Method and tool for planning and dimensioning subsea pipelines for produced fluids

Abstract

The invention relates to a computer implemented method and tool for determining pressure-drop in multiphase pipeline flow where the effective surface roughness, k.sub.eff, of liquid film coated sections of the inner pipeline wall is assumed to be equal to the maximum hydraulic roughness, k.sub.s.sup.max. The maximum hydraulic roughness is further assumed to be proportional to a maximum stable droplet size, d.sub.droplet.sup.max, i.e.: k.sub.eff=k.sub.s.sup.max=K.Math.d.sub.droplet.sup.max, where K is a correlation coefficient. The invention further relates to applying the computer implemented method for designing a pipeline-based fluid transport system for transport of multiphase fluids.

Claims

1. A computer implemented method for determining pressure-drop in a multiphase flow in a non-vertical pipeline having an inner wall facing the multiphase flow, where the multiphase flow at least comprises a continuous gas phase forming a gas/inner wall interface and a continuous liquid phase, and wherein the method comprises: applying a computational fluid dynamic model describing the multiphase flow, and solving the computational fluid dynamic model to determine the pressure-drop in the multiphase flow by applying an effective surface roughness, k.sub.eff, determined by a relation: k.sub.eff=K.Math.d.sub.droplet.sup.max, where K is a correlation coefficient and d.sub.droplet.sup.max is a maximum stable droplet size as determined by a droplet model; applying the effective surface roughness, k.sub.eff, to determine a shear-stress, τ.sub.g; and applying the shear-stress, τ.sub.g, as a boundary condition determining gas flow resistance at the gas/inner wall interface; and optimizing one or more dimensions of the non-vertical pipeline based on the pressure-drop determined by the computational fluid dynamic model.

2. The computer implemented method according to claim 1, wherein the computational fluid dynamic model describes the multiphase flow as a set of fluid fields representing: a set of continuous fluid phases being separated by a large-scale interfaces, at least comprising a continuous liquid phase and a continuous gas phase being separated by a large-scale gas/liquid interface, and a set of dispersed phases, at least comprising a dispersed phase of liquid droplets entrained in the continuous gas phase, and wherein each fluid field is described by Eulerian formulated volume and ensemble averaged turbulent transport equations which are solved independently for each fluid field by applying a volume of fluid method for each fluid field representing a continuous phase and a dispersed flow method for each fluid field representing a dispersed fluid phase.

3. The computer implemented method according to claim 1, wherein d.sub.droplet.sup.max, is given by the relation: $d_{\mathrm{droplet}}^{\max }=0.725{\left(\frac{{\sigma }^{*}}{{\rho }_g}\right)}^{0.6}{\epsilon }^{-0.4}$ where σ*=σ.sub.gl(1+k.Math.Ca), where σ.sub.gl is the surface tension between the liquid and the gas, k is a correlation coefficient, and Ca is a capillary number defined to be: $Ca=\frac{{{\mu }_d\left(\epsilon .Math.d_{droplet}^{\max }\right)}^{\frac{1}{3}}}{{\sigma }_{gl}}$ where μ.sub.d is a dynamic viscosity of the liquid droplet, ρ.sub.g is volumetric density of the continuous gas phase, and ε is an energy dissipation rate per unit mass given by a relation: $\epsilon =\frac{S_g{\tau }_g{u}_g+S_i{\tau }_i\Delta u}{0.25\left(S_g+S_i\right){\rho }_g{D}_{hg}}$ where S.sub.g is a gas perimeter, τ.sub.g is the gas/inner wall shear stress, u.sub.g is a gas velocity, S.sub.i is a interface perimeter, τ.sub.i is an interfacial shear stress, Δu is a gas-liquid slip velocity, and D.sub.hg is a hydraulic diameter for the continuous gas phase.

