温泉科学（J. Hot Spring Sci.），65，234-245（2016）

原著

A Dynamical Model of a Geyser Induced by Gas Inflow
Hiroyuki Kagami1）(Received Dec. 30, 2015, Accepted Feb. 16, 2016)

間欠泡沸泉の動力学モデル
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要

行1)
旨

室内モデル実験の詳細な観察に基づいた，間欠泡沸泉の動力学モデルを構築した．この動力学モデルの数値シミュレーションにより，噴出動力学の様々な地下パラメタ（地下空洞体積，
水柱の長さ（高さ）など）依存性を明らかにした．また，噴出管壁と水の間の摩擦を考慮した動力学モデルに改良した．この改良された動力学モデルの数値シミュレーションにより，噴出管壁と水の間の摩擦によって噴出高さが減衰することが明らかとなった．
キーワード：間欠泡沸泉，動力学モデル，数値シミュレーション，摩擦の効果

Abstract
I constructed a basic dynamical model of a geyser induced by gas inflow throughdetailed observation of the indoor model experiments. And I clarified dependence of
spouting dynamics on various underground parameters (volume of the underground space,length (height) of a water pole and so on) through numerical simulation of the dynamical
model. And I improved the dynamical model, that is, I took effects of friction between thewalls of the spouting pipe and water into account. And I clarified how spouting height wasdamped by friction between the walls of the spouting pipe and water through numericalsimulation of the improved dynamical model.
Key words : geyser induced by gas inflow, dynamical model, numerical simulation, effects offriction

1)

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School of Medicine, Fujita Health University, 1-98 Dengakugakubo, Kutsukake-cho, Totoake, Aichi
470-1192, Japan. 1）藤田保健衛生大学医学部 〒470-1192 愛知県豊明市沓掛町田楽ヶ窪 1-98．
*Corresponding author：Hiroyuki Kagami, E-mail : kagami@fujita-hu.ac.jp, TEL 0562-93-2578, FAX
0562-93-4593

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1.

Introduction

A geyser is defined as a natural spring that sends hot water and steam intermittently into
the air from a hole in the ground. Geysers are classified into two types dependent on inducer.
Namely, one is a geyser induced by boiling and the other is a geyser induced by gas inflow. A
geyser induced by gas inflow spouts due to pressure of underground gas at the temperature
under the boiling point of water. And only a few studies about its mechanism have been
proposed. Iwasaki (1944, 1962) constructed experimentally some geyser models of cold waters
and gases with cavities and estimated spouting time and pause time using the gas supply rate
as a parameter based on the simple calculation of gas balance. But the discussion is too simple
to estimate the spouting or pause time dependent on various underground parameters and does
not discuss spouting dynamics of a geyser induced by gas inflow.
So I derived a static (mathematical) model and a dynamical model of a geyser induced by
gas inflow based on detailed observation of the indoor model experiments and have modified the
dynamical model in diverse ways (Kagami, 2006 ; Kagami, 2007 ; Kagami, 2009 ; Kagami, 2010a ;Kagami, 2010b ; Kagami, 2010c ; Kagami, 2011 ; Kagami, 2012 ; Kagami, 2015a ; Kagami, 2015b).
But I have not discuss results of numerical simulation of the dynamical model in detail in any
papers yet, though the expansion and application of the dynamical model were reported in many
papers as mentioned above. So in this paper we discuss the results of numerical simulation of
the dynamical model in detail for the sake of understanding characteristics of the dynamical
model.

