue89e{{\rho}_{m}^{1/2}\ue8a0\left({\rho}_{w}{\rho}_{g}\right)\ue8a0\left[\begin{array}{c}{P}_{0}{P}_{L}+\\ 0.006945\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e{\rho}_{m}\ue89eL\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta \end{array}\right]}^{1/2}}{\sqrt{\mathrm{Lf}}\ue8a0\left[{\mu}_{w}\ue8a0\left({\rho}_{m}{\rho}_{g}\right)+{\mu}_{g}\ue8a0\left({\rho}_{w}{\rho}_{m}\right)\right]}& \left(D\ue89e\text{}\ue89e12\right)\end{array}$

With the fluid mixture density obtained from the rootsolving procedure just described, the Reynolds number can then be determined using either Eq. D11 or D12. With the two independent measurements available (differential pressure and temperature) two parameters of the problem can be resolved. These are the mixture density and the superficial velocity. A mixture density can be derived from the constituitive relationships of the problem, including the capillary flow relationship and the differential pressure measurements. The pressure and temperature also provides a means of computing the mixture density under these conditions as a function of the fluid composition using an accurate and robust equationofstate.
Evaluation of ThreePhase Flow Metering Parameters

The evaluation of the unknown metering parameters in a threephase system (oil, gas, and water) using distributed temperatures and pressures is by far the most difficult to implement in a stable numerical solution procedure due to the complex relationships between the slip velocities, holdups, mixture viscosities, and superficial velocities of the phases present in the system. The unknowns in the threephase metering analysis include the holdups of all three phases, their superficial velocities, as well as the mixture superficial velocity, the mixture density, viscosity, Reynolds number and friction factor, in addition to the wateroil and gasliquid slip velocities of the system. There are a total of 13 unknowns in the threephase metering analysis problem. Therefore, a total of 13 independent relationships are required to properly resolve the unknowns in the threephase flow metering analysis using distributed temperature and pressure measurements.

As was demonstrated with the twophase flow problems above, the holdup relationship is the first fundamental independent relationship that is used to construct the system of equations in the analysis. The threephase holdup relationship is given by Eq. E1.

1=Y _{o} +Y _{w} +Y _{g } (E1)

The fluid mixture density is defined in the threephase analysis with Eq. E2.

ρ_{m}=ρ_{o} Y _{o}+ρ_{w} Y _{w}+ρ_{g} Y _{g } (E2)

The fluid mixture dynamic viscosity is commonly evaluated using the model presented in Eq. E3.

μ_{m}=μ_{o} Y _{o}+μ_{w} Y _{w}+μ_{g} Y _{g } (E3)

An alternate expression for estimating the fluid mixture dynamic viscosity has been proposed by Hagedorn and Brown. The Hagedorn and Brown fluid mixture viscosity model is given in Eq. E4.

$\begin{array}{cc}{\mu}_{m}={\left(\frac{{\mu}_{o}\ue89e{Y}_{o}+{\mu}_{w}\ue89e{Y}_{w}}{{Y}_{o}+{Y}_{w}}\right)}^{\left({Y}_{o}+{Y}_{w}\right)}\ue89e{\mu}_{g}^{{Y}_{g}}& \left(E\ue89e\text{}\ue89e4\right)\end{array}$

One fluid mixture relationship that has been found to characterize the kinematic viscosity of the threephase system reasonably well is given by Eq. E5. The kinematic viscosity is defined as the ratio of the dynamic viscosity to the fluid mixture density.

$\begin{array}{cc}{\gamma}_{m}=\frac{{\mu}_{m}}{{\rho}_{m}}=\frac{{\mu}_{o}\ue89e{Y}_{o}+{\mu}_{w}\ue89e{Y}_{w}+{\mu}_{g}\ue89e{Y}_{g}}{{\rho}_{o}\ue89e{Y}_{o}+{\rho}_{w}\ue89e{Y}_{w}+{\rho}_{g}\ue89e{Y}_{g}}& \left(E\ue89e\text{}\ue89e5\right)\end{array}$

The simultaneous solution of Eqs. E1 through E5 can be used to develop expressions for the three fluid phase holdups and the mixture viscosity, expressed in terms of the unknown mixture density. The mixture kinematic viscosity is a sum of the kinematic viscosities of the individual phases, the gas holdup can then be evaluated as a function of the mixture density and dynamic viscosity.

The water holdup may then be evaluated using Eq. E6 as a function of the mixture density and viscosity, and the gas holdup. The oil phase holdup can subsequently be computed from the fundamental holdup relationship given in Eq. E1 using the results of the gas holdup and E6.

