DamkÃ¶hler Number Input to Transport-Limited Chemical Weathering

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Damköhler number input to transport-limited chemical weathering calculations Fang Yu, and Allen Gerhard Hunt ACS Earth Space Chem., Just Accepted Manuscript • Publication Date (Web): 30 Jan 2017 Downloaded from http://pubs.acs.org on January 31, 2017

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ACS Earth and Space Chemistry

Damköhler number input to transport-limited chemical weathering calculations Fang Yu1 Allen Gerhard Hunt1 1. Department of Earth & Environmental Sciences, Wright State University [email protected]; [email protected] Fawcett Hall 265, 3640 Colonel Glenn Hwy, Dayton, OH 45435-0001 Abstract Chemical weathering of the silicate minerals in the Earth’s crust is the dominant influence on variability of atmospheric carbon on time scales of about 1Ma and up, lending significance to the ability to predict such weathering rates. Field weathering rates at such large time scales tend to be much slower than laboratory values, however. It has been proposed that the discrepancy occurs because field weathering rates are solute transport-limited, rather than reaction kinetics-limited. In order to assess the relative importance of reaction kinetics and solute transport, the Damköhler number, DaI, a ratio of a transport time to a reaction time can be used. Unfortunately, even in those experiments that suggest importance of transport limitations, the traditional calculation of DaI yields values close to 1 and an extreme sensitivity of experimental reaction rates on DaI. We develop a new method of calculating DaI based on the theory of non-Gaussian solute transport, appropriate for heterogeneous porous media. Our results are that weathering rates are less sensitive to the calculated DaI, and that DaI is now equally sensitive to input parameters, generating a much more straightforward assessment of the relevance of transport limitations to chemical weathering. Our most important single inference appears to be that for field conditions at larger time scales (roughly decadal and up), chemical weathering is practically always transport-limited. Keywords: Damköhler Number, Transport-limited, Chemical Weathering, Percolation Theory, Solute Transport Introduction Chemical weathering is a process by which minerals that form in a deep subsurface environment are altered when exposed to the near surface environment and, as such, is a fundamental component of the rock cycle of geology. Although the Earth’s crust contains a very wide range of minerals, the dominant constituents of oceanic and continental crust, basalt and granite, respectively, consist mainly of aluminosilicates. The classic, Urey, reaction1 is a short-hand representation that indicates that one mole of CO2 from the atmosphere is drawn down for every mole of silicate mineral weathered, CO2 + CaSiO3 ↔ CaCO3 + SiO2. This reaction is one of the key inputs to the modulation of the global carbon cycle, and thus climate change2, over time scales of about 1Myr to tectonic time scales of roughly 200Myr. Consequently, there is a high motivation to being able to predict the rate at which this reaction proceeds over such time scales. It has been recognized at least since 2003, however, that chemical weathering rates measured in the laboratory bear no resemblance to those inferred from the field3. In particular, field derived values are as much as 5 orders of magnitude lower than laboratory rates.

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While there has been considerable discussion addressing the causes of the discrepancy between field and laboratory silicate weathering rates, including the potential role of solute transport4-8, only very recently has it been possible actually to derive a single expression that links reaction rates on the two vastly different time scales9-11. The theoretical approach that led to this reconciliation asserts that virtually all field measurements are limited by the efficiency of the solute transport to deliver the reaction reagents (e.g., CO2) to, and remove the products (e.g., SiO2 and CaCO3) from, the weathering front11. While the derivation proves successful, its presentation lacks the needed comparison of the time scales of advection, τA, and reaction, τR. Such a comparison is conventionally given in terms of the Damköhler number12, DaI = τA / τR. However, the usual calculation12 of this ratio is flawed. There the calculation of τR is based on the observed reaction rate, rather than its well-mixed value. Such a choice places the large advection time in the denominator of DaI, rather than where it belongs, if the rate is slowed by transport, in the numerator. This paper first fixes the inconsistency in the calculation of DaI, and then re-examines data from oft-referenced papers in this context. This correction allows a consistent assessment of the controlling factors on chemical weathering rates at the Earth’s surface. The result is that essentially all weathering reactions at the Earth’s surface on time scales exceeding days are transport-limited as previously contended10,11, though many laboratory measurements are limited by the kinetics. As already shown, the prediction of chemical weathering rates as a function of time also generates the soil production rate, as well as the sequestration rates of both carbon and nitrogen within the soil11, making this a solution of considerable relevance. The link between chemical weathering and soil production is partly a matter of definition. Chemical weathering of the bedrock13, or unconsolidated surface materials from, e.g., fluvial14 or glacial15 deposition, landslides16, treethrow17, etc., is generally understood to be the fundamental process that initiates the formation of soil. In particular, the bottom of the subsoil, or Bw horizon, is called the weathered depth, with the subscript w indicating weathering, and referring to the presence of oxides (particularly of iron), giving this soil horizon its typically red color18. Thus, soil depth, when measured to the Bw horizon, should serve as an indicator for the total volume of weathered material, while the soil production (which is the temporal derivative of the soil depth), will be proportional to the chemical weathering rate11. Then the lowest order approximation for soil carbon and nitrogen sequestration rates, as pointed out in [19], will be that they are proportional to the soil production rate. We discuss first the prevalence20 of non-Gaussian solute transport and heavy-tailed solute distributions2124 , and within this context the only (percolation) theoretical formulation that allows explicit predictions2528 . Then we pass to the present theoretical description of heavy-tailed solute distributions from spatially variable advection in terms of percolation theory29-32. A brief summary of evidence for the relevance of such heavy-tailed solute distributions to geochemistry is given, followed by a calculation of the Damköhler number. We emphasize that our theoretical calculations of non-Gaussian distributions give predictions of chemical weathering scaling (and related properties) when conditions are clearly in the transport-limited regime. However, as will be seen, our results are not accurate when such reactions are limited by the kinetics of well-mixed fluids, nor in an intermediate regime, when part of the medium is kinetically limited and part is transport-limited. A more comprehensive theoretical approach that can treat quantitatively such a cross-over as well as the transport-limited regime is yet to be developed. But our new calculation of the Damköhler number does at least give an objective means to predict a length scale, and thus also a time scale, at which chemical weathering becomes transport limited.