2.
2. 1

Model

A basic dynamical model of a geyser induced by gas inﬂow
From the results of the indoor model experiments of the geyser induced by gas inflow, I

understood that a beginning of spouting is made by the loss of surface tension supporting alump of water packed in a pipe leading to a spouting exit. Namely, in the model experiments
the underground situation shown in Fig. 1 is assumed. A spouting hole is deep and leads to a
space where gas and water are supplied at a constant rate at the deep position under the
ground. Before a beginning of spouting pressure of gas in the space is supported by surface
tension on the lower interface between water and gas (and gravity acting on the mass of a lumpof water packed in the hole (pipe) and the pressure of the atmosphere). But when a value ofpressure of gas in the space becomes larger than a threshold, the surface tension comes not tobe able to support pressure of gas in the space. Then a lump of water packed in a pipe leading
to a spouting exit begins to move up to the exit on the ground. In a basic dynamical model of ageyser induced by the inflow of gas, the dynamics of a lump of water packed in the pipe isdiscussed.
When the pressure of gas in the space just before a lump of water’s beginning to move up
to the exit on the ground is put as pi, pi is represented as ;


(1)
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Fig. 1

An illustration of a geyser induced by inflow of gas.

where p0 represents the pressure of the atmosphere, ρ represents density of water, g represents
gravity acceleration, H represents length of a lump of water packed in the pipe from the lower
interface between water and gas to the upper one and fk represents pressure due to surface
tension on the lower interface between water and gas. And fk is represented as ;(2)

where γ represents a coefficient of surface tension, α represents contact angle and S represents
an area of a cross section of the pipe filling a lump of water.
When a lump of water packed in the hole begins to move up, fk is regarded as fk → 0. Then
when an upper direction of a vertical line is regarded as a plus direction of x-axis, an equation of
motion of the lump of water is written as ;
(3)where p represents the pressure of gas in the underground space. Here, x is regarded as aposition of the lower interface between water and gas of the water pole and friction between thewalls of the pipe and water is ignored.
When it is assumed that gas in the underground space is ideal gas and changes
isothermally,


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where V represents volume of gas filled in the underground space and n represents molar
number of it is realized.
From equation (4),(5)

is derived.

When it is assumed that x＝0 and V＝V0 just before the lump of water begins to move up,
we can write V as ;(6)

From equation (6),(7)

is derived.
From the assumption that gas is supplied at a constant rate in the underground space,(8)where β is constant is derived. From equation (8), n can be represented as;(9)where n0 represents molar number when p＝pi and V＝V0. On this account we can write using
equation (1) as ;(10)Applying equation (6) - (9) to equation (5),(11)

is derived. And from equation (3),(12)

is derived. From equation (11) and (12) we can get(13)x, that is, a position of the lower interface between water and gas of the water pole moves
obeying equation (13).
In general, the effects of surface tension are smaller with decreasing space size. Therefore,
in this model, it is assumed that the actual gate connecting the spouting pipe and theunderground space is enough small and doesn’t resemble the expansion of the shape shown in
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Fig. 1. It is thought that large volume of the underground space consists of the sum of small
volume of the underground small caves which are connected each other by a network. This
assumption is indirectly supported through video observations inside the conduits of erupting
geysers by Belousov et al. (2014).
2. 2 An improved dynamical model in which effects of friction working between the wall＇s
surface along the edge of the paths of water and water are taken into accountIn the improved dynamical model, I take the effects of friction between the walls of the
spouting pipe and water into account. The concrete methods of introduction of the effects offriction are explained in the following.
Taking effects of friction between the walls of the spouting pipe and water into accountmeans that we regard a water pole packed in the spouting pipe as viscous fluid. If we formallyassume the water pole as viscous fluid, we have to consider friction between water and water inthe water pole. That means considering dynamics of viscous fluid and a subject will becomemuch complicated.
So I consider pseudo-friction effects in which only friction between the walls of the spouting
pipe and water is taken into account. Actually, because viscous fluid flowing in a circular pipe
obeys Poiseuilleʼs law which argues friction between the walls of the spouting pipe and water islargest in the circular pipe, the pseudo-friction effects are not beside the mark very much.
In the beginning, in numerical experiments solving an equation of motion of the water poleclarify its velocity at a moment. From the velocity and an area of cross section of the pipe S,
flux of fluid Vf is derived.
On the other hand, from Poiseuilleʼs law distribution of velocity u, of fluid is represented as :(14)

where a represents a radius of the spouting pipe, r represents length from the center of a crosssection of the spouting pipe to the direction perpendicular to the wall of the pipe and B iscoefficient. Calculating flux VB of fluid from equation (14),(15)

is derived. From equation (15),(16)

is derived.
While, from equation (14),
is derived. Therefore we can get

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
On the other hand, inner friction force f is written as :

