Permeability of Petroleum Reservoir Rock
In addition to being porous, a
reservoir rock must be permeable—that is, it must allow fluids
to flow through its pore network at practical rates under
reasonable pressure differentials. Permeability is defined as:
1.
The
ability of a formation to transmit fluids
2.
The
ability of rocks to allow the circulation of fluids contained in their
pores.
3.
The
ability of the rock to allow a fluid with which it is saturated to flow through its pores.
Permeability is a measure of the
ease with which the fluid flows in the porous medium. Figure 4.1
represents a schematic showing the difference between low and high permeability
rocks in a reservoir. The permeability of a rock measured when it is 100%
saturated with a single phase (water, oil, or gas) is often called “single-phase
permeability”, “absolute permeability,” or just “permeability”.
If there are two fluids flowing in a rock, then it relates to another concept
known as “relative permeability”.
Permeability is part of RCAL and is
considered as a flow or transport property that helps in understanding the flow
in the reservoir. The concept of permeability was first introduced by French
civil engineer Henry Darcy in 1856 when he performed an experiment on sand
filtrates and analysed the concept of permeability. Darcy’s law for a single
phase (liquid) is expressed as:
Where q - is the flow rate [m3/s],
k - is the permeability [m2], A - is the core cross-sectional area
perpendicular to the flow [m2], L is the length of the core [m], dP
is the pressure difference across the core [Pa or N/m2], and µ is
the viscosity of the injected fluid [Pa.s or N/m2.s].
This equation is the linear form of
Darcy’s law for incompressible fluid, which is discussed further in the
following sections.
Applications of Permeability
Permeability is a parameter that
describes the flow in porous media. From Darcy’s law, we can estimate the
production flow rate from the reservoir to the surface. This can be done by
determining the permeability of the reservoir through laboratory experiments on
core samples extracted from the same reservoir, as well as determining all the
parameters associated with Darcy’s law.
 |
Figure 4.1: Schematic showing the
cross section at the micro-scale of (a) a lower permeability rock and (b) a
higher permeability rock. The fluid flow is much easier in rock (b) compared to
rock (a). |
Validity of Darcy’s Law for Single-Phase Permeability
Darcy’s law for single-phase flow is
valid under some conditions, which include:
The core sample used needs to be
100% saturated with a single phase (water, oil, or gas). If the system consists
of more than one fluid, then we need to consider relative permeability.
The flow has to be laminar. Flow can
be characterized as either laminar or turbulent. Laminar flow is defined as the
“slow,” uniform flow while turbulent flow is defined as the “fast,” chaotic
flow (Figure 4.2). In
order to determine whether the flow is laminar or turbulent, a dimensionless number
is used, known as the Reynolds number. This number is obtained from the
following equation:
Re=ρvD/μ
where Re - is the Reynolds number
[dimensionless], ρ - is the density of the fluid [kg/m3], v - is the
velocity of the fluid [m/s], D - is the pipe diameter [m], and µ - is the
viscosity of the fluid [Pa.s].
A flow with Reynolds number of 2100
or less is considered laminar.
This equation is mainly used in pipes as they
have a fixed diameter; however, for a porous medium, an average grain diameter
is used. Since it is difficult to compute Reynolds number in porous media, another
technique is used to determine whether the flow is laminar or turbulent. This technique is used when measuring
permeability in the laboratory and will be discussed in the following sections.
It is important to mention that in a reservoir; the flow is generally laminar.
The flow has to be steady-state
flow. Steady-state flow means that whatever enters the system leaves the system,
or that there is no volumetric change over time. This is true for fluid flow in
core samples if the same amount of fluid enters and leaves the system over a certain
period. If the flow is unsteady-state, then Darcy’s law is invalid.
 |
Figure 4.2: Schematic showing the
two different types of flow: (a) laminar flow (a flow that is uniform, smooth,
and occurs at low flow rates) and (b) turbulent flow (a flow that is chaotic
and occurs at high flow rates compared to laminar flow). |
The flow in porous media has to be laminar in order for Darcy’s law
to be valid.
Darcy’s Law Under Different Boundary Conditions
In this section, we will derive
Darcy’s law for different boundary conditions by starting from the differential
form of Darcy’s law:
where q is the flow rate [m3/s], k is the permeability [m2], A is the cross-sectional area perpendicular to
flow [m2], dx is the change
in length [m], dP is the pressure
difference across the core [Pa], and µ is
the viscosity of the injected fluid [Pa.s].
