Bubbles in Magmas
How do bubbles grow by decompression in
silicic magmas? How fast do they rise?
How permeable are magmas?
Application of the ideal
gas law to bubble
growth, bubble rise,
porosity and
permeability of magma
Core Quantitative Issue
Forces
Supporting Quantitative Issues
Ideal Gas Law
Porosity and Permeability
Iteration
Prepared to offend unwary volcanology students
Chuck Connor – University of South Florida, Tampa
1
Preview
This module presents calculations for decompression growth of bubbles in silicate
magmas, bubble rise, and development of porosity and permeability in magma.
Slides 3-7 give some background on bubbles in magmas and discuss how bubbles are intimately
related to magma flow and dynamics.
Slide 8 states the first problem. How do isolated bubbles grow by decompression and rise in magma?
Slides 9 and 12 analyze this problem. The solution involves the ideal gas law and resolving the forces
acting on a rising bubble.
Slide 13 illustrates a spreadsheet that calculates a solution to this problem.
Slide 14 states the second problem. Given many bubbles, such as 109 bubbles per cubic meter, how
does porosity and permeability develop in magma?
Slides 15-18 analyze this second problem. The solution lies in considering the quantitative
relationships between pressure, magma density, bubble volume, and porosity.
Slides 19-20 illustrate a spreadsheet to solve this problem, using an iterative solution.
Slide 21 summarizes the module and points to open issues in modeling bubble behavior in silicate
magmas.
Slide 22 gives the end of module assignments
Slides 23-27 provide endnotes and additional references.
2
Background
Why do bubbles form in
silicate magmas?
Volatiles, mainly water and carbon dioxide,
are completely dissolved in magmas under
high pressure conditions deep within the
earth. In other words, magmas under high
pressure are under-saturated with these
volatile compounds. As the magma rises,
the solubility of volatiles decreases, the
magma becomes saturated or perhaps
super-saturated with respect to volatiles,
and bubbles begin to grow. Exactly where
on the journey to the surface this happens
depends on the amount of volatiles
dissolved in the magma, the solubility of
the each specific volatile in the magma,
and additional factors, such as the
presence of crystals that may help bubbles
form.
Photo by J.B. Judd
This bubble in basaltic magma was
photographed just as it burst, having
risen to the surface of the Mauna Ulu
lava lake during the 1969 eruption. The
bubble diameter at the surface is
approximately five meters. [Photo
courtesy of the USGS].
More about bubbles in general
(including dubious
applications of bubble
physics!):
http://en.wikipedia.org/
wiki/Liquid_bubble
3
Background
Why are bubbles important in volcanology?
Nucleation of bubbles and
bubble growth in some magmas
accelerates ascent and causes
explosive eruptions.
Consider a volcanic conduit with cross-sectional
area:
a  r 2
and mass flow Q. As pressure decreases higher
in the conduit, bubbles grow. The bubbles lower
the density of the ascending magma.
Conservation of mass dictates:
Q
Q  rau
where r is the density of the magma (including
bubbles) and u is the ascent velocity of the
mixture (assuming the bubbles rise at the same
speed as the rest of the magma). So as density
drops, ascent velocity must increase, in order for
Q to remain constant.
r
An explosive eruption at St. Augustine
volcano, Alaska, during 1986. In this
eruption sustained mass flow of magma
from a magma reservoir at depth,
coupled with bubble nucleation and
expansion, causes a sustained explosive
eruption (photo courtesy of the USGS).
Be sure you understand the concept
of conservation of mass. Check the
units of Q. The mass flow is the same
at the top and bottom of the conduit,
but density and velocity are different.
4
Background
Why are bubbles important in volcanology?
At many volcanoes, magma
degases passively – in such
cases the rate of gas escape
from the magma exceeds the
rate of magma rise, at least at
the surface.
Photo by P.C. LaFemina
Image: © Stromboli
online
Passive
degassing at Stomboli volcano, Italy
Passive release of volcanic gases into the atmosphere is
a dramatic end-product of bubble nucleation and rise in
magmas. Such degassing is the most common form of
volcanic activity and is the primary source of Earth’s
atmosphere (Masaya volcano, Nicaragua – one of
Earth’s most persistently degassing volcanoes).
