View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Universidad de Nariño Journal of Volcanology and Geothermal Research 139 (2005) 59–71 www.elsevier.com/locate/jvolgeores A numerical model for the dynamics of pyroclastic flows at Galeras Volcano, Colombia G. Co´rdoba Engineering Faculty, Universidad de Narin˜o, Colombia Accepted 29 June 2004 Abstract This paper presents a two-dimensional model for dilute pyroclastic flow dynamics that uses the compressible Navier–Stokes equation coupled with the Diffusion–Convection equation to take into account sedimentation. The model is applied to one of the slopes of Galeras Volcano to show: (1) the temperature evolution with the time; (2) dynamic pressure change; and (3) particle concentration along the computer domain from the eruption to the impact with a topographic barrier located more than 16 km from the source. Two initial solid volumetric fractions are modeled. For both cases, some of the structures located more distant than 10 km could survive, but in all cases the flow remains deadly. This paper shows that a dynamical model of pyroclastic flows can be implemented using personal computers. D 2004 Elsevier B.V. All rights reserved. Keywords: pyroclastic flow; model; numerical; finite element; hazard 1. Introduction this hazard within its influence area. Investigations by Espinoza (1988) for historical activity of the last Galeras Volcano is regarded as one of the most 500 years at Galeras Volcano from descriptions and active volcanoes in the world. Due to its geological photographs allowed Espinoza (1988) to deduce that characteristics, it is an explosive volcano with pyroclastic flows occurred at least in five occasions, vulcanian type eruptions (Calvache et al., 1997), all of them without geological records (Banks et al., one of its hazards is the production of pyroclastic 1997). According to Banks et al. (1997), the volcano flows (Hurtado and Corte´s, 1997). Geological studies showed very small dilute surge-like flows during its performed by Calvache (1990), Calvache (1995) and active period of 1989. The fact that the historical Calvache et al. (1997) showed several occurrences of pyroclastic flows have not left a geological record is evidence that such pyroclastic flows were composed mainly of a dilute cloud with a negligible basal part. The last version of the Galeras Volcano hazard map E-mail address: [email protected]. (Hurtado and Corte´s, 1997) establishes the run-out 0377-0273/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jvolgeores.2004.06.015 60 G. Co´rdoba / Journal of Volcanology and Geothermal Research 139 (2005) 59–71 distance of the pyroclastic flows based on geological 1996a; Neri et al., 2001). The current development of records. But from the risk assessment point of view, it personal computers enables the exploration of the would also be useful to know some of the dynamical possibility that a fluid dynamics model of pyroclastic parameters like temperature, particle concentration flows can be implemented in a PC, including features and dynamical pressure due to flow evolution in space which were beyond the scope for this kind of and time. This would aide in estimating the devasta- computers a few years ago. The proposed model tion to structures or danger for people located in or focuses on the dilute turbulent companion ash cloud near the flow path (Blong, 1984; Baxter et al., 1995; of a pyroclastic flow (see fig. 6.2 in Sparks et al., 1997) Valentine, 1998, e.g.). which allows one to make some assumptions in order to Several analytical and numerical models based on reduce the number of the equations and variables fluid dynamics have been developed in order to involved. This model could be considered as an describe the above parameter fields. These range from intermediate between the abovementioned models, simple models which assume a non-transient, homoge- due to the fact that only one set of Navier–Stokes neous, one-dimensional or radial flow (Sheridan, 1980; equations is used. Bursik and Woods, 1991, 1996, e.g.) describing the This model is applied to the channelized slope of main physical processes that occur from the column the river basin of the Azufral River, which flows down collapse to emplacement to the so-called supercom- from the active cone (located in the same direction) to puter models (Valentine and Wohletz, 1989; Dobran the Guaitara River canyon, about 16 km away (Fig. and Neri, 1993; Neri and Macedonio, 1996a,b, e.g.) 1). The main part of the Azufral river is confined which model a tansient two-phase, compressible flow within deep and nearly vertical walls which allows us and a multiparticle solid phase (Neri and Macedonio, to assume a 2D approach. Fig. 1. Regional location of Galeras Volcano showing the Azufral River along which the proposed model is applied. Also, some towns under threat can be