Globally there are about 50 volcanoes that erupt every year. During the past century tens of thousands of people have been killed by volcanic flows and hundreds of thousands forced from their homes . Two-phase mass flows containing water and solid particles, called lahars, are common in volcanic regions. They can be initiated by several mechanisms. A volcanic explosion can be accompanied by large plumes and pyroclastic flows consisting of rock and gas that race along the surface of the mountain at speeds as high as 100 m per second . The hot ash can melt snow, creating a muddy mixture that knock down trees and entrain rocks and boulders into the flow. Cotopaxi Volcano in Ecuador is an example of a volcano that has produced many large lahars by this process in the past . Crater lakes on volcanoes are another source of mud flows, a recent example being the 2007 lahar of Ruapehu in New Zealand . A third mechanism for initiating lahars is intense rainfall on hillsides that are devoid of vegetation and exposures of loose material like clay soils or volcanic ash. An example of this type of lahar is the 1998 mudflow at Casita Volcano in Nicaragua that occurred during Hurricane Mitch and caused hundreds of deaths . Lahars can carry constituent particles that are typically from clay to boulder size and can propagate tens of kilometers before coming to rest . As solid particle sediment out of the flowing mass the resulting deposits can be up to one hundred meters thick . However, the typical deposits left after a debris flow passes are on the scale of meters.

In order to develop a complete mathematical model of mud and debris flows, two principal challenges must be overcome: rheology, and scale. First, constitutive relations must be developed to describe granular material including clays, sands, pebbles and rocks, with interstitial water. Second, a computational method must be developed that extends over six orders of magnitude. Neither of these challenges can be fully met at this time. This paper tries to strike a balance between fidelity to the physics of mass flows and computational feasibility. We describe a modeling effort that draws on the wisdom from engineering and geo-science, to postulate constitutive theory and fluid-solid interaction effects, and, through a depth averaging process, results in a system of equations that is computationally tractable.

The modeling effort here has its origins the pioneering work of Savage and Hutter . They began with mass and momentum balance laws based on a Coulomb constitutive description of dry granular material. By scaling and depth averaging, they develop a “thin layer” model for granular flows down inclines. Flow over general topography was addressed in , , and . Comparison of thin layer model results to historic flows is presented in . In , the appropriateness of these thin layer models is considered, for several different types of geophysical flows. Much of the modeling effort is summarized in .

Iverson and his co-workers argue that the presence of interstitial fluid fundamentally alters the behavior of geophysical flows, and fluid effects should be included in the constitutive behavior of the flowing material. Starting with equations of mixture theory and through a careful examination of experiments, these papers developed a system of mass and momentum balance laws for the mixture. Unfortunately in this development an equation for the motion of pore fluid was not readily available. Instead, based on experimental data, a transport equation for the fluid was postulated.

A different approach, based on a fully three-dimensional model of two phase flows, can be found in and .

Pitman and Le rigorously developed a two phase thin layer model of fluid and granular material. They begin with a fully three dimensional model of two phase flows, based on model equations in engineering . The model equations are scaled and depth averaged. The resulting system of equations is not complete, and closure assumptions are required. With these assumptions, the mathematical system is shown to be hyperbolic under common conditions, and thus well posed (see ). The model of includes a drag term, which is the only term describing the interaction of the two phases; the coefficient of drag must be fitted to experiments. That model assumes the fluid is inviscid, and that there is no frictional dissipation in the fluid phase at the basal surface. Both of these features, which are reasonable in bench-scale fluidized bed experiments, are suspect for large mass flows. This paper reconsiders the model equations of and proposes a revision that better represents two phase geophysical flows. Related work can be found in and .

A different approach to modeling mud flows employ a visco-plastic constitutive assumptions ; see also and . In these papers, the choice of visco-plastic flow model drives the subsequent derivation, as well as the required parameter-fitting necessary for the constitutive relations. The process of depth averaging a visco-plastic flow is always difficult. The interface between yielding and non-yielding material is itself a free surface that must be determined. This attribute requires the use of multiple layers in the model system, with all the resulting complexity.

This paper uses a similar framework to that developed in . However, a complete set of model equations for a granular phase and for a fluid phase are written. Phase interaction terms are modeled, and scaling of all terms suggests simplifications that can be made. Depth averaging and closure assumptions completes the derivation.

A note on sign convention: in soil mechanics it is common to consider compressive stresses as positive; in fluid mechanics compression is negative. We caution the reader to observe the sign convention in the equations below.

The final step in the derivation is a depth averaging of the mass and momentum balance equations. That is, for some function f, we compute 1|h-b|∫b(x,y)h(x,y,t)f(x,y,z,t)dz=f‾. Repeated use of the Leibnitz rule is made to interchange integration and differentiation, and boundary conditions are employed to evaluate terms at b and h. In addition, several approximations must be made during the depth averaging process. In what follows, we only briefly sketch the depth averaging process, noting as appropriate those places where approximations are made. provides an estimation of the errors typically made by these assumptions.

The terms of order ϵ are assumed small and we hope to drop all such terms from the model. However argues that diagonal contributions to the solid stress must be retained. Because there is no preferential downslope direction in our applications, and the flow direction may change during a flow, we retain the stress terms in both the x and y directions, dropping only O(ϵ) terms in the z direction; see the discussion in . Other O(ϵ) terms are dropped.

