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# Fluid friction

The pressures in a flowing fluid are calculated assuming the value of the Fanning friction factor is known. Determination of the Fanning friction factor, in fact, may be the most difficult step in this calculation. Fluid friction is studied by the science of rheology.

## Fluid rheology

The science of rheology is concerned with the deformation of all forms of matter, but has had its greatest development in the study of the flow behavior of suspensions in pipes and other conduits. The rheologist is interested primarily in the relationship between flow pressure and flow rate, and in the influence thereon of the flow characteristics of the fluid. There are two fundamentally different relationships:

1. The laminar flow regime prevails at low flow velocities. Flow is orderly, and the pressure-velocity relationship is a function of the viscous properties of the fluid.
2. The turbulent flow regime prevails at high velocities. Flow is disorderly and is governed primarily by the inertial properties of the fluid in motion. Flow equations are empirical.

The laminar flow equations relating flow behavior to the flow characteristics of the fluid are based on certain flow models, namely:

• Newtonian
• Bingham plastic
• Pseudoplastic
• Yield power-law
• Dilatant

Only the first four are of interest in drilling-fluid technology. Most drilling fluids do not conform exactly to any of these models, but drilling-fluid behavior can be predicted with sufficient accuracy by one or more of them. Flow models are usually visualized by means of consistency curves, which are plots either of flow pressure vs. flow rate or of shear stress vs. shear rate.

Shear stress is force per unit area and is expressed as a function of the velocity gradient of the fluid as ....................(1)

where μ is the fluid viscosity and dv/dr is the velocity gradient. The negative sign is used in Eq. 1 because momentum flux flows in the direction of negative velocity gradient. That is, the momentum tends to go in the direction of decreasing velocity The absolute value of velocity gradient is called the shear rate and is defined as ....................(2)

Then, Eq. 1 can be written as ....................(3)

Viscosity is the resistance offered by a fluid to deformation when it is subjected to a shear stress. If the viscosity is independent of the shear rate, the fluid is called a Newtonian fluid. Water, brines, and gases are examples of Newtonian fluid. The shear stress is linear with the shear rate for a Newtonian fluid and is illustrated by Curve A in Fig. 1. The symbol μ without any subscript is used to refer to the viscosity of Newtonian fluid. Most of the fluids used in drilling and cementing operations are not Newtonian, and their behavior is discussed next.

If the viscosity of a fluid is a function of shear stress (or, equivalently, of shear rate), such a fluid is called non-Newtonian fluid. Non-Newtonian fluids can be classified into three general categories:

1. Fluid properties are independent of duration of shear.
2. Fluid properties are dependent on duration of shear.
3. Fluid exhibits many properties that are characteristics of solids.

### Time independent

The following three types of materials are in this class.

#### Bingham plastic

These fluids require a finite shear stress, τy; below that, they will not flow. Above this finite shear stress, referred to as yield point, the shear rate is linear with shear stress, just like a Newtonian fluid. Bingham fluids behave like a solid until the applied pressure is high enough to break the sheer stress, like getting catsup out of a bottle. The fluid is illustrated by Curve B in Fig. 1. The shear stress can be written as ....................(4)

where τy is called the yield point (YP), and μp is referred to as the plastic viscosity (PV) of the fluid. Some water-based slurries and sewage sludge are examples of Bingham plastic fluid. Most of the water-based cement slurries and water-based drilling fluids exhibit Bingham plastic behavior. Drilling muds are often characterized with YP and PV values, but this is for historical reasons and does not necessarily imply that the Bingham fluid model is the best model for all muds.