$\begin{array}{cc}{Y}_{w}=\frac{{\rho}_{m}{\rho}_{o}{Y}_{g}\ue8a0\left({\rho}_{g}{\rho}_{o}\right)}{{\rho}_{w}{\rho}_{o}}& \left(E\ue89e\text{}\ue89e6\right)\end{array}$

The threephase flow metering analysis solution procedure next addresses the issue of the fluid phase and mixture superficial velocities and the two sets of twophase slip velocity relationships that are required in the analysis of a threephase flow system. The slip velocity relationships that are applicable for the oil and water phases in a threephase analysis are given by Eqs. E7 and E8.

$\begin{array}{cc}\phantom{\rule{4.4em}{4.4ex}}\ue89e{V}_{\mathrm{sow}}=\frac{{V}_{\mathrm{so}}\ue8a0\left({Y}_{o}+{Y}_{w}\right)}{{Y}_{o}}\frac{{V}_{\mathrm{sw}}\ue8a0\left({Y}_{o}+{Y}_{w}\right)}{{Y}_{w}}& \left(E\ue89e\text{}\ue89e7\right)\\ {V}_{\mathrm{sow}}=0.6569\ue89e{\left(\frac{{\rho}_{w}{\rho}_{o}}{62.428}\right)}^{0.25}\ue89e\mathrm{exp}\left[\begin{array}{c}0.788\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e\mathrm{ln}\ue8a0\left(\frac{115.5}{{\rho}_{w}{\rho}_{o}}\right)\\ \left(\frac{{Y}_{o}}{{Y}_{o}+{Y}_{w}}\right)\end{array}\right]\ue89e\left(1+\mathrm{0.04`}\ue89e\alpha \right)& \left(E\ue89e\text{}\ue89e8\right)\end{array}$

The gasliquid slip velocity relationships that are applicable in threephase flow analyses are presented in Eqs. E9 and E10.

$\begin{array}{cc}{V}_{\mathrm{sgL}}=\frac{{V}_{\mathrm{sg}}}{{Y}_{g}}\frac{{V}_{\mathrm{so}}+{V}_{\mathrm{sw}}}{{Y}_{o}+{Y}_{w}}& \left(E\ue89e\text{}\ue89e9\right)\\ {V}_{\mathrm{sgL}}=\left[{\left(0.95{Y}_{g}^{2}\right)}^{0.5}+0.025\right]\ue89e\left(1+0.04\ue89e\alpha \right)& \left(E\ue89e\text{}\ue89e10\right)\end{array}$

The threephase mixture superficial velocity is given by Eq. E11.

V _{m} =V _{so} +V _{sw} +V _{sg } (E11)

The mass superficial velocity can best be characterized using a relationship such as the one given in Eq. E12 and a value of the mixture density derived from an accurate equationofstate.

ρ_{m} V _{m}=(ρ_{o} Y _{o}+ρ_{w} Y _{w}+ρ_{g} Y _{g})(V _{so} +V _{sw} +V _{sg}) (E12)

The solution of Eqs. E7 through E12 results in expressions for the phase and mixture superficial velocities and slip velocities that are only functions of the previously determined fluid phase holdups and dynamic viscosity, all of which can be directly related to the fluid mixture density.
One component of the rootsolving procedure basis function is obtained in the form of the Reynolds number, given by Eq. E13.

$\begin{array}{cc}{N}_{\mathrm{Re}}=\frac{124\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e{\rho}_{m}\ue89eD\ue89e\uf603{V}_{m}\uf604}{{\mu}_{m}}& \left(E\ue89e\text{}\ue89e13\right)\end{array}$

Substitution into Eq. E13 for the mixture superficial velocity (Eq. E11) and dynamic viscosity (Eq. E3) results in one component of the basis function of the rootsolving procedure used in the threephase flow metering analysis. The other component of the basis function used in the rootsolving procedure for evaluating the mixture density, satisfying all of the conditions and relationships in the threephase flow metering analysis, is obtained from the capillary flow relationship. Rearranged in terms of the Reynolds number, this relationship is given in Eq. E14. The Fanning friction factor in this expression is evaluated using FIG. 4 for the Reynolds number defined by Eq. E14.

$\begin{array}{cc}{N}_{\mathrm{Re}}=\frac{1722.9\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e{D}^{3/2}\ue89e{{\rho}_{m}^{1/2}\ue8a0\left({\rho}_{w}{\rho}_{g}\right)\ue8a0\left[\begin{array}{c}{P}_{0}{P}_{L}+\\ 0.006945\ue89e\phantom{\rule{0.6em}{0.6ex}}\ue89e{\rho}_{m}\ue89eL\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\mathrm{cos}\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e\theta \end{array}\right]}^{1/2}}{\sqrt{\mathrm{Lf}}\ue8a0\left[{\mu}_{o}\ue89e{Y}_{o}+{\mu}_{w}\ue89e{Y}_{w}+{\mu}_{g}\ue89e{Y}_{g}\right]}& \left(E\ue89e\text{}\ue89e14\right)\end{array}$

With the threephase fluid mixture density resolved with the nonlinear rootsolving mixture described herein, the unknown parameters in the problem are recovered by backsubstitution in the analysis procedure. For instance, the Reynolds number can be computed directly using the mixture density and Eqs. E13 or E14. The mixture superficial velocity is determined with Eq. E11 and the mixture dynamic viscosity is obtained with Eq. E3. The oilwater system slip velocity can be evaluated using Eq. E8 and the gasliquid slip velocity can be computed with Eq. E10. The water phase superficial velocity can be evaluated using Eq. E15 and the gas phase superficial velocity can be evaluated with Eq. E16. The oil phase superficial velocity can then be determined by rearranging Eq. E11.

