A spatial <scp>M</scp>arkov model for the evolution of the joint distribution of groundwater age, arrival time, and velocity in heterogeneous media

Massoudieh, Arash; Dentz, Marco; Alikhani, Jamal

doi:10.1002/2017wr020578

Cited by 16 publications

(12 citation statements)

References 44 publications

(69 reference statements)

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“…We consider now a more complex model to account for the velocity dependence in the evolution of s‐Lagrangian velocity PDF. Specifically, we consider the evolution of v s ( s ) based on an Ornstein‐Uhlenbeck process (Massoudieh et al, ; Morales et al, ). The s‐Lagrangian velocity here is obtained according to the bijective map

v false(s false) = F false[w false(s false) false],

from the Gaussian process w ( s ).…”

Section: Markov Models For the Evolution Of Lagrangian Velocitiesmentioning

confidence: 99%

Upscaling and Prediction of Lagrangian Velocity Dynamics in Heterogeneous Porous Media

Hakoun

Comolli

Dentz

2019

Water Resources Research

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The understanding of the dynamics of Lagrangian velocities is key for the understanding and upscaling of solute transport in heterogeneous porous media. The prediction of large‐scale particle motion in a stochastic framework implies identifying the relation between the Lagrangian velocity statistics and the statistical characteristics of the Eulerian flow field and the hydraulic medium properties. In this paper, we approach both challenges from numerical and theoretical points of view. Direct numerical simulations of Darcy‐scale flow and particle motion give detailed information on the evolution of the statistics of particle velocities both as a function of travel time and distance along streamlines. Both statistics evolve from a given initial distribution to different steady‐state distributions, which are related to the Eulerian velocity probability density function. Furthermore, we find that Lagrangian velocities measured isochronally as a function of travel time show intermittency dominated by low velocities, which is removed when measured equidistantly as a function of travel distance. This observation gives insight into the stochastic dynamics of the particle velocity series. As the equidistant particle velocities show a regular random pattern that fluctuates on a characteristic length scale, it is represented by two stationary Markov processes, which are parametrized by the distribution of flow velocities and a correlation distance. The velocity Markov models capture the evolution of the Lagrangian velocity statistics in terms of the Eulerian flow properties and a characteristics length scale and shed light on the role of the initial conditions and flow statistics on large‐scale particle motion.

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v false(s false) = F false[w false(s false) false],

from the Gaussian process w ( s ).…”

Section: Markov Models For the Evolution Of Lagrangian Velocitiesmentioning

confidence: 99%

Upscaling and Prediction of Lagrangian Velocity Dynamics in Heterogeneous Porous Media

Hakoun

Comolli

Dentz

2019

Water Resources Research

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“…It evolves according to the Boltzmann‐type equation

\frac{\partial f false(t, v; ℓ false)}{\partial t} - v \frac{\partial f false(t, v; ℓ false)}{\partial ℓ} = - v κ_{c} f false(t, v; ℓ false) + v κ_{c} p_{ℓ} false(v false) true \int_{0}^{\infty} normald v^{'} f false(t, v^{'}; ℓ false) .

with the initial condition f ( t , v ; ℓ = 0) = δ ( t ) p 0 ( v ); see Appendix B and Massoudieh et al (). The arrival time density f ( t ; ℓ ) = ⟨ δ [ t − t ( ℓ )]⟩ is obtained from f ( t , v ; ℓ ) by marginalization:

f false(t; ℓ false) = true \int_{0}^{\infty} normald v f false(t, v; ℓ false) .

…”

Section: Stochastic Particle Motionmentioning

confidence: 99%

“…with the initial condition f(t, v; = 0) = (t)p 0 (v); see Appendix B and Massoudieh et al (2017). The arrival time density f(t; ) = ⟨ [t − t( )]⟩ is obtained from f (t, v; ) by marginalization: The BTC F(t, x 1 ) is given in terms of f (t, v; ) as…”

