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

*Reviewer 1:*Anonymous

*Reviewer 2:*Anonymous

**Round 1**

*Reviewer 1 Report*

The manuscript „Upscaling Mixing in Highly Heterogeneous Porous Media via a Spatial Markov Model“ is certainly an interesting piece of work along the research line of the authors. It is of interest for the community working on mixing processes in porous media and it entails enough novelty to merit publication. I found it clear and well written although some minor suggestions for improvement are suggested:

1. Introduction: the introduction could be enriched referring to works showing the importance of incomplete mixing for reactive solute transport of Ginn (2018, Modeling Bimolecular Reactive Transport With Mixing‐Limitation: Theory and Application to Column Experiments, WRR), the work of Oates, P. (2007, Upscaling reactive transport in porous media: laboratory visualization and stochastic models, Ph.D. thesis, Mass. Inst. Technol., USA.) and Chiogna and Bellin (2013, Analytical solution for reactive solute transport considering incomplete mixing within a reference elementary volume, WRR). Moreover, in lines 37-44, only the classical concept of mixing enhanced by large flow heterogeneity is presented, although it has been recently shown that flow topology plays also an important role without the need for large permeability contrasts (see for example Trefry et al., 2018 https://arxiv.org/abs/1808.03641; Ye et al., 2018, Effect of Anisotropy Structure on Plume Entropy and Reactive Mixing in Helical Flows, TiPM)

2. System Setup: the author should verify that the Reynold’s number is in the appropriate range to apply Darcy’s law. They should give the proper units to all parameters they apply. For example, the mean velocity is 1 and the constant isotropic diffusion coefficient 10^{-2 }(please correct the name in the paper: it must be a diffusion coefficient and not a dispersion coefficient since it does not depend on the velocity as it is assumed to be constant. This is impossible for transport in porous media). Since they are combined to build the Peclet number I assume they have the same units. Since the typical diffusion coefficient in porous media is 10^{-9}m^{2}/s it means that the flow velocity is something like 10^{-7}m/s and not 1 m/s, otherwise we would not obtain a Pe=800.

3. If I am not wrong, in the mixing literature the letter M is for the reactor ration (normalized dilution index). It is confusing to use it for Eq. 4 and I suggest to change it where needed.

*Author Response*

We thank the reviewer for their thoughtful review and comments. The reviewer’s comments are in bold, while our responses are in standard font.

**The manuscript „Upscaling Mixing in Highly Heterogeneous Porous Media via a Spatial Markov Model“ is certainly an interesting piece of work along the research line of the authors. It is of interest for the community working on mixing processes in porous media and it entails enough novelty to merit publication. I found it clear and well written although some minor suggestions for improvement are suggested:**

We are pleased that the reviewer finds our manuscript to be well written and a worthwhile contribution.

**1. Introduction: the introduction could be enriched referring to works showing the importance of incomplete mixing for reactive solute transport of Ginn (2018, Modeling Bimolecular Reactive Transport With Mixing****‐****Limitation: Theory and Application to Column Experiments, WRR), the work of Oates, P. (2007, Upscaling reactive transport in porous media: laboratory visualization and stochastic models, Ph.D. thesis, Mass. Inst. Technol., USA.) and Chiogna and Bellin (2013, Analytical solution for reactive solute transport considering incomplete mixing within a reference elementary volume, WRR). Moreover, in lines 37-44, only the classical concept of mixing enhanced by large flow heterogeneity is presented, although it has been recently shown that flow topology plays also an important role without the need for large permeability contrasts (see for example Trefry et al., 2018 ****https://arxiv.org/abs/1808.03641****; Ye et al., 2018, Effect of Anisotropy Structure on Plume Entropy and Reactive Mixing in Helical Flows, TiPM)**

Thank you for your suggestions. We have now cited these works in the revised manuscript in lines 21-23 and 45-49.

**2. System Setup: the author should verify that the Reynold’s number is in the appropriate range to apply Darcy’s law. They should give the proper units to all parameters they apply. For example, the mean velocity is 1 and the constant isotropic diffusion coefficient 10 ^{-2 }(please correct the name in the paper: it must be a diffusion coefficient and not a dispersion coefficient since it does not depend on the velocity as it is assumed to be constant. This is impossible for transport in porous media). Since they are combined to build the Peclet number I assume they have the same units. Since the typical diffusion coefficient in porous media is 10^{-9}m^{2}/s it means that the flow velocity is something like 10^{-7}m/s and not 1 m/s, otherwise we would not obtain a Pe=800.**

It is not technically a diffusion coefficient. It is a dispersion coefficient that is assumed constant, as is commonly done and which we make more clear now between lines 107 and 108. We are building on Le Borgne’s work original benchmark here.

All of the parameters presented are deliberately in arbitrary units and defined in terms of dimensionless numbers such as the Peclet number.

The Reynolds number would be hard to define in the setting we consider. By virtue of the fact that we are solving Darcy’s law we are assuming that flow at the pore scale is a low Reynolds number flow at the pore scale (but this is not resolved, as our work represents the porous domain through an equivalent continuum).

**3. If I am not wrong, in the mixing literature the letter M is for the reactor ration (normalized dilution index). It is confusing to use it for Eq. 4 and I suggest to change it where needed.**

M is used to refer to the integral of the squared concentration in the following works:

Tanguy Le Borgne, Marco Dentz, Diogo Bolster, Jesus Carrera, Jean Raynald de Dreuzy, and Philippe Davy. Non-Fickian mixing: Temporal evolution of the scalar dissipation rate in heterogeneous porous media. Advances in Water Resources, 33(12):1468–1475, 2010.