$\begin{array}{cc}{V}_{\mathrm{sw}}={Y}_{w}\ue8a0\left[{V}_{m}{V}_{\mathrm{sgL}}\ue89e{Y}_{g}\frac{{V}_{\mathrm{sow}}\ue89e{Y}_{o}}{{\left({Y}_{o}+{Y}_{w}\right)}^{2}}\right]& \left(E\ue89e\text{}\ue89e15\right)\\ {V}_{\mathrm{sg}}={Y}_{g}\ue8a0\left[{V}_{m}+{V}_{\mathrm{sgL}}\ue8a0\left({Y}_{o}+{Y}_{w}\right)\right]& \left(E\ue89e\text{}\ue89e16\right)\end{array}$
Example Computational Results

The results of an example computation of multiphase flow velocities, holdup, slip velocities, and mixture density and viscosity for a pressure traverse in a vertical section of wellbore production tubing using the computational methodology disclosed in this invention is presented in the following discussion. The fluids considered in this theoretical example include a 40° API hydrocarbon liquid (oil) with a density of 45.923 lbs/cu ft and a dynamic viscosity of 0.487 cp, produced formation water with a salinity of 40,000 ppm that has a density of 65.762 lbs/cu ft and a dynamic viscosity of 0.271 cp, and a natural gas mixture that has a density at downhole wellbore conditions of 2.456 lbs/cu ft and a dynamic viscosity of 0.014 cp.

Simulated temperature and pressure measurements are modeled for two spatial positions in a vertical section of the wellbore for multiphase flow metering purposes, at wellbore depths of 10,000 and 10,100 ft. The temperature in the wellbore at 10,000 ft of depth was assumed to be 240° F. and the flowing wellbore pressure at that depth was assumed to be 1,000 psia. At 10,100 ft of depth, the corresponding temperature was modeled to be 241.8° F. and the wellbore flowing pressure was assumed to be 1,025 psia. The production tubing (flow conduit) in this section of the wellbore in this example is 2 ⅜in OD tubing which has an internal diameter of 1.995 inches and a relative roughness of 0.004.

An example of the output results obtained using a computer program consisting of the computational methodology described in this invention disclosure is presented in the following summary table. Note that in this synthetic example there is threephase flow in the wellbore. In fact, there is upward flow of gas while there is fallback (downward flow) of the hydrocarbon liquid (oil) and water phases. The Reynolds number indicates that the flow conditions are in the transition flow regime range (not quite fully developed turbulent flow) and the pipe friction is relatively low due to the moderate Reynolds number and the relatively low relative roughness of the conduit.

The gas holdup obtained for these conditions indicates that gas occupies 36% of the wellbore flow area, with water present in about 30%, and oil occupying about 34% of the flow area or volume. The volumetric flow rates obtained in the analysis are presented in the summary tables as well. Note that the gas volumetric flow rate includes the free gas present in the flowstream, as well as the solution gas dissolved in the oil and water phases at downhole conditions.


Wellbore Segment Computed Results: 



Oil holdup = 
0.337 

Water holdup = 
0.298 

Gas holdup = 
0.364 

Oil superficial velocity = 
−0.009 ft/s 

Water superficial velocity = 
−0.123 ft/s 

Gas superficial velocity = 
0.062 ft/s 

Liquid superficial velocity = 
−0.131 ft/s 

Mixture superficial velocity = 
−0.069 ft/s 

Oilwater slip velocity = 
0.244 ft/s 

Gasliquid slip velocity = 
0.378 ft/s 

Mixture density = 
35.999 lbs/cu ft 

Mixture dynamic viscosity = 
0.250 cp 

Mixture kinematic viscosity = 
0.007 cpcu ft/lbs 

Reynolds number = 
2456.4 

Fanning friction factor = 
0.01238 

Oil flow rate = 
−2.54 STB/D 

Water flow rate = 
−38.84 STB/D 

Gas flow rate = 
5.62 Mscf/D 



Referring again to FIG. 1, it is understood that computer system 10 may be implemented as any type of computing infrastructure. Computer system 10 generally includes a processor 12, input/output (I/O) 14, memory 16, and bus 17. The processor 12 may comprise a single processing unit, or be distributed across one or more processing units in one or more locations, e.g., on a client and server. Memory 16 may comprise any known type of data storage, including magnetic media, optical media, random access memory (RAM), readonly memory (ROM), a data cache, a data object, etc. Moreover, memory 16 may reside at a single physical location, comprising one or more types of data storage, or be distributed across a plurality of physical systems in various forms.