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Solute Transport Theory Solute transport as developed within percolation theory generates fat-tailed arrival time distributions, of approximately power-law form, like the Continuous Time Random Walk (CTRW)33-36, or the Fractional Advection Dispersion equation (FADE)37-40. In contrast to both of these techniques, however, percolation theory does not yield an arbitrary power, but a value that is specified mostly by the flow conditions41 and, to a lesser extent, the properties of the medium31. The theoretical procedure generates the entire arrival time distribution as well29-32. According to Sahimi42, however, this procedure may be simplified by applying the argument that, whenever the dominant flow paths are describable using percolation theory (in the form of critical path analysis), solute transport follows basic percolation scaling laws. These two distinct perspectives lead to somewhat different predictions, but the difference is not great, and the simplification produced by using a completely analytical scaling law is worth the slight degradation in accuracy of its theoretical predictions at long time scales, particularly since it enables a simple calculation of a dimensionless number used to assess the relevance of transport vis-à-vis kinetic limitations, called the Damköhler number12. We applied29-32 the framework of critical path analysis43,44, developed within percolation theory45-48, to a network representation of the medium49-51 in order to quantify all distinct solute transport paths that can be connected through the system, and the fluid fluxes along those paths. Critical path analysis and concepts from percolation theory are relevant to fluid flow in all disordered media, because the tendency for flowing fluids to exploit paths of least resistance52-55 is quantifiable using percolation theory. Knowing the topology of each system of paths27,41,56,57 together with the associated fluid fluxes, made it possible to calculate the variations in the characteristic velocity along the path, and thus the total solute transport time29-32. This probabilistic treatment of spatially variable advection, which later incorporated molecular diffusion30, generated the solute arrival time distribution as a function of transport distance in agreement with simulation58, and predicted32 1) the entire observed solute arrival time distribution 59,60 as a function of saturation, 2) the typical system crossing time31 for conducting particles61-66as a function of length, 3) [30] the known35,67-70 scaling of the dispersion coefficient with Peclet number, 4) and the range of observed values of the dispersivity in 2200 experiments31, as well as the Gelhar “rule of thumb71” over 10 orders of magnitude of length scale. Although previous work developed full solute arrival time distributions, and thus the solute flux, a simple scaling theory, which gives nearly the same results for a solute velocity (anticipated by Sahimi42) is sufficient for the proposed research. Three scaling relationships together10, water transport distance, solute transport along optimal paths confined near the surface, and solute transport through bulk three dimensional media, limit crop height (x), natural vegetation height (x), and soil depth (x), respectively, as functions of time (ca. 8000 data points). From percolation theory, the solute transport time, t, increases more rapidly than linearly with transport distance, x, in particular, as a power of x equal to the percolation backbone fractal dimensionality, Db27. Db = 1.8741 for wetting conditions, or of full saturation, and for which the network of pores supporting the fluid flow is fully three-dimensional; thus the advection time is calculated as,

= ( ) .

(1)

with L0 a particle size and L0/t0 = v0, a pore-scale fluid velocity. Relevance of Solute Transport to Chemical Weathering



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Our recent study9 assumed a proportionality of chemical weathering rates to solute transport velocities, and thus to fluid flow rates. That this is an experimental fact can be seen from Fig. 1, digitized from [8]. This fundamental result indicates transport limitations of these reactions. If, in contrast, the reaction were controlled by surface reaction rates, the rate would not be affected by the flow magnitude, as depicted in Fig. 2. With increasing length and time scales and for certain classes of non-Gaussian transport21, solute transport limitations become increasingly important to chemical weathering. In particular, the requirements for Gaussian transport can fail if either the first moment, or the second moment (a less restrictive condition) of the solute arrival time does not exist21. In the former case, the velocity of the centroid of the spatial solute distribution is a negative power of the time, and this is precisely the prediction of percolation theory above. In reality, of course, reaction rates are not always proportional to fluid flow rates. If the fluid flux is large enough, the sample small enough, and the mixing adequate, reaction rates are limited only by reaction kinetics. Given that, in our theoretical framework, solute velocities decline as a power of the time, for long enough time periods (or large enough distances), transport limitations must set on. This means that we expect a cross-over from kinetic controlled reactions to transport-limited reactions with increasing time (or length) scale, as shown schematically in Fig. 2. The actual dependence of a weathering rate on system length will, of course, only approach these two predictions asymptotically, but the possibility to predict reaction rates at large length and time scales must depend sensitively on the ability to determine the length scale at which the two predictions cross-over. In our framework, the calculation of the Damköhler number is the means to calculate this length scale, as well as the corresponding time scale. Calculation of Damköhler number The basis for our calculation of the Damköhler number is provided by an analogy to the Peclet number, which determines the relative importance of advection to diffusion72. The diffusion time is calculated without reference to advection, while the advection time is calculated neglecting diffusion, and the ratio evaluated. Of course, when both advection and diffusion are important, similarly to when transport begins to limit weathering reactions, neither process is completely independent from the other. Existing calculations12 of the Damköhler number define DaI as, DaI =

=

∗ ,

.

(2)

∗ !