4. The computer implemented method according to claim 1, wherein the multiphase flow is a two-dimensional flow and the effective surface roughness k.sub.eff for liquid film covered gas/inner wall interface is given by the correlation:
k.sub.eff=0.052.Math.min[(1+33.Math.Ca),6.6].sup.0.6d.sub.Hinze where d.sub.Hinze is $d_{Hinze}=0.725{\left(\frac{{\sigma }_{gl}}{{\rho }_g}\right)}^{0.6}{\epsilon }^{-0.4},$  σ.sub.gl is a gas-liquid surface tension, ρ.sub.g is gas density, and ε is an energy dissipation rate per unit mass given by a relation: $\epsilon =\frac{S_g{\tau }_g{u}_g+S_i{\tau }_1\Delta u}{0.25\left(S_g+S_i\right){\rho }_g{D}_{hg}},$  where S.sub.g is a gas perimeter, τ.sub.g is a gas-wall shear stress, u.sub.g is a gas velocity, S.sub.i is an interface perimeter, τ.sub.i is an interfacial shear stress, Δu is a gas-liquid slip velocity, and D.sub.hg is a hydraulic diameter for the continuous gas phase, and Ca is a capillary number defined to be: $Ca=\frac{{{\mu }_d\left(\epsilon .Math.d_{Hinze}\right)}^{\frac{1}{3}}}{{\sigma }_{gl}},$  where μ.sub.d is a dynamic viscosity of a liquid droplet.

5. The computer implemented method according to claim 1, wherein the multiphase flow is a three-dimensional flow comprising a first and a second liquid, and the correlation coefficient, K, is: $K=0\text{.052}.Math.{\left(\min \left(1+33.Math.\mathrm{Ca},6.6\right)+33.Math.\left(\frac{{\mu }_l^{eff}}{{\mu }_l^{pure}}-1\right).Math.\mathrm{Ca}\right)}^{\frac{3}{5}},$ where Ca is a capillary number: $Ca=\frac{{{\mu }_l^{\mathrm{eff}}\left(\epsilon .Math.d_{Hinze}\right)}^{\frac{1}{3}}}{{\sigma }_{eff}},$ where σ.sub.eff is an effective surface tension of a mixed first and second liquid: ${\sigma }_{\mathrm{eff}}=\frac{{\alpha }_1{\sigma }_1+{\alpha }_2{\sigma }_2}{{\alpha }_1+{\alpha }_2},$  where α.sub.1 is a fraction of pipeline wall covered by the first liquid, α.sub.2 is a fraction of pipeline wall covered by the second liquid, σ.sub.1 is a surface tension of the first liquid, σ.sub.2 is a surface tension of the second liquid, and where μ.sub.l.sup.eff is an effective dynamic viscosity of the mixed liquid: μ.sub.l.sup.eff=μ.sub.l.sup.pure(1−φ).sup.−2.5, where φ is a droplet concentration of the first liquid being in suspension in second liquid phase, or alternatively, the droplet concentration of second liquid being in suspension in the first liquid phase.

6. The computer implemented method according to claim 1, wherein the non-vertical pipeline has an inclination angle in a range of from −75° to +75°, where a positive inclination angle corresponds to the pipeline ascending in the flow direction, a negative inclination angle corresponds to the pipeline descending in the flow direction, and an inclination angle of 0° corresponds to the pipeline being horizontal relative to the earth gravitational field.

7. A computer implemented method for designing a pipeline-based fluid transport system for transport of multiphase fluids, where the method comprises: applying the method according to claim 1 to numerically predict the pressure-drop when loading an intended flow rate volume of the multiphase fluid through a set of pipeline segments having different inner diameters and inclination angles towards the earth gravitational field; and optimizing, the pressure-drop determined by the computational fluid dynamic model, a pipeline diameter and a pipeline trajectory/path in a terrain/seabed with aim of minimising material use in the pipeline-based fluid transport system and energy consumption for operating the pipeline-based fluid transport system.

8. A computer, comprising a processing device and a computer memory, the computer memory is storing a computer program comprising processing instructions which causes a computer to perform the method according to claim 1 when the instructions are executed by a processing device in the computer.