seen. G. Co´rdoba / Journal of Volcanology and Geothermal Research 139 (2005) 59–71 61 2. Mathematical model expressed in the Einstein summation convention (Lai et al., 1974). From the point of view of physics, the dynamics of a pyroclastic flow is controlled by the mass, 2.1.1. Mass momentum and energy balance equations. It is a Bb Bðbu Þ Bðw h q Þ þ j ¼À s s s d ð1Þ multiphase and multicomponent mixture, which Bt Bx Bx 2j carries gas (including water vapor), liquid drops of j j water and magma, clasts and fine ash. The pyro- 2.1.2. Momentum clastic flows at Galeras Volcano that occurred in the B B B B last 500 years were clearly dominated by fine ash as ðbuiÞ ðbuiujÞ p sij þ ¼À þðb À qgÞgi þ was pointed out above. As demonstrated by Sparks Bt Bxj Bxi Bxj B and Wilson (1976) and Neri and Macedonio (1996b), ðwshsqsuiÞ À d2j ð2Þ fine ash carried by some pyroclastic flows could Bxj comprise most of the flow in weight and volume. Thus for the model it is assumed that the dilute flow 2.1.3. Energy is composed of dry gas and a solid phase consisting The energy equation expressed in its internal (for simplicity) of one size spherical particles of 1 energy form: mm diameter. First, this size lets us assume a thermal equilibrium between particles and surrounding gas BðbeÞ Bðbeu Þ B BT Bu Bs u (Woods and Bursik, 1991). Additionally, for small j j ij j B þ B ¼ B kT B À p B þ B particles, the Biot number, which relates the thermal t xj xj xj xj xi conductivity with the heat transfer coefficient (Fan ð3Þ and Zhu, 1998), will be less than the unity, therefore only one equation is needed for the energy balance. 2.1.4. Convection–Diffusion equation Moreover, for Biot numbers less than 0.1, the For small particles in a dilute suspension, and temperature distribution in the solid can be consid- neglecting cohesive forces, the concentration of ered uniform with a maximum error of 5% (Fan and particles can be approximated by the convection– Zhu, 1998). diffusion equation (Larock and Schamber, 1981; Also, for very small particles, the two phases will Chippada et al., 1993; Hermann et al., 1994): move nearly at equal velocities (Stewart and Wendr- off, 1984). Thus it will be assumed that the velocity B B B B B of the solid phase will be the same as the gas phase bhs bhsuj hs ðqswshsÞ B þ B ¼ B lt B À B d2j ð4Þ and the material can be treated as a continuum with t xj xj xj xj bulk flow properties (Freunt and Bursik, 1998) or the so-called two phase flow model with equal velocities where b = bulk density, u = velocity field, p = pressure, (Stewart and Wendroff, 1984). In this way, only one lt= turbulent eddy viscosity, g = gravity force, hs = set of Navier–Stokes equations will be used. How- volume fraction of solids, qs = density of solids, ever, it is considered that the change in the qg = gas phase density, ws=characteristic settling concentration of particles and its correspondent velocity, kT = bulk thermal conductivity, cp = specific change in bulk density are a consequence of heat of the mixing, T = bulk temperature, dij = Kro- sedimentation by means of the Convection–Diffusion neker’s delta and: equation. e ¼ cpT ð5Þ 2.1. Government equations 2.2. Constitutive equations The following equation system includes mass, momentum and energy conservation, coupled with The above equations system is closed by the the diffusion–convection equation, all equations following constitutive equations. 62 G. Co´rdoba / Journal of Volcanology and Geothermal Research 139 (2005) 59–71 The Reynolds stress tensor sij is: 2.3. Equations of state As usual in this kind of models, we assume an ideal Bui Buj 2 Buj ÀÀÀÀÀ sij ¼ l þ À dij À buiVujV ð6Þ gas law: Bxj Bxi 3 Bxj p ¼ bRT ð11Þ The fluctuating part of the tensor can be approxi- mated using the Boussinesq analogy (Wilcox, 1998): where R is the gas constant of the mixture, R ¼ R , R Mg being the universal gas constant and the molecular ÀÀÀÀÀ Bui Buj 2 Buj À buiVujV ¼Àlt þ À dij : ð7Þ weigh of the gas is assumed as the same of the air Bxj Bxi 3 Bxj Mg=28.964 kg/kg mol. For a dilute flow, and neglecting particle interactions, A bulk thermal conductivity is proposed, which the characteristic settling velocity can be approxi- relates the volumetric fractions of solids and gas with mated by (Sparks et al., 1997): their respective thermal conductivities, as: 1 k ¼ h k þð1: À h Þk : ð12Þ 4 q gs 2 T s s s g w ¼ s ð8Þ s 3 C b d The bulk density can be estimated in a similar manner as (Stewart and Wendroff, 1984): where: s=size of the pyroclasts and Cd is the Drag coefficient given by (Dobran and Neri, 1993): b ¼ hsqs þð1 À hsÞqg ð13Þ ( 24 ½1: þ 0:15Re0:687; if Re b1000 where the gas density is calculated using the C ¼ p p d Rep Boussinesq approach (Zienkiewicz and Taylor, 1994): 0:44; if Repz1000 q q ¼ a ð14Þ g 1: þ hðT À T Þ where Rep is the Reynolds number based on particle a diameter.