DOI: 10.3303/CET2290077 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 23 January 2022; Revised: 14 March 2022; Accepted: 7 May 2022 
Please cite this article as: Palazzi E., Currò F., Vairo T., Fabiano B., 2022, Critical Issues in Short-cut Modelling of Massive Carbon Dioxide 
Releases, Chemical Engineering Transactions, 90, 457-462  DOI:10.3303/CET2290077 
  

 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 90, 2022 

A publication of 

 
The Italian Association 

of Chemical Engineering 
Online at www.cetjournal.it 

Guest Editors: Aleš Bernatík, Bruno Fabiano 
Copyright © 2022, AIDIC Servizi S.r.l. 
ISBN 978-88-95608-88-4; ISSN 2283-9216 

Critical Issues in Short-cut Modelling of Massive Carbon 
Dioxide Releases 

Emilio Palazzi, Fabio Currò, Tomaso Vairo, Bruno Fabiano* 
DICCA – Civil, Chemical and Environmental Engineering Department - University of Genoa, Via Opera Pia, 15 – 16145 
Genoa, Italy 
brown@unige.it 

In case of accidental gaseous releases, the detailed analysis of rather complicated situations (partial 
confinement, irregular shapes, unsteady-state) usually requires the use of sophisticated integral models and/or 
CFD calculations, but when conservative results are enough, especially in preliminary hazard assessment, 
analytical models can be conveniently applied. This paper presents a very simple analytical model accounting 
also for wind effect on accidental release evolution, suitable to be adopted as short-cut evaluation tool in case 
of accidental carbon dioxide semi-continuous release at high speed (jet), and nearly instantaneous releases 
from high-pressure systems. It may allow a preliminary evaluation of emergency actions, in case of a massive 
and deadly release of carbon dioxide from a storage site, or due to natural event, once properly defined the 
source term and refined for possible thermal effects. 

1. Introduction 
CCS is a key reduction strategy to limit CO2 emissions, but its development is strongly limited mostly by high 
costs, social and political acceptability, thus requiring to carry out an exhaustive perspective plan to cover all 
uncertainties linked to it (Viebahn et al., 2015). Examples include the storage scenarios and the need of an 
appropriate analysis of the site and consequence modelling, to prevent or mitigate possible leakages (Hansson 
& Bryngelsson, 2009). The development of a simplified bow-tie approach for CO2 storage may rely on 11 main 
hazardous events (Le Guénan et al., 2011), i.e,: 
• Leakage via an operational well 
• Mechanical disruption nearby the injection well 
• Mechanical disruption at the storage complex scale 
• Expected lateral extent exceeded 
• Leakage due to sealing deficiency of the caprock 
• Leakage via existing faults 
• Leakage via an abandoned well 
• Accumulation in a secondary reservoir 
• Flow modifications 
• Disruption by a later activity 
• Disruption by a natural earthquake. 

However, these methods often require precise probability values to support the evaluation and, in practical 
applications, there are often many knowledge limitations. As stated by Vairo et al., (2021), due to incomplete 
statistics and knowledge, there will be some uncertainties on the likelihood and interdependence of root risk 
events in Fault Tree (FT) and events in Event Tree (ET), which may lead to unrealistic results. Generally 
speaking, the consequences of CO2 leaks can be divided into two groups: global and local. The global ones are 
connected with leakage into the atmosphere, not considering space and time so reducing the effectiveness of 
geological storage and leading to an increase in CO2 concentration into the atmosphere. Local consequences 
can be short or long term and can fall into three categories, namely (i) health and safety, (ii) environmental and 
(iii) equity. For the first two, there are natural and anthropogenic analogues (such as incidents in NG pipelines), 

457



which may help in developing risk assessment related to CO2 geological storage. Mazzoldi et al. (2013) 
evidenced by simulation runs a peculiar feature of CO2 plumes and consequent affected from ruptured high-
pressure pipelines: more than 25% of the CO2 contour at 1 m above the ground is composed of a rather flat 
tongue approximately 20 m wide and extending nearly 100 m from the break. A step forward was proposed by 
Li et al. (2014), who performed experimental runs on leakage from a supercritical CO2 pipeline, observing the 
multiphase jet and the formation of a dry ice bank outside of the nozzle. The leakage flow, after a brief transient 
state, reaches a steady value, while the amount of carbon dioxide, still inside the pipeline keeps decreasing. 

