Comments on the paper HESS-2023-131 entitled: “Adjoint subordination to calculate backward travel time probability of pollutants in water with various velocity resolutions”; by Y. Zhang et al.

The study proposes both a theoretical framework and applications to backtracking particles in a context of non-Fickian solute transport within diverse compartments of surface and subsurface water flows.

The concept of backtracking, mainly developed to retrieve transit time distribution of solutes reaching a given location, is not new. However, it is here developed in a context where sub- and super-diffusion could occur. A partially homogenized transport equation to mimic both sub- and super-diffusion could be that of an advection dispersion equation (ADE), complemented with fractional derivatives of the concentration with respect to both time and space coordinates. Sub-diffusion occurs mainly in systems where the solute is reversibly trapped by the porous medium, resulting in a squared displacement spread of solute proportional to time to power  γ (< 1). For its part, super-diffusion results from preferential high-velocity pathways with the consequence of a solute displacement spread to the power α (< 2) proportional to time.

The resulting equation simulating transport is a fractional derivative equation, subordinated to the flow velocity field, named as S-FDE, for which backtracking is theoretically grounded in an Adjoint (to concentration) state equation. The authors develop this adjoint S-FDE, which has some physical meaning if it is solved backward in time and over a reversed flow field. The development of the adjoint S-FDE is complemented by changes of boundary conditions compared to that of the forward problem. Those are documented by the authors for a simplified 1-D sweeping flow. Then, the authors propose diverse numerical test cases to solve the adjoint S-FDE in a Lagrangian framework moving particles over space within a reversed flow field and back in time.

First of all, I must acknowledge that the paper is very well crafted, not to say excellent. Two reasons for that.

• The mathematics are sound, clear and concise, even if a few shortcuts may persist. Nothing wrong in that because there is nothing that could not be retrieved by any attentive reader analyzing the paper in depth.
• The test cases are duly selected to show that an S-FDE and its adjoint companion, are what I could name a smart adaptive Physics. The fractional coefficients evolve according to the weakly versus highly resolution degree of the velocity field. Weakly-resolved fields tend to lower the fractional coefficients, when highly resolved fields render fractional coefficients close to 1 and 2, resulting in an “quasi” ADE mimicking solute transport. In short, the S-FDE and its adjoint report on an up-scaled Physics adapted to the prior knowledge we have on the system. The demonstration is clear in the paper and puts dots on the I and crosses on the T regarding the versatility of a S-FDE.

After perusing the manuscript twice (in truth, 2.5 times!), I only denoted a very few very minor points (minuscule points?) that could be easily cured within half an hour.

• I guess that h in line 163, is some kind of objective function (as in an inverse problem). What is its form for an adjoint seeking the changes of the system if the source mass M0 is changed?
• Table 1. First row. A typo I think. Change reference to Eq. 1a into 13a, and ref. to 11a into 14a.
• Lines 514-515. Probably a typo again mixing units in cm/s and m/s. Otherwise I do not understand how the value reported in line 514 would enter the range reported in line 515.
• 7, up left plot. Change “moile” in the posted caption by “mobile”.
• Line 495. In my opinion the notion of backward location probability (BLP) is not fully clear. As far I understand a BLP is simply a “standard” backtracking, then post-processed to get particle densities over elementary surfaces of volumes, then normalized so that the sum of these densities over the domain is one? If I am right, I suggest to mention it as such in the manuscript.
• Line 532. In essence, I do not see why the calculation of BLP for non-point pollutant differs from that of the test case before and held over a homogeneous radial flow field. The radial flow field backtracked from a pumping well should result in a single location of the source. The complex flow field of the KRAA, backtracked from a single monitored location should result in multiple probable sources. In both cases, the calculation via the adjoint S-FDE should not change. It is not what is suggested by the sentence in line 532.

The manuscript develops an adjoint subordinated fractional-dispersion equation (S-FDE)

in order to estimate release times and source locations of contaminants in aquifers and

rivers. The author first (Section 2) present the three-dimensional forward S-FDE and then derive its

adjoint following the approach of Neupauer and Wilson and using fractional-order integration

by parts, and a fractional-order extension of the divergence theorem. Then, a Lagrangian

backward solver is presented based on the developments of the lead authors. The solver

is validated by comparison to finite difference solutions of the S-FDE. In Section 3,

the developed backward tracking methodology is then applied to three field scenarios to estimate the

release history of pollutants, and groundwater age. Section 4 discusses extensions of the

proposed method to identify pollutant source locations, and multi-scale subordinated models, relevant

for fractured media. This is an interesting contribution that adds to the literature on S-FDEs

and source and release time identification in aquifers and rivers. In the following, I list

 a few comments and recommendations:

Comments:

- Line 103: Could the authors give a physical explanation of the meaning of the space-fractional

advection term and the subordination to the velocity field? This is important because in field

applications, solute transport is typically advection-dominated. 

- Line 126 and following: The detailed derivations could be moved to an appendix.

- Lines 123-124: It is not clear what the authors mean here. Molecular diffusion should

model hydrodynamic dispersion? I assume the space fractional derivative should account

for dispersion. This should be clarified. 

- Lines 146 and 152: When the authors refer to fractional-order integration and

integration by parts of the spatial derivatives, do they mean the use of the divergence

theorem and its fractional-order extensions? This should be clarified.

- Section 2.2: This section refers extensively to previous work by the lead author. It

would be instructive for the reader if the authors could provide the Lagrangian equations

that are implemented in the solver.  

- Lines 297 and 302: What is meant by relatively homogeneous/heterogeneous? How are the K-fields

generated and how are they characterized (log-K variance, correlation length, etc.)?

- Line 300: Do the authors add diffusion to capture hydrodynamic dispersion? This needs to be clarified.