Pete 314 1.1

PETE 314 — Transport Processes in Petroleum

Production

� Fluid Mechanics: fluid statics; stress and strain, mass,

energy, momentum balances; friction losses, turbulent

flow, Reynolds Number (Moody diagram); Newtonian/Non-

Newtonian fluids; flow in porous media (Darcy's law and

Non-Darcy flow).

� Other transport phenomena: Heat conduction in solids;

Heat convection in fluids. Diffusion of substances. (Fluid

mechanics revisited: Transport of momentum.)

� Flow and heat transfer processes in reservoirs and wells.

Pumps, compressors, heat exchangers.

Pete 314 1.2

Introduction to Fluid Mechanics

� Fluid Mechanics is concerned with the behavior

of fluids at rest (statics) and in motion

(dynamics)

� Distinction between solids and fluids:

• A solid is “hard” and not easily deformed. A fluid is

“soft” and deforms easily.

• Fluid is a substance that alters its shape in response

to shear force however small and deforms

continuously while acted on.

• Fluids tend to take the shape and form of the

container.

Pete 314 1.3

Some applications of fluid mechanics

1. Hydraulics: the flow of water in rivers, pipes, canals, pumps, turbines.

2. Aerodynamics: the flow of air around airplanes, rockets, projectiles.

3. Meteorology: the flow of the atmosphere.

4. Particle dynamics: the flow of fluids around particles, the interaction of particles and fluids (i.e., dust settling, slurries, pneumatic transport, fluidized beds, air pollutant particles, corpuscles in our blood).

5. Hydrology: the flow of water and water-borne pollutants in the ground.

6. Reservoir engineering: the flow of oil, gas, and water in petroleum reservoirs.

7. Multiphase flow: coffee percolators, oil wells, carburators, fuel injectors, combustion chambers, sprays.

8. Combinations of fluid flow: with chemical reactions in combustion, with electromagnetic phenomena in magnetohydrodynamics, with mass transport in distillation or drying.

9. Viscosity-dominated flows: lubrication, injection molding, wire coating, lava, and continental drift.

Pete 314 1.4

BASIC IDEAS IN FLUID MECHANICS

1. The principle of the conservation of mass.

2. The first law of thermodynamics (the principle of the

conservation of energy).

3. The second law of thermodynamics.

4. Newton’s second law of motion, which may be

summarized in the form F = ma.

Each of these four ideas is a generalization of experimental data. None of them can

be deduced from the others or from any other prior principle. Rather, they stand on

their ability to predict correctly the results of any experiment ever run to test them.

Pete 314 1.5

Stresses

(Force normalized by area)

� Tensile

� Compressive

� Shear

Unit of stress:

1 Pa = 1 N/m2

1 psi = 1 lbf/in2

Pete 314 1.6

Reminder: Units of force and Newton’s 2nd Law

F = m · a

� SI system: Base dimensions are Length, Time, Mass, Temperature

A Newton is the force which when applied to a mass of 1 kg produces

an acceleration of 1 m/s2.

Newton is a derived unit: 1N = (1kg).(1m/s2)

� BG system: Base dimensions are Length, Force, Time, Temperature

A slug is the mass which produces an acceleration of 1 ft/s2 when a

force of 1lb is applied on it:

Slug is a derived unit: 1slug=(1lb) (s2)/ (ft) = 32.174 lbm

� EE system: Base dimensions are Length, Time, Mass, Force and

Temperature

The pound-force (lbf) is defined as the force which accelerates

1pound-mass (lbm) with 32.174 ft/s2.

Pete 314 1.7

Reminder: Dimensional homogeneity

� All theoretically derived equations are dimensionally homogeneous:

dimensions of the left side of the equation must be the same as those

on the right side.

� All equations must use consistent units: each term must have the same

units. Answers will be incorrect if the units in the equation are not

consistent. In strict SI system there is no need for conversions

� In English system calculations we often need:

� In Pete we use an even less coherent “system” of units, so we need

more conversion factors. For instance

2sN

mkg 11

⋅

⋅=

ft

in1441

2

2

=

22/sft 25,037

Btu/lbm1 =

ft lbm 32.2

s lbf1

s lbf

ft slug1

s lbf

ft lbm 32.21

2

22 ⋅

⋅=

⋅

⋅=

⋅

⋅=

kJ 4.18

kcal

J 4.18

cal

J 1055

Btu

lbfft 778

Btu1 ===

⋅=

md

ft10 1.06

md

m10 9.871

2-142-16 ×=

×=

Pete 314 1.8

Flow

� Continuous deformation

� Change of velocity of particles with respect to location

� If all particles move with the same velocity, there is no flow

(example: translation or rotation)

water

Movement but not flow! Movement but not flow!