Here "#$ is the advection time, "% the reaction time, L the column length, v the fluid flow velocity, Vp the pore volume description of fluid flow rate, C an equilibrium concentration, &'()* the measured reaction rate normalized by surface area, and + , the surface area. Note that the subscript -./01 makes Eq. (2) specific to [12]. This procedure underestimates the advection time by assuming that the solute velocity is constant rather than diminishing with transport distance through the column. It also has adverse consequences in conceptualization by using the measured reaction rate: 1) reporting measured reaction rates in terms of the measurement-derived DaI does not test theory against experiment, although it did a fair job of collapsing data to a single relationship (our scaling procedure does so even more effectively), 2)


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representing reaction-rate slowing as an increase in the factor in the denominator ("% ) makes DaI decrease with increasing distance rather than increase, as would be consistent with slowing advective transport and increasing the numerator ("#$ ). In a practical sense, wide ranges of physical variables, such as column lengths and velocities, become compressed into narrow ranges of DaI, accentuating the apparent dependence of reaction rates on such quantities, leading to false conclusions regarding the suitability of using DaI as a proxy for system length9. Since observed reaction rates decrease with increasing length scales in accord with the limitations provided by transport, our calculation of DaI will be performed for conditions at the column end, rather than the entire column. DaI is proposed here to be best calculated as, DaI =

=

∗ ( )2.34 ∗ , ∗ !

.

(3)

Here L is the column length, Lp a particle size or pore separation, v the fluid flow velocity, Vp the pore volume description of fluid flow rate, C an equilibrium concentration, and RMg the surface-normalized reaction rate under well-mixed conditions (horizontal lines in Fig. 2). In general the well-mixed rate will be denoted by Rm. + , the surface area. We note that our calculation substitutes (L/Lp)1.87 for (L/Lp) in the numerator of the published calculation12. In summary, we have calculated the advection time in the absence of a reaction, and the reaction time in the absence of limitations due to the interaction. Individually, these correspond to the asymptotic lines in Fig. 2. While a complete calculation of the reaction rate certainly will require the ability to address the interaction of kinetics and transport, the determination of the length scale at which the dominant influence crosses over from reaction kinetics to transport can be found from the intersection of the lines in Fig. 2. Materials and Method The Damköhler number was re-calculated using the proposed method (Eq.3) based on the published experimental data from [12]. The flow-through experiment was done at 3 column lengths: 5cm, 10cm and 22cm. Only the dataset from the mixed column was used to calculate DaI. Flow rates q ranging from 0.18m/d to 36m/d were used to study the dissolution rates of magnesite at different flow rates. The magnesite specimen was ground and sieved to grain sizes between 354 µm and 500 µm. The geometric mean of the grain sizes was taken as the typical particle size. Thus we approximated Lp = 400 µm in Eq (3). Porosity of the material ranged between 0.39 and 0.43. The well-mixed reaction rate &'( = Rm = 10-9 mole/m2/s and BET surface area = 1.87m2/g according to [12]. Results Scaling of Magnesite Dissolution Rate with Distance The overall reaction rates of each column in all three column-lengths increase as flow rate increases (Figure 3a). The exponent of the scaling of reaction rate with fluid velocity decreases from 0.82 to 0.71 to 0.68 as column length reduces from 0.22m to 0.05m. Our previous study9 shows that the surface reaction rate is proportional to solute velocity (itself proportional to fluid rate) if the reaction is transport-limited. That the scaling of reaction rate with solute velocity is more likely to be linear (follow the red dash line in



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Figure 3a) at longer column length indicates that with increasing length scales, advection time keeps increasing and solute transport limitation becomes increasingly important to the overall chemical reaction rate of the entire column. One can expect a transport-limited scaling of the overall reaction rate when the advection time is so large comparing with reaction time that the limiting effect of surface reaction kinetics can be negligible. The divergence of the reaction rate from linear scaling at the same column length as solute velocity increases also indicates that slowing down solute velocity has similar effect as extending distance on the limitation of chemical reaction rate. The transition of the limiting effect on chemical reaction rate as distance and flow rate increase is more clearly shown in Figure 3b by comparing the theoretical transport-limited reaction rates predicted by percolation theory and the measured rates. Here, transport-limited reaction rate is calculated as, &5 =

6 7 & ( )8. 6 % 7

(4)

where, &5 is the predicted reaction rate, q the flow rate, 9% a reference flow rate, &% a reference reaction rate, : the distance from the inlet of the column, and :% a reference distance. Eq.4 was derived from the basic solute transport equation (Eq.1). The reference point was chosen as the reaction at slowest flow rate (0.18m/d) and at the longest column length (0.22m), because at this point, reaction is clearly limited by solute transport (Figure 3a), and the reaction rate at higher flow rate should proportionally increase, given that reaction rate is calculated as proportional to solute velocity. As flow rate increases, observed reaction rates become more divergent from prediction, indicating a lower importance of solute transport in limiting the overall reaction rate. Interestingly, the reaction rate approximately reaches steady state at the fastest flow rate and shortest distance (36m/d, 0.05m), meaning that the 36m/d flow rate is just fast enough to eliminate altogether the influence of solute transport at the shortest travel distance of 0.05m. Damköhler Number Values Values of the Damköhler number calculated using the proposed method (Eq.3) were compared with the measured reaction rates. In order to simplify the figure of DaI at various column lengths and flow rates, corresponding distances of reaction rates at various flow rates were normalized to flow rate q = 0.18m/d (Figure 4) and q = 36m/d (Figure 5), consistent with a proportionality of the reaction rate to the flow rate. Only data points (q = 0.18m/d and q = 0.36m/d at 3 column lengths, and q = 1.8m/d at L = 0.22m) that are transport-limited (Figure 3b) could be normalized consistently because in that case the reactions conform to our predictions (Eq. 4). In Figure 4, DaI based on our calculation increases from 448 to 33,844 as scaled distance extends from 0.022m to 0.22m at flow rates q = 0.18m/d, and shows a large contrast to the original calculation proposed in [12] et al (Eq.2). All original DaI are equal to or less than 1, implying that reactions are limited by surface reaction, revealing a conflict with the fact that the distance dependence of all reaction rates shown here follows exactly that of the transport-limited reaction rates (Figure 3b). Again, such a discrepancy is a result from the underestimation of the original calculation of DaI by considering the solute velocity as a constant and by calculating reaction time using measured rates. As distance decreases,


