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A transient depth-averaged lava flow model with a Herschel-Bulkley rheology accounting for three phases

This study presents a transient depth-averaged model for three-phase lava flows with Herschel-Bulkley rheology, demonstrating that a temperature-dependent crystallinity closure yields more accurate predictions than a simplified relaxation approach while revealing that pulsatory dynamics arise from slope irregularities and that gas bubbles significantly modulate viscosity through shear-thinning behavior.

Original authors: Julie Binard, Alain Burgisser, Enrique D. Fernández-Nieto, Gladys Narbona-Reina

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Julie Binard, Alain Burgisser, Enrique D. Fernández-Nieto, Gladys Narbona-Reina

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a river of molten rock, but instead of just being hot soup, it's a chaotic, three-ingredient smoothie: liquid magma, solid crystal chunks, and gas bubbles. This paper builds a new digital simulator to predict how this "lava smoothie" flows down a mountain, specifically looking at how those crystals and bubbles change the way the lava moves.

The Big Discovery: It's All About the Temperature
The researchers found that the best way to predict where lava goes is to track its temperature carefully. They tested two different ways to model how the lava hardens (crystallizes) as it cools.

The first method was like a simple timer: "After X minutes, the lava hardens this much." It was easy to calculate but ignored the messy reality of heat escaping into the air or ground.

The second method was like a full weather station: it tracked four different ways heat leaves the lava (radiation, air convection, ground conduction, and internal friction). The paper suggests that this "full weather station" approach yields more accurate results than the simple timer. When they compared their simulation to real data from the Mauna Ulu eruption, the temperature-tracking model did a better job of predicting the flow's behavior.

The "Pulsing" Lava
One of the coolest things the model predicts is that lava doesn't always flow smoothly like water from a hose. When lava gets trapped in a narrow, steep channel, it doesn't just glide; it pulses.

Think of it like a traffic jam on a steep hill where cars keep stopping and starting. The lava fills up a section of the channel, gets stuck, then suddenly breaks free and surges forward, only to get stuck again. The paper suggests this "fill-then-breakout" cycle creates a series of lumps that travel down the slope. Because of this, the lava flow is never truly steady; it's always beating like a heart, dominated by these surges.

The Role of Bubbles and Crystals
The lava in this model is a three-phase suspension:

Liquid Magma: The base fluid.
Crystals: Solid bits that form as it cools.
Bubbles: Gas pockets trapped inside.

The paper argues that you can't just treat bubbles as hard, round marbles floating in the soup. In the conditions they simulated, bubbles act like shape-shifters. When the lava moves slowly, the bubbles stay round and make the lava thicker (more viscous), slowing it down. But when the lava moves fast, the bubbles stretch out like long sausages, which actually helps the lava flow faster.

The paper explicitly rules out the idea that bubbles always just make lava thicker. Instead, they found that bubbles can modulate the lava's thickness by a factor of 2 (making it twice as thick or half as thick depending on speed). If you use a simplified model that treats bubbles as hard spheres, you miss this whole "stretching" effect and only see a change in the overall thickness.

What About the Crystals?
As the lava cools, crystals form. The paper notes that these crystals make the lava thicker and can even give it a "yield stress"—a point where it acts like a solid until you push hard enough to make it move (like toothpaste in a tube).

The researchers tested a specific formula for how "flowy" the lava is (called the flow index, n). They found that the more crystals and bubbles you have, the more the lava likes to thin out when you push it (shear thinning). They proposed a new formula for this behavior that fits experimental data better than older models, especially when the crystal concentration is high.

How Sure Are They?
It's important to note that these findings come from numerical simulations and comparisons with existing data, not from a new physical experiment in a lab. The authors suggest that their multi-parametric temperature model is superior, but they are working within the realm of computer modeling.

They also point out that their model works best for certain conditions. For instance, if the lava is packed with so many bubbles that it becomes a foam (more than 70% gas), or if the crystals are packed so tightly that they touch and lock together (above a packing fraction of roughly 0.8), the rules might change. Their current model focuses on the "sweet spot" where the lava is still flowing but has a significant mix of crystals and bubbles.

The Bottom Line
This paper presents a new, depth-averaged model (a 2D slice of a 3D world) that treats lava as a complex mix of liquid, crystals, and bubbles. By tracking temperature carefully and accounting for how bubbles stretch and crystals form, the model suggests that lava flows are often pulsating, surging events rather than steady streams. While it's a simulation, it offers a more nuanced and accurate picture of how real lava, like that from Mauna Ulu, behaves on a steep, irregular slope.

