The Failure of Monte Carlo Radiative Transfer at Medium to High Optical Depths

Computer simulations of photon transport through an absorbing and/or scattering medium form an important research tool in astrophysics. Nearly all software codes performing such simulations for three-dimensional geometries employ the Monte Carlo radiative transfer method, including various forms of biasing to accelerate the calculations. Because of the probabilistic nature of the Monte Carlo technique, the outputs are inherently noisy, but it is often assumed that the average values provide the physically correct result. We show that this assumption is not always justified. Specifically, we study the intensity of radiation penetrating an infinite, uniform slab of material that absorbs and scatters the radiation with equal probability. The basic Monte Carlo radiative transfer method, without any biasing mechanisms, starts to break down for transverse optical depths above ~20 because so few of the simulated photon packets reach the other side of the slab. When including biasing techniques such as absorption/scattering splitting and path length stretching, the simulated photon packets do reach the other side of the slab but the biased weights do not necessarily add up to the correct solution. While the noise levels seem to be acceptable, the average values sometimes severely underestimate the correct solution. Detecting these anomalies requires the judicious application of statistical tests, similar to those used in the field of nuclear particle transport, possibly in combination with convergence tests employing consecutively larger numbers of photon packets. In any case, for transverse optical depths above ~75 the Monte Carlo methods used in our study fail to solve the one-dimensional slab problem, implying the need for approximations such as a modified random walk.Comment: Accepted for publication in the ApJ; 13 pages, 6 figure

A hierarchy of models for simulating experimental results from a 3D heterogeneous porous medium

In this work we examine the dispersion of conservative tracers (bromide and fluorescein) in an experimentally-constructed three-dimensional dual-porosity porous medium. The medium is highly heterogeneous ($\sigma_Y^2=5.7$), and consists of spherical, low-hydraulic-conductivity inclusions embedded in a high-hydraulic-conductivity matrix. The bi-modal medium was saturated with tracers, and then flushed with tracer-free fluid while the effluent breakthrough curves were measured. The focus for this work is to examine a hierarchy of four models (in the absence of adjustable parameters) with decreasing complexity to assess their ability to accurately represent the measured breakthrough curves. The most information-rich model was (1) a direct numerical simulation of the system in which the geometry, boundary and initial conditions, and medium properties were fully independently characterized experimentally with high fidelity. The reduced models included; (2) a simplified numerical model identical to the fully-resolved direct numerical simulation (DNS) model, but using a domain that was one-tenth the size; (3) an upscaled mobile-immobile model that allowed for a time-dependent mass-transfer coefficient; and, (4) an upscaled mobile-immobile model that assumed a space-time constant mass-transfer coefficient. The results illustrated that all four models provided accurate representations of the experimental breakthrough curves as measured by global RMS error. The primary component of error induced in the upscaled models appeared to arise from the neglect of convection within the inclusions. Interestingly, these results suggested that the conventional convection-dispersion equation, when applied in a way that resolves the heterogeneities, yields models with high fidelity without requiring the imposition of a more complex non-Fickian model.Comment: 27 pages, 9 Figure