Fluids can be either compressible or
incompressible. “Fluid” is a term that refers to liquids and gases.
Compressible fluids are fluids that change volume due to a change in pressure,
such as gases. Liquids, on the other hand, are considered incompressible
because their volumes change negligibly with a change in pressure. An example
of compressible fluids in real-life applications is scuba tanks, where air is
compressed inside the tank. The actual volume of the compressed air in the
tank, under atmospheric pressure, is much larger than the actual volume of the
tank itself. However, if we wish to fill the same tank with water (which is an
almost incompressible fluid), the volume of water that fills the tank will approximately
be the same inside and outside the tank. When we consider flow in porous media,
we need to consider whether the flow is compressible or incompressible, as this
will change the governing equations. In addition, we need to consider whether
the flow is linear in Cartesian coordinates (most laboratory experiments) or
radial coordinates (reservoir conditions).
Case 1: Linear solution of Darcy’s Law for incompressible fluid
Let us assume we have the system shown in Figure 4.3, and we inject it with an incompressible fluid such as water. Based on the system, we know that the length is L, the inlet pressure is P1, and the outlet pressure is P2. In order for the flow to occur from the inlet to outlet, P1 needs to be greater than P2, because fluid flows from high pressure to low pressure. This is analogous to the movement of most of the other flows, such as the flow of electrical charges from high to low voltage, and heat transfer from high to low temperature. Our homes also have a real-life example of such flows. In a vacuum cleaner, for the particles to flow to the dust collector, a pressure lower than the atmospheric pressure is required inside the appliance. The vacuum cleaner achieves this by creating a partial vacuum inside the machine for the particles or dust to flow towards it.
Darcy’s law is usually written with
a negative sign. This is because dP = P2 – P1 and since P2
is smaller than P1, a negative sign will make the overall equation
positive. For simplicity, we will use the following differential form of
Darcy’s law:
Where dP = P1 – P2
and P1 > P2.
Now, we can rearrange the equation
to obtain:
Taking the integral with the
boundary limits as shown in Figure 4.3:
Then, the equation becomes:
Finally, we divide both sides by L:
Equation is the final form of
Darcy’s law for an incompressible linear system.
 |
Figure 4.3: Schematic showing a
cylindrical core sample used for permeability measurement, where a cross
section is taken to display the boundary limits of the linear system. |
Radial solution of Darcy’s Law for incompressible fluid
In this case, the only difference is that the flow occurs in a
radial manner as opposed to linear. This flow is more representative of a
reservoir. One of the changes is that the area perpendicular to the flow is the
circumference of the circle (2πr) multiplied by the thickness of the reservoir
(h) (in contrast to πr2 in the linear flow for a cylindrical-shaped
object). The second change is that instead of the flow varying in the
x-coordinate, it will vary in the r-coordinate and thus instead of having
dP/dx, we will have dP/dr. Figure 4.4 shows the radial system of
the reservoir.
We start with the differential form of Darcy’s law and substitute the area perpendicular to flow, which is the circumference of
the circle, and change the coordinates from linear to radial. The equation
becomes:
q=(2kπrh/μ)*(dP/dr)
Then, we rearrange the equation to obtain:
q (dr/r)=(2πkh/μ)*dP
Now, we take the integral with the boundary limits in agreement
with Figure 4.4, thus obtaining:
where re is the reservoir’s outer radius [ft], rw
is the well radius [ft], Pe is the reservoir’s outer pressure
[psia], and Pwf is the wellbore flowing pressure [psia].
Then, we integrate the equation with the given boundary condition to obtain:
This is the final form of Darcy’s
law for incompressible radial system.
Dealing with gases is different from
liquids as gases are compressible fluids. Hence, we need to account for the
change in volume when deriving the equation.
In addition, equations for Darcy’s
law for different systems can also be modified to account for the gravity term
in vertical systems. Nevertheless, these equations are rarely used in petroleum
engineering, as the flow in the reservoir and in laboratory experiments is
mainly horizontal.
 |
Figure 4.4: Schematic showing (a) a
radial system where the flow occurs from the outer boundary to the wellbore
region and (b) a zoomed in section of one part of the reservoir to clearly
explain the system. |
Unit Systems
When using Darcy’s law, three main unit systems can be used to find
permeability. These units are explained in Table 4.1. It is
important to note that since permeability values are very small in m2,
a unit of Darcy [D] or milli Darcy [mD] is used, named in honor of Henry Darcy.