Technical Summary: A Transient Depth-Averaged Lava Flow Model with Herschel–Bulkley Rheology Accounting for Three Phases

Problem Statement
Lava flows exhibit complex dynamics driven by significant variations in rheology as magma cools, crystallizes, and degasses. Existing deterministic models often simplify these processes by treating lava as a single-phase fluid or by neglecting the explicit coupling between temperature, crystallization, and gas content. While shallow-water (depth-averaged) models offer a balance between computational speed and physical completeness, many current implementations fail to explicitly account for the three-phase nature of lava (melt, crystals, and gas bubbles) or the non-equilibrium dynamics of crystallization. Furthermore, the influence of gas bubbles on rheology is often oversimplified, and the feedback between viscous heating, cooling, and flow velocity is not always fully integrated in a transient framework. This study addresses the need for a more physically complete, yet computationally tractable, one-dimensional depth-averaged model that explicitly resolves the interactions between melt, crystals, and bubbles under Herschel–Bulkley rheology.

Methodology
The authors develop a transient, one-dimensional depth-averaged model based on the conservation of mass and momentum for a three-phase suspension. The rheology is defined by the Herschel–Bulkley law, which incorporates a yield stress ( $\tau_c$ ), a consistency factor (apparent viscosity $\eta$ ), and a flow index ( $n$ ).

The core of the methodology involves deriving constitutive relationships for these rheological parameters based on a synthesis of existing literature (specifically Mueller et al., Truby et al., and Mader et al.):

Rheological Closures: The apparent viscosity combines effects of particle concentration (using a Krieger-Dougherty type relationship) and bubble concentration (dependent on the capillary number). The yield stress is modeled as a function of crystal concentration, bounded by a maximum physical limit. A novel relationship for the flow index $n$ is proposed, which accounts for shear-thinning behavior induced by both particle aspect ratio and bubble concentration, introducing a critical solid volume fraction threshold below which the flow remains Newtonian.
Two Evolutionary Closures: The study proposes and compares two distinct approaches for modeling the evolution of the crystal volume fraction ( $\Phi_p$ $Φ_{p}$ ):
- Model 1 (Relaxation Approach): Treats the crystal fraction as a transported quantity subject to a relaxation term toward an equilibrium state with a characteristic crystallization time ( $\tau_{crys}$ ). This approach minimizes complexity by decoupling crystallization from explicit heat transfer mechanisms.
- Model 2 (Thermal Approach): Couples the crystal fraction to the lava temperature via a prescribed temperature-crystallinity relationship. The temperature evolution is governed by a depth-averaged advection equation that explicitly includes four heat transfer mechanisms: radiation, convection, conduction, and viscous heating.
Numerical Implementation: The derived system is discretized using a well-balanced numerical scheme involving hydrostatic reconstruction and semi-implicit discretization to handle the stiff source terms and non-linear rheology.

Key Contributions

Three-Phase Rheological Framework: The paper presents a generalized rheological model that explicitly integrates the effects of both solid crystals and gas bubbles on viscosity, yield stress, and flow index, moving beyond single-phase approximations.
Dual Closure Strategies: The introduction of two distinct closures for crystal evolution allows for a direct comparison between a simplified relaxation model and a complex thermodynamic model. This highlights the trade-offs between computational efficiency and physical fidelity regarding heat transfer.
Transient Dynamics: Unlike many steady-state models, this framework captures the transient nature of lava flows, specifically the evolution of flow parameters over time and space.
Validation and Calibration: The rheological parameters, particularly the flow index, are calibrated against experimental data from Truby et al. (2015), ensuring the model's consistency with laboratory observations of three-phase suspensions.

Results
The models were tested numerically and applied to a case study of a channeled lava flow from the Mauna Ulu eruption.

Model Intercomparison: The comparison between Model 1 and Model 2 against real lava flow data suggests that the multi-parametric evolution of temperature (Model 2) yields more accurate predictions of flow parameters than the relaxation-based closure (Model 1).
Transient Behavior: The model successfully predicts that confined lava traveling down an irregular, steep slope does not reach a strict steady state. Instead, the flow exhibits pulsatory behavior characterized by a series of cascading "fill-then-breakout" lumps. These pulses dominate local dynamics, preventing the system from stabilizing into a uniform flow.
Role of Bubbles: The study finds that in the conditions explored, gas bubbles modulate viscosity by a factor of approximately 2. They induce shear-thinning behavior, which decelerates slow flows and accelerates fast flows. When a simplified rheology treating bubbles as hard spheres was used, the only dynamic parameter affected was the bulk viscosity, failing to capture the shear-thinning nuances.
Stationary Solutions: Theoretical analysis of stationary solutions reveals that for Model 2, a constant lava flux can be maintained if viscous heating counteracts cooling, a balance dependent on flow velocity and depth.

Significance
The paper claims that its primary significance lies in providing a more robust physical description of lava flow dynamics by explicitly resolving the three-phase nature of the fluid and the coupling between thermal and rheological evolution. By demonstrating that the transient nature of the flow leads to pulsatory dynamics rather than steady states, the model offers a more realistic representation of flow behavior on irregular terrain. Furthermore, the comparison of closures indicates that while simplified relaxation models are computationally cheaper, the inclusion of explicit heat transfer mechanisms (Model 2) is superior for accurate prediction of flow emplacement. The work bridges the gap between probabilistic hazard models and complex 3D deterministic models, offering a depth-averaged tool that retains critical physical processes such as disequilibrium crystallization and bubble-induced rheological changes.

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