#### Pseudoplastic

These fluids exhibit a linear relationship between shear stress and shear rate when plotted on a log-log paper. This is illustrated by Curve C in Fig. 1. This fluid is also commonly referred to as a power-law non-Newtonian fluid. The shear stress can be written as ....................(5)

where K is the consistency index, and n is the exponent, referred to as power-law index. A term μa is defined that is called the apparent viscosity and is ....................(6)

Note that apparent viscosity and effective viscosity as defined by different authors are not always defined in the sense used here, so read with caution. The apparent viscosity decreases as the shear rate increases for power-law fluids. For this reason, another term commonly used for pseudoplastic fluids is "shear thinning." Polymeric solutions and melts are examples of power-law fluid. Some drilling fluids and cement slurries, depending on their formulation, may exhibit power-law behavior.

#### Yield power law

Also known as Herschel-Bulkley fluids, these fluids require a finite shear stress, τy, below which they will not flow. Above this finite shear stress, referred to as yield point, the shear rate is related to the shear stress through a power-law type relationship. The shear stress can be written as ....................(7)

where τy is called the yield point, K is consistency index, and m is the exponent, referred to as power-law index.

### Dilatant

These fluids also exhibit a linear relationship between shear stress and shear rate when plotted on a log-log paper and are illustrated as Curve D in Fig. 1. The shear stress expression for dilatant fluid is similar to power-law fluid, but the exponent n is greater than 1. The apparent viscosity for these fluids increases as shear rate increases. For this reason, dilatant fluids are often called "shear-thickening."

Quicksand is an example of dilatant fluid. In cementing operations, it would be disadvantageous if fluids increased in viscosity as shear stress increased.

#### Time dependent

These fluids exhibit a change in shear stress with the duration of shear. This does not include changes because of reaction, mechanical effects, etc. Cement slurries and drilling fluids usually do not exhibit time-dependent behavior. However, with the introduction of new chemicals on a regular basis, one should test and verify the behavior.

#### Solids characteristic

These fluids exhibit elastic recovery from deformation that occurs during flow and are called viscoelastic. Most of the cement slurries and drilling fluids do not exhibit this behavior. However, as mentioned earlier, new polymers are being introduced on a regular basis, and tests should be conducted to verify the behavior.

The unit of viscosity is Pascal-second (Pa-s) in the SI system and lbf/(ft-s) in oilfield units. One Pa-s equals 10 poise (P), 1,000 centipoise (cp), or 0.672 lbf/(ft-s). The exponent n is dimensionless, and consistency index, K, has the units of Pa-sn in the SI system and lbf/(secn-ft–2) in oilfield units. One Pa-sn equals 208.86 lbf/(secn.ft–2). The yield point for Bingham fluids is often characterized in units of lbf/(1,00ft2), and plastic viscosity is usually given in centipoise.

### Viscometry

The rheology parameters of the fluids, μ and μp , and τo , K, and n, are determined by conducting tests in a concentric viscometer. This consists of concentric cylinders with one of them rotating, usually the outer one. A sample of fluid is placed between the cylinders, and the torque on the inner cylinder is measured. Assuming an incompressible fluid, with flow in the laminar flow regime, the equations of motion can be solved for τ to give ....................(8)

where

τ = shear stress, Pa;

MT = torque, N-m;

L = length, m;

and

In a concentric viscometer, torque, MT, is measured at a different rotational speed of the outer cylinder. Shear stress is then calculated from Eq. 8 , and shear rate is given by

and ....................(9)

where

Rb = radius of inner cylinder (bob), m;

Rc = radius of outer cylinder (cup), m;

κ = ratio of radius of inner cylinder to outer cylinder;

and

Ω0 = angular velocity of outer cylinder.

Shear stress and shear rate are then analyzed to determine the rheology model.

A number of commercially available concentric cylinder rotary viscometers are suitable for use with drilling muds. They are similar in principle to the viscometer already discussed. All are based on a design by Savins and Roper, which enables the plastic viscosity and yield point to be calculated very simply from two dial readings, at 600 and 300 rpm, respectively. They are referred to in the industry as the direct-indicating viscometer and typically are called Fann viscometers.

The underlying theory is as follows: Eqs. 8 and 9 are combined to give ....................(10)

where avs and bvs are constants that include the instrument dimensions, the spring constant, and all conversion factors; ω is the rotor speed in revolutions per minute (rpm).