2. Instantaneous release short-cut evaluation 
In developing a short-cut approach, we started from the consideration by Chow et al. (2009) for an instantaneous 
release scenario under 22 different combinations of wind, topography, atmospheric stability and release 
strength, summarized as follows:  
• In wind absence, CO2 spreads laterally over flat terrain and a concentrated CO2 plume tends to hug the 

ground, migrating towards the sites with the lowest elevations. 
• Owing to density effects, CO2 spreads more quickly than a neutrally-buoyant gas so that (although counter-

intuitive) it resulted that ground level concentrations can drop in a shorter time for a dense gas under calm 
conditions. 

• CO2 acts as a dense gas under no, or weak wind conditions, but in wind presence from 2 m·s-1, the behavior 
is affected by transport and diffusion phenomena sweeping away the plume. 

As a result, in some circumstances, with stable atmospheric conditions and low wind velocities, CO2 might pond 
in a populated topographic low near a large CO2 release from a pipeline or a storage site and be lethal at a 
distance considerably larger than the danger zone for natural gas explosions. However, there is no accord to 
the safety distance even from a pipeline of natural gas or carbon dioxide, while the definition of the critical dose 
Dc requires a proper refinement, as detailed in Palazzi et al., 2016a and CO2 toxicity threshold summarized in 
Table 1. The critical concentration yC may be used to identify, by means of an appropriate model of atmospheric 
dispersion, the distance from the release rC, beyond which y<yC and the unwanted event could not theoretically 
happen. On the other hand, this effect can rise just if the duration of the exposition at the concentration yC is not 
lower than the critical value τC. 
The evolution in time of an instantaneous release under no wind conditions, is schematically depicted in Figure 
1; after a rapid initial dilution, which is necessary to complete the sublimation (SV) of the CO2, the amount of 
motion runs out, because of the greater density respect to the air and because of the friction with the ground. A 
dense cloud is formed, subject to shedding by gravity; it dilutes with the air while radially propagates itself. 
Because the mechanism of formation of the cloud is different if compared to the one relative to the 
semicontinuous vertical jet, it is necessary to modify the hypothesis concerning the characteristics of the cloud 
after the initial phase of dispersion, even to take into account the fact that, under certain situations here treated, 
the above-mentioned hypotheses would be too precautionary. 

Table 1: Carbon dioxide levels of toxicity. 

 

Figure 1: Simplified time evolution of an instantaneous release, from V0 to VL.  

Concentration yC Exposure Time τC [s] Effects Source Dose DC [s] 
0.25 60 Death (Mazzoldi et al, 2013) 15 
0.10    600 (1) Death (Mazzoldi et al, 2013) 60 
0.04           1,800 IDLH NIOSH 72 

(1) Precautionary assumption, in the given range: 600 – 750  (Mazzoldi et al., 2013) 

458



Table 2: Transition point from jet phase to cloud diffusion under the hypothesis of Van Ulden, for different 
values of wind velocity.    

u [m·s-1] u* [m·s-1] vr* [m·s-1] mrL [kg] r*=βi·mr1/2·vr*-1 [m] rr [m] 

2 0.132 0.264 
9.5·105 4.3·103 1.7·103 
1.8·109 1.9·106 6.4·104 

10 0.66 1.32 
4.8·107 6.3·103 3.3·103 
6.6·1010 2.3·105 9.6·104 

20 1.32 2.64 
3.1·108 8.0·103 4.9·103 
3.5·1010 2.7·105 1.3·105 

 
The simplified approach here discussed relies on previous simulation results validated by Palazzi et al, (2016a, 
2016b) obtained under no wind conditions, as follows:   

𝑣𝑣𝑟𝑟 =
1
𝑟𝑟 �

𝜌𝜌0 − 𝜌𝜌𝑎𝑎
𝜌𝜌𝑎𝑎

𝑔𝑔
𝜋𝜋
𝑚𝑚𝑟𝑟
𝜌𝜌0𝑤𝑤0

 (1) 

𝑟𝑟2 = 𝑟𝑟02 + 2𝛽𝛽𝑖𝑖𝑚𝑚𝑟𝑟1 2⁄ 𝑡𝑡 ≅ 2𝛽𝛽𝑖𝑖𝑚𝑚𝑟𝑟1 2⁄ 𝑡𝑡 (2) 

