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1. Dimensionless Group Analysis

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Table of Content

1. Dimensionless Group Analysis ..................................................................................................................... 1

1.1. Introduction ....................................................................................................................................................................................... 4

1.2. Methods for the Derivation of Dimensionless Groups ................................................................................ 4

1.1.1. Dimensional Analysis ................................................................................................................................................... 4

1.2.1.1. The Buckingham's Pi-Theorem ................................................................................................................. 5

1.2.2. Inspectional Analysis ..................................................................................................................................................... 7

1.3. Conventional Dimensionless groups ........................................................................................................................... 8

1.3.1. Ratio of Convection and Diffusion .................................................................................................................... 8

1.3.2. Dimensionless Groups Involving Surface Tension ........................................................................... 13

1.3.3. Dimensionless Groups Involving Gravity ................................................................................................ 14

1.3.4. Dimensionless Numbers in Natural Convection ................................................................................. 15

1.3.5. Other Dimensionless Groups ............................................................................................................................... 16

1.4. Dimensionless Groups in Petroleum Engineering ........................................................................................ 19

1.4.1. Dombrowski-Brownell number ......................................................................................................................... 19

1.4.2. Stability number .............................................................................................................................................................. 20

1.4.3. Lake number....................................................................................................................................................................... 21

1.4.4. Capillary end-effect number ................................................................................................................................. 22

1.4.5. Capillary-to-gravity number ................................................................................................................................. 22

1.4.6. Gravity number ................................................................................................................................................................ 23

1.5. Dimensionless Groups for Fluid Flow in Fractured Porous Medium .......................................... 23

1.5.1. Fracture Capillary Number .................................................................................................................................... 23

1.5.2. Modified Diffusive Capillary Number ........................................................................................................ 24

1.5.3. Fracture Diffusion Index ......................................................................................................................................... 24

1.5.4. Matrix-Fracture Diffusion Number ................................................................................................................ 25

1.5.5. Dimensionless Time .................................................................................................................................................... 26

1.6. New Dimensionless Groups as Combination of Dimensionless Numbers .............................. 29

1.6.1. Grattoni et al. Number (2001) ............................................................................................................................. 29

1.6.2. Kulkarni and Rao Number (2006) ................................................................................................................... 29

1.6.3. Rostami et al. Number (2009) ............................................................................................................................. 30

1.6.4. Rostami et al. Number (2010) ............................................................................................................................. 30

1.7. Dimensionless Groups for Scaling EOR Processes ..................................................................................... 31

1.8. Application ..................................................................................................................................................................................... 35

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1.9. References ....................................................................................................................................................................................... 40

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1.1. Introduction

Dimensionless groups are products of suitable powers of important variables which have zero

net dimension. They are widely used in mathematics, physics and engineering. The

dimensionless groups are useful for several reasons. They reduce the number of variables

needed for description of the problem. They can thus be used for reducing the amount of

experimental data and at making correlations. They predicting the effect of changing one of

the individual variables in a process (which it may be impossible to vary much in available

equipment) by determining the effect of varying the dimensionless group containing this

parameter (this must be done with some caution, however). They make the results

independent of the scale of the system and of the system of units being used. They simplify

the scaling-up or scaling-down of results obtained with models of systems by generalizing the

conditions which must exist for similarity between a system and its model. They deduce

variation in importance of mechanisms in a process from the numerical values of the

dimensionless groups involved; for instance, an increase in the Reynolds number in a flow

process indicates that molecular (viscous) transfer mechanisms will be less important relative

to transfer by bulk flow (inertia effects), since the Reynolds number is known to represent a

measure of the ratio of inertia forces to viscous forces.

1.2. Methods for the Derivation of Dimensionless Groups

Dimensionless groups can be attained in two ways: dimensional analysis and inspectional

analysis. The dimensional analysis approach is based on the Buckingham's Pi-theorem.

Inspectional analysis is better since it takes advantage of the problems' full mathematical

specification based on physical laws, and reveals a higher degree of similarity than

dimensional analysis.

1.1.1. Dimensional Analysis

Dimensional analysis is a mathematical technique making use of the study of dimensions. In

dimensional analysis, from a general understanding of fluid phenomena, one first predicts the

physical parameters that will influence the flow, and then, by grouping these parameters in

dimensionless combinations, a better understanding of the flow phenomena is made possible.

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First step in dimensional analysis is to identify the transport properties (fluid velocity, heat

influx, mass influx, etc.) and the material properties (viscosity, thermal conductivity, specific

heat, density, etc.) which are relevant to the problem, and list out all these quantities along

with their dimensions. In general, if a problem contains n important variables, and there

groups contain m fundamental dimensions, then there are n-m dimensionless groups. This

important principle is called the Buckingham Pi Theorem.

It should be emphasized that dimensional analysis does not provide a complete solution to

fluid problems. It provides a partial solution only. The success of dimensional analysis

depends entirely on the ability of the individual using it to define the parameters that are

applicable. If one omits an important variable, the results are incomplete and this may lead to

incorrect conclusions.

On the other hand, if one includes a variable that is totally unrelated to the problem, an

additional insignificant dimensionless group will result.

1.2.1.1. The Buckingham's Pi-Theorem

There are several methods of reducing a number of dimensional variables into a smaller

number of dimensionless groups. The scheme given here was proposed in 1914 by

Buckingham and is now called the Buckingham pi theorem. The name pi comes from the

mathematical notation , meaning a product of variables. The dimensionless groups found

from the theorem are power products denoted by 1, 2, 3, etc. The method allows the pi's

to be found in sequential order without resorting to free exponents.

The first part of the pi theorem explains what reduction in variables to expect:

If a physical process satisfies the PDH1 and involves n dimensional variables, it can be

reduced to a relation between only k dimensionless variables or s. The reduction j= n - k

equals the maximum number of variables which do not form a pi among themselves and is

always less than or equal to the number of dimensions describing the variables.

Take the specific case of force on an immersed body contains five variables F, L, U, and

described by three dimensions {MLT}. Thus n = 5 and j 3. Therefore it is a good guess that

we can reduce the problem to k pi's, with k = n - j 5 -3 = 2. On rare occasions it may take

more pi's than this minimum.