9. A computer implemented method for determining pressure-drop in a multiphase flow in a non-vertical pipeline having an inner wall facing the multiphase flow, where multiphase flow at least comprises a continuous gas phase forming a gas/inner wall interface and a continuous liquid phase, and wherein the method comprises: applying a computational fluid dynamic model describing the multiphase flow, and solving the computational fluid dynamic model to determine the pressure-drop in the multiphase flow by: determining a first fraction of the gas/inner wall interface considered being covered by a liquid film and a second fraction of the gas/inner wall interface considered not being covered by a liquid film, for the first fraction; applying an effective surface roughness, k.sub.eff,1, determined by a relation: k.sub.eff,1=K.Math.d.sub.droplet.sup.max, where K is a correlation coefficient and d.sub.droplet.sup.max is a maximum stable droplet size as determined by a droplet model, and applying the effective surface roughness, k.sub.eff,1, to determine a shear-stress, τ.sub.g,1, and for the second fraction; applying an effective surface roughness, k.sub.eff,2, defined to be equal to a surface roughness of a dry inner wall of the pipeline, and applying the effective surface roughness, k.sub.eff,2, to determine a shear-stress, τ.sub.g,2, and applying the shear-stresses, τ.sub.g1 and τ.sub.g2 as boundary conditions determining a gas flow resistance at the gas/inner wall interface; and optimizing one or more dimensions of the non-vertical pipeline based on the pressure-drop determined by the computational fluid dynamic model.

10. The computer implemented method according to claim 9, wherein the computational fluid dynamic model describes the multiphase flow as a set of fluid fields representing: a set of continuous fluid phases being separated by large-scale interfaces, at least comprising a continuous liquid phase and a continuous gas phase being separated by a large-scale gas/liquid interface, and a set of dispersed phases, at least comprising a dispersed phase of liquid droplets entrained in the continuous gas phase, and wherein each continuous fluid field is described by Eulerian formulated volume and ensemble averaged turbulent transport equations which are solved for each continuous fluid field, and each of the dispersed phases is described by “slip relations”, which are expressions that describe a velocity difference between dispersed fields and an associated continuous field.

11. The method according to claim 10, wherein the determination of the first fraction of the gas/inner wall interface considered being covered by a liquid film and the second fraction of the gas/inner wall interface considered not being covered by a liquid film is obtained by: determining a liquid droplet concentration profile, C.sub.in gas.sup.droplet(y), in the continuous gas phase as a function of vertical distance above the large-scale interface separating the continuous liquid phase and the continuous gas phase, determining a critical droplet concentration profile C.sub.crit.sup.droplet by a relation: log.sub.10 C.sub.crit.sup.droplet=−min[max(250.Math.σ.sub.gl,4.0),9.0], where □.sub.gl is a gas-liquid surface tension, and applying the determined liquid droplet concentration profile, C.sub.in gas.sup.droplet(y) at the gas/inner wall interface and define the part of the gas/inner wall interface where the C.sub.in gas.sup.droplet(y)≥C.sub.crit.sup.droplet is satisfied to be the first fraction, and similarly define the part of the gas/inner wall interface where the C.sub.in gas.sup.droplet(y)<C.sub.crit.sup.droplet is satisfied to be the second fraction.

Description

LIST OF FIGURES

(1) FIG. 1 is a diagram showing measured pressure drops in a pipeline segment for various flows as compared to predicted pressure-drop in a similar pipeline segment made by a prior art computational fluid dynamic model available under the trademark LedaFlow 2.4. The experimental measurements are presented in Kjolaas et al. [2].

(2) FIG. 2 is a drawing schematically illustrating the assumed droplet induced formation of a liquid film on the gas/inner wall interface.

(3) FIG. 3 is a diagram showing the ratio between surface roughness at the gas/inner wall interface and the droplet size determined by Hinze droplet model [10] plotted against capillary number. The dotted line marked “Equation (13)” corresponds to eqn. (7) herein.

(4) FIGS. 4a) and 4b) display diagrams showing measured and predicted pressure drops for different two-phase flows at different gas flow velocities. The markers represent measured values, the dashed lines represent the predictions made by a prior art computational fluid dynamic model, while the solid lines represent the predictions made by a similar computational fluid dynamic model adapted according to the present invention.

