Dissolution and precipitation processes in reactive transport in porous media are ubiquitous in a multitude of contexts within the field of Earth sciences. In particular, the dynamic interaction between the reactive dissolution and precipitation processes and the solute transport is of interest as it is capable of giving rise to the emergence of preferential flow paths in the porous host matrix. It has been shown that the emergence of preferential flow paths can be considered to be a manifestation of transport self-organization in porous media as these create spatial gradients that distance the system from the state of perfect mixing and allow for a faster and more efficient fluid transport through the host matrix. To investigate the dynamic feedback between the transport and the reactive processes in the field and its influence on the emergence of transport self-organization, we consider a two-dimensional Darcy-scale formulation of a reactive-transport setup, where the precipitation and dissolution of the host matrix are driven by the injection of an acid compound, establishing local equilibrium with the resident fluid and an initially homogeneous porous matrix, composed of a calcite mineral. The coupled reactive process is simulated in a series of computational analyses employing the Lagrangian particle-tracking (LPT) approach, capable of capturing the subtleties of the multiple-scale heterogeneity phenomena. We employ the Shannon entropy to quantify the emergence of self-organization in the field, which we define as a relative reduction in entropy compared to its maximum value. Scalability of the parameters, which characterize the evolution of the reactive process, with the Peclet number in an initially homogeneous field is derived using a simple one-dimensional ADRE model with a linear adsorption reaction term and is then confirmed through numerical simulations, with the global reaction rate, the mean value, and the variance of the hydraulic-conductivity distribution in the field all exhibiting dependency on the reciprocal of the Peclet number. Our findings show that transport self-organization in an initially homogeneous field increases with time, along with the emergence of the field heterogeneity due to the interaction between the transport and reactive processes. By studying the influence of the Peclet number on the reactive process, we arrive at a conclusion that self-organization is more pronounced in diffusion-dominated flows, characterized by small Peclet values. The self-organization of the breakthrough curve exhibits the opposite tendencies, which are observed from the perspective of a thermodynamic analogy. The hydraulic power, required to maintain the driving head pressure difference between the inlet and outlet of the field, was shown to increase with the increasing variance, as well as with the increasing mean value of the hydraulic-conductivity distribution in the field, using a simple analytic model. This was confirmed by numerical experiments. This increase in power, supplied to the flow in the field, results in an increase in the level of transport self-organization. Employing a thermodynamic framework to investigate the dynamic reaction–transport interaction in porous media may prove to be beneficial whenever the need exists to establish relations between the intensification of the preferential flow path phenomenon, represented by a decline in the Shannon entropy of the transport, with the amount of reaction that occurred in the porous medium and the change in its heterogeneity.

Dissolution and precipitation processes in reactive transport in porous media play an important role in multiple contexts in the field of Earth sciences, such as geological

These wormholes that funnel the flow can be regarded as preferential flow paths, ubiquitous in heterogeneous porous media, where most of the transport is concentrated. The importance of preferential flow paths in determining the transport properties of the porous media is widely recognized as they allow rapid solute transport and alter residence times

The findings discussed above have brought us closer to classification of the emerging preferential flow paths in reactive transport in porous media as an embodiment of self-organization. Before addressing the specific problem of the emergence of preferential flow paths, it is in order to first discuss some basic concepts related to the phenomenon of self-organization in physical systems. Self-organization refers to a broad range of pattern formation processes, occurring through interactions internal to the system without intervention by external directing influences

Within the thermodynamic framework,

While the thermodynamic framework refers to physical entropy, as introduced by

Among the family of numerical methods employed to simulate transport and reaction problems in porous media, the Lagrangian particle approach continues to gain more prominence in the recent decades, along with the continuing development of powerful computers

To investigate the dynamic interaction between the transport and the reactive process in an initially homogeneous porous medium and its influence on the emergence of transport self-organization in the medium, we consider a two-dimensional Darcy-scale formulation of a reactive-transport setup, where precipitation and dissolution in the medium are driven by the injection of an acid compound, establishing local equilibrium with the resident fluid and an initially homogeneous porous medium, composed of calcite mineral. The coupled reactive process is simulated in a series of computational analyses where the low-pH water is injected into an initially homogeneous domain, at first in equilibrium with the resident fluid (high-pH water). We employ a Lagrangian particle-tracking (LPT) approach, capable of capturing the subtleties of the multiple-scale heterogeneity phenomena, along the lines of