1 Principle of Dimensional Homogeneity: If an equation truly expresses a proper relationship between variables in a physical process, it will be dimensionally homogeneous; i.e., each of its additive terms will have the same dimensions.

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The second part of the theorem shows how to find the pi's one at a time:

Find the reduction j, then select j scaling variables which do not form a pi among themselves.

Each desired pi group will be a power product of these j variables plus one additional variable

which is assigned any convenient nonzero exponent. Each pi group thus found is

independent.

To be specific, suppose that the process involves five variables

1 = f (2, 3, 4, 5) (1-1)

Suppose that there are three dimensions {MLT} and we search around and find that indeed j

= 3. Then k = 5- 3= 2 and we expect, from the theorem, two and only two pi groups. Pick out

three convenient variables which do not form a pi, and suppose these turn out to be 2, 3 and

4. Then the two pi groups are formed by power products of these three plus one additional

variable, either 1 or 5:

1= (2)a(3)

b(4)c 1=M

0L0T0 2= (2)a(3)

b(4)c 5=M

0L0T0 (1-2)

Here we have arbitrarily chosen 1 and 5, the added variables, to have unit exponents.

Equating exponents of the various dimensions is guaranteed by the theorem to give unique

values of a, b, and c for each pi. And they are independent because only 1 contains 1 and

only 2 contains 5. It is a very neat system once you get used to the procedure.

Typically, six steps are involved:

1. List and count the n variables involved in the problem. If any important variables are

missing, dimensional analysis will fail.

2. List the dimensions of each variable according to {MLT} or {FLT}.

3. Find j. Initially guess j equal to the number of different dimensions present, and look for j

variables which do not form a pi product. If no luck, reduce j by 1and look again. With

practice, you will find j rapidly.

4. Select j scaling parameters which do not form a pi product. Make sure they please you and

have some generality if possible, because they will then appear in every one of your pi

groups. Pick density or velocity or length. Do not pick surface tension, e.g., or you will form

six different independent Weber-number parameters and thoroughly annoy your colleagues.

5. Add one additional variable to your j repeating variables, and form a power product.

Algebraically find the exponents which make the product dimensionless. Try to arrange for

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your output or dependent variables (force, pressure drop, torque, power) to appear in the

numerator, and your plots will look better. Do this sequentially, adding one new variable each

time, and you will find all n - j = k desired pi products.

6. Write the final dimensionless function, and check your work to make sure all pi groups are

dimensionless.

1.2.2. Inspectional Analysis

Inspectional analysis is a similar method for obtaining dimensionless groups to study the

mechanistic behavior of a process. However, IA transforms the dimensional space to

dimensionless space variable-by-variable instead of making decision based on the primary

dimensions of variables (Ruark, 1935; Bear, 1972; Shook, 1992; Djuro, 2002). Inspectional

analysis combines differential equations of the physical process together with its initial and

boundary conditions and then transforms the whole system into dimensionless forms. This

can be done in the following steps:

1. Formulation of all governing equations together with the initial and boundary conditions

2. Transformation to dimensionless space

3. Primary elimination and secondary substitution

4. Redundancy elimination

5. Independence testing

Both methods have their own advantages and limitations. Inspectional analysis usually gives

scaling groups whose physical meanings are readily apparent, whereas the physical meanings

of groups from dimensional analysis may be quite obscure (Ruark, 1935; Bear, 1972; Sonin,

1997; Gharbi et al, 1998; Djuro, 2002; Hernandez and Wojtanowicz, 2007). For example,

without knowing a differential equation, it is easy to define intuitively a similarity group as a

ratio of the capillary to viscous forces, or the viscous to gravity forces. On the other hand, IA

requires mathematical equations for the process under study. If such equations are

unavailable, inspectional analysis cannot begin.

Dimensional analysis is useful in providing some guidance for setting up experiments to

initiate the study and it is especially useful when people dont know much about their

research objectives. Therefore, the dimensional analysis method is more universal than the

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inspectional analysis when very little theory is available, dimensional analysis can be relied

upon to provide initial guidance in setting up experiments.

The inspectional analysis method is used more frequently in reservoir engineering because

the underlying physical laws are known and expressed in the form of partial differential

equations and boundary conditions (Shook et al, 1992). Principles of scaling in petroleum

studies were outlined by Rapoport (1955) for the immiscible displacement of oil by water.

Interestingly, different dimensionless groups have been derived by various authors, to

describe the same phenomenon (Croes and Schwarz, 1955; Rapoport, 1955). In many cases

these dimensionless groups were reportedly obtained from IA, but the procedure used was not

explained.

1.3. Conventional Dimensionless groups

Dimensionless groups can be classified into different types.

1.3.1. Ratio of Convection and Diffusion

Diffusion is a molecular process, where material is transported down concentration gradients

due to molecular motion. The molecules in a fluid are in constant motion which is random

with equal probability in all directions. Consequently, at equilibrium, there is no net motion

of mass in any one direction. However, if there is a spatial variation of the concentration of a

species or the thermal energy of the molecules, this random motion will transport mass or

heat down the gradient. Consequently, diffusive transport of mass, heat and momentum have

some common characteristics. The diffusive transport (per unit time) of a quantity such as

mass or heat through a material is directly proportional to the cross sectional area, inversely

proportional to the length and directly proportional to the difference in concentration or

temperature across the material.

1. For transport of mass, the mass flux is related to the concentration difference by Fick's law.

If a concentration difference c is maintained between two ends of a slab of length l, the

mass flux jc per unit area, which has units of (mass / area / time), is

= (1-3)

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Here, just from dimensional analysis, it is easy to see that the diffusion coefficient has units

of L2T-1.

2. For the transport of heat, the heat flux is related to the temperature difference by Fick's

law. If the temperature difference between two surfaces is T, then the heat flux je (energy/

area / time) is given by

= (1-4)

where k is the thermal conductivity. It is possible to define a diffusion coefficient for heat

transfer in the following form. Consider a volume which is heated up by a flux incident on it

from its boundaries. The difference in temperature T can be expressed in terms of the

difference in the specific energy between the two sides as T = (E/cv) where E is the

specific energy (per unit volume). With this, the equation for the heat flux can be written as

= (1-5)

It is obvious that the above equation has the same form as the mass flux equation, with a

thermal diffusivity Dt = (k/cp) which has units of L2T-1.