In this module you will concentrate on passive degassing
5
Background
What are the possible mechanisms for passive degassing?
In passive degassing (i.e., gas flow from the
volcano but very little or no eruption of magma),
bubbles of gas must move toward the surface and
escape. Consider two methods of accomplishing
this without erupting large amounts of magma.
Method 1: Individual
bubbles rise through the
magma due to buoyancy
force. As they rise they grow
primarily by decompression.
When these bubbles reach
the surface they break,
releasing gas and perhaps
throwing volcanic bombs as
the wall of the bubble bursts
into the air. Such a
mechanism requires a low
viscosity magma and will not
sustain high gas flux.
Method 2: Bubbles ascend the
conduit together with magma (Q).
As bubbles rise they grow by
decompression. Bubble density
(number of bubbles per cubic
meter) is high enough for
bubbles to touch and connect as
they ascend and grow. This
connection creates permeability
and allows gas to escape. As the
bubbles degas, they lose
pressure and collapse,
increasing the density of the
magma and causing it to sink,
perhaps at the margins of the
conduit (q). Viscous magmas
may degas in this manner, as
long as mass flow is sustained
from depth. Such a mechanism
can sustain very high gas flux.
6
Background
How permeable is a vesicular magma?
Images courtesy of Margherita Polacci .
The difference between explosion
and passive degassing depends on
the rate of ascent of magma, and
often on the permeability of the
magma. Once the magma becomes
permeable, the volcano can literally
“lose steam”.
Studies of the porosity and
permeability of volcanic rocks,
therefore, provide important clues
about the dynamics of magma ascent
and eruption. Computer imaging and
visualization of volcanic rock pore
space in 3D is one technique used to
better understand these processes.
What are the units of permeability?
7
Problem 1
Given a bubble of water vapor at some depth in a lava lake, what is
the change in radius of the bubble and rise rate of the bubble as it
ascends through the lava lake toward the surface?
In this example you will
specify properties of the
gas bubble (e.g., moles of
water vapor in the bubble)
and the magma
(temperature, viscosity,
density) in order to
quantify the bubble ascent.
First you will consider only
individual bubbles in
isolation; later you will
consider groups of
ascending bubbles.
Start a spreadsheet by entering the
physical constants and given information
for the problem. Calculate the number of
moles water vapor.
8
Designing a Plan, Problem 1, Part 1
Given a mass of water vapor
in a bubble in magma, what is
the bubble radius and ascent
rate?
You will need to
• convert the mass of water
vapor into moles
• use the ideal gas law to
calculate the volume and
radius of the bubble
•Calculate the buoyancy
force, viscous drag, and
terminal velocity of the
bubble.
Give answer in units of
meters for radius and meters
per second for rise rate of the
bubble.
Notes:
(1) In this example you will only consider
the affects of decompression. Leave
the problems of bubble nucleation
and diffusion of gas into bubbles from
the magma for another time!
(2) At this point, only consider an
isolated bubble (method 1 for passive
degassing on slide 6).
(3) In order to fully solve the problem,
you will need to resolve all the forces
acting on an individual bubble.
9
Designing a Plan, Problem 1, Part 2
Using the ideal gas law to
find bubble volume
Water vapor, and any other
gas, is compressible. That
is, at higher pressures a
given number of moles of
water vapor takes up less
volume than at lower
pressures. The volume
occupied by a gas at some
pressure and temperature is
reasonably approximated by
the ideal gas law.
PV = nRT
Where P is pressure, V is
volume, n is moles of the
gaseous substance, R is the
gas constant and T is
temperature.
What is the volume of a water vapor bubble (n=1 mol)
at 10 m depth in a lava lake (T= 1473 °K, r = 2500
kg m-3)?
1. Visualize the problem:
What is the
bubble
radius with
fixed T, n?