In particular, we are interested in the influence of the transport Peclet number on the reactive process and the emergence of self-organization of the transport in the porous medium. To investigate this relation, we simulate a number of reactive-transport scenarios for different values of advective to diffusive transport rates, characterized by the Peclet number. This is achieved by applying different values of the inlet–outlet hydraulic-pressure-head-drop boundary condition to the field. We employ Shannon entropy along the lines of

The key component in dissolution–precipitation reactions is the stoichiometric equilibrium between the reactants and the products. In the system under investigation, the simulated reactive-transport scenario is that of an injected acid compound (low-pH water), establishing local equilibrium with the resident fluid (high-pH water) and the porous medium, composed of calcite mineral. This fairly common setting in the field of geosciences constitutes the specific case of

To investigate the dynamic coupling between the transport and reactive processes in the porous medium, we consider a two-dimensional field of dimensions

The solute transport across the field is simulated using a Lagrangian particle-tracking approach

The injected

At first, no

The transport part of the model has been validated against the well-known case of one-dimensional instantaneous injection in a homogeneous medium, for which an analytical solution exists

In the system under investigation, the kinetic reactive process operates according to an algorithm developed to mimic the actual chemical dissolution–precipitation reactions that take place in practice. After all existing

After that, the cell porosity

The Peclet number plays an important role in our reactive system. As a usual practice, Peclet number is calculated based on an Eulerian length scale, such as the mean grain or pore diameter in the pore-scale simulation or the characteristic correlation length of the heterogeneous porous media for the case of the Darcy-scale simulation

To investigate the influence of the Peclet number of the transport on the evolution of the reactive process and the emergence of transport self-organization in the field, we simulate a number of reactive-transport scenarios for different values of the Peclet number. This is achieved by applying different values of the inlet–outlet hydraulic-head-drop boundary condition

The computational setting described in the previous section mimics the dynamics of a coupled dissolution–precipitation reactive process in a calcite porous medium, leading to the emergence of heterogeneity in an initially homogeneous field. Previous studies have shown that self-organization of the solute transport in the field is expected to emerge in such a situation in the form of preferential flow paths that lead to solute concentration gradients in the direction transverse to flow

Section S1 in the Supplement presents a simple heat transfer example that illustrates how an open thermodynamic system can be maintained in a stationary non-equilibrium state through an inflow of energy. Following the same concept, one can generalize this finding for the system under investigation, represented by a snapshot of the reactive field at a specific computational time, where the non-reactive solute transport self-organization in the field in terms of emergence of preferential flow paths is the outcome of the coupled reactive process that introduces heterogeneity in an initially homogeneous field. In our system, energy influx occurs in the form of hydraulic power, supplied to the flow to overcome the hydraulic resistance of the field by the applied hydraulic-pressure-head-drop boundary condition

To identify the driving mechanism that leads to an emergence of heterogeneity in an initially homogeneous porous medium followed by self-organization of the solute transport in the field, let us consider two limiting cases related to the nature of the transport mechanism in the field, as applicable to our reactive-transport setting, described in Sect.

Following the above discussion, we argue that, for the transport self-organization to emerge in an initially homogeneous field undergoing a dissolution–precipitation reactive process as defined in our reactive setup, it is necessary for the transport mechanism to include both diffusive and advective contributions. Here, the stochastic diffusion leads to local concentration variations in

In an attempt to derive an analytical justification for this, let us relate the heterogeneity, emerging in an initially homogeneous field due to the coupled reactive-transport process, to the global reaction rate in the field. The reasoning behind this is that, in the initially homogeneous field, reaction that occurs in the field is the driving force behind the emergence of heterogeneity there; heterogeneity, in turn, is responsible for the emergence of transport self-organization in the field. While some of the reaction events that occur may cancel each other out in terms of dissipating or precipitating some of the calcite in a computational cell, the remaining part of reaction is useful in creating heterogeneity. Thus, we may speculate that the reaction rate and the emergence of heterogeneity in the field are directly related. Consider, for simplicity, a 1D advection–diffusion reaction equation (ADRE):

Having obtained a qualitative understanding of the subject of self-organization in the context of reactive transport of the porous medium, we shall now seek a way to characterize this phenomenon quantitatively using the concept of Shannon entropy, also referred to as information entropy. Shannon entropy was introduced originally in the field of communication theory, whose fundamental problem is formulated as “reproducing at one point either exactly or approximately a message, selected at another point”