From our point of view these are related to fluid statics!

Pete 314 1.9

Description & Classification of Fluid

Motions

� Viscous and Inviscid Flows

� Laminar and Turbulent Flows

� Compressible and Incompressible Flows

� Internal and External Flows

Internal Flows

• Flows completely bounded by solid surfaces.

Examples: Flow in ducts and vessels; flow in pumps, fans, and compressors.

• Open-channel flow

The internal flow of liquids in which the ducts does not flow full.

Examples: Flow in rivers, irrigation ditches, and aqueducts.

External Flows

• Flow over bodies immersed in an unbounded fluid.

Examples: Flow over a submerged submarine, flow over an airplane, and

flow over a gulf ball, etc.

Pete 314 1.10

Fluid types: Gas vs liquid

If the fluid is a gas: it will expand readily, filling all the space vacated by the piston; gases can expand without limit to occupy space made available to them.

If the fluid is a liquid: as the piston is raised, the liquid can expand only a small amount, and then it can expand no more. What fills the space between the piston and the liquid? Part of the liquid must turn into a gas by boiling, and this gas expands to fill the vacant space. (When the molecules separate more than this distance, they cease behaving as a liquid and behave as a gas)

Because of their closer molecular spacing, liquids normally have higher densities, viscosities, refractive indices, etc., than gases. This frequently leads to quite different behaviors of liquids and gases.

Pete 314 1.11

Properties of Fluids

� Fundamental approach: Study the behavior of

individual molecules when trying to describe the

behavior of fluids

� Engineering approach: Characterization of the

behavior by considering the average, or

macroscopic, value of the quantity of interest,

where the average is evaluated over a small

volume containing a large number of molecules

• Treat the fluid as a continuum: Assume that all the fluid

characteristics vary continuously throughout the fluid

Pete 314 1.12

Measures of Fluid Mass and Weight

Density of a fluid, ρ (rho), is the amount of mass per unit volume of a substance:

ρ = m / Vo

� For liquids, weak function of temperature and pressure

� For gases: strong function of T and P

from ideal gas law: ρ = P M / (R T)

where R = universal gas constant, M = molecular mass (In strict SI the unit of

molecular mass is kg/mol)

In strict SI the gas constant “is really constant” :

In the English system :

)T,P(ρ=ρ

R mol-lb

Btu 1.987

R mol-lb

ftpsi 10.73R

oo

3

⋅=

⋅

⋅=

K mols

mkg 8.314

K mol

mN 8.314

K mol

J 8.314

K mol

mPa 8.314R

2

3

⋅⋅

⋅=

⋅

⋅=

⋅=

⋅

⋅=

Pete 314 1.13

DensityV

m =ρ

Pete 314 1.14

Pete 314 1.15

Specific gravity

Pete 314 1.16

Viscosity (resistance to flow)

Shear

stress=γ&

Pete 314 1.17

Newtonian vs non-Newtonian rheology

)(στ f=

Pete 314 1.18

Pete 314 1.19

s ft

lbm 106.72 cP 1

s Pa 0.001 cP 1

s cm

g 0.01 cP 1

4-

⋅×=

⋅=

⋅=

( )cP 0100s m

kg 1 s Pa 1 =

⋅=⋅

Units of viscosity

� The viscosity is the slope of the line of shear stress versus shear rate

so its SI unit is 1 Pa / (1/s) = 1 Pa · s

� The customary unit of viscosity is the poise , however it is too large a

unit for most common fluids.

� By sheer coincidence the viscosity of pure water at about is 0.01

poise; for that reason the common unit of viscosity in the US is the

centipoise.

Pete 314 1.20

Kinematic viscosity

Exercise: What is the kinematic viscosity of air in strict SI ?

Pete 314 1.21

Surface tension

Pete 314 1.22

Units

1 N/m = 1 kg / s2

(lbf/in)

dyne/cm

1 dyne/cm = 0.001 N/m

Pete 314 1.23

Exercise: Water at normal conditions has approx 73 dyne/cm.

What would be the required mass in the previous

example if the fluid were water?

