We consider a 3D numerical model of the flow of a two-phase incompressible immiscible viscous fluid which approximates lava extrusion (one phase) from a volcanic vent into the air (another phase) and the subsequent lava flow. The model domain \(\Omega = [0,{{l}_{1}}] \times [0,{{l}_{2}}] \times [f({{x}_{1}},{{x}_{2}}),f({{x}_{1}},{{x}_{2}}) + {{l}_{3}}]\) is limited from below by the topography \(f({{x}_{1}},{{x}_{2}})\) of the studied terrane, where l1 and l2 are horizontal dimensions and l3 is the height of the model area. In this domain, we study the flow of a viscous Newtonian inhomogeneous incompressible fluid in the field of gravity. In Cartesian coordinates, this flow is described by the non-stationary Navier–Stokes equation (Ismail-Zadeh and Tackley, 2010; Tsepelev et al., 2019) with the initial condition \({\mathbf{u}}(t = 0,{\mathbf{x}}) = 0\):
$$\begin{gathered} \frac{{\partial (\rho {\mathbf{u}})}}{{\partial t}} + \left\langle {{\mathbf{u}},\nabla } \right\rangle (\rho {\mathbf{u}}) \\ = - \nabla p + \nabla \cdot {\text{(}}\eta (\nabla {\mathbf{u}} + \nabla {{{\mathbf{u}}}^{T}})) + {\mathbf{G}} \\ \end{gathered} $$
(1)
and continuity equation
$$\nabla \cdot {\mathbf{u}} = 0,$$
(2)
where \({\mathbf{u}} = ({{u}_{1}},{{u}_{2}},{{u}_{3}})\) is the velocity vector, \({\mathbf{G}} = (0,0, - g\rho )\) is the vector of external mass forces, g is the gravitational acceleration, p is the pressure, \(\rho \) is the density, \(\eta \) is the viscosity, \({\mathbf{x}} = ({{x}_{1}},{{x}_{2}},{{x}_{3}})\) is the spatial variable, and t is the time. The advection of a two-phase fluid with an initial condition \(\alpha (t = 0,{\mathbf{x}}) = {{\alpha }_{0}}({\mathbf{x}})\) is described by the equation:
$$\frac{{\partial \alpha }}{{\partial t}} + \nabla \cdot (\alpha {\mathbf{u}}) = 0,$$
(3)
where \(\alpha (t,{\mathbf{x}}) \in [0,1]\) determines the volume fraction of the fluid at point \({\mathbf{x}} \in \Omega \) at time t. At the initial time, the model domain is filled with air and therefore \({{\alpha }_{0}}({\mathbf{x}}) = 0.\) The density and viscosity are then described by the following equations:
$$\begin{gathered} \rho (t,{\mathbf{x}}) = {{\rho }_{l}}\alpha (t,{\mathbf{x}}) + {{\rho }_{a}}(1 - \alpha (t,{\mathbf{x}})), \\ \eta (t,{\mathbf{x}}) = {{\eta }_{l}}\alpha (t,{\mathbf{x}}) + {{\eta }_{a}}(1 - \alpha (t,{\mathbf{x}})), \\ \end{gathered} $$
(4)
where: \({{\rho }_{a}}\) and \({{\eta }_{a}}\) are the density and viscosity of air, \({{\rho }_{l}}{\text{ and }}{{\eta }_{l}}\) are the lava density and viscosity.
Although thermal effects have an impact on the formation of lava crusts and on lava flow, it was shown that at high magma discharge rate, the crust is fairly thin compared to the lava flow thickness (~3–5%) (Tsepelev et al., 2019). The lava crust cracks, drifts with lava flow driven by gravity until its thickness is small, and does not significantly affect the advance of the flow (Tsepelev et al., 2016). This model considers lava flow due to gravity without thermal effects.
No slip condition \({\mathbf{u}} = 0\) is prescribed at the lower and side boundaries of the model domain. At the part of the lower boundary that contains the vent, it is prescribed that \({\mathbf{u}} = {{{\mathbf{u}}}_{0}},{\text{ }}\rho = {{\rho }_{l}},{\text{ }}\eta = {{\eta }_{l}}\), where \({{{\mathbf{u}}}_{0}}\) is the rate of magma extrusion. At the upper boundary of the model domain, we specify the conditions \(\nabla {\mathbf{u}} = 0,{\text{ }}p = 0,{\text{ }}\rho = {{\rho }_{a}},{\text{ }}\eta = {{\eta }_{a}}\).
The Yellowstone Summit Lake lava flows contribute to the present-day terrain topography of the region. To model the lava flows in the geological past, the present terrain topography should be transformed to account for the thickness of the erupted lavas. Namely, let \(F({{x}_{1}},{{x}_{2}})\) be the present topography of the studied region; the lava thickness \(Z(x_{1}^{v},x_{2}^{v})\) reaches its maximum 230 m (Loewen et al., 2017) in the volcano vent and linearly decreases with the distance from the vent, and \(\left( {x_{1}^{v},x_{2}^{v}} \right)\) are the coordinates of the intersection of lines A–A', B–B' and C–С' which corresponds to the location of the volcano vent. Hence the model topography is calculated by the formula \(f({{x}_{1}},{{x}_{2}}) = F({{x}_{1}},{{x}_{2}})\) – \({{Z(x_{1}^{v},x_{2}^{v})} \mathord{\left/ {\vphantom {{Z(x_{1}^{v},x_{2}^{v})} {(1 + {\text{ }}r({{x}_{1}},{{x}_{2}}))}}} \right. \kern-0em} {(1 + {\text{ }}r({{x}_{1}},{{x}_{2}}))}}\) where \(r({{x}_{1}},{{x}_{2}}) = {{\sqrt {{{{\left( {{{x}_{1}} - x_{1}^{v}} \right)}}^{2}} + {{{\left( {{{x}_{2}} - x_{2}^{v}} \right)}}^{2}}} } \mathord{\left/ {\vphantom {{\sqrt {{{{\left( {{{x}_{1}} - x_{1}^{v}} \right)}}^{2}} + {{{\left( {{{x}_{2}} - x_{2}^{v}} \right)}}^{2}}} } D}} \right. \kern-0em} D}\), D = 20 km. Figure 1b illustrates the model topography calculated in this way.
Thus, the lava flow modeling problem is reduced to solving Eqs. (1)–(4) with the above initial and boundary conditions in the model domain \(\Omega \). In the numerical modeling, all variables are reduced to the dimensionless form with the time scale \(t* = {{{{\eta }_{l}}} \mathord{\left/ {\vphantom {{{{\eta }_{l}}} {({{\rho }_{l}}gh)}}} \right. \kern-0em} {({{\rho }_{l}}gh)}}\), length scale h, and velocity scale \({h \mathord{\left/ {\vphantom {h {t{\text{*}}}}} \right. \kern-0em} {t{\text{*}}}}\).