3. The relation for the momentum diffusivity is a little more complicated. It turns out that the

diffusion of momentum is due to viscous effects, and the diffusivity for momentum transfer is

the "kinematic viscosity" = (/), which has units of L2T-1.

Convective transport takes place due to the mean flow of a fluid. For example, if the flow has

a velocity U and the difference in concentration between two points is c, the rate of

transport of material per unit area is equal to the Uc. Consequently, the ratio between the

rate of transport due to convective and diffusive effects is (Ul/D), where l is the length scale

in the problem. Many of the dimensionless numbers provide ratios of convective to diffusive

transport, or ratio of diffusive transport of two different quantities, as shown in Table 1.1.

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Table 1.1: Dimensionless numbers as ratios of forces

Dimensionless number Ratio Expression

Reynolds number Momentum convectionMomentum difussion

Prandtl Number Momentum difussionHeat diffusion !" = #$

Schmidt number Momentum difussionMass diffusion ! =

Peclet Number Heat convectionHeat diusion "

Reynolds number: It is named after Osborne Reynolds (18421912), a British engineer who

first proposed it in 1883. The Reynolds number is always important, with or without a free

surface, and can be neglected only in flow regions away from high-velocity gradients, e.g.,

away from solid surfaces, jets, or wakes. The Reynolds number is the ratio of the inertial to

the viscous forces:

&' = Momentum convectionMomentum difussion =

(1-6)

Re > 1, convection (or inertial) forces are dominant. But one should be really cautious

here: in this case, the viscous forces cannot be neglected everywhere in the flow, only away

from interfaces. In particular, in areas close to interfaces and boundary layers, the viscous

forces are of the same order as the inertial forces, because in the vicinity of interfaces, very

strong gradients may exist.

The critical Reynolds number denotes the boundary between laminar and turbulent flow (it is

2000 for flow down a pipe). Low Re indicate laminar flow ("creeping" flows typically have

Re < 1). Large Reynolds numbers indicate turbulent flow.

The Modified Reynolds number is used to determine the flow regime of the fluid within the

porous medium. Modification to the fluid velocity term (v) and the characteristic linear

dimension (D) are required. When considering flow within the bed the appropriate velocity is

the interstitial, which can be related to the superficial velocity V by (v = V/). The

characteristic linear dimension was deduced by Kozeny and is the volume open to the fluid

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flow divided by the surface area over which it must flow (i.e. product of volume of solids and

specific surface area per unit volume)

= ()*()+1 *./ = *+1 *./ (1-7)

Thus, the Modified Reynolds number (Rem) can be expressed as:

&'0 = *+1 *./ 1* = 1+1 *./ (1-8)

Conceptually, the number still represents the ratio of inertial to viscous forces in the fluid and

provides a means to assess when the inertial effects become significant. The conventionally

applied threshold to indicate significant turbulence is 2, whereas for the flow Reynolds

number the conventional threshold is about 2000. It is important to note that the density term

in above equation is the density of the fluid: the turbulences described are that of the fluid,

the particles do not move in a packed bed.

Prandtl Number: It is named after Ludwig Prandtl, who introduced the concept of boundary

layer in 1904 and made significant contributions to boundary layer theory. Prandtl number is

the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity.

23 = Momentum difussionHeat diffusion = !" = #$ (1-9) The Prandtl numbers of fluids range from less than 0.01 for liquid metals to more than

100,000 for heavy oils (Table 1.2). Note that the Prandtl number is in the order of 10 for

water.

The Prandtl numbers of gases are about 1, which indicates that both momentum and heat

dissipate through the fluid at about the same rate. Heat diffuses very quickly in liquid metals

(Pr 1) and very slowly in oils (Pr 1) relative to momentum. Consequently the thermal

boundary layer is much thicker for liquid metals and much thinner for oils relative to the

velocity boundary layer.

Table 1.2: Typical ranges of Prandtl numbers for common fluids.

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Fluid Pr

Liquid metals 0.004-0.03

Gases 0.7-1.0

Water 1.7-13.7

Light organic fluids 5-50

Oils 50-100000

Glycerin 2000-100000

Schmidt number: It was named after the German engineer Ernst Heinrich Wilhelm Schmidt

(1892-1975). Schmidt number is a dimensionless number defined as the ratio of momentum

diffusivity (viscosity) and mass diffusivity, and is used to characterize fluid flows in which

there are simultaneous momentum and mass diffusion convection processes.

/ = Momentum diusionMass diffusion = ! = (1-10)

Where is the kinematic viscosity or (/), D is the mass diffusivity, is the dynamic

viscosity of the fluid and is the density of the fluid. It physically relates the relative

thickness of the hydrodynamic layer and mass-transfer boundary layer. The heat transfer

analog of the Schmidt number is the Prandtl number.

Peclet and PecletII Numbers

The Peclet and PecletII numbers are analogous to the Reynolds number but for heat and mass

transfer, respectively.

The Peclet number is the ratio,

2' = Heat convectionHeat diusion = "

(1-11)

Using the Prandtl number, we can write the Peclet number as a function of Reynolds number:

2' = &'. 23 (1-12)

Likewise, for mass transfer,

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2'55 = Mass convectionMass diusion (1-13)

1.3.2. Dimensionless Groups Involving Surface Tension

There are three important dimensionless groups involving surface tension, which are the

Weber number, the Capillary number and Bond number.

Weber number: It is named after Moritz Weber (18711951) of the Polytechnic Institute of

Berlin, who developed the laws of similitude in their modern form. The Weber number is the

ratio of inertial and surface tension forces,

6' = 78)9 (1-14)

The Weber number is important only if it is of order unity or less, which typically occurs

when the surface curvature is comparable in size to the liquid depth, e.g., in droplets,

capillary flows, ripple waves, and very small hydraulic models. If We is large, its effect may

be neglected.

