And Open the Mouth Of Hell;

Estimation of the Duration of a Super Volcano Eruption by Simulation of Fluid Flow

This paper estimates the duration of eruption of a super volcano based on the depletion of a magma reservoir from 15 km depth. The reservoir is modeled as a perfect solution; depletion is through a single vent, subject to erosion. The model is run at 3 rates of erosion; 0.1, 1.0 and 10% of eruptive flow, and two initial diameters, 10 & 132 m dia. A 14º tube, 50 Km long is also modeled @ 132 m Dia

The eruption is found to be extremely dependent of erosion rate, and somewhat dependent on initial vent diameter. The angled vent has eruption durations similar to the vertical, but with significantly reduced environmental impacts. With a 10% erosion rate the period is 3 months, 1% is 6, and 0.2% is just short of 17 years.

This paper is the result of a discussion on the Sime~Gen mailing list during the summer of 2004. The question was asked: How could an event that lasted moments affect the whole earth? The counter argument is that the event takes months.

The issue is how long will the volcano continue to erupt? Viewing recent TV specials one might conclude that it would be over in a matter of moments. This did not seem reasonable to the author.

It occurred to the author that with a number of assumptions it would be possible to estimate the time it would take to drain a given volume of magma.

Reference 1 gives the dimensions of the magma chamber as 50 x 25 x5 mi, or roughly 80 x 40 x 8 km. It is located some 15 km below the unbroken surface of the earth. In the case of Yellowstone, this would be reduced by 500 – 1000 m due to previous eruptions.

The calculation of the volume of the chamber was done using the equation of a 3 dimensional ellipse

1 = X² + Y² + Z²

A² B² C²

The values of the ½ height of ¼ of the chamber were calculated on a spreadsheet and summed. The result is the "class" volume is 14,404.47 = 8 x 1,800 (Km)³

The resulting eruption will dump 13 000 – 22 000 (km)³ of volcanic ash onto the surface (depending on bulking factor, taken as .52 - .30)

This paper assumes that NO additional magma enters the chamber during the eruption. This was NOT the case at Thera 3500 years ago (it lasted 40 years), and with some results running to 16 years this is not a good assumption

This is a quick and easy calculation to establish a volume. The correct method, entailing the drafting of the relative thickness of both the magma chamber and the overburden was explored, however time constraints prevented its use, and it is believed that the resulting error is less than the uncertainty from other sources (Erosion rate in particular).

The correct method would develop 2 sinh (hyperbolic) curves of the correct distance apart and the appropriate diameter, and stretch the overburden over them as well. The second function would be to ‘melt’ sufficient overburden in the center of the magma chamber to permit structural collapse at some point.

In truth there is no pressure caused by the volcano. All of the pressure is the result of the weigh of the overlying rock on the liquid melt. Therefore, it is a straightforward calculation to determine the pressure on the top of the magma chamber.

P= ρ*g*h = 2 700*9.82*15000 = 3 977 Bar

=58 000 psi

The author has assumed a slightly low bulk density of crustal rock of 2700, rather than a more typical 3 300 due to the potential for previous layers of volcanic tuft, rather than more solid volcanic crust.

It is further assumed that normal fluid pressure laws apply in liquid magma.

I have assumed that the atmospheric pressure at the discharge location to be 1 bar, which is sufficiently accurate for the purposes of this model.

There is appreciable discussion of the real make up of super volcano magma. The author is not entirely convinced they truly are "Hot Spot" volcanoes on land (this will not stop him from using this theory in his fiction). Clearly, the majority of these volcanoes are associated with contental crust. Most of them are close to major subduction zones.

The author notes that only the Sumatran Super volcanoes are at all close to the subduction faults. More typically, these volcanoes are a substantial distance from the edge of the continental plate.

It is therefore thought probable that real super volcano magma is deeply subducted oceanic crust that has finally melted and is returning to the surface, complete with gods own mixture of ingredients.

The important part about the mix is that its liquidity is caused by the high gas content. The second point to remember is that the volatiles is that they are liquid only under the extreme pressure.

The volatiles are H₂O, H₂S, SO₂, CO₂ and others. The bulk of the non-silicate materials are iron, but there will also be traces of most other mantle metals as this material if not from the mantel certainly has transited a fraction of it.