Section: Continuous Time Random Walkmentioning

confidence: 99%

Linking Structural and Transport Properties in Three‐Dimensional Fracture Networks

et al. 2019

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We investigate large‐scale particle motion and solute breakthrough in sparse three‐dimensional discrete fracture networks characterized by power law distributed fracture lengths. The three networks we consider have the same fracture intensity values but exhibit different percolation densities, geometric properties, and topological structures. We considered two different average transport models to predict solute breakthrough, a streamtube model and a Bernoulli continuous time random walk model, both of which provide insights into the flow fields within the networks. The streamtube model provides acceptable predictions at short distances in two of the networks but fails in all cases to predict breakthrough times at the outlet plane, which indicates that particle motion in such fracture networks cannot be characterized by a constant velocity between the inlet and control plane at which the breakthrough curve is detected. Rather, the structure of the network requires that frequent velocity transitions be made as particles move through the system. Despite the relatively broad distribution of fracture radii and relatively small number of independent velocity transitions, the continuous time random walk approach conditioned on the initial velocity distribution provides reasonable predictions for the breakthrough curves at different distances from the inlet. The application of these averaged transport models provides a richer understanding of the link from the fracture network structure to flow and transport properties.

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“…The transition matrix can be determined empirically by sampling velocity transitions along particle trajectories (Le Borgne et al, ), inverse modeling algorithms applied to experimental concentration profiles (Sherman et al, , ), or by parametric models given by analytical Markov models (Dentz et al, ; Hakoun et al, ; Kang, Le Borgne, et al, ; Kang, Dentz, et al, ; Morales et al, ). Here we focus on the CTRW implementation that models the series of particle velocity magnitudes as a Bernoulli process (Carrel et al, ; Dentz et al, ; Holzner et al, ; Hyman et al, ; Kang et al, ; Massoudieh et al, ; Puyguiraud et al, , ), meaning a particle's speed persists from the previous step if a weighted coin lands heads and is resampled if it lands tails. This probability is often found by assuming velocity transitions at a constant rate, inversely proportional to a correlation distance (Dentz et al, ; Hyman et al, ).…”

Section: Introductionmentioning

confidence: 99%

Characterizing the Influence of Fracture Density on Network Scale Transport

et al. 2020

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The topology of natural fracture networks is inherently linked to the structure of the fluid velocity field and transport therein. Here we study the impact of network density on flow and transport behaviors. We stochastically generate fracture networks of varying density and simulate flow and transport with a discrete fracture network model, which fully resolves network topology at the fracture scale. We study conservative solute trajectories with Lagrangian particle tracking and find that as fracture density decreases, solute channelization to large local fractures increases, thereby reducing plume spreading. Furthermore, in sparse networks mean particle travel distance increases and local network features, such as velocity zones where flow is counter to the primary pressure gradient, become increasingly important for transport. As the network density increases, network statistics homogenize and such local features have a reduced impact. We quantify local topological influence on transport behavior with an effective tortuosity parameter, which measures the ratio of total advective distance to linear distance at the fracture scale; large tortuosity values are correlated to slow-velocity regions. These large tortuosity, slow-velocity regions delay downstream transport and enhance tailing on particle breakthrough curves. Finally, we predict transport with an upscaled, Bernoulli spatial Markov random walk model and parameterize local topological influences with a novel tortuosity parameter. Bernoulli model predictions improve when sampling from a tortuosity distribution, as opposed to a fixed value as has previously been done, suggesting that local network topological features must be carefully considered in upscaled modeling efforts of fracture network systems. and topology, and the corresponding flow field. The flow field within an individual fracture is typically highly correlated, commonly causing solute velocity to display persistent, low variability behavior over the in-fracture scale; consequently, the greatest Lagrangian accelerations occur at fracture intersections . As the fracture density increases, solute encounters more intersections on average, and the velocity correlation scale decreases. Furthermore, strong preferential flow paths form within interconnected networks of large fractures and channel a significant portion of mass, enabling solute to persist at high velocities for distances greater than the single fracture scale Sherman et al., 2019). This channelization becomes enhanced in sparse networks, where particles encounter fewer intersections, enabling them to persist on single fractures for longer distances. Resolving all these intranetwork features in 3-D DFN models is still computationally costly, and so upscaled modeling approaches, which account for network variability through effective parameter schemes, while maintaining a parsimonious framework, present an attractive alternative. However, how to properly parameterize network properties, such as velocity correlation and geometry, and inco...

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A spatial Markov model for the evolution of the joint distribution of groundwater age, arrival time, and velocity in heterogeneous media

Cited by 16 publications

References 44 publications

Upscaling and Prediction of Lagrangian Velocity Dynamics in Heterogeneous Porous Media

Upscaling and Prediction of Lagrangian Velocity Dynamics in Heterogeneous Porous Media

Linking Structural and Transport Properties in Three‐Dimensional Fracture Networks

Characterizing the Influence of Fracture Density on Network Scale Transport

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