Capillary number: It is defined as the ratio of the viscous force to the capillary force at the

pore scale (Chatzis and Morrow, 1984). For two-phases, it can be defined as:

: = 79 = 29) (1-15)

where v is the Darcy velocity, is the viscosity of the displacing phase, is the interfacial

tension, k is the medium permeability, P is the pressure difference and L is the characteristic

lenght . Morrow et al. 1988 have shown that variations in the pore size distribution or the

degree of consolidation, produce changes in the value of the capillary number due to change

of trapped oil size distribution. The influence of the pore structure and wettability could be

introduced by using the capillary pressure term defined as:

: =

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where is the contact angle and Ra is the average pore throat radius. For strongly wetting

conditions, cos =1. At constant velocity, the capillary number is naturally constant, but

when the flow rate varies with time it becomes a dynamic parameter.

Capillary number affects the residual saturation of non-wetting phase in two phase flow for

Nc >10-5 (Abrams, 1975). Lefebvre du Prey (1973) have showed that the capillary number

affects the wetting phase relative permeability and residual saturation for Nc >10-7.

Bond number: The Bond number (Hove et al., 1995) is a dimensionless group that measures

the relative strength of gravity or buoyancy in some cases and capillary forces. For the

trapping of the non-wetting phase, it can be expressed as

:@ = A89 (1-17)

where is the density difference between the two fluids, g is the gravitational constant, l is

a characteristic length of the porous medium often taken as the average grain radius, (Morrow

et al., 1988). For a vertical flooding, NB takes into account the balance between gravity and

capillary forces and is directly proportional to the advance of the displacing phase front.

Thus, NB can also be a dynamic parameter depending additionally on the front velocity.

1.3.3. Dimensionless Groups Involving Gravity

Froude number: It is named after William Froude (18101879), a British naval architect

who, with his son Robert, developed the ship-model towing-tank concept and proposed

similarity rules for free-surface flows (ship resistance, surface waves, and open channels).

Froude number is the ratio of inertial and gravitational forces:

B3 = 78A) (1-18)

where U is the velocity, and g is the acceleration due to gravity, and L is the characteristic

length.

Fr >> 1, the body forces can be neglected.

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Fr

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&= = 8#GEAF (1-20)

The "Rayleigh-Benard" instability of a fluid layer heated from below occurs when the

Rayleigh number increases beyond a critical value.

1.3.5. Other Dimensionless Groups

Mach number: The Mach number is named after Ernst Mach (18381916), an Austrian

physicist. The ratio is of fluid speed to speed of sound is relevant in two areas:

H= = (1-21)

It is the most important correlating parameter when fluid velocities are near the local sonic

velocity. Compressible fluids can be treated like incompressible fluids when Ma < 0.3.

Euler number (pressure coefficient): This is named after Leonhard Euler (17071783) and is

rarely important unless the pressure drops low enough to cause vapor formation (cavitation)

in a liquid. The Euler number comes from the ratio between pressure forces and inertial

forces,

7 = ID8 (1-22)

Knudsen Number: If a flow is sufficiently rarefied, a dimensionless ratio called the Knudsen

number is likely to come into prominence. Knudsen number is normally defined as the ratio

between the mean free path of the molecules and a characteristic length. When the mean free

path is very small compared with other lengths involved, as is the case with ordinary densities

and sizes, Knudsen number does not enter; but when the mean free path becomes sufficiently

large, the equations we have used to describe the flow are no longer entirely appropriate.

Knudsen diffusion occurs when gas molecules experience significant interaction with pore

walls. Knudsen diffusion is significant in porous media with very small pores (on the order of

a few micrometers or smaller) or at low pressures. It is characterized by the ratio of the

molecules mean free path divided by the average pore throat radius,

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JK = L3GNO (1-23)

where Kn is the Knudsen number, L is the gas phase molecular mean free path in m, and rpore is the characteristic length scale of the flow path in m, in porous media representing an

equivalent hydraulic radius (Civan 2008). The Knudsen number signifies the degree to which

the flow is non-Darcy. For Knudsen numbers less than 0.01 imply Darcy flow. For Knudsen

numbers higher than 0.01, the flow equations must be modified for Knudsen diffusion and/or

slip flow.

Nusselt Number: It is named after Wilhelm Nusselt, who made significant contributions to

convective heat transfer in the first half of the twentieth century, and it is viewed as the

dimensionless convection heat transfer coefficient. It, therefore, provides a measure of the

convection heat transfer at the surface. It is defined as:

:7 = ) (1-24)

where, h is the heat transfer coefficient, Lc is a characteristic length and k is the thermal

conductivity.

To understand the physical significance of the Nusselt number, consider a fluid layer of

thickness L and temperature difference T=T2-T1. Heat transfer through the fluid layer will

be by convection when the fluid involves some motion and by conduction when the fluid

layer is motionless. Heat flux (the rate of heat transfer per unit time per unit surface area) in

either case will be

QRNK. = (1-25)

and

QRNKS. = ) (1-26)

Taking their ratio gives

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QRNK.QRNKS. = ) = :7 (1-27)

which is the Nusselt number. Therefore, the Nusselt number represents the enhancement of

heat transfer through a fluid layer as a result of convection relative to conduction across the

same fluid layer. The larger the Nusselt number, the more effective the convection. A Nusselt

number of Nu=1 for a fluid layer represents heat transfer across the layer by pure conduction.

Lewis number: It is named after Warren K. Lewis (18821975). Lewis number is

a dimensionless number defined as the ratio of thermal diffusivity to mass diffusivity. It is

used to characterize fluid flows where there is simultaneous heat and mass transfer by

convection.

)' = " = #G (1-28)

where is the thermal diffusivity and D is the mass diffusivity.

The Lewis number can also be expressed in terms of the Schmidt number and the Prandtl

number:

)' = ScPr (1-29)

Sherwood number: It is named in honor of T. K. Sherwood. Sherwood number represents

the ratio between mass transfer by convection and mass transfer by diffusion.