In order to simulate its behavior more conveniently it is modeled as a 3 component perfect solution.

A perfect fluid is one in which the volume of the solution is the same as the sum of the volumes of the various components.

The three components in the simulated magma, and their mass fractions are; water (10%), Iron (10%) and Silica (80%).

The resulting liquid magma has a bulk density of just under 1640 kg/m³. As a result, the residual pressure at the surface (from 15 km) is 1575 bar (22,825 psi) for comparison scuba tanks are filled to 200 bar.

The eruptive fluid, which is mostly steam by volume, is taken as having a Ratio of specific Heats of 1.3 (the same as water).

In the real eruption, the magma first rises up through cracks in the overlying structure until the available residual pressure x area equals the local rock strength by its remaining area. At this point, the eruption begins with a bang, as the remaining rock is blown out. This results in an appreciable volume of magma undergoing violent de-compression, releasing its volatiles, and shredding the still liquid Silica and Iron solution into tiny fragments.

The working model ignores any transients at startup.

It is abundantly clear that there will be a lot of erosion of the tube as this hot enough to melt the rock; sand paper rushes through it at near sonic speed, but how to model this erosion? The author has no ready solution to estimating the actual erosion rate, however it seems unlikely that the amount of material eroded (once the initial gains of the not quite successful co-risers are absorbed) will be a small fraction of the actual fluid carrying away the eroded rock.

The author therefore set up calculations with 10%, 1% and 0.1% on a mass over mass basis of the flow leaving the magma chamber.

It was then a mater of calculating the mass of rock to be eroded before the next "flow Area Step" and allowing the appropriate multiple of that mass to flow by.

The Flow Area Steps were chosen as approximately 2% of the previous in a fixed repetition of units between 1 and 1000.

In truth, 50% of the material leaves the chamber in the last 40% of the eruption duration.

The principal tool for the development of this paper is "Flow of Fluids through Valves Fittings And Pipe – SI version" by Crane Valves Inc. and M$ Excel spread sheet.

The MKS system is used exclusively in actual calculations.

The last variant of Eq. 3-19 in Crane is:

_________

The D’Arcy formula is used to calculate the fluid friction head in a moving fluid. It takes the form:

h_L = f (L/d) v²

2g

= head loss in m of fluid.

f = Moody Friction Factor

L = Length of Pipe / Tube

v = Velocity of Fluid

g = 9.82 m/sec²

Where K for straight pipe is f*(L/d) Given that the tube will grow over time the value of K from L/d will change.

For other obstacles are listed in Crane section A. In this application, there are 2,

K_Entrance = 0.5

K_Exit = 1.0

Total = 1.5 for "Fittings".

It may be noted that the model does not allow for changes in cross section, or direction of the fluid flow.

The Moody Diagram, Plate 2 hard copy ⁱⁱ only, shows the relationship between f and the Reynolds Number (Re), and the relative roughness of the tube = ε/d. An electronic version can be found Ref.

Reynolds number relates viscosity forces to gravitational forces and is calculated as:

Re = dvρ

μ

The value of ε is the root mean squared roughness of the tube, measured in mm, and taken as 100 mm (4”) which is representative of its condition in the ongoing mode.

The net expansion factor is a linear function of K and ∆p/p’₁, and γ the compressibility factor. Crane gives a table listing the Limiting Factors for Sonic Velocity at γ = 1.3 (and a list of substances) on Page A-22.

In summary they very from 0.718 when K>100 to 0.658 when K= 3 (and structural collapse is immanent). For values in between linear interpolation is used.

The spreadsheet file was set up for an initial study at 10 m dia. and 1% erosion rate. The first sheets were devoted to developing the necessary characteristic numbers.

At the point where additional modeling was begun, it was found much easier to start a new file and adjust the calculation rather than repeat it on a new sheet.

Where calculating from a larger base diameter the unnecessary rows were hidden.

In part due to the complexity of the calculation, and in part due to past experience in doing this type of calculation in this type of program each step of the calculation of the flow at a particular time was given its own column.

The first steps calculated fixed values preparatory to the flow calculation and these occupied the upper rows. From there the Reynolds Number and Relative roughness (ε/d) were calculated.