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DaI diminishes. One can expect DaI < 1 if the distance is small enough that the rate is only limited by surface reaction kinetics. In Figure 5, transport distances were normalized to the highest flow rate q = 36m/d. DaI decreases dramatically from 1.65×107 to 11 as distance reduces from 102 m to 0.05m. Notice that at 0.05m, the author stated a well-mixed reaction rate, suggesting that at this point, reaction is limited by surface reaction itself and DaI should be equal to or less than 1. However, in our calculation, DaI = 11. Possible reasons could be: 1) the mixed control of the reaction by both solute transport and surface reaction. As the solute is transported from the inlet to the outlet of the column, the reaction passes through a surface reaction-limited regime (DaI ≤ 1) and a transport-limited regime (DaI >1). Even within 0.05m, there could be a small region near the outlet that is transport-limited, causing the overall DaI to exceed 1, 2) One order of magnitude overestimation of DaI of our calculation. Given the fact that the reaction rate still decreases as column length increases from 0.05m to 0.1m (Figure 3 and 5), DaI should not be less than 1 at column length = 0.1m (DaI = 37 in our calculation). Consequently, any overestimation of DaI cannot exceed a factor of 40. In Figure 5, theoretical reaction-rate scaling at flow rate = 36m/d is shown. According to the authors12, the well-mixed reaction rate is about 10-9 mole/m2/s. Theoretically, reaction is only limited by surface reaction within short distance and it is not affected by solute velocity (flat line shown in Figure 5). As solute transport distance increases, solute velocity is slowing down with a power of 1.87, and the limiting factor of chemical reaction switches from surface reaction to solute transport when advection time is larger than reaction time. Here, there is no abrupt change of measured reaction rates from the well-mixed value to the theoretical scaling. Such curvature is caused by the method the reaction rate was calculated. As the overall reaction rate of the entire column, the measured rate is an averaged value of reaction rates limited by both surface reaction and solute transport. Discussion As an important parameter indicating the limiting factor of chemical reaction rate, our calculation of the Damköhler number shows a similar decreasing trend as reaction rate increases but with large discrepancy for individual values comparing with previous results calculated in [12] (Figure. 6). Comparing with previous results that range from 0.17 to 1, our DaI shows a much wider range (11.5 to 33,843.85) as reaction rate changes for two orders of magnitude. The wide range of DaI from our calculation results from the advection time in the numerator that increases with a power of 1.87 rather than increasing linearly with distance and the reaction time in the denominator that is diminished by using the well-mixed reaction rate instead of the measured rate. For the specific case in which the well-mixed reaction rate (and thus reaction time) is a constant, the Damköhler number is proportional to advection time, thus scaling as the 1.87 power of column length. When the reaction rate is transport-limited, one can derive the scaling of the reaction rate with the Damköhler number by, =>? ∝ #$ = & ∝ D = E

A

( )BC

(5)

8BC

∝F

∝ =>?

2GHC HC

(6)



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where D is the solute velocity. Using Db = 1.8768 for conditions of full saturation in three-dimension, reaction rate limited by solute transport scales as the power of -0.465 with Damköhler Number (lines in Figure 6). To examine further our calculation of the Damköhler number, theoretical scaling of transport-limited reaction rates at various flow velocities and corresponding DaI were plotted in Figure 7. At lower flow rates, DaI are close to 1 when reaction rates reach the well-mixed value, generating consistent results of the Damköhler number and reaction rate scaling in that, as the Damköhler number increases past 1, the reaction rate slows relative to its well-mixed value. At rapid flow rate q = 36m/d, the discrepancy increases. That is because as the flow rate increases, the scaling of reaction rate tends to diverge earlier from the theoretical scaling as distance decreases (as shown in Figure. 7), making a shorter length for reaction rate to meet the well-mixed value, thus resulting in a smaller corresponding DaI. It is this mixedcontrol regime that degrades the accuracy of our Damköhler number estimation. The proposed method works reasonably well when the reaction is transport-limited. Attempts were also made to predict the overall reaction rate of the entire column (the curvature shown in Figure 5) by combining the limiting effects from surface reaction and solute transport together. If one combines effects of advection and reaction by simply adding the times of each process together, one can generate the following expression for the reaction rate, &I =

JK

(7)

2 2G H

LB#M C

This expression correctly reproduces the Damköhler number dependence at large DaI, while also generating the well-mixed reaction value, Rm, in the limit of small DaI, but it does not properly reproduce the mixed-condition regime and thus is not a candidate for the correct result. Another approach we developed was to separate the entire column into two sections by "cut-off " length where DaI = 1. Within that length, reaction rate equals well-mixed reaction rate, and beyond that length, reaction rate was proposed to follow the scaling predicted by percolation theory. Since reaction rate was calculated from the concentration of the solution in the experiment12 as, ) ∗Q &I = NOP R

(8)

!

where &I is the overall reaction rate of the column, /STE is the concentration at the outlet, Q is the flow rate, and + , is the surface area, /STE in Eq. 8 could be substituted by the sum of the two concentrations of the two sections, and one can generate, ]GHC

G2

H H &I = (U8B ) &V (1 − =Y + =[\ C ) =[\ C

(9)

