# Upscaling Mixing in Highly Heterogeneous Porous Media via a Spatial Markov Model

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## Abstract

**:**

## 1. Introduction

## 2. System Setup

**v**is the velocity $\left(\right)$, $p\phantom{\rule{3.33333pt}{0ex}}\left[M{L}^{-1}{T}^{-2}\right]$ is pressure, and $\mu \phantom{\rule{3.33333pt}{0ex}}\left[M{L}^{-1}{T}^{-1}\right]$ is viscosity. System (1) is solved using a finite volume method [57]. At the boundaries we impose permeameter like conditions: no flux across the lateral boundaries and constant head at the upstream and downstream boundaries. The mean velocity can then be rescaled to any desired value, so for convenience we choose $\overline{{v}_{x}}=1$. Figure 1b shows the natural log of the absolute value of the velocity $v=\sqrt{{v}_{x}^{2}+{v}_{y}^{2}}$, where ${v}_{x}$ and ${v}_{y}$ are the velocity fields in the horizontal and vertical directions, respectively. The flow domain was generated to be large enough so that when we simulate transport the solute will not interact with the boundaries. Domain lengths in the x and y directions equal to ${L}_{x}=8000$ and ${L}_{y}=2000$, respectively, were deemed sufficient for this purpose.

## 3. Upscaled Model: Spatial Markov model

#### 3.1. Model Parameterization

#### 3.2. Model Mean Longitudinal Transport

#### 3.2.1. Choice of Cell Length

## 4. Downscaling Procedure to Predict Mixing

#### 4.1. Downscaling the Streamwise Location x

#### 4.2. Downscaling the Spanwise Location y

#### 4.2.1. Method 0

#### 4.2.2. Method 1

#### 4.2.3. Method 2

#### 4.2.4. Method 3

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The natural log of the permeability field $\kappa $ and (

**b**) the natural log of the absolute value of the velocity v = $\sqrt{{v}_{x}^{2}+{v}_{y}^{2}}$, where ${v}_{x}$ and ${v}_{y}$ are the velocity fields in the horizontal and vertical flow directions, respectively. The solid black line on each of the fields indicates the location of the solute line injection and the small squares indicate regions which we have zoomed in on so the flow features could be seen more clearly on the right side of the figure.

**Figure 2.**The integral of the squared concentration, M, versus time for the fully resolved random walk model and the different versions of the upscaled model described in Section 4.

**Figure 3.**The dilution index, E, versus time for the fully resolved random walk model and the different versions of the upscaled model described in Section 4.

**Figure 4.**Diagram illustrating the parameterization step for the upscaled model. Each particle moves by random walk over a distance of two cell lengths and the particle’s initial y position (${y}_{0}$), y position at the inlet of the second cell (${y}_{1}$), and travel times through the first and second cells (${\tau}_{1}$ and ${\tau}_{2}$, respectively) are recorded. The red circles represent an example of a single particle trajectory.

**Figure 5.**Breakthrough curves measured at distances 48$\lambda $, 96$\lambda $, 144$\lambda $, and 192$\lambda $ downstream from the inlet for both the fully resolved model (solid lines) and the Spatial Markov model (dotted lines) with cell lengths of (

**top**) 6λ, (

**middle**) 12λ, and (

**bottom**) 24λ. From these results, it was determined that a cell length of 24λ was an appropriate choice for our upscaled model.

**Figure 6.**Illustration of the downscaling procedure in the x direction. A particle is in Cell $n+1$ and will travel through that cell over an amount of time ${\tau}^{(n+1)}$. We want to determine the particle’s x location at time ${t}^{\ast}$.

**Figure 7.**Histogram of particle x locations at times (

**a**) t = 1, (

**b**) t = 20, (

**c**) t = 40, (

**d**) t = 60, (

**e**) t = 80, and (

**f**) t = 100 for both the fully resolved particle tracking simulations (solid black line) and the Spatial Markov model with the downscaling method described in Section 4.1 (dashed blue line). The bin size selected for the histogram is the same as the fully resolved grid resolution.

**Figure 8.**Illustration of the downscaling procedure Method 1 in the y direction. A particle is in Cell $n+1$ and will travel through that cell over an amount of time ${\tau}^{(n+1)}$. The particle’s x location is determined using the method described in Section 4.1. We want to predict the particle’s y location at time ${t}^{\ast}$. In this method, the downscaled y location is equal to each particle’s initial y position, ${y}_{0}$.

**Figure 9.**Illustration of the downscaling procedure Method 2 in the y direction. A particle is in Cell $n+1$ and will travel through that cell over an amount of time ${\tau}^{(n+1)}$. The particle’s x location is determined using the method described in Section 4.1. We predict the particle’s y location at time ${t}^{\ast}$ using Method 2.

**Figure 10.**(

**a**) The distribution of ${y}_{0}$ separated into ${N}_{z}=20$ equiprobable zones and (

**b**) the distribution of y values for each zone calculated in the parameterization step.

**Figure 11.**Illustration of the parameterization step for downscaling Method 3. For simplicity, the schematic only shows 4 zones, but in our simulations we separate the domain into ${N}_{z}=20$ equiprobable zones based on the initial y positions, ${y}_{0}$. As is depicted here, we measure the particle y positions at ${N}_{t}=20$ equally spaced locations in x across the first cell in order to capture the trajectory of each particle. We illustrate this with four sample trajectories for particles in Zone 2. The particle trajectories defined by ${y}_{t,1}$, ${y}_{t,2}$, …, ${y}_{t,{N}_{t}}$ may have y positions outside of their defined zone, but this does not change the set of y values associated with each zone ${Y}_{j}$ to which they contribute. The set ${Y}_{j}$ contains all y positions ${y}_{0}$, ${y}_{t,1}$, ${y}_{t,2}$, …, ${y}_{t,{N}_{t}}$ = ${y}_{1}$ for every particle that had ${y}_{0}$ in that zone.

**Figure 12.**Particle locations for the fully resolved simulation, Method 0, Method 1, Method 2, and Method 3 at $t=$ (

**a**) 1, (

**b**) 10, and (

**c**) 100.

**Table 1.**The mean absolute error of ${\mathrm{log}}_{10}\beta $, where $\beta $ is either $M\left(t\right)$ or $E\left(t\right)$, for each of the upscaled methods as defined by Equation (15).

ϵ(M) | ϵ(E) | |
---|---|---|

Method 0 | 0.5895 | 0.4888 |

Method 1 | 0.1275 | 0.0893 |

Method 2 | 0.1121 | 0.1217 |

Method 3 | 0.0352 | 0.0646 |

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**MDPI and ACS Style**

Wright, E.E.; Sund, N.L.; Richter, D.H.; Porta, G.M.; Bolster, D.
Upscaling Mixing in Highly Heterogeneous Porous Media via a Spatial Markov Model. *Water* **2019**, *11*, 53.
https://doi.org/10.3390/w11010053

**AMA Style**

Wright EE, Sund NL, Richter DH, Porta GM, Bolster D.
Upscaling Mixing in Highly Heterogeneous Porous Media via a Spatial Markov Model. *Water*. 2019; 11(1):53.
https://doi.org/10.3390/w11010053

**Chicago/Turabian Style**

Wright, Elise E., Nicole L. Sund, David H. Richter, Giovanni M. Porta, and Diogo Bolster.
2019. "Upscaling Mixing in Highly Heterogeneous Porous Media via a Spatial Markov Model" *Water* 11, no. 1: 53.
https://doi.org/10.3390/w11010053