Then, ....................(11)

where θ1 and θ2 are dial readings taken at ω1 and ω2 rpm, respectively. PV is the conventional oilfield term for plastic viscosity, thus measured. Then, the yield point is determined. ....................(12)

YP is the conventional oilfield term for yield point, thus measured. The numerical values of avs , bvs , ω1 , and ω2 were chosen so that ....................(13)

and ....................(14)

Apparent viscosity μa may be calculated from the Savins-Roper viscometer reading as

and ....................(15)

where θ is the dial reading at ω rpm. Typical viscometer results are shown in Fig. 2. Notice that real fluids are not ideally any of the models shown, but generally are pretty close to one model or another. The selection of the model may be motivated by a particular fluid velocity of interest. For instance, fluid 6 in Fig. 2 would be modeled well by a yield-power law for rpm below about 100.

### Fanning friction factor correlations

Flow in pipes and annuli are typically characterized as laminar or turbulent flow. Laminar flow often can be solved analytically. Correlation for turbulent flow is usually developed empirically by conducting experiments in a flow loop. Typical data will look like those that are shown in Fig. 3. Experimental data are usually analyzed and correlated through the use of two dimensionless numbers: f, the Fanning friction factor, and Re, the Reynolds number. The relationship between the friction factor, f, and Reynolds number for Newtonian fluids is given in Fig. 4, with the pipe roughness given in Fig. 5. This figure is based on the experimental results of Colebrook. The relationship between friction factor f vs. Re for pseudoplastic fluids is shown in Fig. 6. This figure is based on the experimental results of Dodge and Metzner. Here non-Newtonian fluids usually assume this pseudoplastic friction factor for turbulent flow.

The pressure drop per unit length for flow through a duct is given by ....................(16)

where f is Fanning friction factor, Δz is the length, v is the velocity, ρ is the density, Dhyd is a characteristic "diameter," and ΔP is the pressure drop. The friction factor depends on Reynolds number, Re, and the roughness of the pipe. The Reynolds number, Re, is defined as ....................(17)

where ρ is the density of the fluid, v is the average velocity, D is a characteristic length (e.g., pipe diameter), and μ is a characteristic viscosity. Correlations for friction factor, f, in both laminar and turbulent flow regime and for critical Reynolds number are available for a number of fluids and geometries. However, in critical situations, it is recommended that flow-loop tests be conducted and data compared with calculations that are based on fundamental equations for flow. For example, experimental data in laminar flow should be compared with estimated values from correlation such as Eq. 20. However, some solid-laden polymers are known to exhibit what is known as shear-induced diffusion, in which solids migrate away from the walls to the center of the pipe. These fluids show deviation in calculated and experimental values in laminar flow. Correlations should be modified as needed to reflect this behavior. Several polymers are known to exhibit drag reduction in turbulent flow. Theoretical prediction of polymer-flow behavior is not yet good enough, so flow-loop data are almost always needed.

Commonly used Fanning friction correlations are summarized in the next section. Correlations are provided for three geometric configurations: pipe flow, concentric annular flow, and slit flow. For each case, the ΔP and Re are defined for the specific geometry and flow model. The laminar flow equations for annular flow are approximate for Newtonian and power-law flow in annuli with low-clearance but reasonably accurate and much simpler than the exact solutions. Note that for low clearance annuli, the slit flow model provides almost as accurate a result as the concentric annular model but can also be modified to account for eccentric annuli.

### Rheological model 1: Newtonian fluids

#### Pipe flow

Frictional pressure drop: ....................(18)

Reynolds number: ....................(19)

where Di is the pipe inside diameter (ID).

Laminar flow: ....................(20)

for Re < 2,100. Turbulent flow: ....................(21)

for Re > 3,000, and k is the absolute pipe roughness in the same units as D.

#### Annular flow

Frictional pressure drop: ....................(22)

Reynolds number: ....................(23)

where Do is the annulus outside diameter (OD), and Di is the ID.