/ = Convective mass transportDiffusitive mass transport = ) (1-30)

where h is the mass transfer coefficient, Lc is the characteristic length, and D is the diffusion

coefficient. In mass transfer, the Sherwood number plays the role the Nusselt number plays in

heat transfer.

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Stanton number: It is named after Thomas Edward Stanton. The Stanton number is the ratio

of heat transferred into a fluid to the thermal capacity of fluid.

/Z = Actual heat ^lux to the ^luidHeat ^lux capacity of the ^luid ^low

=

#$< (1-31)

where h is the convection heat transfer coefficient, is the density of the fluid, Cp is

the specific heat of the fluid and v is the velocity of the fluid.

Biot number: It is named after J. B. Biot who, in 1804, analyzed the interaction between

conduction in a solid and convection at its surface. Biot number is the ratio of the convection

at the surface to conduction within the body.

bc = Convection at the surface of the bodyConduction within the body = ) (1-32)

where h is the convective heat transfer coefficient, LC is the characteristic length and k is

the thermal conductivity of the body. A small value of Bi indicates that the inner resistance of

the body to heat conduction is small relative to the resistance to convection between the

surface and the fluid. As a result, the temperature distribution within the solid becomes fairly

uniform.

1.4. Dimensionless Groups in Petroleum Engineering

1.4.1. Dombrowski-Brownell number

In 1954, Dombrowski and Brownell presented a general correlation between residual wetting-

phase saturation and the dimensionless viscous and gravitational forces that cause

desaturation. The dimensionless number they presented generally called the Dombrowski-

Brownell number, NDB. The DombrowskiBrownell number is defined as the ratio of the

gravity force to the capillary force at the pore scale.

:e@ = AJ9 (1-33)

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where is the fluid density difference, g is the effective gravitational constant, K is the

medium permeability and is the interfacial tension. Dombrowski and Brownell presented

wetting-phase saturation data that suggested a critical dimensionless number of 10-3. Above

that number, the residual wetting-phase saturation was significantly reduced and approached

zero. In two-phase flow, this number affects the residual saturation of non- wetting phase for

NDB>10-5 (Morrow and Chatzis, 1984) and residual saturation of wetting phase for NDB>10

-2.

NDB can be interpreted as a microscopic version of the Bond number. Typical values for

macroscopic Bond number suggest that no correction is needed in residual oil saturation in

order to scale between lab and field condition. However, typical values for microscopic Bond

number, NDB, suggest that capillary forces cannot be neglected on the scale of fluid flow.

This is true primarily when saturation variation occurs in the sample, although this shows up

nowhere in the definition of NDB. In fact, NDB can be used as an indicator when saturation

variation expected. When present, it must be accounted for in the interpretation of relative

permeability, especially in the range of low saturation (Edward, 1998).

1.4.2. Stability number

Peters and Flock (1981) developed a dimensionless stability number for quantitative

prediction of the onset of instability in a displacement process. Their stability number for a

cylindrical system is given by:

:f = +H 1.+<

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instability in both oil-wet and water-wet porous media. If in a given cylindrical system the

computed value of the stability number exceeds the critical value of 13.56, then the

displacement will be unstable. In addition, the magnitude of the stability number provides

quantitative information regarding the severity of instability. The higher the stability number,

the more severe is the degree of instability. If the stability number for a displacement is less

than the critical value, then the displacement will be stable.

1.4.3. Lake number

Beginning with a partial differential equation describing two-phase flow, Rapoport and Leas

identified a group of variables whose value, it was argued, indicates when a flow is stabilized,

i.e., becomes independent of capillary pressure gradient effects. The dimensional group they

identified is (Lv), where v is the speed of the injected phase and is its viscosity.

In the literature, the Rapoport and Leas (RL) scaling group has often been used to select the

rate of water flood. Lake defined a dimensionless group that included the RL criterion, Lv,

interfacial tension, contact angle, end-point relative permeability to the water phase and

geometrical parameters of the porous medium. The Lake number, NL, is defined as

:i = jk g7)Og k9 cos (1-37)

The appeal of the dimensionless Lake number is that it incorporates many of the important

parameters pertaining to immiscible two-phase flow, including the pore geometry factor, K

(it employs the inverse of this factor). Its apparent lack of success with the present set of data

may be due to the use of a constant pressure drop boundary condition instead of constant flow

rate conditions.

Lake found that a NL of 3 corresponded to a scaling group of 1 in Rapoport and Leas data.

The advantage of the dimensionless number is that it also includes permeability of the rock.

For small NL, capillary pressure (Pc) will cause shock waves to spread out. The limit of rate

criteria above which relative permeability is negligibly affected by capillary forces in a1-D

wateroil displacement, when the core is water wet, is NL equal to 3.

22

1.4.4. Capillary end-effect number

The capillary end effect can be represented by a dimensionless flow parameter, Nc,end, the

capillary end-effect number (Mohanty and Miller, 1991). It interpreted as the ratio of the

capillary force to the viscous force at the system or macroscopic scale. Nc,end is the ratio of a

characteristic capillary pressure, Pci, to the viscous pressure drop across the core, p; ie.,

:,KS = 2oI 9q+*/./0.1,

as Nc,end increases, both oil and water relative permeabilities decrease. When Nc,end <

0.1, it does not affect relative permeability.

1.4.5. Capillary-to-gravity number

Saedi (2007) defined capillary-gravity number as the ratio of the threshold capillary pressure

to the pressure differential created by centrifugation in the core sample for centrifugal

displacement:

:s = 2tu30v8) (1-39)

Larger values of capillary-gravity number imply a larger capillary end effect. While Ncg is

directly related to threshold capillary pressure, it is implicitly a function of permeability.

Hagoort (1980) used the capillary to gravity number and showed that for Ncg > 103 the

capillary effect is important and needs to be taken into account. Saeedi (2007) performed a

large number of centrifuge experiments and numerical investigation and stated that if the

capillary to gravity number, Ncg, is maintained in order of 102 or smaller, the capillary end-

effect becomes negligible.

23

1.4.6. Gravity number

It is the ratio of gravity forces to viscous forces.