C

This expression reproduces the Damköhler number dependence at small DaI, but it does not properly reproduce the dependence of reaction rate at large DaI and thus is not a candidate for the correct result either. Unfortunately, both approaches, adding the advection and diffusion times on the one hand, or adding the well-mixed and less mixed concentrations on the other, are unsuccessful at predicting the reaction rate under mixed-control regime, although they do both generate a curved region connecting the


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two straight line regions in Fig. 2. Thus a more comprehensive theoretical approach that can predict the cross-over from kinetic controlled reactions to transport-limited reactions is yet to be developed. In this discussion we wish also to point out that, in at least one important factor, typical experimental conditions tend to accentuate the relevance of kinetic limitations, whereas field conditions tend to emphasize transport limitations. In particular, the flow velocities of [12] range from 0.18m/day (= 65m/yr) to 36m/day (= 13,410m/yr). However, under typical field conditions flow rates will not exceed infiltration rates, which range from a fraction of a meter per year to 10m/yr. This discrepancy in flow rates implies that typical field values of the Damköhler number are likely to be roughly a factor 1000 larger than those in the experiments of [12]. Finally, we reiterate that the physical characteristic that underlies both the strong dependence on length and time scales of such reactions as weathering rates, and the non-Gaussian transport which we use to predict such rates, is the heterogeneity of the pore space. There is thus no requirement to base one’s approach on the particular avenue of the non-Gaussian transport, and other general approaches to treat heterogeneity can be used, such as the partitioning of the pore space into dual domain, or multi-domain models73. A general comparison of both the theoretical relationships between these diverse methods and their relative efficacy is not available. We do contend, however, that our treatment should have the advantage of greater parsimony. Conclusions The analysis of the experimental data12 shows that chemical reaction rates may be controlled by either surface reaction kinetics or solute transport. At small length scale (or rapid flow rate), the rate is limited only by reaction kinetics, and not affected by solute velocity. As the distance increases (or flow rate decreases), solute transport takes place in limiting the reaction rate, forming a mixed-control of chemical reaction within the column. Only when the distance is long enough (or flow rate is slow enough) that the limiting effect of surface reaction kinetics is so small comparing with that of solute transport that the reaction rate would follow the transport-limited scaling. The calculation of the Damköhler number based on percolation theory is consistent with the scaling of reaction rate. The apparent one order magnitude overestimation of the Damköhler number at the highest flow rate might be due to the fact that we are calculating the overall DaI at the end of the column. However, our calculation shows a good estimation at slow flow rate when the entire column is more likely to be limited by solute transport. Given that the infiltration rate (thus the flow rate for chemical weathering11) in the field (between 5×10-6m/d and 0.01m/d) is much slower than the slowest flow rate in the presented experiment (q = 0.18m/d), the proposed method in calculating the Damköhler number, and corresponding predictions of solute transport-limited weathering rates, can be expected to be useful in most field experiments. References 1. Urey, H. C. On the Early Chemical History of the Earth and the Origin of Life. Proc. Natl. Acad. Sci. U.S.A. 1952, 38 (4), 351–363, DOI: 10.1073/pnas.38.4.351. 2. Berner, R. A. Weathering, Plants, and the Long-Term Carbon-Cycle. Geochim. Cosmochim. Acta. 1992, 56 (8), 3225–3231, DOI: 10.1016/0016-7037(92)90300-8.



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3. White, A. F.; Brantley, S. L. The Effect of Time on the Weathering Rates of Silicate Minerals: Why Do Weathering Rates Differ in the Lab and in the Field? Chem. Geol. 2003, 202 (3-4), 479–506, DOI: 10.1016/j.chemgeo.2003.03.001. 4. Molins, S.; Trebotich, D.; Steefel, C. I.; Shen, C. An Investigation of the Effect of Pore Scale Flow on Average Geochemical Reaction Rates Using Direct Numerical Simulation. Water Resour. Res. 2012, 48 (3), DOI: 10.1029/2011wr011404. 5. Dentz, M.; Gouze, P.; Carrera, J. Effective Non-Local Reaction Kinetics in Physically and Chemically Heterogeneous Media. J. Contam. Hydrol. 2011, 120–121, 222–236, DOI: 10.1016/j.jconhyd.2010.06.002. 6. Raoof, A.; Hassanizadeh, S. M. Upscaling Transport of Adsorbing Solutes in Porous Media. Transp. Porous Media. 2010, 13 (5), 395–408, DOI: 10.1615/jpormedia.v13.i5.10. 7. Noniel, C.; Steefel, C. I.; Yang, L.; Ajo-Franklin, J. Upscaling Calcium Carbonate Precipitation Rates from Pore to Continuum Scale. Chem. Geol. 2012, 318, 60–74, DOI: 10.1016/j.chemgeo.2012.05.014. 8. Maher, K. The Dependence of Chemical Weathering Rates on Fluid Residence Time. Earth Planet. Sci. Lett. 2010, 294 (1-2), 101–110, DOI: 10.1016/j.epsl.2010.03.010. 9. Hunt, A. G.; Ghanbarian-Alavijeh, B.; Skinner, T. E.; Ewing, R. P. Scaling of Geochemical Reaction Rates via Advective Solute Transport. Chaos. 2015, 25 (7), 075403. DOI: 10.163/1.4913257. 10. Hunt, A. G. Spatio-Temporal Scaling of Soil Depth and Vegetation Growth. Vadose Zone J. 2016, 15 (2). DOI:10.2136/vzj2015.01.0 01. 11. Hunt, A. G.; Ghanbarian, B. Percolation Theory for Solute Transport in Porous Media: Geochemistry, Geomorphology, and Carbon Cycling. Water Resour. Res. 2016, 52 (9), 7444–7459, DOI: 10.1002/2016wr019289. 12. Salehikhoo, F.; Li, L.; Brantley, S. Magnesite Dissolution Rates at Different Spatial Scales: The Role of Mineral Spatial Distribution and Flow Velocity. Geochim. Cosmochim. Acta. 2013, 108, 91–106, DOI: 10.1016/j.gca.2013.01.010. 13. Navarre-Sitchler, A.; Brantley, S. L. Basalt Weathering Across Scales. Earth Planet. Sci. Lett. 2007, 261 (1-2), 321–334, DOI:10.1016/j.epsl.2007.07.010. 14. White, A. G.; Blum, A. E.; Schulz, M. S.; Bullen, T. D.; Harden, J. W.; Peterson, M. L. Chemical Weathering Rates of A Soil Chronosequence on Granitic Alluvium: I. Quantification of Mineralogical and Surface Area Changes and Calculation of Primary Silicate Reaction Rates. Geochimi. Cosmochim. Acta. 1996, 60 (14), 2533–2550, DOI: 10.1016/0016-7037(96)00106-8. 15. Mavris, C.; Egli, M.; Pltze, M.; Blum, J. D.; Mirabella, A.; Giacci, D.; Haeberli, W. Initial Stages of Weathering and Soil Formation in the Morteratsch Proglacial Area (Upper Engadine Switzerland). Geoderma. 2010, 155 (3-4), 359–371, DOI: 10.1016/j.geoderma.2009.12.019.