Laminar flow: ....................(24)

for Re < 2,100. Turbulent flow: ....................(25)

for Re > 3,000, and k is the absolute pipe roughness in the same units as D.

### Rheological model 2: Bingham plastic fluids

#### Pipe flow

Frictional pressure drop: ....................(26)

Reynolds number: ....................(27)

where Di is the pipe ID, and μp is the plastic viscosity.

Laminar flow: ....................(28)

for Re < ReBP1, where Turbulent flow: ....................(29)

for Re > ReBP2 , where: For He < = 0.75 × 105 , A = 0.20656, and B = 0.3780. For 0.75 × 10 5 < He< = 1.575 × 10 5 , A = 0.26365, and B = 0.38931. For He > 0.75 × 10 5 , A = 0.20521, B = 0.35579, and He = τoρD2/μp2 .

#### Annular flow

Frictional pressure drop: ....................(30)

Reynolds number: ....................(31)

where Do is the annulus OD; Di is the ID; and μp is the plastic viscosity.

Laminar flow: ....................(32)

for Re < ReBP1 , where: Turbulent flow: ....................(33)

for Re > ReBP2, where: For He < = 0.75 × 105 , A = 0.20656, and B = 0.3780. For 0.75 × 105 < He< = 1.575 × 105 , A = 0.26365, and B = 0.38931. For He > 0.75 × 105 , A = 0.20521, B = 0.35579, and He = τoρ(Do2Di2)/ μp2 .

#### Slit flow

Frictional pressure drop: ....................(34)

Reynolds number: ....................(35)

where Do is the annulus OD; D i is the ID; and μp is the plastic viscosity.

Laminar flow: ....................(36)

for Re < ReBP1 , where: Turbulent flow: ....................(37)

for Re > ReBP2 , where: For He < = 0.75 × 10 5, A = 0.20656, and B = 0.3780. For 0.75 × 105 < He< = 1.575 × 105 , A = 0.26365, and B = 0.38931. For He > 0.75 × 105 , A = 0.20521, B = 0.35579, and He = τoρ(Do2Di2)/(1.5μp)2 .

### Rheological model 3: power law fluids

#### Pipe flow

Frictional pressure drop: ....................(38)

Reynolds number: ....................(39)

where Di is the pipe ID. Laminar flow: ....................(40)

for Re ≤ 3,250 – 1,150n .

Turbulent flow: ....................(41)

for Re ≥ 4,150 – 1,150n.

#### Annular flow

Frictional pressure drop: ....................(42)

Reynolds number: ....................(43)

where Do is the annulus OD, and Di is the ID.

Laminar flow: ....................(44)

for Re ≤ 3,250 – 1,150n.

Turbulent flow: ....................(45)

for Re ≥ 4,150 – 1,150n.

#### Slit flow

Frictional pressure drop: ....................(46)

Reynolds number: ....................(47)

Laminar flow: ....................(48)

for Re ≤ 3,250 – 1,150n.

Turbulent flow: ....................(49)

for Re ≥ 4,150 – 1,150n.

### Rheological model 4: yield power law (YPL) fluids

#### Pipe flow

Fictional pressure drop: ....................(50)

Reynolds number: ....................(51)

where

Laminar flow: ....................(52)

for Re ≥3,250 – 1,150n.

Turbulent flow: ....................(53)

for Re ≥ 4,150 – 1,150n.

#### Slit flow

Frictional pressure drop: ....................(54)

Reynolds number: ....................(55)

where

and ....................(56)

Laminar flow: ....................(57)

for Re ≤ 3,250 – 1150n .

Turbulent flow: ....................(58)

for Re ≥ 4,150 – 1,150n .