(5) FIG. 5 displays diagrams showing measured and predicted normalized pressure drops for different three-phase flows at different gas flow velocities and water cuts. The markers represent the measured values, the dashed lines represent the predictions made by a prior art computational fluid dynamic model, while the solid lines represent the predictions made by a similar computational fluid dynamic model adapted according to the present invention.

(6) FIG. 6 is a facsimile of FIG. 14 of US 2010/0059221 showing an example of a production system.

VERIFICATION OF THE INVENTION

(7) The invention is demonstrated by first providing an example embodiment of a simple mathematical model for a two-phase flow showing how the mathematical model may be adapted to incorporate the correction of the shear forces according to the invention at the gas/wall interface. And further by presenting a comparison of predictions of the pressure drop made by a commercial computational fluid dynamic model marketed under the trademark “LedaFlow 2.4” and by a new version of the same commercial computational fluid dynamic model having incorporated the correction of the shear forces according to the invention and which is just being released under the trademark “LedaFlow 2.5”. The predictions are compared to experimental measurements of the pressure drop.

Example Embodiment of a Mathematical Model Having Incorporated the Invention

(8) The example embodiment of the computational fluid dynamical model is a so-called “steady-state point model”, which predicts the pressure drop and phase fractions given a set of input parameters, which are: 1. Pipe inner diameter D, wall surface roughness k.sub.s and inclination θ. 2. The volumetric flow rates for each continuous fluid phase. 3. The thermodynamic properties of each continuous fluid phase (comprises at least densities, viscosities and surface tensions of the fluid phases).

(9) The example embodiment of computational fluid dynamic model is called a “point model” because it only provides predictions locally in a system. By “locally”, we mean a region small enough such that we can assume that all the fluid properties are constant in that region. The example embodiment further assumes the following: 4. The flow consists of one liquid phase and one gas phase (two-phase flow), both of which are considered incompressible. 5. The flow is stratified with liquid droplets, which means that the liquid flows as a continuous liquid phase at the bottom of the pipe, and possibly also as droplets entrained in the gas above. 6. The interface between the continuous liquid phase at the bottom of the pipe and the gas is assumed to be flat. 7. The flow is steady and fully developed, meaning that the flow does not change in time or in space. Mathematically, this is equivalent to assuming that all temporal and spatial derivatives in the mass/momentum equations are zero. 8. The liquid droplets travel at the same velocity as the gas.

(10) The mass balance equations for the example may be given as:

(11) $\begin{array}{cc}\frac{Q_g}{A}={\alpha }_g{u}_g& \left(16\right)\\ \frac{Q_l}{A}={\alpha }_l{u}_l+{\alpha }_{\mathrm{droplet}}{u}_g& \left(17\right)\end{array}$
where Q.sub.g and Q.sub.l are the volumetric gas and liquid flow rates, A is the cross section area of the pipe, α.sub.g, α.sub.l and α.sub.droplet are the volumetric fractions of the gas, the liquid film at the bottom of the pipe, respectively, and the liquid droplets, u.sub.g is the gas velocity, and u.sub.l is the velocity of the continuous liquid phase at the bottom of the pipe.

(12) The momentum equations for the gas zone (continuous gas phase and liquid droplets) and the continuous liquid phase for this model are:

(13) $\begin{array}{cc}\left({\alpha }_g+{\alpha }_{\mathrm{droplet}}\right)\frac{\mathrm{dp}}{\mathrm{dx}}+\frac{S_g{\tau }_g}{A}+\frac{S_i{\tau }_i}{A}+\left({\alpha }_g+{\alpha }_{\mathrm{droplet}}\right){\rho }_{g}g\mathrm{sin\theta }=0& \left(18\right)\\ {\alpha }_l\frac{\mathrm{dp}}{\mathrm{dx}}+\frac{S_l{\tau }_l}{A}-\frac{S_i{\tau }_i}{A}+{\alpha }_l{\rho }_{l}g\mathrm{sin\theta }=0& \left(19\right)\end{array}$

(14) Here, dp/dx is the pressure gradient, ρ.sub.g is the gas density, pi is the liquid density, S.sub.g is the gas-wall perimeter, S.sub.l is the film-wall perimeter, S.sub.i is the width of the gas-liquid interface, τ.sub.g is the gas-wall shear stress, τ.sub.l is the liquid film shear stress, and τ.sub.i is the shear stress between the gas and liquid at the interface. From geometrical considerations, it can be shown that the parameters α.sub.l and S.sub.l are related as follows for a flat continuous liquid layer in a circular pipe:

(15) $\begin{array}{cc}{\alpha }_l=\frac{1}{2\pi }\left[\frac{S_l}{D}-\sin \frac{S_l}{D}\right]& \left(20\right)\end{array}$

(16) Furthermore, the relationship between the interface width S.sub.i and the film perimeter S.sub.l is:

(17) $\begin{array}{cc}{S}_i=Dcos\frac{s_l}{2D}& \left(21\right)\end{array}$

(18) Finally, we have that the sum of the wall perimeters S.sub.g and S.sub.l must equal the total pipe perimeter.

(19) The shear stresses can be expressed as follows:

(20) $\begin{array}{cc}{\tau }_g=\frac{1}{2}{f}_g{\rho }_g.Math.u_g.Math.u_g& \left(22\right)\end{array}$$\begin{array}{cc}{\tau }_l=\frac{1}{2}{f}_l{\rho }_l.Math.u_l.Math.u_l& \left(23\right)\end{array}$$\begin{array}{cc}{\tau }_i=\frac{1}{2}{f}_i{\rho }_g.Math.u_g-u_l.Math.\left(u_g-u_l\right)& \left(24\right)\end{array}$

(21) Here, f.sub.g, f.sub.l and f.sub.i are the friction factors for the gas, the liquid film and the interface, and these friction factors require closure laws.

(22) The example model incorporates the inventive determination of the gas flow resistance by applying the Haland model [13] as closure law for determination of the shear stress at the gas/inner wall interface, i.e. to determine the gas friction coefficient, f.sub.g, from the effective surface roughness, k.sub.eff, as defined in relations (1) or preferably by relations (11) to (13) given above.

(23) In most prior art models, the k.sub.eff parameter takes on the value for a dry pipeline wall, which is part of the user input to the model. However, in the example model, the value of k.sub.eff is determined as given in equations (1) and (3) to (7) given above to take into account for droplets depositing on the wall, creating a liquid film at the gas/inner wall interface which has a roughness that is significantly different from the effective surface roughness of a dry pipe wall.

(24) For the continuous liquid phase, the example model applies the same closure law, which in this case may be given as:

(25) $\begin{array}{cc}\frac{1}{\sqrt{f_l}}=-3.6\log \left[\frac{6.9}{R{e}_l}+{\left(\frac{k_{\mathrm{eff},l}}{3.7D_{\mathrm{hl}}}\right)}^{1.11}\right]& \left(25\right)\end{array}$

(26) Here Re.sub.l is the Reynolds number of the continuous liquid, which is usually expressed as:

(27) $$\begin{gathered} \begin{array}{cc}R{e}_l=\frac{{\rho }_l{D}_{\mathrm{hl}}{u}_l}{{\mu }_l} \\ \text{where:}& \left(26\right)\end{array} \end{gathered}$$$\begin{array}{cc}{D}_{\mathrm{hl}}=\frac{4{\alpha }_{l}A}{S_l}& \left(27\right)\end{array}$

(28) However, for the liquid-wall shear stress, k.sub.eff,l, takes on the value provided as user input for the surface roughness of the pipeline wall.

(29) There are many proposed closure laws for the interfacial friction factor in the literature which may be applied to close the model over the gas-liquid interface. The example model applies a correlation proposed by Andritsos & Hanratty [14]:

(30) 0$\begin{array}{cc}{f}_i={\left(-3.6\log \frac{6.9}{{\mathrm{Re}}_g}\right)}^{-2}\left(1+15\frac{h_l}{D}\max \left(\frac{u_g}{5}-1,0\right)\right)& \left(28\right)\end{array}$

(31) Here, h.sub.l is the centre line depth of the continuous liquid phase.