Nicole L Sund, Giovanni M Porta, and Diogo Bolster. Upscaling of dilution and mixing using a trajectory based Spatial Markov random walk model in a periodic flow domain. Advances in Water Resources, 103:76–85, 2017.

We prefer to keep our notation consistent with these existing publications.

*Reviewer 2 Report*

In this manuscript, the authors aim to extend the Spatial Markov model to predict effective solute mixing in an idealized heterogeneous porous medium. They develop four down-scaling methods within the upscaled model to predict measures of mixing and dilution of a solute moving through the 2D heterogeneous porous medium. Finally, the authors compare the results of each methods with a fully resolved random walk simulation and find good matches for most of their proposed methods.

I find this manuscript very interesting and well written. However, I have some concerns for this paper before it can be published in Water (see below).

(i) The *constant isotropic* dispersion coefficient *D*=10^-2 is used in the study of this paper. However, recent investigations (see below) indicate that the dispersion is grain-size and velocity dependent, and significantly affects the convective mixing of solute in porous media. Can the authors make some comments on that?

Effect of dispersion on solutal convection in porous media, Geophys. Res. Lett. 45, 9690-9698, 2018;

Rayleigh-Darcy convection with hydrodynamic dispersion, Phys. Rev. Fluids 3, 123801, 2018;

Three-dimensional structure of natural convection in a porous medium: Effect of dispersion on finger structure, Int. J. Greenh. Gas Control 53, 274-283, 2016

(ii) From the definition of equation (4), I think *M*(t) should increase with time, because initially there is no solute in the system (*M*(t=0) = 0). Why is it monotonically declined with time (figure 2)?

(iii) Figure 12(b): there is some discontinuous structure around the maximum x for the results of Method 3. Can the authors make some comments on that?

*Author Response*

We thank the reviewer for their thoughtful review and comments. The reviewer’s comments are in bold, while our responses are in standard font.

**In this manuscript, the authors aim to extend the Spatial Markov model to predict effective solute mixing in an idealized heterogeneous porous medium. They develop four down-scaling methods within the upscaled model to predict measures of mixing and dilution of a solute moving through the 2D heterogeneous porous medium. Finally, the authors compare the results of each methods with a fully resolved random walk simulation and find good matches for most of their proposed methods. **

**I find this manuscript very interesting and well written. However, I have some concerns for this paper before it can be published in Water (see below).**

Thank you for your accurate summary. We are pleased the reviewer finds our manuscript to be well written and an interesting contribution.

** (i) The constant isotropic dispersion coefficient D=10^-2 is used in the study of this paper. However, recent investigations (see below) indicate that the dispersion is grain-size and velocity dependent, and significantly affects the convective mixing of solute in porous media. Can the authors make some comments on that?**

**Effect of dispersion on solutal convection in porous media, Geophys. Res. Lett. 45, 9690-9698, 2018;**

**Rayleigh-Darcy convection with hydrodynamic dispersion, Phys. Rev. Fluids 3, 123801, 2018;**

**Three-dimensional structure of natural convection in a porous medium: Effect of dispersion on finger structure, Int. J. Greenh. Gas Control 53, 274-283, 2016**

This is a fair point and so we now make it more explicitly clear between lines 107 and 108 that we assume a constant value of dispersion coefficient for parsimony, but also discuss possible implications, including by citing the suggested papers. Given that several papers have recently shown that even typical velocity based dispersion models have problems when predicting mixing and mixing driven reactions (e.g. Massimo Rolle’s experimental work) we prefer to focus on the simpler model, which neglects the dependence of dispersion on velocity.

**(ii) From the definition of equation (4), I think M(t) should increase with time, because initially there is no solute in the system (M(t=0) = 0). Why is it monotonically declined with time (figure 2)?**

** **

Our solute line injection is introduced into our system at t=0, so there is solute in our system at this time. This is not a continuous injection, so we are not adding any more solute mass into our system after t=0. M(t) monotonically decreases in time because our solute line injection is mixing over time due to diffusion and advection by the non-uniform flow field. M describes the amount variability in the concentration field over time. As the concentration field becomes more uniform over time due to mixing, we observe a decrease in the value of M(t).

** **

**(iii) Figure 12(b): there is some discontinuous structure around the maximum x for the results of Method 3. Can the authors make some comments on that?**

Well observed! The discontinuous structure observed for Method 3 in Figure 12b is due to the fact that at each step in the Spatial Markov model (SMM) each particle selects a new y value from the set Y_{j }associated with that particle’s zone number *j*. This means that as a particle enters a new SMM cell it selects a new y value that is somewhere between the minimum and maximum values of Y_{j}, resulting in the observed structure as particles make small jumps in the y-direction within the range of Y_{j}. Figure 12b shows the point in time where we have particles beginning to enter the second SMM cell, so this effect is more visible. We have added a comment on this in lines 298-303 to clarify this point.

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**Round 2**

*Reviewer 1 Report*

The authors properly addressed my comments and the paper can be published in the present form.

*Reviewer 2 Report*

The authors have addressed all my concerns and I think the revised manuscript could be published in “Water” in the present form.