𝛽𝛽𝑖𝑖 =
(𝜌𝜌0 − 𝜌𝜌𝑎𝑎)
𝜌𝜌𝑎𝑎𝜌𝜌0

𝑔𝑔
𝜋𝜋𝑤𝑤0

 (3) 

Following the reasoning of van Ulden (1974) the transition from the jet phase to cloud corresponds to the 
attainment of the condition vr = 2u*, i.e., from Eq. (1): 

𝑣𝑣𝑟𝑟 =
𝛽𝛽𝑖𝑖
𝑟𝑟

(𝑚𝑚𝑟𝑟)1 2⁄  (4) 

As shown in Table 2, under the following conditions: T=  273 K, P= 10 MPa, r0=0, βi=1.20, the transition point is 
never attained, being r < r* and vr   *= 2·u*.   
In case of dispersion under no wind conditions, the relevant contribution to turbulent diffusion is due to the term 
vr, while in case of moderate windy conditions it is assumed that after the jet phase transition, the relevant cloud 
dilution effect is connected to the wind velocity u. Under the simplifying assumptions of flat terrain, no 
obstructions, no local concentration fluctuation, no chemical reaction, the unsteady – state behavior of a nearly-
instantaneous carbon dioxide release, near the ground, can be described by Eq(5). 
𝑑𝑑𝑑𝑑
𝑑𝑑𝑡𝑡

= 𝑘𝑘𝜋𝜋𝑟𝑟2𝑢𝑢 (5) 

From Eq (2), one can write: 

𝑑𝑑𝑡𝑡 =
𝑟𝑟

𝛽𝛽𝑖𝑖𝑚𝑚𝑟𝑟1 2⁄
𝑑𝑑𝑟𝑟 (6) 

and 

𝑑𝑑𝑑𝑑 =
𝑘𝑘𝜋𝜋𝑢𝑢

𝛽𝛽𝑖𝑖𝑚𝑚𝑟𝑟1 2⁄
𝑟𝑟3𝑑𝑑𝑟𝑟 (7) 

By integration, one can easily obtain: 

𝑑𝑑 − 𝑑𝑑0 =
1
4

𝑘𝑘𝜋𝜋𝑢𝑢
𝛽𝛽𝑖𝑖𝑚𝑚𝑟𝑟1 2⁄

(𝑟𝑟4 − 𝑟𝑟04) (8) 

For the purposes of the dose calculation, it is fundamental to determine the variations in the carbon dioxide 
concentration, y(r), during the dilution of the release. In a conservative way, the dose calculation is referred to 
the CO2 concentration on the axis of the cloud ya that is assumed according to van Ulden (1974) and original 
wind tunnel experimental runs at the laboratory scale on jet scenarios, corresponding to the double of the mean 
concentration. Being: 

𝑑𝑑 − 𝑑𝑑0 =
𝑚𝑚𝑟𝑟
𝜌𝜌𝑎𝑎
�1
𝑤𝑤
− 1

𝑤𝑤0
�                             (9) 1

𝑤𝑤
− 1

𝑤𝑤0
= 𝑀𝑀𝑎𝑎

𝑀𝑀𝑟𝑟
�1
𝑦𝑦
− 1

𝑦𝑦0
�  (10) 

Taking in account Eq(8) one can obtain: 

𝑦𝑦 = 𝑦𝑦0[1 + 𝜉𝜉2(𝑟𝑟4 − 𝑟𝑟04)]−1 (11) 

being by definition: 

𝜉𝜉2 =
1
4
𝑘𝑘𝜋𝜋𝜌𝜌𝑎𝑎𝑦𝑦0𝑢𝑢
𝛽𝛽𝑖𝑖𝑚𝑚𝑟𝑟3 2⁄

 (12) 

459



From Eq. (11), it is possible calculating the cloud radius as follows: 

𝑟𝑟 = �𝑟𝑟04 +
1
𝜉𝜉2
�

1
𝑦𝑦
−

1
𝑦𝑦0
��
1
4�

 (13) 

On the basis of the previous simplifying assumptions, the dose can be calculated as: 