:w = A< (1-40)

where, v is the Darcy velocity, is the density difference between the displacing and

displaced phases, is the viscosity diference, g is the gravity, k is the medium

permeability.

The gravity number being a combination of the Bond and capillary numbers appears to be a

better basis for comparison of laboratory and field data as it includes the gravity and viscous

forces thereby indicating the relative effects of drainage versus displacement.

1.5. Dimensionless Groups for Fluid Flow in Fractured Porous

Medium

1.5.1. Fracture Capillary Number

Putra (1999) proposed parameters that affect the viscous and capillary forces can be grouped

into a Fracture Capillary Number (FCN) equation. In naturally fractured reservoirs, fluid

displacement in fracture network occurs due to its higher conductivity compared to the matrix

while an exchange of fluids occurs between matrix and fracture system. The fluid transfer

process is controlled by flow of water under naturally imposed pressure gradients (viscous

force) and the spontaneous movement of water into the matrix under capillary forces

(imbibition). The parameters that affect this process can be grouped into a dimensionless

equation, called the fracture capillary number. This equation is used to determine the critical

injection rate and is defined as the ratio between the viscous force that is parallel to the

fracture direction and the capillary force that is perpendicular to the fracture direction. The

viscous force is defined as a function of water velocity, water viscosity and fracture volume

and is assumed to occur only in the fracture. The capillary force that occurs only in the matrix

is defined as a function of interfacial tension, contact angle and matrix volume. Thus, the

fracture capillary number can be written below:

24

:x yR =

25

Trivedi and Babadagli (2006) proposed a dimensionless group, Fracture Diffusion Index

(FDI), as a ratio of viscous forces in the fracture and diffusion forces in the matrix using the

analogy to capillary imbibition controlled immiscible displacement in fractured porous

medium.

B = qx

26

where o is the solute viscosity, uT is the flow velocity, b is the fracture width, kf is the

fracture permeability, is the density difference, g is the gravity force, Dm is the Effective

diffusion coefficient in matrix for matrix-fracture, L is the length of fracture, m is the matrix

porosity, r is the matrix width, Pe is the Peclet number defined as:

2' = 70 (1-46)

Arm is the aspect ratio for matrix defined as:

(30 = k03) (1-47)

and Ng is the gravity number defined as:

:s = xAN7 (1-48)

The expression for the (NM-FD) proposed is similar some extent to fracture capillary number,

Nf,Ca, where the capillary forces are replaced by diffusive forces in the matrix with addition of

gravity factor.

1.5.5. Dimensionless Time

Mattax and Kyte presented an equation for scaling of imbibition data under the following

conditions: (1) the sample shape and boundary conditions are identical (2) the oil/water

viscosity ratio is duplicated (3) gravity effects can be neglected (4) initial fluid distributions

are duplicated (5) the capillary pressure functions arc directly proportional and (6) the

relative permeability functions are the same.

This group expresses the ratio of capillary force to viscous resistance, which is a form of

inverse capillary number.

27

t = tjk L8 (1-49)

where t is the time, k is the permeability, is the fractional porosity, is the interfacial

tension, w is the water viscosity, L is a characteristic length. The characteristic length was

not further defined by Mattax and kyte.

Ma et al. modified the Mattax and Kyte scaling law to define a dimensionless time for

countercurrent imbibition in which gravity has no effect and the viscosities are finite:

Ze = Zj* 9qgN1)8 (1-50)

Lc is the characteristic length determined by the size, shape, and boundary conditions of the

sample defined by Ma et al. as:

B = 11 (f (1-51)

) = j 1B (1-52)

where V is the volume of the sample, A is the area open to flow, l is the distance from the

open surface to the no-flow boundary and the sum is over all open surfaces of the block. Ma

et al. found that plots of the fraction of recoverable oil produced as a function of the

dimensionless time, Eq. 1, for cores of different size with different fluid viscosities, boundary

conditions, and interfacial tensions all lay approximately on the same universal curve.

Zhou et al. used another equation to scale imbibition experiments in water-wet diatomite

where the viscosity ratio was varied by four orders of magnitude:

Ze = Zj* 9)8 LOgLON1

H + 1H (1-53)

28

where r*(=kr*/) is a characteristic mobility and M*(=rw * /ro * ) is a mobility ratio. They

used endpoint relative permeabilities to calculate r* and M*.

Eqs. (1-50) and (1-53) were developed for strongly water-wet rocks, where it is assumed that

the oil/water contact angle is close to zero. To account for larger oil/water contact angles in

mixed-wet media, Gupta and Civan and Cil et al. presented the following expression based

on Eq. (1-50):

Ze = Zj* 9 cos ~qgN1)8 (1-54)

Zhou et al. used a similar approach by defining an apparent dynamic advancing contact

angle, AD, to match their experimental results. They defined this angle as:

~e = cos Ze,fgg+0.5.Ze+0.5. (1-55)

where the subscript vsww refers to very strongly water-wet conditions and tD(0.5) is defined

as the dimensionless time, Eq. 1, for one half of the total recovery. This gives a dimensionless

time given by Eq. 5 with =AD.

Xie and Morrow tested Eq. 1 on 32 weakly water-wet Berea sandstone samples. They

suggested that when capillary forces are sufficiently small, gravity segregation will make a

significant contribution to oil recovery; therefore, this force must be included in scaling laws

for weakly water-wet rocks:

Ze = Z *qgN)8 +2|+~. +A)8) . (1-56)

where Pc is a representative imbibition capillary pressure proportional to /K/, f() is a

wettability factor, and LH is the vertical height of the sample.

29

1.6. New Dimensionless Groups as Combination of Dimensionless

Numbers

1.6.1. Grattoni et al. Number (2001)

A new dimensionless group N is proposed by Grattoni et al. (2001) which is a linear

combination of the three-phase Bond number and capillary number. This group represents the

relative strength of the combined effects of gravity and viscous forces to capillary forces.

This new group can be expressed as:

: = :@ + (+Ss.: (1-57)

where d is the viscosity of displaced phase and g is the viscosity of displacing phase. They

developed a linear relationship between this new group and the total recovery based on the

experimental investigations to include the pore scale effects.