An examination of the Moody Diagram shows that for values of Re>>10⁷ you are at the end of the graph and fluid friction is entirely a function of relative roughness (ε/d).

A good first approximation to this is:

f = f(ε/d₁ ) * Ln (ε/d₁)

Ln (ε/d_step)

The author then when through and corrected the value of f at the fixed values from the end of the Moody Diagram.

The result is a reasonably close approximation of f, and thus of K (= f*L/d) throughout the system of time steps.

The author notes that in the multipule runs he can not be absolutely sure all such corrections were installed at the correct locations.

As is typical of the computation business it proved easier to run a new file for each condition.

The lower erosion rates required the deletion of "area Steps" while the higher rates required the addition of time steps.

The "tilt" run is a look at the major mitigation strategy, having the exit at a low angle relative to the ground. The duration is similar to the vertical runs with the same assumed erosion rate.

5.4. End State

The results summarized in Table 1 below. There are two significant results for each run, the duration of the eruption in days, converted to years for clarity, and the final diameter of the tube.

Once the eruption is over there will be a period of several years of poor crops, and wide spread famine, assuming there are survivors left to starve.

BELOW - You may click to the file from this table or the references at the back except for the 1.44 erosion rate runs which are not included

6.1. The Good, The Bad, and The Ugly

It appears that the erosion rate has been bracketed, and the author does not recommend conducting the experiment unless absolutely necessary.

The results at 1% and those forced to 5½ months (165 days) suggest that the year of the volcano will be that, and that there is a reasonable outside chance that civilization will survive, particularly if it has several years in which to plan. Joseph’s time in Egypt makes a good model for this.

Unlike a boloid, this event does not impose saver worldwide thermal shocking and indeterminate overpressures.

It is clear from the results that anything like a 10% erosion rate will result in almost immediate structural collapse (the 47 Km dia. tube is larger than the reservoir and so over states the duration.

7.1. Threat Count

Plate 3 mapsupereruptcopy shows a map of the world with 18 known super volcanoes, and 2 Traps volcanoes (Traps Volcanoes are NOT part of this study. Recent programming [2007 April 12 – no reference] suggest they are anti-pole to major Asteroid Hits).

The Smithsonian Institution’s Global Volcanism Program lists 1500 volcanoes world wide, of which 84 are marked as Calderas.

7.2. The New Map

The source of Plate 3 (in the hard copy of this paper only) is unknown. After an unsuccessful search of the internet for it, the author turned to his own resources to deploy an alternative.

The Smithsonian institution’s list of volcanoes was taken and sorted for Calderas. A world map was found on the internet and used as the basis for an AutoCad drawing. The world map used had grid lines at 10 degrees of latitude & longitude (but they were not uniformly space on the jpeg used). The lines were re-drawn in AutoCad and the continental and island outlines splined in.

Two phenomenon became apparent in the process of producing this image:

The following noteworthy clusters can be seen;

To show a cluster the diameter of the circle was increased, or if appropriate, a single ellipse was drawn around the multiple circles (Italy, Sumatra).

Due to its historical importance Krakatoa / Krakatau was moved southeast of its calculated position. The author also notes that the Alaskan Calderas appear to be at sea, but have landside names (probably due to poor base map).

The author has grave doubt about the explosive nature of the volcanoes sited in Iceland and Ethiopia, they are on spreading centers, not the edges of continents subducting ocean bottom (which is an excellent source of water & organic material).

Two additional volcanoes are shown on the map, Tambora (1816) and Santorini or Thera, which erupted about 1650 BC, and one of them, is the largest eruption in recorded history. They are shown as Yellow triangles.

7.3. Toba, Tombora, and Krakatoa

The last Super Volcano Eruption was Lake Toba, Sumatra 72 000 years ago. Recent reports indicate that the human population prior to this eruption could have been as high as 500 M. There is also evidence of casualties of this eruption in Sumatra.

DNA studies have looked at this event and it appears that there were remarkably few survivors, perhaps less than 100 000 total.

The debate about whether this was or was not a Super Volcano eruption will continue for a while. However, BBC Horizons has identified this eruption as one of the great crisis points in history.