The definition of information entropy, given by Shannon, is equivalent to physical entropy in statistical mechanics as defined by Gibbs, where the logarithm in Eq. (

To characterize the emergence and development of transport self-organization in an initially homogeneous field as the dissolution–precipitation reactive processes in the field evolve, we adopt a straightforward use of the Shannon entropy, in a similar vein as

Following the thermodynamic framework, as applied to the system under investigation, energy must be invested in the heterogeneous field to maintain an ordered state of solute transport there. This energy comes in the form of hydraulic power that enables the flow to overcome the hydraulic resistance of the porous medium under the applied hydraulic-pressure-head-drop boundary condition

In an attempt to better understand the relation between the degree of heterogeneity of the porous medium and the hydraulic power required for the fluid to overcome the resistance of the medium under the inlet–outlet hydraulic-head-drop boundary condition

We emphasize that the presented study does not intend to construct a complete thermodynamic formalism for the problem under investigation, such as the one that is presented in

Based on the reactive-transport algorithm described in Sect.

We begin by examining the evolution of the reactive process, as depicted by the snapshots of the relative hydraulic conductivity

Evolution of the relative hydraulic conductivity

The influence of the Peclet number on the evolution of the reactive process in the field is characterized in Fig.

Influence of Peclet number on the evolution of the reactive process in the field over time:

To verify the dependency of the global reaction rate on the reciprocal of the Peclet number, as suggested by the theoretical result obtained in Sect.

Figure

Figure

Similar trends are observed in Fig.

The dependence of the parameters, depicted in Fig.

The relatively small scale of the phenomenon presented in this study must be pointed out. The initial state of the porous medium is completely homogeneous, and the heterogeneity that develops is relatively minor. The trends presented could have been more pronounced quantitatively had the simulation been allowed to run longer. The simulation was stopped soon after pore volume time was reached due to computational time limitations since running the LPT code with a large number of particles consumes considerable computational resources. Had the model been allowed to run longer, the mean conductivity and the heterogeneity trends are expected to have further increased, while the dependency of the reactive-transport evolution on the reciprocal of Peclet is expected to have persisted as long as the degree of heterogeneity of the field remained moderate (see Sect.

To characterize the emergence and development of transport self-organization in our model as the dissolution–precipitation reactive processes in the field advance, we adopt a straightforward use of the Shannon entropy, similarly to

The emergence of transport self-organization in the reacting field is evident from the Fig.

Evolution of the transport self-organization in the field for

The deviations of the decimal logarithm of the relative non-reactive tracer concentration

The observations from Fig.

Self-organization of the transport in the field via normalized Shannon entropy

In this context, it is interesting to consider the findings of

The Peclet number scalability is also exhibited in the Shannon entropy of transport self-organization. Assuming again the separation of variables in

Here, again, the relatively small scale of the presented phenomenon must be pointed out, although the apparent tendencies are clear. Had the model been allowed to run longer, a further increase in heterogeneity would lead to a more pronounced decrease in the Shannon entropy of the transport. The dependency on the reciprocal of the Peclet number is expected to persist as long as the degree of heterogeneity of the field remains moderate (see Sect.

As a next stage in the analysis of transport self-organization in the reactive field, we turn to the particle breakthrough curve in an attempt to analyze the influence of self-organization of the field's transport on the particle arrival times. Figure

Breakthrough curve self-organization:

The Shannon entropy of the breakthrough curves was calculated by dividing the arrival time span into 1000 bins and employing Eq. (

A picture emerges where the Shannon entropy of the arrival times increases with the passage of time, reflecting a larger uncertainty and a declining order in the temporal distribution of travel times. This corresponds to an increase in both the field's heterogeneity and the self-organization of transport as the reactive process in the field evolves. We also observe that the level of self-organization in arrival times directly correlates with the Peclet number as the normalized breakthrough entropy

The Peclet number scalability is also exhibited in the Shannon entropy of the breakthrough curve, from approximately

The evolution of the reactive process in an initially homogeneous field is accompanied by the emergence of heterogeneity and, consequently, transport self-organization in the field. This demands that increased energy is supplied to the field to maintain an increasingly ordered state that emerges there. This energy is supplied in the form of hydraulic power, required for the flow to overcome the hydraulic resistance of the porous medium. In our model, the hydraulic power is supplied due to the inlet–outlet hydraulic-head-drop boundary condition applied to the porous medium, which works similarly to a pump.

