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**A multi-scale particle-tracking framework for dispersive solute transport modeling.**
*(English)*
Zbl 1405.65137

Summary: Particle-tracking simulation offers a fast and robust alternative to conventional numerical discretization techniques for modeling solute transport in subsurface formations. A common challenge is that the modeling scale is typically much larger than the volume scale over which measurements of rock properties are made, and the scale-up of measurements have to be made accounting for the pattern of spatial heterogeneity exhibited at different scales. In this paper, a statistical scale-up procedure developed in our previous work is adopted to estimate coarse-scale (effective) transition time functions for transport modeling, while two significant improvements are proposed: considering the effects of non-stationarity (trend), as well as unresolved (residual) heterogeneity below the fine-scale model. Rock property is modeled as a multivariate random function, which is decomposed into the sum of a trend (which is defined at the same resolution of the transport modeling scale) and a residual (
representing all heterogeneities below the transport modeling scale). To construct realizations of a given rock property at the transport modeling scale, multiple realizations of the residual components are sampled. Next, a flow-based technique is adopted to compute the effective transport parameters: firstly, it is assumed that additional unresolved heterogeneities occurring below the fine scale can be described by a probabilistic transit time distribution; secondly, multiple realizations of the rock property, with the same physical size as the transport modeling scale, are generated; thirdly, each realization is subjected to particle-tracking simulation; finally, probability distributions of effective transition time function are estimated by matching the corresponding effluent history for each realization with an equivalent medium consisting of averaged homogeneous rock properties and aggregating results from all realizations. The proposed method is flexible that it does not invoke any explicit assumption
regarding the multivariate distribution of the heterogeneity.

### MSC:

65M75 | Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs |

76S05 | Flows in porous media; filtration; seepage |

76T20 | Suspensions |

### Keywords:

statistical scale-up; dispersion; multi-scale methods; particle tracking; tracer transport; subsurface heterogeneity
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\textit{V. Vishal} and \textit{J. Y. Leung}, Comput. Geosci. 22, No. 2, 485--503 (2018; Zbl 1405.65137)

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