𝐷𝐷 = � 𝑦𝑦𝑎𝑎(𝑡𝑡)𝑑𝑑𝑡𝑡 =
𝑡𝑡2

𝑡𝑡1
� 2𝑦𝑦(𝑡𝑡)𝑑𝑑𝑡𝑡 = � 2𝑦𝑦(𝑟𝑟)

𝑟𝑟
𝛽𝛽𝑚𝑚𝑟𝑟1 2⁄

𝑑𝑑𝑟𝑟
𝑟𝑟2

𝑟𝑟1

𝑡𝑡2

𝑡𝑡1
 (14) 

And taking into account Eq(5), it follows 

𝐷𝐷 = � 2𝑦𝑦0[1 + 𝜉𝜉2(𝑟𝑟4 − 𝑟𝑟04)]−1
𝑟𝑟

𝛽𝛽𝑚𝑚𝑟𝑟1 2⁄
𝑑𝑑𝑟𝑟

𝑟𝑟2

𝑟𝑟1
 (15) 

After straightforward algebraic manipulations, one can finally calculate: 
 

𝐷𝐷 =
𝑦𝑦0

𝛽𝛽𝑖𝑖𝑚𝑚𝑟𝑟1 2⁄
1

𝜉𝜉�1 − 𝑟𝑟04𝜉𝜉2
�𝑎𝑎𝑟𝑟𝑎𝑎𝑡𝑡𝑔𝑔

𝜉𝜉
�1 − 𝑟𝑟04𝜉𝜉2

�
2𝛽𝛽𝑚𝑚𝑟𝑟1 2⁄

𝑢𝑢
− 𝑟𝑟1�

2

− 𝑎𝑎𝑟𝑟𝑎𝑎𝑡𝑡𝑔𝑔
𝜉𝜉

�1 − 𝑟𝑟04𝜉𝜉2
𝑟𝑟2� (16) 

 
where  r=r(y) is easily calculated by means of Eq. (13). 
The value of r0 is equal to 0, in the case of no wind condition (model a). In the case of wind presence, it can be 
calculated under the simplifying hypothesis that at the end of initial expansion phase, the cloud, can be 
approximated by an equivalent equilateral cylinder (model b). 

𝑟𝑟0 = �
1

2𝜋𝜋𝑤𝑤0𝜌𝜌0
�
1
3�

𝑚𝑚𝑟𝑟
1
3�  (17) 

Following a different simplified approach (model c), r0  is estimated be by imposing that the initial expansion 
phase ends when dilution velocities along the down and transversal wind direction are equal, so that it follows:   

𝑟𝑟0 = �
2𝛽𝛽𝑖𝑖

𝜋𝜋𝑤𝑤0𝜌𝜌0𝑢𝑢
�
1
3�

𝑚𝑚𝑟𝑟
3
8�  (18) 

3. Results and discussion 
We applied the framework making reference to the different working hypotheses previously outlined in Table 2 
and adopting as illustrative examples 3 wind velocities, namely 2, 10, 20 m·s-1, even if the case of high wind 
velocity may be of scarce practical interest. The explored operative conditions are summarized in Table 3, 
selected from operative range commonly adopted in storage and transport activities, derived from literature 
(Webber, 2011), as follows: 10 ≤ pi ≤ 20 [MPa], 273 ≤ Ti ≤ 323 [K].  

Table 3: Explored operative conditions. 

 Run I Run II Run III Run IV 
Ti [K] 273 273 323 323 
pi [MPa] 10 20 10 20 

 
In this regard, because of the high Δp value, the velocity of the fluid in the efflux section is quite high (100 to 
250 m s-1). The physico-chemical properties of CO2 and, in particular, the critical point (pc=7.3825 MPa; 
Tc=304.21 K), and the sublimation point at atmospheric pressure (Ts=194.65 K), imply the possible presence of 
three phases, although not simultaneously. Performed calculations provide only an estimate aiming at 
evaluating the sensitivity of attained results on the different possible calculations of initial cloud dimension, r0.  