Figure

Hydraulic power in the field and its relation to heterogeneity and transport entropy:

The confirmation of the proposition put forth in Sect.

Finally, Fig.

Our computational study tackles the quantitative characterization of the transport self-organization that emerges in an initially homogeneous calcite porous medium due to the dynamic interaction between the reactive precipitation–dissolution processes and the solute transport. Our work leads to the following key conclusions:

As the reactive particles advance and react in an initially homogeneous field, heterogeneity is introduced into the field in the form of local dissolution–precipitation areas, evolving further downstream and intensifying with time. A dissolution area is located in the immediate vicinity of the inlet, where the chemical equilibrium is tilted towards dissolution; this correlates with experimental and simulation observations. The global reaction rate, normalized by the distance from the inlet sampled by the flow, is shown to be approximately constant in time. The influence of the Peclet number is exhibited by an increase in the global reaction rate with the decrease in Peclet, thus confirming that the diffusive transport mechanism, responsible for the mixing in the field, enhances the reactive process. This can be explained by the fact that diffusion, being stochastic in nature, allows particles to sample regions away from the path suggested by the advection mechanism, thus allowing a better chance for reaction. The net reaction is tilted towards dissolution as the mean conductivity value in the field grows monotonously with time, as well as with the Peclet number, which is reasonable due to the influx of low-pH fluid at the inlet of the field. Similar trends are observed in the evolution of the variance of hydraulic conductivity in the field. Scalability of the parameters that characterize the evolution of the reactive process in the field with the reciprocal of the Peclet number was derived using a simple one-dimensional ADRE model with a linear adsorption reaction term and then confirmed through numerical simulations, with the global reaction rate, the mean value, and the variance of the hydraulic-conductivity distribution in the field all exhibiting dependency on the reciprocal of the Peclet number. We thus state that reactive-transport scenarios for lower Peclet values, corresponding to the dominant diffusive transport mechanism, coincide with an increased global reaction rate in the field and, thus, an increase in the field's heterogeneity. The dependence of the parameters that characterize the reactive process in the field on the reciprocal of the Peclet number confirms that this dimensionless number is indeed the driving force behind the evolution of the reactive process in the porous medium.

As the reactive processes in the field advance, the evolving variations in the hydraulic conductivity of the field create an autocatalytic feedback mechanism that leads to an emergence of finger-like preferential flow paths of a linear shape. These paths interact, competing for the available flow, so that eventually some of the paths carry a significantly larger part of the injected particles than the others, as seen from the increasing concentration gradients in the direction transverse to flow. This observation is confirmed by the mean normalized Shannon entropy in the field that decreases with time, signifying an increase in the level of transport self-organization in the field. The influence of the Peclet number on the evolution of transport self-organization is exhibited by a decrease in the mean normalized Shannon entropy of the transport with a decrease in Peclet number, signifying an increase in self-organization. This clearly indicates that diffusion is the dominant mechanism in creating self-organization in the initially homogeneous field. Here the link between the emergence of heterogeneity and transport self-organization in an initially homogeneous field is being drawn as the increase in heterogeneity results in the increase in the level of self-organization of the transport in the field. Peclet number scalability was shown for the mean normalized Shannon entropy curves as well. To conclude, we state that reactive transport at lower Peclet numbers, corresponding to diffusion-dominated flow, results in a higher field heterogeneity and, thus, stronger transport self-organization in the field. An important note should be made regarding the initial state of the field that undergoes the reactive process: in the case of an initially homogeneous field, diffusion acts as self-organization enhancer; in an initially heterogeneous case, diffusion is capable of decreasing the transport self-organization due to its smoothing property.

To switch the observing perspective, the particle temporal breakthrough curve was analyzed in an attempt to understand the influence of self-organization of the field's transport on the particle arrival times. The width of support of the breakthrough curve clearly increases with time as the reactive process in the field evolves, indicating an increasing scatter in arrival times of non-reactive particles; this result is in line with an increase in the field's heterogeneity. A picture emerges where the Shannon entropy of the arrival times increases with the passage of time, reflecting a larger uncertainty and a declining order in the temporal distribution of travel times. We also observe that the level of self-organization in arrival times directly correlates with the Peclet number as the normalized breakthrough curve entropy decreases with an increase in Peclet number, an opposite tendency compared to the entropy of the field transport. While there are other factors that affect the total entropy budget of the problem, such as production of the macroscopic-flow entropy due to hydraulic power dissipation through heat, these two properties are correlated in a self-consistent way. Thus, a thermodynamic analogy can be made: since, according to the second law of thermodynamics, the overall entropy of the system and its surroundings cannot decrease, the decreasing entropy of the system, represented, among other contributions, by the entropy of the transport in the field, needs to be exported outside. This leads to an increase in the entropy of the surroundings, which is reflected, among other factors, by an increase in the temporal breakthrough curve entropy. This analogy should be regarded on a qualitative level only.