(19)



where η represents viscosity coefficient and A represents an area where water keeps in touchwith a wall of a pipe or water.
From
(20)



and equation (16), (18) and (19), we can write friction force fw between the wall of the pipe andwater as :(21)

Furthermore, a direction of friction force fw is opposite to that of velocity of the water pole.

Then a term of friction force fw is added to an equation of motion of the lump of the water(equation (3)). Hereafter the same discussion as that in the basic dynamical model is developed.
Concretely, we can write(22)


Noticing a sign of fw, from equation (3), (21) and (22) we get(23)From equation (23), we get(24)Finally, from equation (11) and (24) we get(25)Second term of equation (25) is a newly added term that represents effects of friction between
the walls of the pipe and water.

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3.
3. 1

Results and Discussion

Results of numerical simulation of the basic dynamical model and discussion
I show some results of numerical simulation of the basic dynamical model of a geyser

induced by gas inflow as follows.
In the beginning, I show the dependence of variation of height of a water pole (x) on length(height) of a water pole (H ) during spouting in Fig. 2. Adopted values of parameters are shownin Table 1. The values are decided based on expected values. fk is calculated using equation (2)in case of α＝30° and S＝1 [m2]. In this case, S represents not an area of a cross section of thepipe filling a lump of water in the experimental system but an expected area of a cross sectionof a pipe connected just to the real underground space. On the other hand, the value of S in
Table 1 represents an expected value (close to the observed value) of a cross section of the
spouting pipe. And the value of V0 represents the
expected sum of small volume of the undergroundsmall caves which are connected each other by a

Table 1 Adopted values of parametersin numerical simulations.

network, as described above. And the value of β isestimated based on the total volume of spouted
water during a spouting mode at a real geyser
induced by the inflow of gas. While spouted water
is removed from the water pole during spouting, thewater is not removed from the water pole before
spouting. From Fig. 2, we see that the higher awater pole (H ) is, the smaller an amplitude of thewater pole’s oscillation is and the longer a spouting
period (a period of the water pole’s oscillation) is.

Fig. 2 Dependence of variation of height of a water pole (x) on length (height)of a water pole (H ) during spouting.

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Then I show the dependence of variation of height of a water pole (x) on length (height) of awater pole (H ) before spouting in Fig. 3. Adopted values of parameters are the same as ones
sown in Table 1. The characteristics seen from Fig. 3 resemble ones seen from Fig. 2. But thecharacteristics are a little different from ones seen from Fig. 2 because there is a loss of a waterpole due to water’s spouting in case of the former.
The difference is shown in Fig. 4. h in a legend of Fig. 3. 4 means a length (height) between
a spouting exit and the upper surface of a water pole at the beginning. Though in case of h＝30
the spouting has not started yet in the figure, in case of h＝30 the spouting has already started.
And the length (height) of water poles (H ) before spouting is same in both cases. From Fig. 4,

Fig. 3 Dependence of variation of height of a water pole (x) on length (height)of a water pole (H ) before spouting.

Fig. 4 Difference of variation of height of a water pole (x) between before
spouting and during spouting.