Surface of lava lake
h
2. Consider the variables: P, n, R, T
Given n, T
R is constant
P=rgh
3. Solve for P
4. Solve for V
5. Evaluate if your answer makes sense. What happens
to the bubble volume at shallower depth (e.g., 5 m)
or greater depth (e.g., 50 m)?
10
Designing a Plan, Problem 1, Part 3
What is the terminal velocity of an
ascending bubble ?
You need to know what
forces are acting on a
bubble in a lava lake
Three forces are acting on the ascending bubble.
Fbuoyancy
Buoyancy force imparted by the density contrast between
the bubble and the magma:
Fbuoyancy
 r magma 
 mbubble g 

r
 bubble 
Gravitational force due to the weight of the bubble:
Fgravity  mbubble g
Viscous force due to the drag imposed by magma flowing
around the rising bubble:
Fviscous  6rhu
Fgravity
Fviscous
Where g is gravity, m is mass, r is density, h is viscosity of
the magma, r is the bubble radius, and u is the ascent
velocity of the bubble
More about buoyancy force
More about viscous force
11
Designing a Plan, Problem 1, Part 4
What is the terminal velocity of an
ascending bubble ?
The net force acting on the bubble is:
Fnet  Fbuoyancy  Fgravity  Fviscous
When the net force is positive, the bubble is accelerating
upward. When the net force is zero, the bubble is not
accelerating, but has reached its terminal velocity.
Recast the equation for net force so you
can solve for the terminal velocity of a
bubble rising in magma. Use the equations
on Slide 11.
12
Carrying Out the Plan, Problem 1: Spreadsheet to Calculate Bubble Radius and Velocity
In a spreadsheet the calculation looks like:
A cell containing
given information
A cell containing a
physical constant
A cell containing a
formula
Using information of
the previous slides,
decide what to enter
in each cell
containing a formula
13
Problem 2
Given a regular distribution of uniformly sized
bubbles, at what depth does an ascending bubbly
magma become permeable?
In the first problem, you considered a single
bubble ascending from deep in a lava lake
to the surface. Now consider a more
common case, in which there are many
bubbles in a given volume of magma. At
what depth does the magma become
permeable so that the gas can escape from
the bubbles and into the atmosphere?
To Solve this problem you will need to:
1) Make assumptions about the number of bubbles and their
distribution in the magma
2) Use the ideal gas law to estimate bubble volume
3) Discover a solution “iteratively”, solving repeatedly for porosity,
pressure, and bubble volume
Learn more about bubbles and permeability
14
Designing a Plan, Problem 2, Part 1
Consider one cubic meter of magma.
In a regular, mono-dispersed pattern,
bubbles of equal volume are spaced
at equal intervals to fill this cubic
meter. In reality, magmas are polydispersive. That is, randomly spaced
bubbles have a size distribution that
reflects the complexities of bubble
nucleation in the magma and the
pressure history of the cubic meter of
magma as it ascends.
If we assume:
1) A regular, mono-dispersed bubble
distribution
2) Ideal gas law behavior of the bubbles
Then it is short work to estimate the
porosity of the 1 m3 of magma for
different hydrostatic pressures.
Prove to yourself that there
will be 109 bubbles per cubic
meter for the above regular,
mono-dispersed bubble
distribution. What is the
porosity of this magma, if the
radius of each bubble is 0.2
mm? Answer: about 3.3%
15
Designing a Plan, Problem 2, Part 2
Although the magma is porous when bubbles of any size
are present, it is not permeable until the bubbles
connect to form a network that allows gas to escape
from the magma.
This happens in the ideal regular, mono-dispersed bubble
distribution when the radius of each bubble grows to
one-half the distance separating bubbles.
Experiments suggest that actual magmas become
permeable at lower porosities than forecast by the
regular, mono-dispersed bubble distribution. With
random bubble nucleation sites and a variety of
bubble-size distributions, Blower (2001) developed
the relation for porosity 30-80 percent:
k  ar2 f  fcr  ,f  fcr
b
where k is permeability (m2), a, b are constants, f is
porosity, fcr is the critical porosity, below which
permeability is zero, and r is bubble radius (m).