& = [:@ + ( Ss :] (1-58)

Then m and A are obtained from linear regression of R versus NB close to the end of the

experiment and R versus NC at the beginning of the experiment.

= &:@ (1-59)

( = s&S: (1-60)

1.6.2. Kulkarni and Rao Number (2006)

Kulkarni and Rao (2006) presented the effect of major dimensionless groups on the nal

recovery based on various miscible and immiscible gas assisted gravity drainage eld data

and laboratory experimental data. The NG versus oil recovery plot did not yield as good a

correlation. However, the individual plots of NC and NB versus recovery resulted in good

30

correlations. Therefore, it was hypothesized that there is some other important mechanistic

parameter that is not well represented in the gravity number, and a mathematical combination

of the NC, NB and NG groups with that mechanistic parameter should yield an improved

correlation parameter. Literate review suggested that two ratios: density and viscosity (gas to

oil), were important for gravity drainage flow. The density ratio was factored into a newly

defined dimensionless group. They have termed this new dimensionless group as "Gravity

Drainage Number" of NGD.

:we = :w + w +: + :@. (1-61)

1.6.3. Rostami et al. Number (2009)

Rostami et al. (2009) conducted a number of forced gravity drainage experiments. It was

found that neither the capillary number, nor the Bond number is sufficient to obtain a

satisfactory correlation with recovery explain the full range of behaviors observed. In order to

account for the effect of viscosity ratio (r = g/o) and proper ratio of the Bond to capillary

number, a combined dimensionless group, NCo, is proposed:

:RN = :@+O.:R@ (1-62)

where subscript Co stands for the term combined, and the parameters A and B are the

scaling factors. They found that the values of A = B = 0.5 provide a good correlation. The

small values of exponent suggest that the effect of the capillary number and the viscosity

ratio is less than the importance of the Bond number. It is found that if recovery at a higher

PV-injection was used, other values of A and B are needed to obtain a good match.

Nevertheless, in all the cases the A and B exponents were smaller than 1.

1.6.4. Rostami et al. Number (2010)

In addition, Rostami et al. (2010) proposed a combined dimensionless group to generalize the

estimation of remaining and residual oil saturations. Examination of residual oil saturations

as a function of Bond and capillary number suggests that none of them individually can

predict the remaining oil saturation. According to their experimental results, residual oil

31

saturation has an opposite relation to the Bond and capillary number. It is also demonstrated

that the effect of capillary number is less than the importance of Bond Number. Therefore,

the existence of the scaling factor is necessary.

:RN = :@:R (1-63)

where A is the scaling factor. This dimensionless group successfully combines the effect of

main forces exist in forced gravity drainage flow.

1.7. Dimensionless Groups for Scaling EOR Processes

Shook et al. (1992) presented dimensionless scaling groups for the waterflood applicable to

represent the two phase flow through homogeneous 2-dimensional Cartesian dipping

reservoir.

Effective Aspect ratio:

&i = ) j (1-64)

Mobility ratio:

HgN = OgN NONN g (1-65)

Dip Angle group:

: = ) tan " (1-66)

Buoyancy number:

:sN = LO8N A cos "7 ) (1-67)

32

Capillary number:

:$ = LO8N )7 qk (1-68)

Where H is the height of reservoir, L is the distance between injection and production wells,

kx is the horizontal permeability, kz is the vertical permeability, koro is the endpoint relative

permeability of oil, korw is the endpoint relative permeability of water, Mwo mobility ratio

(water), uT is the fluid velocity, is the dip angle, is the difference between oil and water

densities, o r2 is the endpoint mobility of oil, o is the viscosity of oil, w is the viscosity of

water, g is the acceleration because of gravity, is the interfacial tension, is the porosity.

Gharbi et al. (1998) found that there are eight independent dimensionless scaling groups that

describe miscible displacement of oil by a solvent in two-dimensional, homogeneous,

anisotropic cross section with constant porosity and dip angle. The eight dimensionless

scaling groups describing the displacement are:

Dispersion number:

:S = ) jJJi (1-69)

Effective Aspect ratio:

&i = ) j (1-70)

Longitudinal Peclet number:

2'i = 7/k)Ji (1-71)

Damkholer number:

:e = Ji)8 (1-72)

33

Viscosity ratio:

H = Nf (1-73)

Dip angle number:

: = " (1-74)

Gravity number:

:s = AN7 (1-75)

Aspect ratio:

(O = ) (1-76)

where KT and KL are the transverse and longitudinal dispersion coefficient, is the chemical

rate constant, kx and kz are the absolute permeability in the x- and in the z-directions, H is the

reservoir thickness, L is the characteristic system length, dip angle.

If the molecular diffusion in the dispersion coefficients is negligible, then the longitudinal

Peclet number and the dispersion number become:

2'i = )"i (1-77)

:S = ) j""i (1-78)

where L and T are the longitudinal and the transverse dispersivity, respectively.

34

Also, Gharbi (2002) made use of dimensionless groups for scaling of miscible ooding in a

2D heterogeneous anisotropic reservoir. They used several dimensionless scaling groups

describing the reservoir heterogeneity presented by Li and Lake (1996).

Global heterogeneity number (for log-normal)

: = 9K (1-79)

Local heterogeneity number (for log-normal)

:K = 1G9K8 (1-80)

Effective correlation length in x-direction

Le = L) (1-81)

Effective correlation length in z-direction

Le = L (1-82)

Hurst exponent

(1-83)

where ln is the standard deviation of log-normal permeability field, Vp variance of

permeability field, x and z are the correlation length in x-and z-directions.

Wood et al. (2006, 2008) introduced ten dimensionless groups to describe CO2 ooding in a

dipping waterooded reservoir, and used them for experimental design purposes to develop

screening criteria applicable to Gulf Coast reservoirs. The dimensionless groups proposed by

Shook (1992) served as the initial basis for the scaling of CO2 flooding.