Moreover, the unit bbl/d in the table indicates barrel (bbl) per day (d), which
is a standard unit to quantify volume in the oilfield units.
|
Table 4.1: Summary of different units
used when dealing with permeability. |
|||
|
Parameter |
SI Units |
Darcy Units |
Oilfield Units |
|
q (flow rate) |
[m3/s] |
[cm3/s] or [cc/s] |
[bbl/d] |
|
L (length) |
[m] |
[cm] |
[ft] |
|
A (area) |
[m2] |
[cm2] |
[ft2] |
|
h (thickness) |
[m] |
[cm] |
[ft] |
|
r (radius) |
[m] |
[cm] |
[ft] |
|
P (pressure) |
[Pa] |
[atm] |
[psia] |
|
k (permeability) |
[m2] |
[D] |
[mD] |
|
µ (viscosity) |
[Pa.s] |
[cP] |
[cP] |
By rearranging Equation, the following equation is obtained,
which can be used to find permeability in SI or Darcy units:
However, for oilfield units, this equation is multiplied by a
conversion factor for linear flow:
and similarly, for radial flow:
Laboratory Measurements of Absolute Permeability
Liquid Permeability
The permeability of core samples is measured using either liquid or
gas. The procedure and the governing equation are different for each case.
Before we start measuring the
permeability of a core sample, we first need to measure its dimensions (length
and diameter) as length and area are part of Darcy’s law. Secondly, we insert
the core sample in a core holder as shown in Figure 4.5. A common
procedure is to vacuum the core using a vacuum pump prior to injecting any
liquid. This is to remove any air from the system and to ensure the flow of
only one phase. Then, a liquid (in this case water) is injected at a specific
rate using a pump. Before injecting the water, we need to ensure that a confining
pressure, similar to the overburden pressure that is squeezing the core sample
from all the sides, is applied. This is done in order to guarantee that the
water is only going through the core sample and not bypassing it as that will
generate errors in the measurement. A general rule of thumb is to apply a
confining pressure (the oil pressure in Figure 4.5) that is at
least 1.5 times higher than the water injection pressure. When water is
injected at a constant flow rate, a wait time is required in order to achieve a
steady- state flow where the inlet and outlet pressures become constant and do
not fluctuate. After the steady-state flow is achieved, we record the flow
rate, inlet pressure, and outlet pressures. We then move on to a new flow rate
and follow the same procedure. After recording a few data points, we can plot
the data in order to find the permeability of the core sample.
If we rearrange Darcy’s law for the
liquid phase, it becomes:
This equation is in the linear form
y = mx+b, where y is the dependent variable (q/A), x is the independent
variable (dP/L), m is the slope (k/µ), and b is the y-intercept. The
y-intercept in this case is 0 since the curve goes through the origin.
If we plot this figure with our
data, a figure similar to Figure 4.6 will be generated. The slope
of that figure will be equivalent to the permeability divided by the viscosity.
In order to obtain the permeability, the slope needs to be multiplied by the
viscosity.
 |
Figure 4.5: Schematic showing the
experimental set-up to measure the liquid permeability of a core sample. In
this system, we inject water through the core sample and use oil to apply a
confining pressure, which should be higher than the water injection pressure. |
Gas Permeability
Dealing with gas is different
because gas is compressible as opposed to liquids. Thus, the governing
equations will be different as well to account for this change in volume.
Measuring the gas permeability has advantages over measuring liquid
permeability. The measurements for gas take less time than those for liquid, and
gas does not wet the core, which means the core can be reused for further
analyses. The disadvantage is that gas permeability requires correction as it
tends to be overestimated compared to liquid permeability. Figure 4.7
shows the typical experimental set-up used to measure gas permeability. Since
it is difficult to control the gas flow rate, the core holder is attached to a
gas cylinder, and a metering valve is used to vary the flow rate. There is a
flow meter at the end of the core to measure the flow rate at atmospheric
conditions. The simplest way to derive the equation for the gas permeability is
to take the average pressure across the core:
where P is the average pressure
across the core [atm].
