See the upload.

This manuscript presents results of numerical simulations of localized porous flow in layered heterogenous media with a jump-discontinuity in the initial porosity and permeability fields. The authors employ a novel numerical method to solve the posed problem, and they show interesting results of a few simulations. The second part of the manuscript shows a potential use case of the results presented in the first part, for calculated element transport and partitioning. I find the topic interesting and relevant for the potential readers of GMD.

Unfortunately, it is not clear for me what big-picture message the manuscript aims to convey. The manuscript superficially touches on several key points, but in my opinion the depth of the discussion is not sufficient to really underpin these key points.

• Is this an article aiming to demonstrate the applicability and usefulness of the adaptive space-time method compared to other, traditional numerical methods?
• Is this an article which investigates the physics of jump discontinuities for localized fluid flow?
• Is this an article aiming to tackle geological problems related to fluid flow and element transport?

One of the key issues I found is that the authors use a novel numerical method, referring to submitted but not accepted manuscripts, without demonstrating the applicability of the method. Moreover, due to the small number of simulations presented and the extremely narrow parameter space explored, it is hard to judge the generality of the results. This would be acceptable if the authors would specifically target a given geological problem, but the geological description is very general while some of the chosen parameter values are far from typical (i.e. 0.1% background porosity). Finally, the figures have room for improvement (e.g. reducing empty spaces, balancing font size, positioning labels more intuitively).

To summarize, the work presented is worth publishing, but the big-picture message needs further maturation, and the presentation style needs further polishing. Therefore, I recommend accepting the manuscript after major revision.

Some specific remarks:

• Please include a benchmark figure comparing the result obtained by the novel method and with at least one traditional method for homogenous and discontinuous background porosity.
• Please demonstrate the suitability of the novel space-time method by validating it with a discontinuous analytical solution. I understand that such a solution might not exist for porous fluid flow channeling, but there are many other physical processes with known discontinuous solutions.
• Reading the manuscript, I have the impression that the authors consider their novel method superior to all traditional methods because it can handle discontinuities. This is an obvious advantage compared to (non-adaptive) staggered grid finite difference solvers. Nevertheless, it would be instructive to see how much numerical diffusion affects the results. It is not clear to me what is the advantage of the space-time method compared to finite element or finite volume methods with fitted meshes? Those methods were designed for and are regularly used for problems with discontinuous material properties.
• Please describe the boundary conditions used for the simulations. It seems like that in Figures 4-9 the anomalies either touch the boundaries or get very close to them. Please demonstrate that boundary effects are not distorting the results.
• Maybe choosing absolute porosity and pressure values is not the best way to show the results. I would include subplots showing porosity and pressure change compared to the initial values with a diverging colormap centered at zero.
• I find it confusing that the on many figures colorbars and their corresponding labels are on opposite sides of the figures.
• Choosing 0.1% background porosity with 0.1% as the maximum initial perturbation is extremely peculiar. This is two orders of magnitude lower than the typical range for most porous fluid flow applications. Please either demonstrate that the results are applicable for a wide porosity range or specify from the beginning the geological settings and processes for which these values are representative.
• The concentration equation is not just a transport equation. It is a reactive transport equation, as it implicitly forces a constant concentration ratio, implying mass/element transfer between the two phases. Please expand the description of the chemical model, and make sure the element partitioning is mentioned from the beginning.
• Equation 2 seems to imply that C must be conserved (if this is not the case, please explain it in the manuscript). Comparing the results of Figures 8-9, it seems that the total amount of C in the domain is different for the two models. Is that a boundary effect? If that is the case, please choose a model configuration where boundary effects are negligible.