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30. Hunt, A.G.; Skinner, T. E. Incorporation of Effects of Diffusion into Advection-Mediated Dispersion in Porous Media. J. Stat. Phys. 2010, 140 (3), 544–564, DOI: 10.1007/s10955-010-9992-x. 31. Hunt, A. G.; Skinner, T. E.; Ewing, R. P.; Ghanbarian-Alavijeh, B. Dispersion of Solutes in Porous Media. Eur. Phys. J. B. 2011, 80 (4), 411–432, DOI: 10.1140/epjb/e2011-10805-y. 32. Ghanbarian-Alavijeh, B.; Skinner, T. E.; Hunt, A. G. Saturation Dependence of Dispersion in Porous Media. Phys. Rev. E. 2012, 86, 066316, DOI: 10.1103/physreve.86.066316. 33. Kläfter, J.; Silbey, R. Derivation of the Continuous-Time Random Walk Equation. Phys. Rev. Lett. 1980, 44 (2), 55–58, DOI: 10.1103/physrevlett.44.55. 34. Berkowitz, B.; Klafter, J.; Metzler, R.; Scher, H. Physical Pictures of Transport in Heterogeneous Media: Advection-Dispersion, Random-Walk, and Fractional Derivative Formulations. Water Resour. Res. 2002, 38 (10), DOI: 10.1029/2001WR001030. 35. Bijeljic, B.; Muggeridge, A.; Blunt, M. J. Pore-Scale Modeling of Longitudinal Dispersion. Water Resour. Res. 2004, 40 (11), W11501, DOI: 10.1029/2004wr003567. 36. Margolin, G.; Berkowitz, B. Application of Continuous Time Random Walks to Transport in Porous Media. J. Phys. Chem. B. 2000, 104 (16), 3942–3947, DOI: 10.1021/jp993721x. 37. Meerschaert, M. M.; Benson, D. A.; Baumer, B. Multidimensional Advection and Fractional Dispersion. Phys. Rev. E. 1999, 59 (5), 5026–5028, DOI: 10.1103/physreve.59.5026. 38. Benson, D. A.; Wheatcraft, S. W.; Meerschaert, M. M. Application of a Fractional DdvectionDispersion Equation. Water Resour. Res. 2000, 36 (6), 1403–1412, DOI: 10.1029/2000wr900031. 39. Pachepsky, Y.; Benson, D.; Rawls, W. Simulating Scale-dependent Solute Transport in Soils with The Fractional Advective-dispersive Equation. Soil Sci. Soc. Am. J. 2000, 64 (4), 1234–1243. 40. Krepysheva, N.; Di Pietro, L.; Neel, M. C. Space-Fractional Advection Diffusion and Reflective Boundary Condition. Phys. Rev. E. 2006, 73 (2), 021104 Part 1, DOI: 10.1103/physreve.73.021104. 41. Sheppard, A.P.; Knackstedt, M. A.; Pinczewski, W. V.; Sahimi, M. Invasion Percolation: New Algorithms and Universality Classes. J. Phys. A: Math. Gen. 1999, 32 (49), L521–L529, DOI: 10.1088/0305-4470/32/49/101. 42. Sahimi, M.,1994, Applications of Percolation Theory, Taylor & Francis, London. 43. Pollak, M. A Percolation Treatment of DC Hopping Conduction. J. Non-Cryst. Solids. 1972, 11, 1–24, DOI: 10.1016/0022-3093(72)90181-0. 44. Ambegaokar, V. N.; Halperin, B. I.; Langer, J. S. Hopping Conductivity in Disordered Systems. Phys. Rev. B. 1971, 4 (8), 2612–2620, DOI: 10.1103/physrevb.4.2612. 45. Shante, V. K. S.; Kirkpatrick, S. Introduction to Percolation Theory. Adv. Phys.1971, 20 (85), 325– 357, DOI:10.1080/00018737100101261.