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we see that spouting period becomes shorter after spouting began. For substantial length(height) of water poles (H ) becomes shorter after spouting begins.
Next, I show the dependence of variation of height of a water pole (x) on pressure due tosurface tension on the lower interface between water and gas ( fk) in Fig. 5. Adopted values ofparameters are the same as ones shown in Table 1 except for the value of fk and H＝100 [m].
From Fig. 5, We see that the larger pressure due to surface tension on the lower interface
between water and gas ( fk) is, the larger an amplitude of the water pole’s oscillation is. For fk
represents strength to push up a water pole. On the other hand, spouting period does notdepend on fk because fk has an effect on only strength pushing up a water pole.
Next, I show the dependence of variation of height of a water pole (x) on volume ofunderground space (V0) in Fig. 6. Adopted values of parameters are the same as ones sown in
Table 1 except for the value of V0 and H＝100 [m]. Incidentally it may be thought that fk ispressure due to not only above-mentioned surface tension but also other power. From Fig. 6, we
see that the larger volume of underground space (V0) is, the larger an amplitude of the waterpole’s oscillation is and the longer a spouting period is. Namely, the volume of undergroundspace (V0) affects both an amplitude of the water pole’s oscillation and a spouting period greatly.

Fig. 5 Dependence of variation of height of a water pole (x) on
pressure due to surface tension on the lower interface
between water and gas (fk).

Fig. 6 Dependence of variation of height of a water pole (x) on
volume of underground space (V0 [m3]).

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間欠泡沸泉の動力学モデル

Results of numerical simulation of the improved dynamical model
I show the results of numerical simulation of the improved dynamical model of a geyser

induced by the inflow of gas as follows.
The difference of variation of height of a water pole (x) from start to 8000 second after
between in case of friction’s existing (this improved model) and in case of no friction is shown in
Fig. 7. From Fig. 7, we see that if there is friction, the amplitude of the water pole’s oscillation
becomes smaller as time passes. Then the same graph from 8000 second after to 16000 second
after from the start is shown in Fig. 8. We cannot see downward movement of a water poleFig. 7 Difference of variation of height of a water pole (x) from start to 8000
second after between in case of friction’s existing and in case of no friction.

Fig. 8 Difference of variation of height of a water pole (x) from 8000 second
after to 1600 second after since start between in case of friction’s
existing and in case of no friction.

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almost after 9000 second after from the start if there is friction. This variation of height of awater pole resembles spouting dynamics of a geyser that involves no water pole’s oscillation.
3. 3Application of the dynamical model to a real geyser induced by gas inﬂow
We can estimate the values of underground parameters through comparing spouting

dynamics of real geyser induced by gas inflow with that of numerical simulation of the
dynamical model. Concretely we select values of underground parameters as spouting dynamics
of numerical simulation of the dynamical model fits that of real geyser induced by gas inflow,that is, as a spouting period, amplitude of a water pole’s oscillation and so on of the numericalsimulation fit those of real geyser when we do numerical simulation of the dynamical model. It
is thought that the selected values of underground parameters are close to those of the real
geyser induced by gas inflow. In fact, values of underground parameters of a few real geysers
were already estimated due to this method by me (Kagami, 2006 ; Kagami, 2012a).

4.

Conclusion

I constructed a basic dynamical model of a geyser induced by gas inflow through detailed
observation of the indoor model experiments. And I clarified dependence of spouting dynamics
on various underground parameters (volume of the underground space, length (height) of awater pole and so on) through numerical simulation of the dynamical model.
And I improved the dynamical model, that is, I took effects of friction between the walls of
the spouting pipe and water into account. And I clarified how spouting height was damped by
friction between the walls of the spouting pipe and water through numerical simulation of the
improved dynamical model.
Acknowledgments
The author would like to express my gratitude to Professor Hidenori Yoshida of Kagawa
University, who checked above equations and gave a lot of valuable comments, suggestions andcaring advice. The author would also like to express my gratitude to Professor Naoki Iwamoto,
Professor Tetsuo Hattori and Professor Nobuyuki Sasaki of Kagawa University, who gave a lotof valuable comments, suggestions and caring advice.
References
Belousov, A., M. Belousova, A. Nechayev (2013) : Video observations inside conduits of erupting
geysers in Kamchatka, Russia, and their geological framework: Implications for the geyser
mechanism. Geology, 41, 387-390.
Iwasaki, I. (1944) : Geochemical study of geyser. J. Chem. Soc. Japan, 65, 640-644 (in Japanese).
Iwasaki, I. (1962) : Geochemical investigation of Geysers in Japan. Bull. Tokyo Institute ofTechnology, 46, 1-60.
Kagami, H. (2006) : Explanations of spouting dynamics of a geyser (periodic bubbling spring) andestimation of parameters under it based on a combined model combining the mathematicalmodel (a static model) and the improved dynamical model of one. Advances in Geosciences,244