We use the average pressure because
the flow rate varies across the core and an average value would be more
representative of the flow in the core. We can use Boyle’s law as shown in
Equation We divide both volumes by time (t) in order to convert it to
flow rate. Thus, the equation becomes:
We compare the average flow rate
across the core to the flow rate at atmospheric conditions; since the
atmospheric pressure is 1 atm, it can be eliminated from the equation.
If we rearrange Equation, it
becomes:
Now, we substitute this equation in
Equation:
Next, we can move to the other side
of the equation and substitute Equation in it to obtain:
Then, by rearranging the equation
and factorizing it, we obtain:
For simplicity, Pa is substituted as
1 atm when using Darcy’s units and thus the equation becomes:
Similar to liquid permeability, we
can rearrange the equation to find the gas permeability across the core after
acquiring several data points (Figure 4.8). When dealing with
gases, it is more common to reach a higher flow rate than liquids, because
gases have lower viscosity that can result in turbulent flow, making Darcy’s
law invalid. This can be seen from the plot. A laminar flow, which might also
be referred to as “Darcy’s flow,” will follow the slope from the origin and,
once the slope deviates, enter the turbulent flow regime or “non-Darcy’s flow”.
The data that falls in the turbulent flow regime has to be omitted from the
analysis.
When comparing gas permeability to
liquid permeability, gas permeability tends to be higher than the latter. This
is due to a gas slippage at the pore walls known as the Klinkenberg effect.
This gas slip makes the permeability higher than what it should be; therefore,
it is not representative of the actual value. Fortunately, this can be
corrected by computing the gas permeability (kg) at every
data point and then plotting it
against the inverse of P, as shown in Figure 4.9. The y-intercept
of this line is the equivalent liquid permeability (kL). The x-axis
of the plot is the inverse of P, so
a zero value of x represents infinite pressure. At infinite pressure, gas can
be considered to behave as a liquid. One factor that affects this slippage is
gas molecular weight. As the gas molecular weight increases, the slippage
decreases since gas becomes heavier and closer to liquid. This effect is shown
in Figure 4.10.
 |
Figure 4.7: Schematic showing the
experimental set-up for measuring the gas permeability of a core sample. |
Flow in Layered Systems
The objective of understanding the
flow in a layered system is to find the average permeability across that system
with beddings of different permeability. The concept is similar to an
electrical circuit; the average permeability will vary if the flow is in
parallel or series to the beddings in the system. In addition, it could also
vary if the flow is linear or radial. We will now examine each case
individually.
Case 1: Linear Flow in Parallel
Let us assume a system similar to Figure
4.11 which has three beddings with different permeabilities parallel to
each other, and we try to find the average permeability in that system. First,
we can see a constant pressure difference across the system; however, the flow
rate will be different across each layer as the flow rate is a function of the
permeability of that bedding. Therefore, we can say that the total flow rate
(q) is equal to:
q = q1 + q2 + q3
We also know that the summation of
the thickness of each layer is equal to the total thickness of the system:
h = h1
+ h2 + h3
The total flow rate of the system is equal to:
Then, we substitute for the flow
rate from Darcy’s law in Equation:
Now, we can factor the common
parameters in the equation which will lead to:
kh= k1h1 + k2h2 + k3h3
Finally, we can generalize the
equation to obtain:
This equation can be used to
estimate the average permeability in a linear system where the beddings are
parallel to each other.
 |
Figure 4.11: Schematic showing a
linear system with parallel beddings of different permeabilities. |
Case 2: Linear Flow in Series
Let us assume a system shown in Figure 4.14 which has
three beddings in series with each other; each bedding has a different
permeability. Here, the same flow rate passes through all these rocks:
q = q1 = q2 = q3
However, the pressure across each rock is different, as shown in Figure
4.14 and explained in Figure 4.15. It can be said that:
P1−P2 = ∆P1 +∆P2 +∆P3
Furthermore, the total length is composed of the length of each
rock:
L = L1 + L2 + L3
We know that the flow rate across the entire system is equal to:
Now, we will substitute Darcy’s law
in Equation by making ∆P of each respective equation as the subject to
obtain:
We can now remove the common parameters and the equation becomes:
We can also represent this equation
in the following general form:
This equation can be used to estimate
the average permeability in a linear system, where the beddings are in series
with each other.
 |
Figure 4.12: Schematic showing a
linear system with beddings of different permeabilities in series. |
 |
Figure 4.13: Schematic showing the
pressure profile across a composite system with beddings in series with each
other. |
Radial Flow in Parallel
The flow in parallel systems in
radial orientation (Figure 4.14) is similar to that in linear
orientation, as will be proven below.