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46. Kesten, H. Percolation Theory for Mathematicians. Prog. Probab. Stat. 1982, 2, 423 pp. Birkhauser, Boston, Mass. 47. Stauffer, D.; Aharony, A. Introduction to percolation theory, 2nd ed.; Taylor and Francis: London, UK, 1994. 48. Hunt, A. G.; Ewing, R. P. Lecture Notes in Physics. Percolation Theory for Flow in Porous Media; Springer: Berlin, 2009. 49. Fatt, I. The Network Model of Porous Media. I. Capillary Pressure Characteristics. Petr. Trans. AIME. 1956, 207, 145–159. 50. Fatt, I. The Network Model of Porous Media. II. Dynamic Properties of a Single Size Tube Network. Petr. Trans. AIME. 1956, 207, 160–163. 51. Fatt, I. The Network Model of Porous Media. III. Dynamic Properties of Networks with Tube Radius Distribution. Petr. Trans. AIME. 1956, 207, 164–177. 52. Shah, C. B.; Yortsos, Y. C. The Permeability of Strongly Disordered Systems. Phys. Fluids. 1996, 8 (1), 280–282, DOI: 10.1063/1.868835. 53. Bernabé, Y.; Bruderer, C. Effect of the Variance of Pore Size Distribution on the Transport Properties of Heterogeneous Networks. J. Geophys. Res. 1998, 103, 513–525, DOI: 10.1029/97jb02486. 54. Friedman, S.P.; Seaton, N. A. Critical Path Analysis of the Relationship Between Permeability and Electrical Conductivity of Three-Dimensional Pore Networks. Water Resour. Res. 1998, 34 (7), 1703– 1710, DOI: 10.1029/98wr00939. 55. Hunt, A. G. Applications of Percolation Theory to Porous Media with Distributed Local Conductances. Adv. Water Resour. 2001, 24 (3-4), 279–307, DOI: 10.1016/s0309-1708(00)00058-0. 56. Sahimi, M. Flow and Transport in Porous Media and Fractured Rock: From Classical Methods to Modern Approaches; Wiley-VCH: Weinheim, Germany, 2011. 709pp, DOI:10.1002/9783527636693. 57. Sahimi, M. Fractal and Superdiffusive Transport and Hydrodynamic Dispersion in Heterogeneous Porous Media. Transp. Porous Media. 1993, 13 (1), 3–40, DOI: 10.1007/bf00613269. 58. Liu, Z.; Wang, X.; Mao, P.; Wu, Q. Tracer Dispersion between Two Lines in Two-Dimensional Percolation Porous Media. Chin. Phys. Lett. 2003, 20 (11), 1969–1972, DOI: 10.1088/0256307x/20/11/019. 59. Jardine, P. M.; Jacobs, G. K.; Wilson, G.V. Unsaturated Transport Processes in Undisturbed Heterogenous Porous Media. I. Inorganic Contaminants. Soil Sci. Soc. Am. J. 1993, 57 (4), 945–953, DOI: 10.2136/sssaj1993.03615995005700040012x. 60. Cherrey, K. D.; Flury, M.; Harsh, J. B. Nitrate and Colloid Transport Through Coarse Hanford Sdimentsunder Steady State, Variably Saturated Flow. Water Resour. Res. 2003, 39 (6), 1165, DOI: 10.1029/2002wr001944.



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61. Bos, F. C.; Guion, T.; Burland, D. M. Dispersive Nature of Hole Transport in Polyvinylcarbazole. Phys. Rev. B. 1989, 39 (17), 12633–12641, DOI: 10.1103/physrevb.39.12633. 62. Pfister, G.; Griffiths, C. H. Temperature-Dependence of Transient Hole Hopping Transport in Disordered Organic Solids – Carbazole Polymers. Phys. Rev. Lett. 1978, 40 (10), 659–662, DOI: 10.1103/physrevlett.40.659. 63. Pfister, G. Pressure-Dependent Electronic Transport in Amorphous As2Se3. Phys. Rev. Lett. 1974, 33 (25), 1474–1477, DOI: 10.1103/physrevlett.33.1474. 64. Scher, H. Time-Dependent Electronic Transport in Amorphous Solids - As2Se3. Phys. Rev. B. 1977, 15, 2062–2083, DOI: 10.1103/physrevb.15.2062. 65. Tiedje, T. In Semiconductors and Semimetals; Pankove, J. Eds.; (Academic, New York) Vol. 21C p. 207. 1984. 66. Scher, H.; Shlesinger, M.; Bendler, J. Time-Scale Invariance in Transport and Relaxation. Phys. Today. 1991, 44 (1), 26–34, DOI: 10.1063/1.881289. 67. Gist, G. A.; Thompson, A. H.; Katz, A. J.; Higgins, R. L. Hydrodynamic Dispersion and Pore Geometry in Consolidated Rock. Phys. Fluids A. 1990, 2 (9), 1533–1544, DOI: 10.1063/1.857602. 68. Rigord, P.; Calvo, A.; Hulin, J. Transition to Irreversibility for the Dispersion of a Tracer in PorousMedia. Phys. Fluids A. 1990, 2 (5), 681–687, DOI: 10.1063/1.857721. 69. Bijeljic, B.; Blunt, M. J. Pore-Scale Modeling and Continuous Time Random Walk Analysis of Dispersion in Porous Media. Water Resour. Res. 2006, 42 (1), W01202. 70. Yu, D.; Jackson, K.; Harmon, T. Dispersion and Diffusion in Porous Media under Supercritical Conditions. Chem. Eng. Sci. 1999, 54 (3), 357–367, DOI: 10.1016/s0009-2509(98)00271-1. 71. Gelhar, L. W.; Welty, C.; Rehfeldt, K. R. A Critical Review of Data on Field Scale Dispersion Aquifers. Water Resour Res. 1992, 28 (7), 1955–1974, DOI: 10.1029/92wr00607. 72. Saffman, P.G. A Theory of Dispersion in a Porous Medium. J. Fluid Mech. 1959, 6, 321–349. 73. Flühler, H.; Durner. W.; Flury, M. Lateral Solute Mixing Processes: A Key to Understanding Field Scale Transport of Water and Solutes. Geoderma, 1996, 70 (2-4), 165 –183.