Pete 314 1.24

Pressure: compressive force divided by area

Ordinary fluids cannot permanently resist shear forces, so the water begins to flow and finally flows away.

If we really wanted to squeeze the water, we would put it in some container that would prevent its flowing out to the side.

The pressure at a point in a fluid at rest is the same in all directions.

The usual definition of pressure in a solid is as follows: Pressure at a point is the average of the compressive stresses measured in three perpendicular directions.

22

2

2 sm

kg 1

m

s

mkg

1 m

N1 Pa 1

⋅=

⋅

==

ft

lbf144

in

lbf1 psi 1

22==

Pete 314 1.25

Absolute and gauge pressure

-7.25 psig

Pete 314 1.26

2

2

s

m 3.43

kg

s

m kg

4.56

15.46==

Pete 314 1.27

Problems

1. Calculate the density of drilling mud that is 50 wt. % water, 50 wt. %

BaSO4 (barite). Use SGbarite = 4.49.

2. Why are specific gravities most often referred to the density of water

at 4 oC instead of 0 oC?

3. How many U.S. gallons are there in a cubic mile?

4. The total proven oil reserves of the U.S. are roughly 30 x 109 bbl.

How many cubic miles is this?

5. What is the density of Hg in strict SI units?

6. Estimate your mass, height, average body radius. Approximating your

body as a cylinder, calculate your volume. Then calculate your density

from the above obtained variables. Now compare the obtained density

to any other information you might have (e.g. is it easy to swim?)

Pete 314 1.28

6.

2 cm3 gray fluid is poured into

one tube of an U-tube of diameter

0.5 cm. The difference of the two

levels becomes 5 cm. Find the

Specific Gravity of the gray fluid. Hint: First find l from the volume of

the gray fluid

2 cm3

l A

B C

D

Pete 314 1.29

(1 atm = 101,325 Pa)

7.

Pete 314 1.30

8. (Example 1.2.)Concentric-cylinder (“cup and bob”, e.g.

Fann or Brookfield) viscometer also

called a Couette viscometer. An inner

cylinder (the bob) rotates inside a

stationary outer cylinder (the cup). The

shaft that drives the bob is

instrumented to record both the angular

velocity and the applied torque. What is

the shear rate and the shear stress?

Hint: Consider this viscometer as simply

the device in Fig. 1.4, wrapped around

a cylinder.

D1= 25.15 mm

D2= 27.62 mm

L = 92.27 mm

Rotation rate: N = 10 rpm

Observed torque: Γ = 5×10-6 N·m

(book error)

Pa 0545.0m

N0545.0

m)1027.92(m)1015.25(

)2 /m1015.25/(m N 105/

s

166.10

0

s

m1013.17

m101.23550

233

36

1

1

13

11

3

12

==××

××=

Γ==

=−

=×==

×=−=

−−

−−

−

−

ππτ

γπ

LD

r

A

F

∆r

VNDV

)D(D.∆r

&(not exact)

Pete 314 1.31

1. Fluid mechanics is the study of forces and motions in fluids.

2. Fluids are substances that move continually when subjected to a shear force as long as the force is applied. Solids are substances that deform slightly when subjected to a shear force and then stop moving and permanently resist the force. (There are, however, intermediate types of substances.)

3. Fluid mechanics is based on the principle of the conservation of matter, the first two laws of thermodynamics, Newton’s laws of motion, and careful experiments.

4. Gases have weak intermolecular attractions and expand without limit. Liquids have much stronger intermolecular attractions and can expand very little. With increasing temperature and pressure, the differences between liquids and gases gradually disappear.

5. Density is mass per unit volume. Specific gravity of liquids is density / (density of water at ). Specific gravity of gases is density / (density of air at the same T and P).

6. Viscosity is a measure of a fluid’s resistance to flow. Most simple fluids are represented well by Newton’s law of viscosity. The exceptions (non-Newtonian fluids) are generally complex mixtures, some of which are of great practical significance. Kinematic viscosity is viscosity divided by density.

7. Surface tension is a measure of a liquid’s tendency to take a spherical shape, caused by the mutual attraction of the liquid’s molecules.

8. Pressure is compressive force divided by area. It is the same in all directions for a fluid at rest and practically the same in all directions for most moving fluids.

9. Much of fluid mechanics can be based either on force and momentum, or on energy. We base most of fluid mechanics on energy.