(32) To calculate the droplet fraction α.sub.droplet, another closure law is needed. The example model applies a model suggested by Ishii & Mishima [15] to calculate the total droplet fraction α.sub.droplet:

(33) $\begin{array}{cc}{\alpha }_{\mathrm{droplet}}=\frac{Q_l}{A.Math.u_g}\tan h[7.25.Math.{10}^{-7}{\mathrm{We}}^{1.25}{\mathrm{Re}}_{\mathrm{liq}}^{0.25}\mathrm{where}& \left(29\right)\end{array}$$\begin{array}{cc}\mathrm{We}=\frac{{\rho }_g{\mathrm{Du}}_g^2}{\sigma }{\left(\frac{{\rho }_l-{\rho }_g}{{\rho }_g}\right)}^{\frac{1}{3}}\mathrm{and}& \left(30\right)\end{array}$$\begin{array}{cc}{\mathrm{Re}}_{\mathrm{liq}}=\frac{Q_l{\rho }_{l}D}{A{\mu }_l}& \left(31\right)\end{array}$

(34) The example model is now complete as an example embodiment of the invention according to the first aspect.

(35) However, the example model may further comprise a model for determination of the droplet concentration and thus be an example embodiment of the invention according to the second aspect. The example embodiment, for the sake of simplicity, assumes that the droplet concentration drops linearly with the vertical distance above the gas-liquid interface, and that it reaches a value of zero at the top of the pipe and that the pipe is shaped as a rectangular duct, the local concentration of droplets C.sub.in gas.sup.droplet(y), may be given by:

(36) $\begin{array}{cc}{C}_{\mathrm{in}g\mathrm{as}}^{\mathrm{droplet}}\left(y\right)=2{\alpha }_{\mathrm{droplet}}\left(1-\frac{y}{h_g}\right)& \left(32\right)\end{array}$
where y is the vertical distance from the interface, and h.sub.g is the distance from the gas-liquid interface to the top of the pipe.

(37) Eqn. (32) may then be applied together with eqn. (2) to determine whether the effective surface roughness, k.sub.eff, at the gas/inner wall interface is to be determined by relation (1) or preferably by relations (12) to (14) given above, or to be defined to be equal to a surface roughness of a dry inner wall of the pipeline.

(38) Comparison of Measured and Predicted Pressure Drops

(39) The invention according to the invention is further verified by a comparison of predicted pressure-drops for various two-phase flows in a pipeline segment by a prior art commercially available computational fluid dynamic model as compared to the predicted pressure-drops by a just released upgraded version of a similar computational fluid dynamic model having incorporated the inventive approach for determining the shear-stresses at the gas/inner wall interface. The predictions are compared to measured pressure-drops for different.

(40) The experimental measurements of the pressure-drop were conducted at the SINTEF Multiphase Flow Laboratory applying their “Large Scale Loop Facility” adapted with a 94-meter-long 8″pipe with an inclination angle 2.5°. The nominal pressure was 60 bara, yielding a gas density of 67 kg/m.sup.3. For the experiments, nitrogen was used as the gas phase and Exxsol D60 as oil phase. For the aqueous phase, we used regular tap water with NaOH for corrosion protection, with and without glycerol. The purpose of adding glycerol was to increase the viscosity of the aqueous phase, emulating MEG injection. In the experiments with glycerol, the volumetric concentration was in the range 70-74%. The glycerol experiments were conducted at temperatures 23° C. and 45° C. yielding viscosities of about 42 and 14 cP, respectively, while the experiments without glycerol were conducted at 30° C. When changing the temperature, we also adjusted the pressure such that the gas density was kept the same in all the experiments. Additional details regarding the experimental set-up and execution are presented in Kjolaas et al. [1, 2].

