UBC Theses and Dissertations
A statistical continuum approach for mass transport in fractured media Robertson, Mark Donald
The stochastic-continuum model developed by Schwartz and Smith  is a new approach to the traditional continuum methods for solute transport in fractured media. Instead of trying to determine dispersion coefficients and an effective porosity for the hydraulic system, statistics on particle motion (direction, velocity and fracture length) collected from a discretely modeled sub-domain network are used to recreate particle motion in a full-domain continuum model. The discrete sub-domain must be large enough that representative statistics can be collected, yet small enough to be modeled with available resources. Statistics are collected in the discrete sub-domain model as the solute, represented by discrete particles, is moved through the network of fractures. The domain of interest, which is typically too large to be modeled discretely is represented by a continuum distribution of the hydraulic head. A particle tracking method is used to move the solute through the continuum model, sampling from the distributions for direction, velocity and fracture length. This thesis documents extensions and further testing of the stochastic-continuum two-dimensional model and initial work on a three-dimensional stochastic-continuum model. Testing of the model was done by comparing the mass distribution from the stochastic-continuum model to the mass distribution from the same domain modeled discretely. Analysis of the velocity statistics collected in the two-dimensional model suggested changes in the form of the fitted velocity distribution from a gaussian distribution to a gamma distribution, and the addition of a velocity correlation function. By adding these changes to the statistics collected, an improvement in the match of the spatial mass distribution moments between the stochastic-continuum and discrete models was effected. This extended two-dimensional model is then tested under a wide range of network conditions. The differences in the first spatial moments of the discrete and stochastic-continuum models were less than 10%, while the differences in the second spatial moments ranged from 6% to 30%. Initial results from the three-dimensional stochastic-continuum model showed that similar statistics to those used in the two-dimensional stochastic-continuum model can be used to recreate the nature of three-dimensional discrete particle motion.
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