First, the overall flow rate is the
summation of all the flow rates across the layers, and the
total thickness is the summation of all the thicknesses across all the layers.
The total flow rate for this system
is equivalent to:
Now, if we substitute each flow rate
in each layer in Equation, we will have:
We can now factor the common
parameters to obtain:
The form is indeed the same as the
linear system and can also be represented in the same general form:
This equation can be used to
estimate average permeability in a radial system where the beddings are parallel
to each other.
 |
Figure 4.14: Schematic showing a
radial system with beddings of different permeabilities in parallel. |
Radial Flow in Series
The flow in series for a radial
system is shown in Figure 4.15. We know that the flow rate across
the layers is the same; however, the pressure
difference in each layer is different as shown in Equation. The flow rate
in a radial system is expressed in Equation.
By substituting Darcy’s law in
Equation and making ∆P of each respective equation as the subject, we
obtain:
Then, we eliminate the common
parameters and rearrange the equation to obtain this generic form:
where r(i+1) represents
the outer radius of layer i and ri represents the inner layer.
This equation can be used to estimate the
average permeability in a radial system where the beddings are in series with
each other.
 |
Figure
4.15: Schematic showing a radial system with beddings of different
permeabilities in series. |
The Porosity-Permeability Relationships
The relationship between a
formation’s permeability and porosity depends on the rock type, and
particularly the specific rock formation (Figure 4-15).
As a broad generalization, the
logarithm of the permeability is approximately linear with porosity for most
rock types. However, the precise relationship is found only through the direct
measurement of representative formation rock core samples in the laboratory, or
by drill stem test interpretation.
 |
Figure
4-16: Generic permeability/porosity relationships |
A number of researchers have
developed theoretical relationships between the permeability and porosity by considering
some textural features such as the size, shape, and distribution of pore
channels in the rock. Notable among these are Kozeny (1937) and Carman (1927),
with modifications by Costa (2006), who have developed relationships between
porosity and permeability, such as:
Where k is permeability, ϕ is
porosity, C is the Kozeny constant, and As is internal surface area
per unit bulk volume
For fractured rocks, generalized
formulas have been developed that relate the permeability to a function of the
fracture width.
In many reservoirs, the permeability
is anisotropic—the magnitude of permeability varies as a function of the
direction and degree of grain
alignment. For example, the fluvial deposition of sediments
tends to align grains in the direction of the river’s flow along their long
axis, thus increasing the permeability in that direction. In almost all bedded
reservoirs, the vertical permeability of the rock will be less than the
horizontal permeability.
There are a number of factors that can affect the
porosity-permeability relationship, including:
1.
The size and shape of the pores.
2.
The connectivity of the pores.
3.
The tortuosity of the pore space.
4.
The fluid viscosity.
5.
The fluid density.
6.
The acceleration due to gravity.
The porosity-permeability relationship is an important
concept in the study of porous media. It is used to design and optimize fluid
flow systems, such as oil and gas wells. The relationship is also used to
characterize the properties of porous media, such as soils and rocks.
Effective and Relative
Permeabilities
Effective permeability and relative permeability are two important
concepts in petroleum engineering. They are used to describe the flow of fluids
in porous media, such as oil reservoirs.
Effective Permeability
The
permeability of a rock to a fluid phase (oil, gas, or water) in the presence of
other fluid phases. It is a measure of the ability of that phase to flow in the
presence of the others. The effective permeability of a fluid phase will
decrease as the saturation of that phase decreases. It is measured in Darcy’s
or millidarcys and is therefore the dimensional equivalent of absolute
permeability, hence:
ko =
effective permeability to oil, darcys or md
kw =
effective permeability to gas, darcy or md
kg =
effective permeability to gas, darcys or md
Individual
values of ko, kw, kg may vary from zero up to
the absolute value, k:
Effective
permeability is a function of:
1.
The
prevailing fluid saturation
2.
The
rock-wetting characteristics
3.
The
geometry of the pores of the rock.