(41) The commercially available computational fluid dynamic model is sold under the trademark “LedaFlow 2.4” by LedaFlow Technologies DA. The LedaFlow 2.4 model describes thus the multiphase fluid as consisting of continuous fluid phases being separated by large-scale interfaces and of dispersed fluid phases suspended in one or more of the continuous fluid phases. The LedaFlow 2.4 applies the typically prior art approach for determining the shear-stresses at the gas/inner wall interface by assuming the effective surface roughness being equal to the surface roughness of a dry pipeline wall.

(42) The upgraded version of the computational fluid dynamic model is going to be marketed under the trademark “LedaFlow 2.5” and is similar to the LedaFlow 2.4 model except that the assumptions presented above for the effect on the shear-stresses by the formation of a liquid film on the gas/inner wall are incorporated into the model. The LedaFlow 2.5 model applies the approach according to the second aspect of the invention utilising the critical droplet concentration criteria as given in eqn. (2) above to determine how much of the gas-wall perimeter is dry (with an effective surface roughness equal to k.sub.eff,2), and how much of the gas-wall perimeter is covered by a liquid film (with an effective surface roughness of equal to k.sub.eff,1). The LedaFlow 2.5 model is adapted to apply eqn. (7) to determine k.sub.eff,1 when the multiphase flow is a two-phase flow, and to apply eqn. (8) to determine k.sub.eff,1 when the multiphase flow is a three-phase flow comprising two immiscible liquids and a gas phase.

(43) Two-Phase Flow

(44) FIG. 4a) shows two diagrams presenting measured and predicted pressure drops plotted against the superficial gas velocity USG. In the top graph, the superficial liquid velocity USL is 0.01 m/s, and in the bottom graph USL equals 0.1 m/s. FIG. 4b) shows two diagrams presenting measured and predicted pressure drops plotted against the superficial liquid velocity USL. In the top graph, the superficial gas velocity USG is 9 m/s, and in the bottom graph USG equals 12 m/s.

(45) The predictions are made by the prior art computational fluid dynamic model LedaFlow 2.4 and by the computational fluid dynamic model (LeadFlow 2.5) adapted according to the present invention. The experimental measurements are presented in Kjolaas et al. [1, 2]. The markers represent measured values, the dashed lines represent the predictions made by LedaFlow 2.4, while the solid lines represent the predictions made by LedaFlow 2.5.

(46) As seen from the figures, the prior art model (LedaFlow 2.4) systematically under-predicts the pressure drop at these conditions, while the model adapted according to the present invention (LedaFlow 2.5) is in much better agreement with the measurements.

(47) Three-Phase Flow

(48) FIG. 5 is a diagram showing the normalized pressure drop plotted against the water cut for three different gas rates (USG=6, 9 and 12 m/s), and two different liquid rates (USL=0.01 and 0.1 m/s). Each graph contains data for one or more fluid systems (with/without glycerol), where the main distinguishing factor is the water viscosity. The markers in the graphs represent the measured values, while the lines are predictions. The dashed lines are predictions obtained with the prior art LedaFlow 2.4 model, while the solid lines are predictions by the model (LedaFlow 2.5) adapted according to the invention. Both models applied the Brinkman emulsion model [12].

(49) FIG. 5 shows that also for the three-phase flow, the model adapted according to the invention is significantly more accurate in prediction the pressure-drop than the prior art model which does not compensate for the liquid film effect on the gas flow resistance.

(50) It is noted that the model adapted according to the invention obtains a maximum pressure drop at a water cut of 50% while the experiments indicate a maximum pressure-drop at a water cut of 80%. The predictions in the high water cut range are thus low compared to experiments for this fluid system. It is assumed that the main reason for this discrepancy is that an oil-continuous emulsion forms on the wall. However, such behaviour is difficult model and is outside the objective of the hydrodynamic modelling approach of the present invention.

(51) Nevertheless, the verification presented above clearly shows that the above presented assumption of describing the effective surface roughness by a liquid droplet model at the gas/inner wall interface when a liquid film may be assumed formed, is a simple and computational effective correction of the shear-stresses at the gas/inner wall interface obtaining significantly improved pressure-drop predictions as compared to prior art models assuming the shear-stresses are due to the surface roughness of the (dry) pipeline wall.

REFERENCES

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