1. Bubbles in Magmas
How do bubbles grow by decompression in silicic magmas? How fast do they rise? How permeable are magmas?
Core Quantitative Issue
Forces
Application of the ideal gas law to bubble growth, bubble rise, porosity and permeability of magma
Supporting Quantitative Issues
Ideal Gas Law
Porosity and Permeability
Iteration
Prepared to offend unwary volcanology students
Chuck Connor – University of South Florida, Tampa

2. 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 analyze this second problem. The solution lies in considering the quantitative relationships between pressure, magma density, bubble volume, and porosity.
Slides 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 provide endnotes and additional references.

3. Why do bubbles form in silicate magmas?
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!):

4. Why are bubbles important in volcanology?
Background
Why are bubbles important in volcanology?
Nucleation of bubbles and bubble growth in some magmas accelerates ascent and causes explosive eruptions. r Q
Consider a volcanic conduit with cross-sectional area:
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:
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.
An explosive eruption at St. Augustine volcano, Alaska, during 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.

5. Why are bubbles important in volcanology?
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.
Image: © Stromboli online
Photo by P.C. LaFemina
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).
Passive degassing at Stomboli volcano, Italy
In this module you will concentrate on passive degassing

6. What are the possible mechanisms for passive degassing?
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 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.
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.

7. How permeable is a vesicular magma?
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?

8. 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.

9. 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?
Give answer in units of meters for radius and meters per second for rise rate of the bubble.
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.
Notes:
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!
At this point, only consider an isolated bubble (method 1 for passive degassing on slide 6).
In order to fully solve the problem, you will need to resolve all the forces acting on an individual bubble.

10. Using the ideal gas law to find bubble volume
Designing a Plan, Problem 1, Part 2
Using the ideal gas law to find bubble volume
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:
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)?
What is the bubble radius with fixed T, n? h
Surface of lava lake
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.

11. 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.
Buoyancy force imparted by the density contrast between the bubble and the magma:
Gravitational force due to the weight of the bubble:
Viscous force due to the drag imposed by magma flowing around the rising bubble:
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
Fbuoyancy
Fviscous
Fgravity
More about buoyancy force
More about viscous force

12. Designing a Plan, Problem 1, Part 4
What is the terminal velocity of an ascending bubble ?
The net force acting on the bubble is:
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.

13. In a spreadsheet the calculation looks like:
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

14. 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:
Make assumptions about the number of bubbles and their distribution in the magma
Use the ideal gas law to estimate bubble volume
Discover a solution “iteratively”, solving repeatedly for porosity, pressure, and bubble volume
Learn more about bubbles and permeability

15. 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 poly-dispersive. 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:
A regular, mono-dispersed bubble distribution
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%

16. 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 percent:
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, mono-dispersed bubble distribution (1 mm between the centers of each bubble pair) when the magma first becomes permeable?

17. 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:
alternatively written:
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

18. 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.
Specify the range of depths
(say m at 0.5 m intervals)
Calculate hydrostatic pressure using the density of bubble-free magma
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
Is the answer stable?
(iterate 1000 times)
Start
Finish
pressure
porosity
bubble volume No
Yes
In excel, you can make a spreadsheet “iterate” by using “circular references”. That is, the answer in each cell depends on the answer in the other.

19. In a spreadsheet the calculation looks like:
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 formula

20. Details about the iteration:
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

21. Some useful starting points for learning more about bubbles in magmas:
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: [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: [a starting point for understanding bubbles in rocks]
Blower, J.D., 2001, Factors controlling permeability-porosity relationships in magma. Bulletin of Volcanology 63: [accessible development of a best-fit statistical model of permeability based on analysis of pyroclasts].

22. End of Module Assignments
Make sure you turn in your spreadsheets, showing the two worked examples.
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.
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).
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 = 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.
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 H20) 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).
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.
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.
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 m3 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 SO2 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!

23. Units of Permeability b a r L More about Darcy’s Law:
The unit of permeability is the Darcy. In practice, permeability is reported in mD, milliDarcy, where 1 mD = 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:
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. b a r L
Prove to yourself using the above equation that the units of permeability are m2.
More about Darcy’s Law:
Return to Slide 7

24. Buoyancy Force More about 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:
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:
where m is mass.
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:
Return to Slide 11

25. 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:
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:
where h is viscosity and a is the radius of the sphere.
More about drag and Stokes:
Return to Slide 11

26. 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: ) 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

27. 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.
Return to Slide 20