In order to validate and verify the proposed approach, we did a series of tests using a one dimensional approach of the proposed equation system. First we test the consistency of the results verifying the expected symmetries. Then we used an exponentially decaying topography with and without obstacles (bumps). They show expected features like how the mathematical model can show a reflected wave when the bump was big enough or the flow splitting in a reflected dense part and bump overpass by a very dilute flow.

Verification of the accuracy of the code was done with analytical solutions of the Dam Break problem and with several experiment results. Among them, we check the deposited pattern predicted by the program with the results shown by Liu . The prediction of the arrival time and the flow depth profile was compared with the experimental results shown by from his recent work done on his large channel facility.

Analytical solutions for shallow water problems are scarce. Only one dimensional analytical approaches are available in the litereture, especially for the well known Dam Break problem (e.g. ). Unfortunately, analytical solutions for geo-mass flows are almost imposible to find due to the complexity of the the non-linear partial differential equations . Such solutions can be obtained only in special cases like the similarity based solutions proposed by Savage and Hutter for dry avalenches. In our test we use the solution proposed by for the Dam Break problem on an incline for pure water. In our program we assume φs→0. Figure  shows a comparison of the Two-Phase-Titan prediction with this analytical solution. At least for the one dimensional case, the program successfuly reproduces analytical solutions for different initial conditions down to very low paricle concentrations (less than 1%).

performed several experiments for geo-mass flows in an inclined channel. He modified the initial volume, the channel slope and the particle concentration to find the final size of the debris flows meassured by their resulting width and length. We reproduced experimental final width and length after the simulation reached the same time corresponding to the duration reported by Liu. Figure  shows the correspondense of the model with the experiments for (a) the width of the deposit and (b) the length. A Pearson correlation shows that 90% of the experimental data for the deposit lenght can be explained with the predictions of Titan-Two-Phase, whereas 80% of the data for the deposit width can be explained by the predictions of the program. This ilustrates the high accuracy of the program in predicting the deposit characteristics for different initial volumes and high initial solid concentrations.

The experiments performed by Iverson and co-workers done on a 95m long artificial channel was used to verify the accuracy of the predictions of the flow front arrival time and the temporal evolution of the flow depth. These flows were unsteady and nonuniform. show time series data for several meassured properties: flow thickness, pore pressure, basal normal stress and arrival time of the front. Raw data sent to us by Dr. Iverson were used to test the Two-Phase-Titan prediction concerning time evolution of the flow depth and arrival times at the check points located at 32 and 66m distance from the lock. As shown in Figures  and , the arrival time and the temporal evolution of the predicted depth fits very well within the confidence interval of the experiments.

The range of concentrations that the program cope with, are from φs=10-8 (almost pure water) to φs=0.65 (maximum packing concentration). Finally, as expected, the program predicts high particle concentrations at the front of the flow and low particle concentrations at the tail of the flow (at some cases, even near pure water concentrations, or φ→0), as can be seen in Fig.  where a longitudinal solids particle distribution predicted by Titan2F is shown. The predictions fits with observation that the tail of the flow remains very watery. Using predicted concentration of solids, the density is assessed (Eq. ) and together with the speed of the flow, the dynamic pressure distribution is calculated as well (Eq. ). Figure  shows longitudinal and cross distributions of the dynamic pressures after 14s simulation. As the flow velocity at lateral limits of the flow from the end of the channel are very low, the dynamic pressure shows to be low as well. Knowledge of the dymaic pressure information is of vital importance in risk analysis as structural damage and risk for human life can be assessed from it.

Verification with actual mud flows has been done as well, showing very good fit with field data. For example, shows that the Titan2F predictions are within 10% of the data shown by for the highly channeled mud flow at Ruapehu, New Zealand. In addition, the theory was tested against field data assessed by for the 2006 Vazcun creek lahar at Tungurahua volcano, Ecuador, as shown by , where Titan2F predictions about velocity are within 10% and within 15% for the meassured super-elevation.

In this paper we present a new computational two phase tool for lahar hazard assessment that has no constrains on initial volumentric particle concentration. The program computes space–time evolution of the particle concentration, flow depth, velocity field and dynamic pressure at each ponit of the computational grid.

The model is valid for two phase flows whose phases consist in solids and water. However, the phenomenological approach used for the interphase drag model assumes an average diameter of the solids, which mean individual boulders or particles cannot be tracked. In addition, the model is depth averaged, assuming thin layer and shadow water approaches. Thus, our model correctly predicts the dynamic of gravity driven flows providing the depth averaged values for the particle concentration, flow and phases velocities and flow depth in a three dimensional topography. In order to model other kind of geophysical mass flows, adjustments to the code must be done, for example pyroclastic flows can be modeled if the flow density of the fluid phase is appropriately addressed (eg. thermocoupled air density using ideal gas law and an additional equation for temperature).

The proposed mathematical approach allows to study the whole range of flow behaviour. Regions with almost pure fluid to regions of friction dominated flows are correctly described by the algorithm. Using this information, dynamic pressure is deduced, which becomes a very useful tool for risk assessment.

The highly non-linear coupled equation system makes the time step very small. The use of this neew tool on natural terrains or detailed Digital Elevetion Models (DEMs) requires higher computational power than the one provided for home PCs. The use of a high velocity Work Station with multiple cores is adviced.

Important processes that are not addressed by this tool include the effect of turbulence, incorporation of solid material from the bed of the channel, and incorporation of water into the flow from existing water bodies. Nevertheless, this two-phase flow is an important step forward in forming an aceptable computational model for simulating a hazardous natural phenomena. Currently, we are applying this tool for real lahar hazard assessments like in .