The relationship between the scaling groups and their effects on the overall performance of

immiscible gas-driven gravity drainage were investigated by Jadhawar and Sarma (2008)

35

through the use of uncertainty assessment techniques. They used dimensionless scaling

groups of previous studies for dipping reservoirs (Gharbi et al., 1998; Shook et al., 1992) and

modified them to implement sensitivity analysis for the horizontal type reservoir under

investigation. To consider the GAGD-EOR process dependent constant pressures condition

of the injection wells and the production wells, and the highly compressible nature of the

injectant (CO2), they modified gravity number to make it applicable to the horizontal type

reservoir.

Gravity number (based on the gas injection rate):

:s = LONA7 ) (1-84)

Gravity number (based on the gas injection and oil production pressures):

:sG = A2 (1-85)

1.8. Application

The GAGD process extends the highly successful gravity stable gas floods in pinnacle reefs

and dipping reservoirs to horizontal type reservoirs. To allow for scalability of the laboratory

experiments, the reproduction of the various multiphase mechanisms and fluid dynamics,

which have been found to be influential in the success of the gravity stable gas floods is

crucial. Literature reviews of multiphase mechanics and fluid dynamics, suggests that

dimensionless characterization of flood parameters to generate analogous field scale

multiphase processes into the laboratory, is one of the most effective and preferred scaling

tools (Kulkarni, 2004).

Dimensional analysis becomes especially necessary for better understanding and performance

characterization of novel processes like the GAGD. The dependant and independent variables

used in this analysis are shown in Table 1.3 along with their fundamental dimensions.

The various dimensionless groups obtained after the analysis are included as Table 1.4. It is

important to note that the Buckingham-Pi analysis does not rank the dimensionless groups

obtained in order of relative importance as controlling variables of the process. Literature

36

review, experimentation and inspectional analysis are required to determine the order of

controlling groups of variable(s).

Table 1.3: Dependant and Independent Variables used for Buckingham-Pi Analysis

Inspectional analysis of the various dimensionless groups obtained via Buckingham-Pi

analysis suggested the use of five dimensionless groups for the detailed analysis of the

GAGD process namely, Capillary, Bond, Gravity, Dombrowski-Brownell, and N (new

Grattoni et al.s group).

Table 1.4: Dimensionless Groups Obtained Using Buckingham-Pi Analysis

37

Nine commercial gas gravity drainage field applications were considered for the

identification and characterization of various multiphase mechanisms, fluid dynamics and

calculation of the range of various dimensionless groups probable in the GAGD process. The

protocol followed to facilitate their calculation is included as Figure 1.1.

Calculation of these dimensionless numbers for field projects involved the use of various well

logs (for thickness, net-to-gross values, OWC, GOC and grain size), field maps (for Darcy

velocity), grain size classification systems (for Bond number), production / injection data

(for New Grattoni et al.s group), bottom hole pressure survey plots (for PVT simulations),

compositions of injected / produced fluids (for PVT simulations), and PVT simulations (for

fluid properties predictions).

Figure 1.1: Protocol for Calculation of Dimensionless Groups for Field.

All the experiments are designed to mimic or resemble the multiphase mechanisms operative

in the field processes by keeping synonymous dimensionless values obtained from the field

projects.

The grain size of the glass beads and fluid properties in the experimental model are selected

to generate similar values of the selected dimensionless numbers. Variables in the

dimensionless numbers were varied to capture the entire spectrum of dimensionless number

values obtained from the field projects. From Bond number equation we can see that, NB is

directly proportional to the absolute permeability of the porous medium and the density

38

difference between the reservoir fluids. Absolute permeability of a consolidated porous

medium is a strong function of the grain diameter and is given by the Carman-Kozeny

(Carman, 1937) equation:

= $8*E72+1 *.8 (1-86)

In above equation, DP is the grain diameter, is the tortuosity and is the porosity of the

porous medium. In order to obtain favorable and realistic Bond numbers, fluid-fluid

interaction parameters (interfacial tension) are also important. On the other hand, different

capillary numbers were obtained by varying the fluid-fluid systems, for example Decane-

CO2 and Paraffin-CO2 in the physical model.

However, the ranges of capillary number obtained through selection of different fluid-fluid

system were not as large in comparison to the ranges obtained through selection of different

gas injection rates. Therefore different gas injection rates, using a mass flow controller, were

used to generate capillary numbers of various orders of magnitude.

The operating rages of these field projects were duplicated in the laboratory for 1-D

corefloods and 2-D physical model experiments, by proper fluid and operating condition

selections. The values of the dimensionless groups obtained for the laboratory experiments

have also been included in Table 1.5.

Table 1.5: Dimensionless Number Ranges Obtained for Field Applications and Laboratory

Studies

39

In order to scale-up the laboratory run time to a given prototype field, the following

dimensionless time expression was used. The expression for the dimensionless time (td) for

gravity drainage processes was obtained from the literature (Miguel et al., 2004) and is

expressed as:

ZS = ONN + AA.*N+1 /NO /go. Z (1-87)

where k is the absolute permeability of the porous medium, Koro is the end-point relative

permeability to oil, is the density difference between the displaced phase and the

displacing phase, g is the acceleration due to gravity, h is the thickness of the porous medium,

is the porosity, o is the oil viscosity, Sor and Swi are the residual oil and connate water

saturation respectively, td is the dimensionless time, and t is real time.

Above equation enables the scale-up of the run time (in minutes) in the experimental models

to time required in a prototype reservoir to reach similar recoveries.

The results obtained from the physical model and immiscible core flood experiments are

compared with data obtained from the gravity drainage field projects. For effective

comparisons, as well as to account for the relative variations of the Bond and Capillary

numbers in each of these floods, a single comparison parameter is required.

The gravity number is a combination of Bond and capillary numbers, and incorporates the

relative variations of the major reservoir forces, namely the gravity, capillary and viscous

forces. Therefore, the Gravity number appeared to be more appropriate for the comparison of

laboratory and field data.

This is very encouraging, since the data for this comparison are obtained from vastly varied

sources, such as from the atmospheric pressure, homogeneous 2-D sand packs, to the highly

heterogeneous and high-pressure field flood projects. These findings indicate that the

performance of the GAGD process appears to be well characterized by the use of the gravity

number.

40

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