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4, 191-197.
Kagami, H. (2007) : Verification of the combined model of a geyser (periodic bubbling spring) by
underground investigation of Kibedani geyser. Advances in Geosciences, 6, 203-213.
Kagami, H. (2009) : A modified combined model of a geyser induced by inflow of gas : considering
evaporation effect of gas dissolved in hot spring water. Advances in Geosciences, 11, 37-43.
Kagami, H. (2010a) : An extended dynamic model of a geyser induced by an inflow of gas (2) :effects of various shapes and repeated expansions and contractions in an undergroundwatercourse. Data Science Journal, 9, IGY110-IGY120.
Kagami, H. (2010b) : An extended dynamical model of a geyser induced by inflow of gas :considering effects of a complicated underground watercourse. Advances in Geosciences, 17,103-111.
Kagami, H. (2010c) : An extended dynamical model of a geyser induced by inflow of gas in caseof plural underground gas supply sources. Advances in Geosciences, 17, 113-121.
Kagami, H. (2011) : An extended dynamical model of a geyser induced by inflow of gas in case ofplural underground gas supply sources (2). Mathematical Modelling in Civil Engineering, 7,
NO. 2 BIS, 29-34.
Kagami, H. (2012a) : Three-source dynamical model for estimating parameters for irregularly
spouting geysers induced by gas inflow. Advances in Geosciences, 29, 145-156.
Kagami, H. (2012b) : A dynamical model of a geyser induced by boiling : an application of the
dynamical model of a geyser induced by inflow of gas. “Water resources and wetlands”
Conference proceedings, Tulcea, Romania, 14-16 September 2012, 55-58.
Kagami, H. (2015a) : A static model of a geyser induced by gas inflow. Proceedings of 2015
Second International Conference on Mathematics and Computers in Sciences and in
Industry, Sliema, Malta, 17-19 August 2015.
Kagami, H. (2015b) : Protection of geysers as tourism and environmental resources : Applicationsof the dynamical model of a geyser induced by gas inflow. Proceedings of World Engineering
Conference and Convention 2015, Kyoto, Japan, 29 November-2 December 2015.
A part of this paper, namely, derivation of a basic dynamical model of a geyser induced by
gas inflow and an improved dynamical model in which effects of friction working between thewall’s surface along the edge of the paths of water and water are taken into account was alreadypublished in “Proceedings of The 38th Conference of Sciete Internationale des Techniques
Hydrothermales and The 56th Annual Meeting of the Balneological Society of Japan” (2003) and“Data Science Journal” (2010), though results of numerical simulations of the basic and improved
dynamical model were not published in the proceedings and the journal. And I summarized apart of my doctoral thesis on this paper after making the revision.
1. Kagami, H. (2003) : Proceedings of The 38th Conference of Sciete Internationale desTechniques Hydrothermales and The 56th Annual Meeting of the Balneological Society ofJapan, 55-60.
2. Kagami, H. (2010) : Data Science Journal, 9, IGY110-IGY120.
3. Kagami, H. (2015) : Doctoral thesis, Kagawa University, HakuOtsu-No. 5（博乙第 5 号）．

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