Table 4: Comparison between values of critical released mass and critical distance (model a, b, c).  

u [m·s-1] 
Run II Run III Run IV 

mrc [kg] 
rC [m] mrc [kg] 

rC [m] mrc [kg] 
rC [m] 

(a) (b) (c) (a) (b) (c) (a) (b) (c) 
2 1.2·106 895 892 876 3.2·106 538 536 528 9·105 851 848 834 
2 5.0·109 27,684 27,729 27,556 4.1·108 8471 8466 8408 5.0·109 17,043 17,033 16,930 
10 5.6·107 3351 3320 3231 1.9·107 2319 2301 2253 5.6·107 3393 3367 3392 
20 3.4·108 6439 6355 6170 1.3·108 4705 4653 4553 3.4·108 6683 6661 6453 

460



It can be noticed that the maximum difference of the critical distance (defined as the maximum distance of 
release at which the exposition to CO2 can provoke serious effects utilizing the refined dose approach) 
calculated under the three hypotheses is nearly 4%. The most conservative estimate corresponds to the simple 
and drastic approximation r0  = 0, as schematically shown in Table 4. 

Table 5: Comparison between critical distance rC [m] obtained by jet release under no wind condition and the 
instantaneous massive release scenario. 

 Jet scenario u=0 Instantaneous release u =10 m·s-1 Instantaneous release u=20 m·s-1 
Run I 6268 3361 6459 
Run II 5900 3351 6479 
Run III 2394 2319 4705 
Run IV 3873 3393 6683 
 

 

 

 

 

 

 

 

Figure 2: Footprints of the Aloha simulations A and B.  

 

 

 

 

 

 

 

Figure 3: Footprints of the Aloha simulations C and D.  

Table 5 allows an immediate comparison between the most conservative results obtained by a jet release under 
no wind condition (Palazzi 2016a) and the instantaneous massive release scenario.  

Table 6: Aloha simulations conditions adopting as reference different CO2 threshold values. 

 Instantaneous release mrc [kg] Wind speed u [m·s-1] Temperature   Ti [K] Pressure pi [MPa] 
Run A 1.2·106 2 273 20 
Run B 5·109 2 273 20 
Run C 5.6·107 10 273 20 
Run D 5.6·107 10 323 20 

(I) (II) 
(III) 

(IV) 

(A) (B) 

(C) (D) 

461



At last, we explored some situations by means of Aloha integral model, which is developed to plan emergency 
response in case of an actual accident and is therefore rather conservative. The set-up of Aloha is for general 
purpose and it is considered as a short-cut tool; we used it as a preliminary comparison to roughly highlight the 
model capabilities and degree of conservatism. Simulation conditions are summarized in Table 6, while the 
results of different runs are shown in Figures 2 and 3 with calculated threshold distances presented in Table 7. 

Table 7: Threshold distances calculated with Aloha model adopting as different CO2 reference values.  

Aloha run 0.04 (mol/mol) IDLH 0.10 (mol/mol) 0.25 (mol/mol) 
A 1.4 km 1 km 0.7 km 
B 14.8 km 12.3 km 11.1 km 
C 8.6 km 6 km 4.1 km 
D 9.2 km 6.6 km 4.5 km 

4. Conclusions 
The approach relies on strict simplifying assumptions on dispersion behaviour according to criteria of 
representativeness, simplicity, reliability and caution, taking into account, in particular, that carbon dioxide 
accumulation phenomena on the flat soil are excluded, so that the concentration in the release is overvalued. 
Upon proper refinement and validation, also in connection with a sound definition of the critical dose based on 
updated results the framework allows obtaining a single type of general correlation for nearly instantaneous 
massive release. Upon further validation with field data and simulation runs with reliable tools and refinement, 
it may help in a preliminary definition of the critical distance of release from an existing industrial activity or 
natural event, limiting the excessive conservatism of free models such as Aloha.  

Nomenclature

D – dose, s 
h - height of the cloud, m 
Ma – air molecular mass, - 
Mr – release molecular mass, - 
mr – release mass, kg 
r – radius of the cloud, m 
r0 – initial radius of the cloud, m 
rc –critical distance, m 
rL – limit distance of cloud spreading, m·s-1 

u – wind speed, m·s-1 
vr – radius grow velocity, m·s-1 
V – cloud volume, m3 
w – mass fraction of CO2 in the release, - 
w0 – initial mass fraction of CO2 in the release, - 
y – CO2 concentration, - 
y0 – initial CO2 concentration, -  
ya –CO2 concentration in the axis of the cloud, - 
ρ0 – release density, kg·m-3 

References 

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uncertainties and possibilities, Energy Policy, 37(6), 2273-2282.   

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	Critical Issues in Short-cut Modelling of Massive Carbon Dioxide Releases