Constants a, b, and fcr are found by regression of
rock laboratory measurements. Blower found that a=
8.27 x 10-6, b = 2.10, and fcr = 0.3 fit some laboratory
measurements well.
What is the porosity of this
magma with regular, monodispersed bubble distribution
(1 mm between the centers of
each bubble pair) when the
magma first becomes
permeable?
16
Designing a Plan, Problem 2, Part 3
The last part of the problem involves the density of the magma. If
there are a lot of bubbles, then the bubbles change the density of
the magma. This should be taken into account in the estimate of
porosity. For example, in the diagram showing decompressing
bubbles, density increases with increasing depth (e.g., r1<
r2<r3<r4).
For the diagram, pressure at depth H is given by:
P  r1  r2  r3  r4 gh
alternatively written:
4
P  gh r i
i 1
Where g is gravity, h is the thickness over which some constant
density is assumed.
In excel:
The formula is:
D5=D4+B5*9.8*C5
17
Designing a Plan, Problem 2, Part 4
The problem now seems circular! The pressure depends on porosity of the overlying layers; but
porosity, in turn, depends on pressure because pressure controls bubble growth by
decompression. The way to solve this problem lies in iteration. Iteration is the repetition of a series
of commands until the answer converges on a stable solution.
Start
pressure
Specify the range of depths
(say 0.5-5 m at 0.5 m intervals)
porosity
bubble volume
Calculate hydrostatic pressure
using the density of bubble-free
magma
No
Calculate the volume of an
individual bubble (Ideal gas law)
Given the number of nucleation
sites and bubble volume,
calculate the porosity
From the porosity, recalculate
the density and pressure
Yes
Is the answer stable?
(iterate 1000 times)
Finish
In excel, you can make a
spreadsheet “iterate” by using
“circular references”. That is, the
answer in each cell depends on the
18
answer in the other.
Carrying Out the Plan, Problem 2: Spreadsheet to Calculate Permeability
In a spreadsheet the calculation looks like:
A cell containing
given information
A cell containing a
physical constant
A cell containing a
normal formula
without circular
reference.
A cell containing
an iterative
formula – one that
does contain a
circular reference.
Using information of
the previous slides,
decide what to enter in
each cell containing a
19
formula
Carrying Out the Plan, Problem 2: Spreadsheet to Calculate Permeability
Details about the iteration:
In this spreadsheet:
the formula in cell E21
uses the result in cell
G21
The formula in cell F21
uses the result in cell
E21
The formula in G21 uses
the result in cell F21
Note from the above that
this is a circular
reference.
To run the spreadsheet
with a circular reference,
go to Tools: Options:
Calculations
Check the iteration box!
Select Maximum
iterations =1000 and
Maximum Change =
0.001
Learn more about the excel iteration set-up
20
What you have done
You have investigated decompression of bubbles rising in magmas both in isolation
and dispersed throughout a volume of magma
Bubbles play a crucial role in volcanology because it is the formation and expansion of bubbles that accelerates flows, and
because gas, ultimately the Earth’s atmosphere itself, is carried to the Earth’s surface in bubbles. Here we have dealt in detail
with the decompression of bubbles, but the topic is even more complex. The physics of bubble nucleation (where, when, and
why bubbles form) is another extremely important facet of the story. Furthermore, bubbles grow by diffusion of gas from the
melt into the bubble. This is another important factor governing the nature of bubble growth. In this module we have assumed
that equilibrium conditions prevail (for example that the bubble pressure will equilibrate with local hydrostatic pressure). In fact
this is not necessarily the case. It is fair to say that the study of bubbles is a rich and active field of research in physical
volcanology, and full models of the fate of bubbles in magmas are not yet developed.
Nevertheless, you have discovered that it is possible to study bubble decompression by considering the thermodynamics
(ideal gas law) and physics (net force acting on bubbles) of bubble ascent using simplifying assumptions. Often such models
are used in the real world to understand basic processes and as a starting point for more physically realistic (and often much
more challenging!) models. Natural processes are complex – learn to simplify!