### Frictional pressure drop in eccentric annulus

The frictional pressure drop in an eccentric annulus is known to be less than the frictional pressure drop in a concentric annulus. For laminar flow of Newtonian fluids, the pressure drop in a fully eccentric annulus is half the pressure drop in a concentric annulus. For turbulent flow, the difference is about 10%. For non-Newtonian fluids, the effect is less but still significant. In deviated wells, the drillpipe should be fully eccentric over much of the deviated wellbore, resulting in reduced fluid friction.

Define the correction factor for eccentricity. ....................(59)

where subscript e denotes eccentric, and subscript c denotes concentric.

Ce for laminar flow is determined based on the methods used by Uner et al. The flow rate through a concentric annulus is given by ....................(60)

where Rr = ri/ro . The flow rate through an eccentric annulus was determined to be ....................(61)

where ....................(62) ....................(63)

and ....................(64)

where δr is the distance between centers of the inside and outside pipes (e.g., δr = 0 for concentric pipes). The geometry of the eccentric annulus is illustrated in Fig. 7.

The function E may be evaluated using a six-coefficient approximation. The function F must be evaluated using numerical methods (e.g., a seven-point Newton-Cotes numerical integration formula). Setting qa and qe equal, then ....................(65)

Because Ce depends only on f, n, and Rr, Ce need be calculated only once, then used for all future frictional pressure drop calculations, as long as the property n does not vary.

Ce for turbulent flow is determined by applying the same techniques to the turbulent velocity profile determined by Dodge and Metzner. ....................(66)

where ....................(67) ....................(68) ....................(69)

and ....................(70)

The volume flow rate through the concentric annulus is given by ....................(71)

where

h = rori ,

and

w = π(ro + ri ). Integrating Eq. 36 gives ....................(72)

where A is the flow area.

The equivalent integral to Eq. 71 for eccentric flow is given by ....................(73)

where ....................(74)

and ....................(75)

The integral in Eq. 73 must be evaluated numerically (e.g., by a seven-point Newton-Cotes numerical integration). Ce can be determined by setting Eq. 72 equal to Eq. 73 and noting that ....................(76)

where vc* is determined from the concentric solution given by Dodge and Metzner. The resulting nonlinear equation must be solved for Ce numerically (e.g., by using Newton ’ s method). Because Ce depends only on f1, f2 , f3 , n, and Rr , Ce need be calculated only once, then used for all future frictional pressure-drop calculations, as long as the properties ρ,K, and n do not vary.

## Sample calculations

The most important consideration in making hydraulic calculations is the use of consistent units. Unfortunately, oilfield units are rarely consistent; in some cases they are unique to the industry. The universal set of consistent units is the SI Metric System of Units. The Society of Petroleum Engineers (SPE) has available a publication: "The SI Metric System of Units and SPE Metric Standard" that contains every conversion factor necessary. Whenever there is a question of units, the safest solution is to convert all units to SI units, solve the problem, and then convert the answer back to the common engineering units.

### Sample problem

#### Geometry

A deviated well kicks off at 3,000 ft and is drilled to total depth (TD) at an angle of 30° to the vertical. The well ’ s total measured depth is 11,000 ft. The well is cased with 72-ppf 13 3/8-in. casing (13.375 × 12.347 in.) set at 3,000 ft. The drillstring consists of 900 ft of 8-in. 147-ppf drill collars (8 × 3 in.), 19 ½-ppf drillpipe (5 × 4.206 in.), a 9 5/8 -in. bit with 3× 13/22-in. nozzles. The undisturbed temperature is 70°F at the surface with a 1.4°F/100-ft gradient. We will neglect the build section and assume the well trajectory is vertical to 3,000 ft measured depth, and deviated at 30° to the vertical from 3,000 ft measured depth to 11,000 ft measured depth. We will assume the open hole is gauge (9.625 in.).

#### True vertical depth

For measured depth < 3,000 ft, Z = z. For measured depth > 3,000 ft, Z = 3,000 ft + (z-3,000)sin(30)≅ 402 + 0.866z, where z is measured depth in feet, Z is true vertical depth in feet.

#### Hydrostatic pressure

1. Assume the wellbore is filled with 8.34 lbm/gal fluid (fresh water). What is the pressure at TD? True vertical depth at TD is 402 + 0.866 × 11,000 = 9,928 ft. Using (Eq. 10 from Static wellbore pressure solutions) and converting to SI units: This pressure is gauge pressure at TD. For absolute pressure, add atmospheric pressure, 14.7 psi:

2. Assume a layered wellbore with 14 lbm/gal mud from surface to 5,000 ft (measured depth) and 9 lbm/gal mud from 5,000 ft to TD. What is the pressure at TD?

For layer 1:

For layer 2:

3. Assume the wellbore is filled with nitrogen with a surface pressure of 2,000 psi. What is the pressure at TD? This problem is much more difficult because the gas density and temperature vary over the length of the wellbore. The pressure change is given by

The temperature distribution is given by T(Z) = 70 + .014 Z, where Z is true vertical depth. Because we need absolute temperature, in Kelvin: (T °F+ 459.67)/1.8 = T°K, The integral of 1/T with respect to Z is

and

#### Frictional pressure loss

4. Assume fresh water is being circulated at 600 gal/min. What is the pressure change inside a single vertical 30-ft joint of drillpipe? Assume the density is 8.34 lbm/gal and the viscosity is 1 cp.

This Reynolds number indicates turbulent flow. To determine the friction factor, first determine the relative roughness k/D. From Fig. 5, the relative roughness is about .0004 for commercial steel. The friction factor is about .011 from Fig. 4. Friction pressure drop is given by

The hydrostatic pressure change per foot is

Total pressure change per length of pipe for flow downward is

The total pressure change in a 30-ft pipe joint is 0.166 × 30 = 4.98 psi.

5. Assume a 10-lbm/gal mud is being circulated at 100 gal/min. What is the frictional pressure change in the annulus outside a single 30-ft joint of drillpipe? Use the Bingham plastic model and assume the plastic viscosity is 40 cp and the YP is 15 lbf/100 ft 2.

6. Repeat Calculation 5, but assume the fluid is a power-law fluid. Remember that PV and YP were determined from the 300-rpm and 600-rpm readings of the Fann viscometer. The equivalent shear stresses are

7. For a flow rate of 600 gal/min, what is the fluid pressure in the bit nozzles? The mud density is 12 lbm/gal. What is the pressure recovery in the annulus?

## Nomenclature

 a = acoustic velocity, m/s αvs , bvs = constants that include the viscometer dimensions, the spring constant, and all conversion factors A = flow area (see subscripts), m2 c = average concentration of cuttings overall ca = cuttings concentration in annular region co = feed concentration of cuttings cp = cuttings concentration in plug region C = compressibility Cd = discharge coefficients for the flow through an area change, dimensionless CD = drag coefficient, dimensionless Ce = pressure drop correction factor for pipe eccentricity, dimensionless Cp = heat capacity at constant pressure, J/kg-K Cv = heat capacity at constant volume, J/kg-K ds = particle diameter, m dv/dr = velocity gradient, s –1 dv/dt = total derivative of velocity with respect to time, Pa/s D = characteristic length in Reynolds number, m De = special equivalent diameter for yield power law fluid, m Deq = equivalent diameter, m Dhyd = hydraulic diameter, m Dh = wellbore diameter, m Di = inside diameter, m Do = outside diameter, m Dp = drillpipe outside diameter, m Dplug = plug diameter, m Ef = Young’ s modulus for the formation, Pa E(k) = complete elliptic integral of the second kind, parameter k ƒ = Fanning friction factor, dimensionless ƒ1 , ƒ2 , ƒ3 = turbulent flow velocity profile parameters, dimensionless ƒi = the fraction of flow in the pipe, ith iteration ƒm = the fraction of flow in the pipe, lower annular pressure ƒp = the fraction of flow in the pipe, higher annular pressure F(ƒ,n, Rr) = eccentric flow function Fd = total viscous drag force on the particle, N g = acceleration of gravity, m/s2 G = mass flow rate density of mixture, kg/m3–s Gs = mass flow rate density of solids, kg/m3–s h = specific enthalpy, J/kg h = total friction pressure drop, Pa/m He = Hedstrom number Hs = holdup of solid particles, volume fraction of solids k = absolute pipe roughness, m k = cp/cv K = consistency index for pseudoplastic fluid, Pa-sn Kb = elastic bulk modulus, Pa L = length of viscometer bob, m m = power-law exponent for Herschel-Bulkley fluids ṁ = mass flow rate, kg/s ṁs = mass flow rate of solid, kg/s MT = torque measured by viscometer, N-m n = power law exponent for pseudoplastic fluids pn = pressure in bit nozzle, Pa pr = pressure in bit annular area, Pa P = pressure, Pa patm = atmospheric pressure, Pa = aerodynamic force exerted on the cuttings by the air, N qa = total volumetric flow rate, m3/s qc = volumetric flow rate through concentric annulus, m3/s qe = volumetric flow rate through eccentric annulus, m3/s Q = heat transferred into volume, W Qc = volumetric flow rate of the cuttings, m3/s Qm = volumetric flow rate of the mud, m3/s ri = inside radius of annulus, m ro = outside radius of annulus, m R = ideal gas constant, m3 Pa/kg-K Rb = radius of inner cylinder (bob) of viscometer, m Rc = radius of outer cylinder (cup) of viscometer, m Re = Reynolds number Rep = particle Reynolds number Rr = ri/ro S = entropy, J/K t = time, s T = absolute temperature, °K u = radial displacement, m v* = characteristic velocity for turbulent flow calculations, m/s v = average velocity, m/s va = average annulus velocity, m/s vmix = mixture velocity, m/s vn = velocity in bit nozzle, m/s vp = plug velocity, m/s vr = velocity in bit annular area, m/s vs = average settling velocity vsa = average cuttings velocity in annular region, m/s vsl = particle slip velocity, m/s vsp = average cuttings velocity in plug, m/s W = buoyant weight or particle, N Ws = buoyant weight of the cuttings, N x = parameter in settling velocity equation y = parameter in settling velocity equation Ya = parameter in settling velocity equation z = measure depth, ft Z = true vertical depth, ft αc = parameter in Bingham fluid friction factor β = = coefficient of thermal expansion, 1/K γ = shear stress, s–1 γe = equivalent shear stress, s–1 δ = ratio of the average particle volume to its cross-sectional area δr = the distance between centers of the inside and outside pipes, m ΔP = pressure drop, Pa Δt = time increment, s Δv = change in velocity, m/s Δz = length of flow increment, m ε = internal energy, J/kg ζ = measured depth integration variable, m θ = viscometer reading, degrees ϑ = integration variable κ = ratio of radius of inner cylinder to outer cylinder λP = Dplug/Deq , the plug diameter ratio μ = Newtonian viscosity of the fluid, Pa-s μa = apparent viscosity, Pa-s μp = plastic viscosity, centipoise ξ = integration variable corresponding to depth z, m ρ = fluid density, kg/m3 = fluid in-mixture density, kg/m3 ρƒ = fluid density in solid/fluid mixture, kg/m3 ρs = solid density in solid/fluid mixture, kg/m3 = solid in-mixture density, kg/m3 τ = shear stress, Pa τw = wall shear stress, Pa τy = yield point, Pa υƒ = Poisson’ s ratio for the formation Φ = angle of inclination from the vertical Φ = viscous dissipation, W Ψ = sphericity ω = rotor speed, rev/min Ωo = angular velocity of outer cylinder

Subscripts

1 = properties inside pipe, surge calculations

2 = properties inside annulus, surge calculations

3 = properties of moving pipe, surge calculation

c = concentric

e = eccentric

n = properties in bit nozzle, surge calculations

o = upstream, initial, or inlet

r = properties in annulus outside bit, surge calculations

Superscripts

- = upstream properties