I/O 14 may comprise any system for exchanging information to/from an external resource. External devices/resources may comprise any known type of external device, including sensors 30, 32, a monitor/display, speakers, storage, another computer system, a handheld device, keyboard, mouse, wireless system, voice recognition system, speech output system, printer, facsimile, pager, etc. Bus 17 provides a communication link between each of the components in the computer system 10 and likewise may comprise any known type of transmission link, including electrical, optical, wireless, etc. Although not shown, additional components, such as cache memory, communication systems, system software, etc., may be incorporated into computer system 10.

Access to computer system 10 may be provided over a network such as the Internet, a local area network (LAN), a wide area network (WAN), a virtual private network (VPN), etc. Communication could occur via a direct hardwired connection (e.g., serial port), or via an addressable connection that may utilize any combination of wireline and/or wireless transmission methods. Moreover, conventional network connectivity, such as Token Ring, Ethernet, WiFi or other conventional communications standards could be used. Still yet, connectivity could be provided by conventional TCP/IP socketsbased protocol. In this instance, an Internet service provider could be used to establish interconnectivity. Further, communication could occur in a clientserver or serverserver environment.

It should be appreciated that the teachings of the present invention could be offered as a business method on a subscription or fee basis. For example, a computer system 10 comprising a multiphase flow analysis system 18 could be created, maintained and/or deployed by a service provider that offers the functions described herein for customers. That is, a service provider could offer to provide wellbore fluid property information as described above.

It is understood that in addition to being implemented as a system and method, the features may be provided as a program product stored on a computerreadable medium, which when executed, enables computer system 10 to provide a multiphase flow analysis system 18. To this extent, the computerreadable medium may include program code, which implements the processes and systems described herein. It is understood that the term “computerreadable medium” comprises one or more of any type of physical embodiment of the program code. In particular, the computerreadable medium can comprise program code embodied on one or more portable storage articles of manufacture (e.g., a compact disc, a magnetic disk, a tape, etc.), on one or more data storage portions of a computing device, such as memory 16 and/or a storage system, and/or as a data signal traveling over a network (e.g., during a wired/wireless electronic distribution of the program product).

As used herein, it is understood that the terms “program code” and “computer program code” are synonymous and mean any expression, in any language, code or notation, of a set of instructions that cause a computing device having an information processing capability to perform a particular function either directly or after any combination of the following: (a) conversion to another language, code or notation; (b) reproduction in a different material form; and/or (c) decompression. To this extent, program code can be embodied as one or more types of program products, such as an application/software program, component software/a library of functions, an operating system, a basic I/O system/driver for a particular computing and/or I/O device, and the like. Further, it is understood that terms such as “component” and “system” are synonymous as used herein and represent any combination of hardware and/or software capable of performing some function(s).

The block diagrams in the figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods and computer program products according to various embodiments of the present invention. In this regard, each block in the block diagrams may represent a module, segment, or portion of code, which comprises one or more executable instructions for implementing the specified logical function(s). It should also be noted that the functions noted in the blocks may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams can be implemented by special purpose hardwarebased systems which perform the specified functions or acts, or combinations of special purpose hardware and computer instructions.
Calibration Using Data From A Retrievable Production Logging Device

Embodiments of the inventive system, method, and program code can also utilize data obtained from a retrievable production logging device to calibrate one or more of the generated wellbore fluid properties. These retrievable production logging devices are typically deployed in the wellbore on wireline, slickline, or coiled tubing. In the case of highly deviated or horizontal wellbores, the production logging devices may be pushed into position using coiled tubing or stiff wireline cable or may be pulled into position using a downhole tractor. Examples of the types of retrievable production logging devices that may be used include the Production Logging Tool, Memory PS Platform, Gas Holdup Optical Sensor Tool, and Flow Scanner Tool, all available from Schlumberger. The calibration process may involve the identification of or confirmation that one or more sensors that are providing inaccurate downhole measurements and elimination/rejection of the data provided by these sensors. Alternatively, the wellbore fluid property generation process and/or results may be adjusted to either match or more closely reflect the data obtained from the retrievable production logging device.

Although specific embodiments have been illustrated and described herein, those of ordinary skill in the art appreciate that any arrangement which is calculated to achieve the same purpose may be substituted for the specific embodiments shown and that the invention has other applications in other environments. This application is intended to cover any adaptations or variations of the present invention. The following claims are in no way intended to limit the scope of the invention to the specific embodiments described herein.