You have performed a calculation of change in pressure, porosity, and permeability in a bubbly magma column using in
iterative solution. Such iterative solutions are common to a host of models describing geologic processes.
Some useful starting points for learning more about bubbles in magmas:
Hurwitz, S., and O. Navon, 1994, Bubble nucleation in rhyolitic melts: experiments at high pressure, temperature, and water
content. Earth and Planetary Science Letters 122: 267-280. [a very comprehensive introduction to essential research about
bubbles in magmas].
Cashman, K.V. and Mangan, M.T. ,1994, Physical aspects of magmatic degassing II. Constraints on vesiculation processes from
textural studies of eruptive products. Reviews in Mineralogy, 30: 447-478. [a starting point for understanding bubbles in rocks]
Blower, J.D., 2001, Factors controlling permeability-porosity relationships in magma. Bulletin of Volcanology 63: 497-504.
[accessible development of a best-fit statistical model of permeability based on analysis of pyroclasts].
21
End of Module Assignments
1.
Make sure you turn in your spreadsheets, showing the two worked examples.
2.
Modify your first spreadsheet to estimate the rate of change in bubble radius and terminal velocity as a function of depth for an isolated bubble
ascending through magma. Use a 10 gm bubble. Be sure to graph your results.
3.
Calculate the terminal velocity of a spherical pumice (density = 600 kg m -3, radius = 10 cm) rising through water at shallow depth. Assume the
bubble is filled with water vapor (10 gm) and grows by decompression (no other gas diffuses into the bubble during ascent, no bubbles coalesce).
4.
Consider a pyroclast falling from an eruption column at a height of 30 km. What is the terminal velocity of the pyroclast as it falls? Assume the
pyroclast is a spherical pumice (density = 600 kg m -3, radius = 0.001 m). Assume constant viscosity for air and that viscous resistance applies (not
turbulence). Use the barometric formula to estimate the change in density of the atmosphere at 1 km increments as the pyroclast falls to earth
(sea level). Be sure to graph your results.
5.
Experiment with your solution to problem 2 (the spreadsheet to calculate permeability). You will note that your spreadsheet does not give correct
solutions if the number of nucleation sites is high, or the mass of bubbles (gm H 20) is high. Why is this the case? How would you go about fixing
this problem? (Note: you are not required to implement this fix, only to discuss it).
6.
Modify the spreadsheet for problem 2 (the spreadsheet to calculate permeability) to show the change in pressure, porosity, and bubble volume at
0.25 m intervals from 5 m depth to 0.5 m depth.
7.
Calculate the change in permeability of magma at 0.5 m depth caused by changing the number of bubble nucleation sites smoothly between 1 x
107 m-3 and 2.5 x 107 m-3, with 2 x 10-7 gm water vapor per bubble (melt density = 2500 kg m -3, temperature = 1300 °K). Use the Blower (2001)
model of permeability. Describe how and why permeability changes with the number of nucleation sites.
8.
Sulfur dioxide (SO2) flux from Stromboli volcano has been measured to be around 4.6 kg s-1. Sulfur (S) in melt inclusions (bits of trapped primitive
magma in crystals) is about 0.23 wt%; in contrast, S in pyroclastic bombs thrown from Stromboli is about 0.03 wt%. The difference is the amount
degassed in bubbles, and degassed into the atmosphere by the volcano as long as sufficient permeability is reached. If the bubble-free magma
density is 2500 kg m -3, approximately how much sulfur degases from 1 m 3 of magma (give answer in kilograms)? How much SO2 degases from 1
m3 of magma (give answer in kilograms)? Then, what flux of magma is required to sustain an SO 2 flux of 4.6 kg s-1? What is your estimate of how
much magma reaches the surface and degases at Stromboli every day? Based on these calculations and information in this module, develop a
conceptual model for this degassing. Describe your model in words and draw appropriate illustrations of the major aspects of the model. Please
thank H. Mader for this question if you happen to meet her!