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1.0E-03

Rb (yr -1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


y = 0.0001x0.9698 R² = 0.8422

1.0E-05

1.0E-07 0.001

0.01

0.1

1

10

100

v0 (m/yr)

Figure 1. (digitized from [8]). Reaction rates plotted against flow rates showing agreement with a simple proportionality (extracted power 0.97 ≈ 1). Note that the flow rates correspond approximately to the variability of infiltration on Earth’s surface, from about 1mm/yr to a few meters per year.



0.0000001 Reaction Rate (mol m-2 s-1) Solute Velocity (m s-1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

T3

1E-10

T2 T1 v3 v2

1E-13 0.000001

v1

0.001 Distance (m)

1

Figure 2. Schematic reaction-rate scaling diagram. Horizontal lines give laboratory reaction rates for wellmixed conditions at temperatures T3 < T2 < T1, and solute velocities for flow rates v3 < v2 < v1. Predicted reaction rates connect horizontal line at given T with diagonal line at given v. In reality, one should expect a smooth cross-over from one region to the other. Proportionality of a reaction rate to a flow velocity occurs only within the transport-limited portion of the diagram. Note that our calculation of the Damköhler number (Eq. (3)) is intended only to be sufficiently accurate to diagnose the point, for a given set of experimental conditions, at which the relevant horizontal and diagonal line cross, not the actual dependence of the total reaction time, and thus the total reaction rate.


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1.E-08

1.E-09 Reaction Rate (mol/m2/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.E-10

L = 0.22m

1.E-11

L = 0.1m L = 0.05m Transport-limited

1.E-12 0.1

1

10

100

Fluid Velocity (m/d)

Figure 3a. Scaling of reaction rate (mol/m2/s) with flow rate ranging from 0.18m/d to 36m/d at various column lengths L (m). Dash line represents the theoretical scaling of transport-limited reaction rate with flow velocity. Data from [12].



1.0E-08

1.0E-09 Reaction Rate (mol/m2/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 23

1.0E-10 Observed: q = 0.18m/d q = 0.36 q = 1.8 q = 18 q = 36 Theory: q = 0.18m/d q = 0.36 q = 1.8 q = 18 q = 36

1.0E-11

1.0E-12 0.01

0.1

1

10

Column Length (m)

Figure 3b. Reaction rate vs. column length ranging from 0.05m to 0.22m. Open symbols are observed reaction rates, lines are predicted scaling of transport-limited reaction rates. Numbers indicate flow rates q (m/d). Data from [12].


Page 19 of 23

1E-08

100000

1E-09 1000

DaI

Reaction Rate (mol/m2/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1E-10

10

Observed: q = 0.18m/d q = 0.36

1E-11

q = 1.8 Theory: q = 0.18m/d Da-calculated Da-original

1E-12

0.1 0.01

0.1

1

Normalized Transport Distance (m)

Figure 4. Reactions limited by solute transport only and scaling of Damköhler number with distance at flow rate = 0.18m/d. Open symbols are reaction rates observed at flow rate q = 0.18,0.36 and 1.8m/d (distances were normalized to q = 0.18m/d). Solid circles are DaI calculated based on proposed method (Eq.3), solid diamonds (Da-original) use the existing calculation12 of DaI (Eq.4). Note that our calculated value of DaI in this figure extrapolates to 1 at a length scale ca. 1mm, where the reaction rate extrapolates to 10-9mol/m2/s, the well-mixed value. Data from [12].



1.0E-08

100,000,000

DaI

1.0E-09 Reaction Rate (mol/m2/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 23

1.0E-10

10,000 Observed: q = 0.18m/d q = 0.36

1.0E-11

q = 1.8 q = 36 Theory: q = 36m/d Well-mixed Rate

1.0E-12 0.01

0.1

1

10

100

1 1000

Da-calculated

Normalized Transport Distance (m)

Figure 5. Theoretical reaction-rate scaling at q = 36m/d (dash line). Solid line is the well-mixed reaction rate (10-9mol/m2/s). Open symbols are observed reaction rates at various fluid velocities (q = 0.18,0.36 and 1.8m/d) normalized to q = 36m/d. Solid circles are Damköhler number calculated based on proposed method (Eq.3). Data from [12].


Page 21 of 23

Theory: q = 0.18m/d q = 0.36 q = 36 Calculated: q = 0.18m/d q = 0.36 q = 36 Original: q = 0.18m/d q = 0.36 q = 36

1.E-08

1.E-09


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.E-10

1.E-11

1.E-12

1.E-13 0

1

10

100

1,000

10,000

100,000 1,000,000

DaI

Figure 6. Dependence of Damköhler number on reaction rate (mole/m2/s) based on previous and proposed methods. Lines are theoretical dependence of Damköhler number on reaction rate when reaction is transport-limited at various flow velocities (m/d). Open symbols are DaI calculated using proposed method here based on actual reaction rates. Calculations of DaI for the solid symbols are based on the previous method12. Numbers indicate flow rates. Data from [12].


Page 22 of 23

1E-09

100,000,000

1E-10

1,000,000

1E-11

10,000

DaI

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



1E-12

1E-13 0.0001

100

1 0.001

0.01

0.1

1

Theory: q = 0.18m/d q = 0.36 q = 36 Observed: q = 0.18m/d q = 0.36 q = 36 Well-mixed Rate Da: q = 0.18m/d q = 0.36 q = 36

10

Column Length (m)

Figure 7. Evaluation of the accuracy of calculated Damköhler number. Single lines are theoretical scaling of reaction rates limited by solute transport. Double lines are scaling of calculated DaI with column length when reaction is transport-limited. Blue dash line is the well-mixed reaction rate (10-9mol/m2/s). Open symbols are measured rates. Numbers indicate flow velocities. Data from [12].


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For TOC only.


DamkÃ¶hler Number Input to Transport-Limited Chemical Weathering

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