The saturations, if known, should be
specified to completely define the conditions at which a given effective
permeability exists.
Unlike the
previously defined permeability, one now exists for each particular condition
of fluid saturation. Symbolically, the effective permeability of the medium to
oil is 60 percent when the fluid saturations are 60 percent oil, 13 percent
water, and 27 percent gas. The saturation succession given above, i.e., oil and
water, is always followed. The gas saturation is understood to be the
difference between the sum of oil and water saturations at 100 percent.
It is necessary
to generalize Darcy's law by introducing the concept of "effective
permeability" to describe the simultaneous flow of more than one fluid. In
the definition of effective permeability, each fluid phase is considered to be
completely independent of the other fluids in the flow network. The fluids are
considered immiscible, so Darcy's law can be applied to each individually.
In the same
manner when, porous medium is partly saturated with gas, then the flow rate of
gas defined by
Relative Permeability
It is the ratio
of the effective permeability of a fluid phase to the absolute permeability of
the rock. It is a dimensionless quantity that describes the relative ability of
a fluid phase to flow in the presence of the other phases. The relative
permeability of a fluid phase will decrease as the saturation of that phase
decreases. Thus, relative permeability can be expressed symbolically
Which are the
relative permeabilities to oil, water, and gas, respectively, when the medium
is saturated with 50 percent oil, 30 percent water, and 20 percent gas, and
what is the permeability at 100 percent saturation of one of the fluid phases.
Relative
permeabilities are influenced by the following factors:
Ø Temperature
Ø Viscous, capillary and gravitational forces
Ø Pore geometry
Since the effective permeabilities may range from zero to k,
the relative permeabilities may have any value between zero and one:
Another widely used parameter is the ratio of the effective (or
relative) permeabilities of water and oil, and gas and oil:
And
These ratios are dimensionless and may vary from zero to infinity.
Two phase flow behavior
Consider the two-phase flow behavior in Figure 4.17:
Water and oil completely occupy the pore space, ensuring that Sw+So=100%
at all times.
assumptions
Ø Rock is originally 100% saturated with oil.
Ø We introduce water into every pore simultaneously, and a water-wet equilibrium
is instantaneously established.
When water is
first introduced, it is adsorbed by the rock and held immobile both on the rock
surfaces and in the small corners around the points where different grains
converge. Region A shows this immobility to be present. But take note that it
basically stays at 1.0 over the same saturation range. As this process goes on,
the water saturation reaches a critical point (Swc) where water
starts to move around. Both water and oil are currently flowing; however, as
water saturation rises (and oil saturation falls), there are both increases and
declines, as seen in area B. When the oil saturation reaches a residual value
(SO), only water flows and the SW continues to climb. At
this point, the oil is no longer mobile. This is the lowest saturation to which
water injections can decrease oil.
If the oil
could be extracted in another way, the price would continue to rise until it
reached the value of one, as shown.
Many
researchers in this field have proposed generalized empirical equations to
relate kro and krw to Sw, Swi and Sor.
Of particular note are those cited in Honarpour, Koederitz and Harvey (1982),
Molina (1983), and Pirson, Boatman and Nettle (1964). A commonly used
approximation gives:
 |
Figure 4.17 Typical two-phase flow behavior |
If a well is
completed above the transition zone where the reservoir is at irreducible water
saturation (krw = 0), then water will not be produced.
However, if a
well completion is contemplated in the transition zone, it is useful to know in
advance how much water cut may be expected. This can be calculated as follows:
Three-Phase Relative Permeability
Three phases
(water, oil, and gas) can flow simultaneously sometimes and in order to
characterize the flow in these cases, three-phase relative permeability is
introduced. The usage of three-phase relative permeability is minimal in the
petroleum industry when compared to two-phase relative permeability. In
addition, laboratory measurement of three-phase relative permeability is very
difficult to perform. Therefore, Stone came up with two models Stone I and
Stone II to estimate three-phase relative permeability from two-phase relative
permeability (oil/water and oil/gas); the details of the two models are not
discussed in this book and can be found elsewhere.
Three-phase
relative permeability is presented in ternary diagrams (Figure 4.18)
where each phase is placed in an apex with gradual decrease in saturation away
from the respective apex. The water, oil, and gas three-phase relative
permeabilities are shown in Figures 4.19–4.21, respectively.
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