They indicate that in addition to the famine that can be expected as the result of a VEI 8 plus eruption major outbreaks of the Black Death (from Ethiopia into the Easter Roman Empire & Britain) occurred.

The documentation on this eruption is sparse, but both Indonesian and Chinese records show it occurred in Feb. 535, and was seen for a very large distance. Further, there is significant evidence that Sumatra and Java was one island until that time.

The Selat Sunda (or Sunda Straight) was formed at this time, with an area some 100 km in diameter settling about 100 m.

This mountain in the Flores Sea (Indonesia again) erupted in the spring of 1816 ‘and froze to death’. It caused massive famine, but there is no report of major plague at the time – possibly due to improved public health.

This mountain is only now coming under study, and has an impressive caldera on top of its cone (some 4-9 miles around and 1 mile deep).

7.4. The Threat

A full-scale super volcano eruption will cause the full three horrors, cold, dark, and toxic dust worldwide. It is questionable whither our civilization will survive.

The key to survival will be getting enough food, and what little bit of extra kit we need together before the eruption starts.

Work on the 535 eruption of Karackatoa by the PBS suggests that in addition to the problems outlined above some endemic diseases (Bubonic Plague) break out of their heat-induced traps.

There are two potential mitigation measures. Both somewhat dependent on how sure we are that there IS going to be a super volcano eruption, and both involve significant "Nuclear Engineering".

8.1. And Open The Mouth Of Hell

How it is to be decided that a magma reservoir is about to erupt is unknown. The political difficulties in setting this up are horrendous – about as bad as the consequences.

Whatever the method it becomes clear to a dominant power that somewhere in the world a super volcano is going to erupt in the next year. IF that volcano is on their or a really close allies territory they can drill down in to the devil’s brew. If not they wait until their harvest is in before using a penetrator.

Modern penetrating rounds are capable of penetrating significant depths from aircraft launches. This penetrator would be orbital.

The result will be that the eruption starts "on time", and the upper part of the tube will not be the restriction. It may be anticipated that if a reasonable picture of the underground structure has been obtained then the last bit bill be the area of maximum probability for a restriction.

A second penetrator might be possible to complete the job, if needed.

8.2. The Tilt Option

This option is only possible when there is a lot of time to do the preparatory work. There are two problems, the line must be drilled, and the drilling will be through ground that is undergoing intense tectonic activity.

It also has the potential to seriously alter the spin of the earth as the Richter 9 Earth quake of 2004 Dec 26 did, depending on the bearing (direction) of the "tilt" tube, but this detail has not been calculated.

It will make a spectacular mess of the area of discharge. This site should be chosen for its fantastic drainage. Given the length of the eruption, it may be expected that a permanent "Low Pressure" will form drawing a continuous storm of rain to the area of the vent.

8.3. The Nuclear Waste

Given the significant toxicity of the volcanic ash in the first place, and the requirement for a minimal number of things to go wrong the devices will be largely fusion to start with, i.e. "Clean" bombs. However, this is a circumstance where few if any of the choices are "desirable".

Also given the immense volumes of dust that the early, contaminated, material will be mixed with – will not matter, because if you breathe it in the sharp edges will kill you anyway.

8.4. The Water Option

The fluid flow model is elementary, and it really should be reviewed to be sure the simple application of theory has no major errors in it.

Some of the inputs of moody friction factor need to be upgraded, and the use of a juried formula for its calculation would be a nice addition.

There is no modeling of the melting of the parent rock around the magma chamber, nor the structural collapse of the roof at a reasonable percentage of discharge.

This becomes important in a dry jet case where it is the entrained air that then rises by convection.

No consideration either in this paper nor in the popular literature has considered what a ferocious source of heat like this one will do to the regional, and for that matter global weather pattern (on a purely heat injection basis).

The assumption that NO additional magma enters the chamber during the eruption must be adjusted, and a flow rate evaluated.

Current reports in the popular literature / on the web, suggest that only 50% of the magma is liquid – at pressure. What this does in terms of the volume to be expelled, will take more geochemistry & physics than the author has at his disposal.

Karen MacLeod, a professional editor, for doing a quick edit on it. She may be reached through the Simegen.com domain.

This paper also skirts the background of a novel of Skinner’s World titled "And Open The Mouth Of Hell" in development by the author.