2. Forfatter Fornavn Etternavn Institusjon Viscosity Viscous Fluids Examples Dependency of Viscosity on Temperature Laboratory exercise Non-Newtonian Fluids Fluid Flow Characteristics Viscometer Select one subject!!

3. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Section 1: Viscous Fluids Horizontal flow of viscous fluids Continuity equation for viscous flow Viscous flow in a cylindrical tube Viscous flow through a porous medium made up of a bundle of identical tubes Exercise: capillary tube viscosity measurement Select one subject

4. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Horizontal flow of Viscous Fluids t < 0 Y V t = 0 V t is small t is large V y x Two large parallel plates of area A Y – a very small distance Lower plate is set in motion with constant velocity The fluid start to move due to motion of the plate After a while the fluid enter a steady state velocity profile To maintain this steady state motion, a constant force F is required

5. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext The force may be expressed: The force F per unit area A is proportional to the velocity V in the distance Y; the constant of proportionally is called the viscosity of the fluid.

6. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Also used is the so called Newton`s equation of viscosity: The shear stress The fluid viscosity The fluid velocity in x- direction The dimensions: And By definition: Explicit:

7. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Typical values of viscosity of some fluids: 2180,2840,17100 2090,3570,3080 2000,4690,8060 1900,6562,3140 1811,0059,8620 1711,79253,000 Viscosity Air, Micro Poise[ p] Viscosity Water, centiPoise[cp] Viscosity Castor Oil, Poise[p] Temperature Celsius

8. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext y X y The flow will have the biggest velocity at the top (the surface ) The flow velocity will approximately be zero at the bottom, due to retardation when liquid molecules colliding with the non-moving bottom A graphic presentation of this phenomena is shown here A liquid is flowing in a open channel Fluid flows in an open channel Continuity equation for viscous flow

9. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Transferred momentum pr. time pr. area The change in fluid flow velocity pr. distance between the two layers Viscosity is here defined as s proportionality constant, similar to what was done in the case of defining absolute permeability J p – is often characterised as the momentum intensity or as the shear stress, The unit :

10. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext y z x S S` The change of momentum intensity the box: Momentum transfer between layers S and S` in Newtonian viscous flow dy is the width of the box and S = S` is the cross-section

11. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext is the sum of all forces acting on the box. Let f symbolise a force of more general origin. The momentum density inside the box is defined by. The change of momentum pr. time inside the box is : Combining with the equation from last page gives us: Substituting:

12. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise s` s r r + dr R BackNext Viscous flow in a cylindrical tube Viscous flow in a tube is characterised by a radial decreasing flow velocity, because of the boundary effect of the tube wall To develop the continuity equation for this example, dr (a thin layer), of liquid is considered at a radius r. Cross-section of viscous flow through a cylindrical tube

13. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext The momentum flux through the cylindrical volume: Here the flux area S is varying and. Defining the change in the momentum density inside the cylindrical volume, as, the general equation for laminar flow is: The momentum intensity j p is redefined according to the geometric conditions in this example: The minus sign show that the flow velocity v x is decreasing when the radius r is increasing

14. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext The continuity equation for viscous flow in a cylindrical tube is: This is under stationary conditions. At a constant pressure drop along the tube, no velocity variation is observed.

15. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext The general solution of the previous equation, found by integrating twice, is: The general constants are found by considering the boundary conditions. For this example the flow is directed along the x-axis 1.Since the maximum flow velocity, v x (r=0) in the centre of the tube is less then infinity; C 1 =0 2.Since the flow velocity is zero along the tube wall; vx(r=R)=0 and The solution is then:

16. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext R -R Velocity profile in cylindrical tube flow This flow velocity profile in the tube is given by the formula above. If the outer force, driving the liquid through the tube is the pressure drop along a tube length then and: Defines a the laminar viscous flow pattern

17. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Viscous flow through a porous medium made up of a bundle of identical tubes The incremental flow rate through a fraction of the cross-section of a capillary tube can be expressed: The total flow rate can be found by integration: For the sake of convenience we may present the last equation in the following form: This is also known as the Poiseuille`s equation, where A is the cross-section of the capillary tube

18. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Considering a porous medium as a bundle of identical capillary tubes, the total flow q p through the medium is: Where is the capillary tube cross-section and A is the cross- section of the porous medium. From the equation above the permeability of the medium where capillary tubes are packed together is found: Where the porosity of the bundle of capillary tubes is given by:

19. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Exercise: Capillary tube viscosity measurement 2R Fluid viscosity may also be estimated by the measuring the volume of fluid flowing through a capillary tube pr. time (as the figure below). Rewriting the Poiseuille`s equation an expression for the dynamic fluid viscosity is written:

20. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Which means that if the relative accuracy in the tube-radius is +/- 2-3 %, then the relative accuracy in the viscosity is about +/- 10 %. A small variation in capillary tube fabrication induces large uncertainty in the viscosity measurement. For horizontal flow through a capillary tube of length and radius R, the time it takes to fill a certain volume is measured. The flow pressure, is fixed during the process, by maintaining a constant fluid level at a certain elevation above the capillary tube. The accuracy in these measurements is strongly related to the fabrication accuracy of the capillary tubes, as can be seen from the formula above. If the relative uncertainty is considered,

21. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Section 2: Fluid Flow Characteristics For laminar flow in a cylindrical tube we have derived Poiseuille`s equation: For turbulent flow in a cylindrical tube, an empirical law, called Fanning`s equation has been found: F – the Fanning friction factor This factor is dependent of the tube surface roughness, but also on the flow regime established in the tube

22. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext The flow regime is correlated to the Reynolds number, which characterises the fluid flow in the tube. The Reynolds number, Re, is defined as a dimensionless number balancing the turbulent and the viscous (laminar) flow [ ]. 2R is the spatial dimension where the flow occur – the diameter of a capillary tube or width of an open channel. v – the average flow velocity – density – the fluid viscosity From experimental studies an upper limit for laminar flow has been defined at a Reynolds number; Re =2000. Above this number, turbulent flow will dominate. (This limit is not absolute and may therefore change somewhat depending on the experimental conditions.)

23. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext In the case of porous flow, the velocity v in the previous equation should be the pore flow velocity, where and R is the pore radius. The Reynolds number for flow in a porous medium is: - the bulk velocity residual oil saturation viscosity porosity fluid density bulk velocity pore dimension Typical parameters for laboratory liquid flow experiments:

24. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Using the previous table we find Reynolds number Re = 1 (ca.), for laboratory core flow, which is far below the limit of turbulent flow. In case of reservoir flow, the ”normal” reservoir flow velocity is ca. 1 foot/day or, which indicate that turbulent liquid flow under reservoir conditions is not very likely to occur. For gas, turbulent flow may occur if the potentials are steep enough. If the formula for Reynolds number and Poiseuille`s law is compared: The only fluid dependent parameters are the density and the viscosity Comparing the Reynolds number for typical values of gas and oil : Demonstrates the possibilities for turbulent when gas is flowing in a porous medium

25. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Section 3: Dependency of Viscosity on Temperature Heavy oil Light oil Temperature, K Viscosity, cp The Viscosity of liquids decreases with increasing temperature. For gas it`s the opposite; viscosity increase with increasing temperature.

26. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Another commonly equation: Where n depends on the type of gas (1 < n < 0,75) Temperature depending viscosity of gases expressed by the Satterland`s equation: Where K and C are constants depending on the type of gas.

27. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Section 4: Non-Newtonian Fluids Viscous-Plastic fluids Bingham (1916) and Shvedov (1889) investigated the rheology of viscous- plastic fluids. These fluids also feature elasticity in addition to viscosity. Equation describing viscous-plastic fluids: Here is the breaking shear stress is the so-called structural viscosity there is no fluid flow Some oils, drilling mud and cements slurries represent viscous-plastic fluids.

28. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Pseudo-Plastic fluids Some fluids do not have breaking shear stress but rather, their apparent viscosity depends on a shear rate: Which means that their apparent viscosity decreases when dv/dy grows:

29. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Section 5: Examples 6.2.1 Water viscosity at reservoir conditions 6.2.4 Rotating cylinder viscosity measurement 6.2.2 Falling sphere viscosity measurement

30. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Water viscosity at reservoir conditions Water viscosity is primarily a function of temperature. Salinity has also a slight influence on. The pure water viscosity is listed in viscous fluids section. Due to correction foe salinity and reservoir temperature, the normal range of viscosity at reservoir conditions is from 0,2 to 1,0 cP. Correlation for estimation of water viscosity at res. temp. : Water viscosity is measured in cP and temperature in Fahrenheit At a res. temp. of, the water viscosity will be: NB! Even if the temperature is above, water at reservoir condition will still be in a liquid phase since the res. Pressure is quite high as compared to surface condition.

31. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise h BackNext Falling sphere viscosity measurement A metal sphere falling in viscous fluid reaches a constant velocity v s Then the viscous retarding force plus the buoyancy force equals the weight of the sphere The force F s is given by Stoke`s law: r – sphere radius V s – thermal speed – viscosity – density of the metal sphere – density of the fluid The weight of the sphere will balance the viscous force plus the buoyancy force at the terminal sphere velocity when the sum of forces acting on the sphere is zero g – the constant of gravitation

32. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Alternatively; measuring the time : Viscosity can be estimated by measuring the falling speed of the metal sphere in a cylindrical tube: ( ) C – characteristic constant, determined through calibration with a fluid of known viscosity

33. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Rotating cylinder viscosity measurement h M Here we will also Look at the motion of a fluid between two coaxial cylinders Due to the elastic force (in the fluid) a viscosity shear will exist there One cylinder is rotating with an angular velocity The other one is kept constant The inner cylinder: – Radius – Constant Angular velocity The outer cylinder: – Radius Held stationary by a spring balance which measures the torque M h – the fluid height level on the two cylinders

34. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext For a certain viscometer, the viscosity as function of angular momentum and angular velocity is: C – the characteristic constant for the viscometer

35. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Section 6: Laboratory exercise

36. UpHome DevelopersFAQReferencesSummary -Viscous fluids -Fluid flow characteristics -Dependency of viscosity on temperature -Non-Newtonian fluids -Examples -Laboratory exercise BackNext Section 6: