Rheology and Hydraulics - wzanews.comwzanews.com/IMG2/book/drilling/Hydraulics/Rheology and hydrology.pdf · Rheology and hydraulics 9-3 Overview Fluid rheology and hydraulics are

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9Rheology and hydraulics

Contents

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3

Rheological terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-3

Flow regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-5

Fluid types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-5

Rheological models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-6Bingham model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-7Power law model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-8

Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-9Herschel-Bulkley (yield-power law [YPL]) model . . . . . . . . . 9-10

Fluid hydraulics calculation terms . . . . . . . . . . . . . . . . . . . . . . . 9-11Reynolds number (N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-11ReCritical Reynolds number (NRec) . . . . . . . . . . . . . . . . . . . . . . 9-11Friction factor (f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-11Hedstrom number (N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-12HeEffective viscosity ( ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-13ePressure drop ()P/)L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-14Eccentricity (,) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-14

Fluid hydraulics equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-15Pump and circulating information . . . . . . . . . . . . . . . . . . . . . . 9-16

Pump output per stroke . . . . . . . . . . . . . . . . . . . . . . . . . . 9-16Pump output per minute . . . . . . . . . . . . . . . . . . . . . . . . . . 9-16Annular velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-17Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-17Circulating times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-18

Bit hydraulics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19

Nozzle area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19Nozzle velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19Bit pressure drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19Bit hydraulic horsepower . . . . . . . . . . . . . . . . . . . . . . . . . 9-19Bit hydraulic horsepower per unit bit area . . . . . . . . . . . . 9-19Percent pressure drop at bit . . . . . . . . . . . . . . . . . . . . . . . 9-19Jet impact force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-19

Calculations for laminar and turbulent flow . . . . . . . . . . . . . . 9-20Methods for Herschel-Bulkley (yield-power

law [YPL]) fluids . . . . . . . . . . . . . . . . . . . . . . . . 9-20Deriving dial readings . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-20API methods for power law fluids . . . . . . . . . . . . . . . . . . 9-21SPE methods for power law fluids . . . . . . . . . . . . . . . . . . 9-24SPE methods for Bingham-plastic fluids . . . . . . . . . . . . . 9-27

Equivalent circulating density . . . . . . . . . . . . . . . . . . . . . . . . 9-30Hole cleaning calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-31

Particle slip velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-31Cuttings transport efficiency calculations . . . . . . . . . . . . . 9-35MAXROP calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-35Cuttings concentration in the annulus for a given

penetration rate . . . . . . . . . . . . . . . . . . . . . . . . . 9-37Annular mud density increase . . . . . . . . . . . . . . . . . . . . . 9-38

List of terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9-38

Rheology and hydraulics

9-3

Overview

Fluid rheology and hydraulics are engineering termsthat describe the behavior of fluids in motion.

This chapter explains rheological terms and identifiesflow regimes. In addition, this chapter compares thedifferent rheological models and discusses theconditions under which they are used. Finally, thischapter explains fluid hydraulics and providescalculations for laminar and turbulent flow.

Rheological terms

The terms and definitions in the following table arerelevant to the discussion of rheology and hydraulics.

Rheologicalterm Symbol Unit(s) Definition

Shear rate sec The change in fluid velocity divided by the gap or( -1width of the channel through which the fluid ismoving in laminar flow.

Shear stress lb/100 ft The force per unit area required to move a fluidJ 2Pa at a given shear rate; shear stress is measured

on oil field viscometers by the deflection of themeter's dial at a given shear speed. The specificdial reading is usually denoted by 2. Example: 2300 describes the dial deflection at300 rpm on the rotational viscometer.

Shear speed rpm The rotational speed on a standard oil fieldviscometer on which the shear stress ismeasured.

Viscosity centipoise A fluid's shear stress divided by thecPPa@sec

corresponding shear rate, or = J/(. Fluidviscosity can be measured at a certain point orover a wide range of shear stress/shear ratemeasurements.

Baroid fluids handbook

Rheologicalterm Symbol Unit(s) Definition

Revised August 1, 1997 9-4

Effective cP The viscosity used to describe fluid flowingviscosity Pa@sec through a particular geometry; as hole

e

geometries change, so does .e

Yield point YP lb/100 ft The force required to initiate flow; the calculatedJy

2

Pa value of the fluid's shear stress when therheogram is extrapolated to the y-axis at ( = 0sec . -1

Note: The YP is a time-independentmeasurement and is usually associated with theBingham model.

Yield stress lb/100 ft The force required to initiate flow; the calculatedJ02

Pa value of the fluid's shear stress when therheogram is extrapolated to the y-axis at ( = 0sec . -1

Note: Yield stress is a time-independent measurement

and is usually denoted in the Herschel-Bulkley (yield-

power law [YPL]) model as J and Bingham model as0YP. It can also be considered a gel strength at zero

time.

Gel strengths none lb/100 ft Time-dependent measurements of a fluid's shear2

Pa stress under static conditions. Gel strengths arecommonly measured after 10-second, 10-minute,and 30-minute intervals, but they can bemeasured for any desired length of time.

Plastic PV cP The contribution to fluid viscosity of a fluid underviscosity Pa@sec dynamic flow conditions. Generally the plastic

viscosity is related to the size, shape, andnumber of particles in a moving fluid. PV iscalculated using shear stresses measured at2600 and 2300 on the FANN 35 viscometer.

Flow index n none The numerical relation between a fluid's shearstress and shear rate on a log/log plot. This valuedescribes a fluid's degree of shear-thinningbehavior.

Consistency K (eq) cP The viscosity of a flowing fluid identical inindex Pa@sec concept to the PV.n

lb/100 Note: Viscous effects attributed to a fluid's yieldft sec2 n stress are not part of the consistency index as

this parameter describes dynamic flow only.Table 9-1: Rheological terms. These terms are useful for understanding rheologicalformulas and calculations.


9-5

Flow regimes

There are three basic types of flow regimes. These are:

C LaminarC TurbulentC Transitional

Laminar flow occurs at low-to-moderate shear rates whenlayers of fluid move past each other in an orderly fashion.This motion is parallel to the walls of the channel throughwhich the fluid is moving. Friction between the fluid and thechannel walls is lowest for this type of flow. Mud rheologicalparameters are important in calculating frictional pressurelosses for muds in laminar flow.

Turbulent flow occurs at high shear rates where the fluidmoves in a chaotic fashion. Particles in turbulent flow arecarried by random loops and current eddies. Frictionbetween the fluid and the channel walls is highest for thistype of flow. Mud rheological parameters are not significantin calculating frictional pressure losses for muds in turbulentflow.

Transitional flow occurs when the flow shifts from laminarflow to turbulent flow or vice versa. The critical velocity of afluid is the particular velocity at which the flow changesfrom laminar to turbulent or vice versa.

Fluid typesThere are two basic types of fluids, Newtonian andnon-Newtonian. Rheological and hydraulic models have



been developed to characterize the flow behavior of thesetwo types of fluids.

Newtonian fluids have a constant viscosity at a giventemperature and pressure condition. Common Newtonianfluids include:

C DieselC WaterC GlycerinC Clear brines

Non-Newtonian fluids have viscosities that depend onmeasured shear rates for a given temperature and pressurecondition. Examples of non-Newtonian fluids include:

C Most drilling fluidsC Cement

Rheological models

Rheological models help predict fluid behavior across a widerange of shear rates. Most drilling fluids are non-Newtonian,pseudoplastic fluids. The most important rheological modelsthat pertain to them are the:

C Bingham modelC Power law modelC Herschel-Bulkley (yield-power law [YPL]) model

Figure 9-1 depicts typical rheological profiles for Bingham-plastic fluids, power law fluids, and Newtonian fluids. Atypical drilling fluid's rheological profile is also included toshow that these rheological models do not characterize non-Newtonian drilling fluids very well. The Herschel-Bulkley(yield-power law [YPL]) model is the most accurate modelfor

Shearstress

0Shear rate

Fluid behavior comparison

Bingham-plasticfluid

Power-law fluid

Newtonian fluid

Typicaldrilling fluid


9-7

Figure 9-1: Fluid behavior comparison. This chart shows that the Bingham, power law,and Newtonian fluid models do not predict the same behavior as a typical drilling fluid.

predicting the rheological behavior of common drillingfluids.

Binghammodel

The Bingham model describes laminar flow using thefollowing equation:

J = YP + (PV ()

Where

J is the measured shear stress in lb/100 ft2

YP is the yield point in lb/100 ft2

PV is the plastic viscosity in cP( is the shear rate in sec-1



Current API guidelines require the calculation of YPand PV using the following equations:PV = 2600 2300YP = 2300 PV, orYP = (2 2300) 2600

Because the model assumes true plastic behavior, theflow index of a fluid fitting this model must have n = 1.Unfortunately, this does not often occur and the modelusually overpredicts yield stresses (shear stresses atzero shear rate) by 40 to 90 percent. A quick and easymethod to calculate more realistic yield stresses is toassume the fluid exhibits true plastic behavior in thelow shear-rate range only. A low shear-rate yield point(LSR YP) can be calculated using the followingequation:

LSR YP = (2 23) - 26

This calculation produces a yield-stress value close tothat produced by other, more complex models and canbe used when the required computer algorithm is notavailable.

Power lawmodel

The power law model describes fluid rheologicalbehavior using the following equation:

J = K (n

This model describes the rheological behavior ofpolymer-based drilling fluids that do not exhibit yieldstress (i.e., viscosified clear brines). Some fluidsviscosified with biopolymers can also be described bypower-law behavior.

n 'log(J2/J1)log((2/(1)

K 'J2(2

n

n 'log 2600

2300

log 600300

n ' 3.32 log 26002300

K ' 511 2300

511n(in eq cP) or

K ' 511 2600

1022n(in eq cP)


9-9

The general equations for calculating a fluid's flowindex and consistency index are:

Where

J is the calculated shear stress in lb/100 ft2J is the shear stress at higher shear rate2J is the shear stress at lower shear rate1n is the flow index( is the shear rate in sec-1

( is the higher shear rate2( is the lower shear rate1K is the consistency index

Example

Using the shear stresses measured at shear rates equal to2600 and 2300, the general equations become:

or

C lb/100 ft 2 secn ' eq cP478.8



Note: The power law model can produce widelydiffering values of n and K. The results depend on theshear-stress/shear-rate data pairs used in thecalculations.

Herschel-Bulkley(yield-powerlaw [YPL])model

Because most drilling fluids exhibit yield stress, theHerschel-Bulkley (yield-power law [YPL]) modeldescribes the rheological behavior of drilling mudsmore accurately than any other model. The YPL modeluses the following equation to describe fluid behavior:

J = J + (K ( )0 n

Where

J is the measured shear stress in lb/100 ft2J is the fluid's yield stress (shear stress at zero shear0rate) in lb/100 ft2

K is the fluid's consistency index in cP or lb/100 ft sec2 n

n is the fluid's flow index( is the shear rate in sec-1

K and n values in the YPL model are calculateddifferently than their counterparts in the power lawmodel. The YPL model reduces to the Bingham modelwhen n = 1 and it reduces to the power law model whenJ = 0. An obvious advantage the YPL model has over0the power law model is that, from a set of data input,only one value for n and K are calculated.

Note: The YPL model requires:

C A computer algorithm to obtain solutions.

C A minimum of three shear-stress/shear-ratemeasurements for solution. Model accuracy isimproved with additional data input.


9-11

Fluid hydraulics calculationterms

Mathematical equations are used to predict the behaviorof drilling fluids flowing through pipes and annulars.Fluid velocities and pressure drops encountered whilecirculating are of particular importance to drillingoperations. Several important terms used in hydraulicscalculations are defined below.

Reynoldsnumber(N )Re

A dimensionless, numerical term governs whether aflowing fluid will be in laminar or turbulent flow. Often,a Reynolds number greater than 2,100 will mark theonset of turbulent flow, but this is not always so.

CriticalReynoldsnumber(N )Rec

This value corresponds to the Reynolds number atwhich laminar flow turns to turbulent flow.

Frictionfactor (f)

This dimensionless term is defined for power law fluidsin turbulent flow and relates the fluid Reynolds numberto a "roughness" factor for the pipe. Figure 9-2 showsthe relationship between Reynolds number and frictionfactor for laminar flow (N < 2,100), and for severalrevalues of n for fluids in turbulent flow (Re > 2,100).

987654

3

2

1987654

3

2

1987654

3

2 3 4 5 6 7 8 91 2 3 4 5 6 7 891 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

1,000,000100,00010,000

Reynolds number, NRe1,000100

0.1

0.01

0.001

0.0001

2

Friction factors for power-law fluid modelFrictionFactor, f

N=1N=0.8N=0.6

N=0.4

N=0.2



Figure 9-2: Friction factors for power law fluids. This graph shows friction factors versusReynolds numbers for power law fluids having different values of n.

Hedstromnumber(N )He

This dimensionless term predicts the onset of turbulentflow for fluids that follow the Bingham model. It iscorrelated with the critical Reynolds number (N ), asRecshown in Figure 9-3.

1

98765

4

3

2

1

9876

9876

5

4

3

2

2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

Hedstrom number, NHe103 104 105 106 107

105

104

103

Critical Reynolds numbers for Bingham-plastic fluids Critical Reynolds

number, NRec


9-13

Figure 9-3: Critical Reynolds numbers for Bingham-plastic fluids. This graph showsHedstrom numbers versus Reynolds numbers for Bingham-plastic fluids.

Effectiveviscosity ( )e

This term describes the viscosity of the fluid flowingthrough a particular geometry. It is different from theviscosity determined from the viscometer because thegeometries or wall gaps have changed. Similarly, thefluid flowing inside the drillpipe and in the annulus willhave different effective viscosities. Power law fluidswill then have different flow indexes (n and n ) andp adifferent consistency indexes (K and K ) as comparedp ato the n and K values calculated from viscometer 2600and 2300.



Pressuredrop ())P/))L)

Frictional forces develop when fluids flow through apipe or an annulus. As a result, fluid energy dissipates.These frictional forces are referred to as pressure drops,and are usually referred to as a pressure per unit length.The longer a pipe or annulus, the greater the pressuredrop. Factors that can affect the magnitude of pressuredrop include:

C LengthC Flow rate (flow regime type laminar or turbulent)C Fluid rheological propertiesC Pipe eccentricityC Pipe/annulus geometryC Pipe roughness, etc.

Eccentricity (,,)

This dimensionless term refers to the position of a pipeinside another pipe. In the oil field it usually refers tothe position of the drillpipe in an annulus. When thedrillpipe lies directly in the middle of the annulus, thedrillpipe's position is concentric and the eccentricityfactor is 0. See Figure 9-4 (a).

As the drillpipe moves to one side of the annulus, thedrillpipe becomes increasingly eccentric. If the sides ofthe drillpipe come in contact with the wall of theannulus, the drillpipe is fully eccentric and theeccentricity factor is 1.0. See Figure 9-4 (b).

Eccentricities of a pipe in an annulus

= 0(a)

= 1(b)


9-15

Figure 9-4: Eccentricities of a pipe in an annulus. As the drillpipe moves to one side ofthe annulus, the drillpipe becomes increasingly eccentric.

In high-angle or horizontal wells, the drillpipe usuallylies on the low side of the hole and its eccentricityfactor is 1 >=>= ,, >=>= 0. If the drillpipe lies on the upperside of the hole, its eccentricity factor is negative 0 >=>=,, >=>= -1. Drillpipe eccentricity can affect pressure dropsin the annulus by reducing the frictional forces of fluidflow. A fully concentric drillpipe in an annulus has thehighest pressure drops.

Fluid hydraulics equationsFluid hydraulic equations have been constructed usingrheological parameters from the Bingham and powerlaw models. Typically, pressure drop calculations forlaminar flow situations made using the Bingham model

Pump output ' efficiency100

(2 liner2 & rod diameter 2) stroke

6176.4

Pump output ' efficiency100

liner2 stroke

4117.6

Pump output, bbl/min (POBPM) 'pump output (bbl/stroke) strokes per minute

Pump output, gal/min (POGPM) ' POBPM 42



parameters overpredict actual pressure drops whilethose made using the power law model parametersunderpredict actual pressure drops. Errors in pressuredrop calculations can produce further errors in othercalculations, such as equivalent circulating density(ECD).

Hydraulic equations have been written using the YPLmodel and their solutions can be calculated using thecomputer programs. Because the YPL model betterpredicts drilling fluid rheological behavior at low shearrates, more accurate values result for pressure drops inlaminar flow, ECDs, etc.

Pump andcirculatinginformation

Pump output per stroke

Duplex pump (bbl/stroke):

Triplex pump (bbl/stroke):

Where

C Efficiency is the percent of volumetric efficiencyC Liner is the pump liner diameter in inchesC Stroke is the pump stroke length in inches

Pump output

Va '1029.4 POBPM

ID 2HOLE & OD2DP

CI ' IDDP2 0.00097144 Li

DI ' (OD2DP & ID

2DP) 0.00097144 Li


9-17

Annular velocity

Annular velocity (V ), ft/min:a

Where

C PO is the pump output in barrels per minuteBPMC ID is the diameter of hole or inside diameter ofHOLE

casing in inchesC OD is drillpipe outside diameter in inchesDP

Volumes

Drillpipe or drill collar capacity

Where

C C is the interval capacity of the drillpipe or drillIcollars in barrels

C ID is the inside diameter of the drillpipe or drillDPcollar in inches

C L is the length of the interval in feet1

Drillpipe or drill collar interval displacement

Where

C D is the interval displacement of the drillpipe orIdrill collars in barrels


C OD is the outside diameter of the drillpipe or drillDPcollar in inches

C L is the length of the interval in feeti

VAnnI ' (IDHOLE2 0.00097144 Li ) & CI & DI

VholeI ' VAnnI % CI

BU (min) 'VAnnTotalPOBPM

TCT (min) 'VAnnTotal % CT % VPits

POBPM



Annular Volume

Where

C C is the interval capacity of the drillpipe or drillIcollars in barrels

C D is the interval displacement of the drillpipe orIdrill collars in barrels

C VAnnI is the annular volume of the interval in barrels

C ID is the diameter of hole or inside diameter ofHOLEcasing in inches

C L is the length of the interval in feeti

C Fluid volume in the hole is the sum of the annularvolume and the volume of fluid inside the drillpipe

Circulating times

Where

C BU is the bottom up time in minutesC PO is the pump output in barrels per minuteBPMC VAnn

Total is the total annular volume in barrels

Where

C TCT is the total circulating time in minutesC PO is the pump output in bll/minBPMC VAnn

Total is the total annular volume in barrelsC C is the total capacity of the drillpipe and drillT

collars in barrelsC V is the total circulated pit volume in barrelsPits

jn

i ' 1

(Jeti2 ) 0.000767

VN (ft/sec) 'POGPM 0.32

AN

PDBit (psi) 'VN

2 D

1120

HHPBit (hp) 'PDBit POGPM

1714

HHP/area 'HHPBit

ABit

PDBitPressPump

100

ImpBit (lbf) 'VN POGPM Dmud

1932


9-19

Bit hydraulics Nozzle area

A (in ) = N2

Nozzle velocity

Bit pressure drop

Bit hydraulic horsepower

Bit hydraulic horsepower per unit bit area

Percent pressure drop at bit

Jet impact force

Where

C D is the mud density in lb/galmudC Press is the pump pressure in psigPumpC PO is the pump output in gal/minGPMC Jet is the nozzle diameter in 32nds of an inchiC A is the area of the bitBit



C A is the total nozzle area in inN2

C V is the nozzle velocity in ft/secNC PD is the bit pressure drop in psiBit

Calculations forlaminar andturbulent flow

Deriving dialreadings

Many sets of equations exist for hydraulic parametersusing the Bingham and power law models. Twocommonly used sets of equations include thosesanctioned by the American Petroleum Institute (API)and those that appear in the SPE Applied DrillingEngineering textbook. Both sets of equations are validfor fluid behavior in laminar and turbulent flow; theequations differ only in the approach to problemsolution. The following sections describe the Bingham,power law, and Herschel-Bulkley (yield-power law[YPL]) models; explain terms used in fluid hydraulicscalculations; and give equations for calculating fluidhydraulics.

Methods for Herschel-Bulkley (yield-power law[YPL]) fluids

Hydraulic calculations for Herschel-Bulkley (yield-power law [YPL]) fluids cannot be solved by simpleequations. For quick solutions, consult the Baroidhydraulics programs using DFG+ software. DOS andWindows versions of the program are available.

The 600 and 300 rpm readings are back-calculated fromthe plastic viscosity and yield-point values as shown:

2300 = Plastic viscosity + yield point2600 = Yield point + 230023 = 10 second gel (using a hand-crank

viscometer)23 = 23 (using a FANN 6-speed viscometer)

np ' 3.32 log(26002300

)

Kp '511 2300

511np

na ' 0.657 log(210023

)

Ka '511 23

5.11 na

Vp '0.408 POGPM

IDDP2

Va '0.408 POGPM

ID 2HOLE & OD2DP


9-21

API methods(June 1995)for powerlaw fluids

Plastic viscosity, yield point, n and K

High shear rate n and K values can be back-calculatedfrom the 600 and 300 rpm readings and are used forcalculations inside the drillpipe.

Low shear n and K values can be back-calculated fromthe 100 and 3 rpm readings and are used forcalculations in the annulus.

Where

C K is the consistency index in the annulus in eq cPaC K is the consistency index in the drillpipe in eq cPpC n is the flow index in the annulusaC n is the flow index in the drillpipep

Fluid velocityInside the drillpipe (ft/sec) =

In the annulus (ft/sec) =

Where


C ID is the diameter of hole or inside diameter ofHOLEcasing in inches


Deff ' IDDP

Deff ' IDHOLE & ODDP

ep ' 100Kp96VpIDDP

np & 1 3np% 1

4np

np

ea ' 100Ka144Va

IDHOLE & ODDP

na& 1 2na% 1

3na

na

NRe '928 Deff V Dmud

e



C PO is the pump output in gal/minGPMC V is the average mud velocity inside the annulus ina

ft/secC V is the average mud velocity inside the drillpipe inp

ft/ sec

Power law for each hydraulic interval

Effective diameter inside the drillpipe (D )eff

Effective diameter in the annulus (D )eff

Effective viscosity ( ) inside the drillpipe, cPep

Effective viscosity in the annulus ( ), cPea

Reynolds number (N )Re

Where

C D is the effective diameter of the hole in incheseffC ID is the inside diameter of the drillpipe or drillDP

collar in inchesC ID is the diameter of hole or inside diameter ofHOLE

casing in inchesC n is the flow index in the annulusaC n is the flow index in the drillpipep

f ' a

(NRe)b

where a ' logn% 3.9350

b ' 1.75 & logn7

f ' 16NRe

PDi 'f V 2 Dmud25.81 Deff

L


9-23


C is the effective viscosity of the liquide C D is the mud density in lb/galmudC V is either V for inside annulus or V for insidea p

drillpipeC V is the average mud velocity inside the annulus ina


ft/sec

Friction factor (f)

If the Reynolds number is greater than 2100 the flow isturbulent and the friction factor is:

If the Reynolds number is less than 2100 the flow islaminar and the friction factor is:

Pressure loss in the interval (PD ), psii

Where

C D is the effective diameter of the hole in incheseffC f is the friction factorC D is the mud density in lb/galmudC V is either V for inside annulus or V for insidea p

drillpipe

Vp (ft/sec) '0.408 POGPM

IDDP2

NRep '89,100 Dmud Vp

2& np

Kp

0.0416 IDDP3 % 1/np

np

PDp 'fp Dmud Vp

2

25.8 IDDP L



SPEmethodsfor powerlaw fluids

Power Law inside the drillpipe for eachhydraulic intervalAverage velocity inside the drillpipe (V )p

Where


C PO is the pump output in gal/minGPMC V is the average mud velocity in the drillpipe inp

ft/sec

Determine whether the flow is laminar or turbulent

1. Determine N from Figure 9-2 using the lowestRecvalues of N that intersect the straight line for aRegiven value of n or N = 2100.Rec

2. Calculate N .Rep

3. If N < N , the flow is laminar. If N $ N , theRep Rec Rep Recflow is turbulent.

Where


C K is the consistency index in the drillpipe, eq cPpC D is the mud density in lb/galmudC n is flow index n inside the drillpipep

Turbulent flow pressure drop

Pressure drop (PD ) inside the drillpipe is then:p

PDp 'Kp Vp

np3 % 1/np0.0416

np

144,000 ID(1 % np)

DP

L

Va (ft/sec) '0.408 POGPM

ID 2HOLE & OD2DP


9-25

Where


C f is the friction factor inside the drillpipepC L is the length of the drillpipe in feetC D is the mud density in lb/galmudC V is the average mud velocity inside the drillpipe inp

ft/sec

Laminar flow pressure drop

Pressure drop inside the drillpipe is then:

Where


C K is the consistency index in the drillpipe in eq cPpC n is flow index n inside the drillpipepC V is the average mud velocity inside the drillpipe inp

ft/sec

Power Law in the annulus for each hydraulicinterval

Determine average velocity in the annulus (V )a


NRea '109,100 Dmud Va

2& na

Ka

0.0208 (IDHOLE&ODDP)

2 % 1/na

na

PDa 'fa Dmud Va

2

21.1 (IDHOLE & ODDP) L

PDa 'K Va

na2 % 1/na0.0208

na

144,000 (IDHOLE & ODDP )(1 % na)

L



1. Determine N from Figure 9-2 using the lowestRecvalues of N that intersect the straight line for aRegiven value of n or N = 2100.rec

2. Calculate N .Rea

3. If N < N , the flow is laminar. If N $ N , theRea Rec Rea Recflow is turbulent.


Pressure drop in the annulus is then:

Where

C K is the consistency index in the annulus in eq cPaC f is the friction factor inside the annulusaC L is the length of the annulus in feetC D is the mud density in lb/galmudC ID is the diameter of hole or inside diameter ofHOLE

casing in inchesC OD is the outside diameter of the drillpipe or drillDP

collar in inchesC PO is the pump output in gal/minGPMC V is the average mud velocity inside the annulus ina

ft/secC n is the flow index in the annulusa



Vp (ft/sec) '0.408 POGPM

IDDP2

NHep '37,000Dmud YPID

2DP

PV 2

NRep '928 Dmud Vp IDDP

PV

PDp 'D0.75 Vp

1.75 PV 0.25

1800 ID 1.25DP L


9-27

SPEmethods forBingham-plastic fluids

Bingham-plastic inside the drillpipe for eachhydraulic interval

Determine average velocity inside the drillpipe (V )p


1. Calculate the Hedstrom number in the drillpipe.

2. Determine N from Figure 9-3 using the calculatedRecHedstrom number.

3. Calculate N .Rep

4. If N < N , the flow is laminar. If N $ N , theRep Rec Rep Recflow is turbulent.



Where

C L is the length of the drillpipe in feetC D is the mud density in lb/galmudC ID is the inside diameter of the drillpipe or drillDP

collar in inchesC PO is the pump output in gal/minGPM

PDp 'PV Vp

1500 ID 2DP%

YP225 IDDP

L

Va (ft/sec) '0.408 POGPM

ID 2HOLE & OD2DP

NHea '24,700Dmud YP(IDHOLE& ODDP )

2

PV 2

NRea '757 Dmud Va (IDHOLE& ODDP )

PV



C V is the average mud velocity inside the drillpipe inpft/sec

C PV is the plastic viscosity in cPC YP is the yield point in lb/100 ft2



Bingham plastic in the annulus for eachhydraulic interval

Determine average velocity in the annulus (V )a


1. Calculate the Hedstrom number in the annulus.

2. Determine N from Figure 9-3 using the calculatedRecHedstrom number.

3. Calculate N .Rea

4. If N < N , the flow is laminar. If N $ N , theRea Rec Rea Recflow is turbulent.

PDa 'D0.75mud Va

1.75 PV 0.25

1396 (IDHOLE& ODDP )1.25

L

PDa 'PV Va

1000(IDHOLE& ODDP )2%

YP200(IDHOLE& ODDP )

L


9-29





Where


C L is the length of the drillpipe in feetC D is the mud density in lb/galmudC OD is the outside diameter of the drillpipe or drillDP

collar in inchesC PO is the pump output in gal/minGPMC V is the average mud velocity in the annulus ina


ft/secC ID is the inside diameter of the drillpipe or drillDP

collar in inchesC ID is the diameter of hole or inside diameter ofHOLE

casing in inches

PDa ' jn

i'1

PDi

ECD 'PDa

jn

i'1Li 0.052

% Dmud

ECD 'PDa

jn

i'1LVi 0.052

% Dmud



Equivalentcirculatingdensity

The following formulas can be used to calculatepressure drop (PD) and equivalent circulating density(ECD).

Where

C PD is the pressure drop in the annulus in psiaC n is the number of intervalsC L is the length of the interval in feetiC LV is the vertical length of the interval in feetiC D is the density of the mud in lb/galmud

The sum of the pressure drops for each annular section(regardless of hole angle) is:

The equivalent circulating density (ECD) for anyvertical wellbore is:

In deviated wellbores, the TVD must be taken intoaccount when calculating ECD values. The aboveequation then becomes:

Vs ' 12.0eff

d Df1 % 7.27 d

DpDf

&1d Dfeff

2

&1

eff '(p(

% PV


9-31

Hole cleaningcalculations

Particle slip velocity

Chien method (1994)

Particle slip velocity calculations under laminarconditions cannot be solved by a single equation. A5-step iterative trial-and-error routine is required.Baroid slip velocity computer programs can solvethe equations in a few seconds; the method isoutlined here.

Slip velocity calculations

The general equation for calculating slip velocity forfalling particles is:

Where

C V is the laminar slip velocity of the particle inscm/sec

C is the effective viscosity of the fluid the particleeffexperiences while falling in poise

C d is the average particle diameter in cmC D is the density of the drilling fluid in g/cmf

3

C D is the density of the particle in g/cmp3

Mud effective viscosity during particle slip

The variable in the above equation is , whicheffdepends on the mud shear rate experienced by theparticle when falling. The following equations are usedto calculate .eff

Bingham plastic model:

eff ' K(n& 1

eff 'J(

% K( n& 1

(p 'Vsd



Power law model:

Herschel-Bulkley model:(yield-power law [YPL] model)

Where


C ( is the shear rate in sec-1

C ( is the settling shear rate in secp -1

C J is the calculated shear stress in lb/100 ft2

Particle settling shear rate

Determine settling shear rate experienced by the fallingparticle from the calculated slip velocity:

Where

C V is the slip velocity of the particle in cm/sec sC ( is the settling shear rate in secp

-1

C d is the average particle diameter in cm

To solve for particle slip velocity, follow these steps:

1. Guess the shear rate experienced by the particlewhen falling.

Note: Chien states that most drilled particlesexperience shear rates of 50 sec or less.-1

2. Calculate .eff3. Using from Step 2, solve for V .eff s4. Using V from Step 3, calculate ( .s p

NRes 'd Vs Df

eff

Vst ' 32.355 d DpDf

& 1


9-33

5. If ( in Step 4 is very close to the shear rate guessedpin Step 1, then the solution is obtained. If ( is notpclose to the shear rate, then reduce the value of theguessed shear rate and repeat Steps 1 through 4.

Note: As the iterative process gets closer to thesolution, the differences between ( from Step 1 andpStep 4 should get smaller. If the differences insuccessive calculations get larger, then increase theguessed shear rate values.

To determine whether drilled cuttings are falling underlaminar or turbulent conditions, first calculate theparticle Reynolds number (N ):Res

Where

C is the effective viscosity of the fluid the particleeffexperiences while falling (poise)

C V is the slip velocity of the particle in cm/sec sC d is the average particle diameter in cmC D is the density of the drilling fluid in g/cmf

3

If N < 10, the particle is falling in laminar slip. If theResN > 100, the particle is falling in turbulent slip, andRescalculations for turbulent slip are made.

Turbulent slip velocity calculations

Particles falling at high velocities can experienceturbulent slip. To determine at which velocity turbulentslip occurs, use the following equation:

Vslip '53.3 (Dcut & Dmud) Diamcut

2 Vann6.65 YP (IDHOLE & ODDP ) % PV Vann

Vs ' 1.06 Diamcut ((Dcut 8.345) & Dmud )

Dmud



Where

C V is the turbulent slip velocity of the particle inst cm/sec

C d is the average particle diameter in cmC D is the density of the drilling fluid in g/cmf

3

C D is the density of the particle in g/cmp3

Alternate method for Bingham-plastic fluidsIf a computer is not available to perform the Chienmethod calculations, the following equations can beused to approximate slip velocities in Bingham-plasticfluids.

Laminar slip velocity calculations

Turbulent slip velocity calculations

Where

C V is the slip velocity of the particle in cm/sec sC V is the slip velocity of the particle in ft/sec slip C V is the annular velocity in ft/secann C ID is the diameter of hole or inside diameter ofHOLE

casing in inchesC Diam is the diameter of the drilled cutting incut

inchesC D is the density of the drilled cutting in sgcutC D is the mud density in lb/galmudC PV is the plastic viscosity in cPC YP is the yield point in lb/100 ft2


TE (%) 'Va& Vslip

Va 100


9-35

Cuttings transport efficiency calculations

Vertical holes

Cuttings transport efficiency in vertical holes iscommonly calculated by:

Where

C V is the slip velocity of the particle in ft/sec slip C V is the annular velocity in ft/seca

In these calculations, the effect of reduced mudviscosity caused by mud flow is usually neglected. It isimportant that V and V have identical units (fora slipexample, ft/min or cm/sec).

High angle or horizontal holes

In deviated or horizontal holes, cuttings transportefficiency is not easy to calculate because the mudvelocity distribution under the eccentric drillpipe andthe corresponding effect of changes in mud shear ratesunder the drillpipe must be considered. To calculatecuttings transport efficiency in deviated orhorizontal holes, use the Baroid hole cleaningcomputer programs.

MAXROP calculations

Calculations can be made to estimate the maximum rateof penetration while maintaining good hole cleaning. Alimit of 5 percent by volume cuttings in the annulus hasbeen recommended in the literature. However, many

MAXROP(ft/hr) 'CC Va TE (ID

2HOLE& OD

2DP )

ID 2HOLE (100& CC) 3600

MAXROP(ft/hr) 'CC Va TE (UD

2& OD 2DP )

(UD 2& ID 2HOLE ) (100& CC) 3600



operators recommend a maximum cuttingsconcentration of 4 percent by volume.

Note: The following calculations assume there is aconcentric drillpipe.

Vertical holes

Where

C CC is the cuttings concentration in the annuluspercent by volume, using 5 maximum

C V is the average annular velocity in ft/secaC TE is the cuttings transport efficiency in percentC ID is the diameter of hole or inside diameter ofHOLE

casing in inchesC OD is the drillpipe diameter in inchesDP

Vertical holes underreamed to a largerdiameter

Where

C CC is the cuttings concentration in the annuluspercent by volume

C V is the average annular velocity in ft/secaC TE is the cuttings transport efficiency in percentC ID is the diameter of hole or inside diameter ofHOLE

casing in inchesC OD is the drillpipe diameter in inchesDPC UD is the underreamer diameter in inches

CC (% v/v) 'ROP ID 2HOLE (100& CC)

Va TE (ID2HOLE& OD

2DP ) 3600

CC (% v/v) 'ROP (UD 2& ID 2HOLE ) (100& CC)

Va TE (UD2& OD 2DP ) 3600


9-37

Cuttings concentration in the annulus for agiven penetration rate

Cuttings concentration at a given penetration rate can bedetermined using the following equations.

Vertical holes

Where


C V is the average annular velocity in ft/secaC ROP is the penetration rate in ft/hrC TE is the cuttings transport efficiency in percentC ID is the diameter of hole or inside diameter ofHOLE

casing in inchesC OD is the drillpipe diameter in inchesDP

Vertical holes underreamed to a largerdiameter

Where


C V is the average annular velocity in ft/secaC ROP is the penetration rate in ft/hrC TE is the cuttings transport efficiency in percentC ID is the diameter of hole or inside diameter ofHOLE

casing in inchesC OD is the drillpipe diameter in inchesDPC UD is the underreamer diameter in inches

MWann '(Df (100& CC)) % (Dp CC 8.345)

100



Annular mud density increase

The annular mud density increase due to cuttings at agiven penetration rate can by calculated by:

Where

C D is the mud density of the drilling fluid in lb/galfC D is the density of the drilled cutting in g/cmp

3


List of termsa Coefficient in friction factor calculationsA Nozzle area of bit, inN

2

A Area of bit Bitb Exponential coefficient in friction factor

calculationsBU Bottoms-up circulating time, minCC Cuttings concentration in the annulus, % v/vC Capacity of drillpipe and drill collars, bblTd Average particle diameter, cmD Diameter, inD Interval displacement of the drillpipe or drilli

collars, bblD Effective diameter of holeeffDiam Diameter of the drilled cutting, in or cmcutECD Equivalent circulating density, lb/gal or sgf Friction factorf Friction factor in the annulusaf Friction factor inside the drillpipepHHP Hydraulic horsepower at the bit, hpbitID Inside diameter drillpipe, inDPID Inside diameter of Hole, inHOLEImp Jet impact force at the bit, lb/ftBitJet Jet nozzle diameter, 32nds iniK Consistency index


9-39

K Consistency index K in the annulus, eq cPaK Consistency index K inside the drillpipe, eqp

cPL Length, ftL Interval length, ftiLV Vertical interval length, ftiLSR YP Low shear rate YP, lb/100 ft or Pa2

n Flow indexn Flow index n in the annulusaN Hedstrom numberHen Flow index n inside the drillpipepN Reynolds numberReN Reynolds number in the annulusReaN Critical Reynolds numberRecN Reynolds number inside the drillpipeRepN Reynolds number of a falling particleResOD Outer diameter of drillpipe, inDPOD Outer diameter (hole diameter), inHOLEPD Total pressure drop in an annulus, psiPD Pressure drop in the annulus, psiaPD Pressure drop at the bit, psi/ftbitPD Pressure drop inside the drillpipe, psipPO Pump output, bbl/minBPMPO Pump output, gal/minGPMPress Pump pressure, psigpumpPV Plastic viscosity (Bingham-plastic model),

cPTCT Total circulating time, minTE Cuttings transport efficiency, %TVD True vertical depth, ft or mUD Underreamer diameter, inV Average mud velocity V in the annulus anda

V in the drillpipe, ft/secpV Interval length, ftiV Average mud velocity in the annulus, ft/secaVAnn

I Annular volume of the interval, bblVAnn

Total Total annular volume of the interval, bblVHole

I Hole volume of the interval, bbl



V Nozzle velocity, ft/secNV Average mud velocity inside the drillpipe,p

ft/secV Volume of pits, bbl or mPits

3

V Slip velocity of falling particle, cm/secsV Turbulent slip velocity of falling particle,st

cm/secV Slip velocity of falling particle, ft/secslipYP Yield point, lb/100 ft2

YPL Yield-power law (Herschel-Bulkley)rheological model

( Shear rate, sec-1

( Shear rate experienced by falling particle,psec-1

, Drillpipe eccentricity2 Viscometer dial reading at a particular

operating speed Effective viscosity, cPe Effective viscosity in the annulus, cPea Effective viscosity experienced by settlingeff

particle, cP Effective viscosity inside the drillpipe, cPepD Density of the drilled cutting, sgcutD Density of drilling fluid, g/cmf

3

D Mud weight, lb/galmudD Density of the particle, g/cmp

3

J Shear stress, lb/100 ft or Pa2

J Yield stress, lb/100 ft or Pay2

J Yield stress at zero shear rate, lb/100 ft or02

Pa

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