The evolution of the reactive process in an initially homogeneous field, accompanied by the emergence of heterogeneity and, consequently, transport self-organization in the field, demands that increased energy is supplied to the field to maintain the increasingly ordered state that emerges therein. This energy is supplied in the form of hydraulic power, required for the flow to overcome the hydraulic resistance of the porous medium. We observe that, for all Peclet numbers, the total dissipated hydraulic power increases with time, along with the increase in heterogeneity and mean hydraulic conductivity of the field, confirming the trends suggested by a simple parallel-channels model. The entropy of the transport in the field decreases with an increase in the hydraulic power, reflecting an increase in transport self-organization. This increasing power is supplied to the field in order to maintain the hydraulic-head-drop boundary condition due to an increase in the field's heterogeneity. Following our thermodynamic framework, we thus argue that the power, required to maintain the driving head pressure drop boundary condition between inlet and outlet, increases with the increasing variance of the hydraulic conductivity in the field due to the evolution of the reactive process therein (although the contribution of an increase in the mean conductivity value also should not be forgotten in this context). This increase in the supplied power results in an increase in the level of transport self-organization in the field.

To conclude, the scenario presented in the paper corresponds to that of an open thermodynamic system that interacts with its surroundings by exchanging matter and energy. Due to the influx of power from outside, this system is kept in a non-equilibrium state that corresponds to a certain degree of internal self-organization and, thus, to a decreased entropy state. This decrease in the entropy of a system corresponds to an increase in the entropy of the surroundings by means of production of entropy in the system through various processes, which is later exported outside. In our system, we investigate the interplay between three thermodynamic parameters: the entropy of transport self-organization within the field; the entropy of the breakthrough curve; and the hydraulic power, dissipated by the flow, which may be viewed as an influx of power required to maintain the current level of the system's self-organization. While there are additional processes that may influence the overall entropy budget, such as production of the macroscopic-flow entropy due to hydraulic power dissipation through heat, we find that these three properties are correlated in a self-consistent way. The contributions of the above-mentioned additional factors are beyond the scope of the current study. The increase in the hydraulic power, dissipated in the field, and the emergence of transport self-organization are both the result of an increase in the heterogeneity of the field (although the contribution of an increase in the value of the mean hydraulic conductivity to the hydraulic power also should not be overlooked). The latter, in turn, can be viewed as a consequence of the energy invested in the field by the dissolution–precipitation reactive process. We emphasize that the presented study does not intend to construct a complete thermodynamic formalism for the problem under investigation. The thermodynamic framework presented in the current study is aimed at providing qualitative dependencies and/or trends between the above-mentioned parameters of interest. It is our intent to arrive at a more complete thermodynamic formalism for the reactive flow in porous media in the course of the research work.

Employing a thermodynamic framework to investigate the dynamic reaction–transport interaction in porous media may prove to be beneficial whenever the need exists to establish relations between the intensification of the preferential flow path phenomenon, represented by the decline in Shannon entropy of the transport, with the amount of reaction that occurred in the porous medium and the change in its heterogeneity. This can be of considerable significance to the implications of reactive-transport interaction in various geophysical applications and can assist, for example, in the estimation of efficient ways to remediate soil contamination, the determination of optimal conditions for

Codes and data are available on a dedicated GitHub repository upon request to Evgeny Shavelzon (eshavelzon@campus.technion.ac.il).

The supplement related to this article is available online at:

ES developed the research methodology, wrote major parts of the simulation code, performed numerical simulations, analyzed data, and wrote the paper. YE supervised the implementation of the research methodology, supplied the initial version of the simulation code, and oversaw the writing of the paper.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

This research has been supported by the Israel Science Foundation (grant no. 801/20).

This paper was edited by Harrie-Jan Hendricks Franssen and reviewed by three anonymous referees.