22
Units of Permeability
The unit of permeability is the Darcy. In practice, permeability is reported in mD, milliDarcy, where 1 mD
= 10-12 m2.
Permeability is a coefficient in Darcy’s law, which can be expressed in terms of the volume flux of a
liquid through a tube packed with permeable material as:
kr 2  Pb  Pa 
Q
h  L 
b
a
r
L
Where Q is the volume flux (m3 s-1), r is a cross sectional area (m2), h is the liquid viscosity (Pa s), and
[Pb-Pa]/L is the pressure gradient (Pa m-1) that drives the flow.
Prove to yourself using the above equation that the units
of permeability are m2.
More about Darcy’s Law:
http://en.wikipedia.org/wiki/Darcy's_law
Return to Slide 7
23
Buoyancy Force
Buoyancy force arises because pressure increases with depth and because pressure acts on all sides
of a body, such as a bubble rising through magma or a pumice rising through water.
The buoyancy force is equal to the weight of the
fluid displaced by the object:
Fbuoyancy  rmagma Vobject g
where r is density, V is volume, and g is gravity.
Note that this means that the object experiences
buoyancy force regardless of the object’s density
(or whether the object rises or sinks in the fluid).
Given the above equation, prove to
yourself that another way to express the
buoyancy force is:
Fbuoyancy
where m is mass.
rmagma

mobject
robject
The object submerged in this vat of magma
experiences buoyancy force. Pressure on the
object (illustrated schematically by the black
arrows) is greater deeper in the vat than
shallower in the vat (remember that P=rgh,
where h is depth). The greater pressure at
depth results in upward force on the object.
More about buoyancy force:
http://hyperphysics.phy-astr.gsu.edu/Hbase/pbuoy.html
Return to Slide 11
24
Viscous Force (Drag Force)
Rising or falling objects are slowed by the resistance (drag) of the fluid they are moving through. If the
object is relatively small and/or the fluid has a high viscosity, then viscosity is the dominant factor
controlling the amount of resistance the object encounters. For bigger objects moving through low
viscosity fluids (such as a block or bomb falling through air), the flow becomes turbulent and there is
more drag than predicted by the viscous force.
Consider two objects (one small and one big) rising through a fluid:
For the comparatively
small object, flow is
laminar, the streamlines
(representing the motion
of the fluid passed the
object) are smooth and
continuous. In this case
the viscous force can be
used to estimate the
terminal velocity.
For the comparatively
large object, flow is
turbulent, the streamlines
are disrupted by the
movement of the object,
and the viscous force will
underestimate the total
drag (that is, the terminal
velocity of the object will
be less than expected).
When flow passed the object is laminar (not turbulent), then the drag force is proportional to velocity:
Fdrag  bu
where b is a constant and u is velocity (the negative sign means the force acts in the direction opposite velocity).
Stokes discovered that for laminar flow around a sphere:
b  6ha
where h is viscosity and a is the radius of the sphere.
More about drag and Stokes:
http://en.wikipedia.org/wiki/Drag
Return to Slide 11
25
Differences in permeability in granular and vesicular rocks
In granular rocks, such as sandstones or pyroclastic deposits, permeability is related to the
shape and connectivity of spaces between individual grains (a). In vesicular magmas and rocks
that cool from these magmas, such as lava flows, permeability is related to the connectivity of
bubbles (b). Blower (2001, Bulletin of Volcanology, 63: 497-504) showed that the width of the
aperture, rap, controls the resistance to flow in linked sets of bubbles. Note that rap is related to
the radius of the bubbles and the distance between them, using geometry rules. Diagram from
Blower (2001).
Return to Slide 14
26
A screen-shot from excel showing the “iteration set-up”
Arrive at
Options by
using the
Tools menu
Note that F9
Key on your
computer runs the
iterative calculation
Sometimes it helps
to click on to an
empty cell on the
spreadsheet, then
press the F9 key,
even if “automatic”
calculations are
selected.
27
Return to Slide 20