This lesson introduces reaction kinetics of mineral dissolution and precipitation and how to set up simulation in well-mixed batch reactors in CrunchFlow.
By the end of this lesson, you should be able to:
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If you have any questions, please post them to our Questions? discussion forum (not e-mail), located in Canvas. The TA and I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you, too, are able to help out a classmate.
The natural subsurface is composed of rocks, soils, as well as other forms of porous materials that contain various types of minerals. Minerals dissolve and precipitate when interacting with water. As minerals dissolve, chemicals in the solid phase transform into ions in water, resulting in a decrease in mineral mass and volume. For example, calcite (CaCO3) dissolves into carbonate species (H2CO3, HCO3-, CO32-) and Ca(II)-containing species in water. In contrast, mineral precipitation occurs when aqueous species transform to become solid phases, therefore leading to mass and volume increase in solid phases while decrease in aqueous concentrations.
Mineral dissolution and precipitation are important in both natural and engineered processes. They influence water chemistry, soil formation, contaminant transport and fate, acid stimulation in reservoir engineering, environmental remediation, and global carbon cycling. For example, acid stimulation accelerates mineral dissolution, therefore increasing reservoir porosity and permeability and enhancing oil production. On the other hand, mineral precipitation during hydrocarbon production results in wellbore clogging. The dissolution of carbonate caprocks can potentially result in CO2 leakage from CO2 storage reservoirs. Alteration of flow fields induced by mineral dissolution and precipitation in geothermal systems could affect the long-term production of geothermal energy. Over geological time scale, the CO2 consumption during chemical weathering (through mineral dissolution) helps maintain the clement conditions for life on earth.
Calcite is one of the most common minerals on Earth's surface. Here we take calcite dissolution as an example to illustrate the Transition State Theory (TST) based kinetic rate law. The dissolution of calcite has been proposed to occur via three parallel reactions (Plummer, Wigley et al. 1978, Chou, Garrels et al. 1989):
Each reaction pathway has its own reaction rate dependence on different chemicals. The rate of reaction (1) dominates under acidic conditions and depends on the activity of hydrogen ion. The rate of reaction (2) dominates under CO2- rich conditions, while the rate of reaction (3) prevails under neutral pH conditions. The overall dissolution rate R (mol/s) is the summation of the rates of all three parallel reactions:
Here , A is the reactive surface area A (m2) k1, k2, and k3 are reaction rate constants (mol/m2/s) for the three parallel reactions, respectively; and are the activities of hydrogen ion and carbonic acid: IAP and Keq are the ion activity products and the reaction equilibrium constants for reactions (1), (2), and (3), respectively.
The equilibrium constants determine mineral solubility and depend on temperature, pressure, and salinity in ways similar to those discussed for aqueous complexation reactions. As indicated by Equation (4), the reaction rates depend on several factors, including the intrinsic mineral properties such as the amount of reactive surface area A and the intrinsic rate constants k, as well as external conditions such as the concentration of “catalyzing” species including H+ and H2CO30, and how far away the reactions are from equilibrium (IAP/Keq).
The rate dependence on pH is illustrated in Figure 1. The rates in the figure are normalized by the amount of surface area and therefore are in the units of mol/m2/s. Under low pH conditions with very fast dissolution rates, the dissolution rates can be transport-controlled even in well-mixed reactors, because the speed of mixing may not be as fast as dissolutin. In contrast, under closer to neutral pH conditions, the rates are much slower and the reactions are often kinetics controllled.
A general TST rate law is as follows:
Here nk is the total number of parallel reactions, kj is the rate constant of the parallel reaction , the term describes the rate dependence on pH, and the term describes the rate dependence on other aqueous species that potentially accelerate or limit reactions. In Equation (4) for calcite dissolution rates, also accelerates calcite dissolution. The affinity term quantifies the distance of solution from the equilibrium state of the mineral reaction j. The exponents m1,j and m2,j describe the nonlinear rate dependency on the affinity term and are normally measured in experiments. Note that Equation (4) is a special case of the general reaction rate law in Equation (5).
Various factors affect reaction rates, including, for example, temperature, salinity, pH, organic ligands. Mineral dissolution rates are often accelerated by the presence of H+ or OH-. As a result, many minerals dissolve fast in acidic and alkaline conditions and slow down under neutral conditions. Figure 2 shows silicate dissolution rates as a function of pH.
The intrinsic dissolution rates of minerals vary significantly. Calcite dissolution is in the fast end of dissolution rate spectrum, while quartz dissolves the slowest. Table 1 shows the lifetime of a 1mm crystal of different minerals, indicating more than 5 orders of magnitude difference in dissolution rates of silicates and quartz.
Mineral | Lifetime |
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Quartz | 34,000,000 |
Muscovite | 2,700,000 |
Forsterite | 600,000 |
K-feldspar | 520,000 |
Albite | 80,000 |
Enstatite | 8,800 |
Diopside | 6,800 |
Nepheline | 211 |
Anorthite | 112 |
The Arrhenius equation is used to quantify the reaction rate dependence on temperature.
Here AF is the pre-exponential factor and is usually considered as a constant that is independent of temperature: the activation energy is always positive; here R is the ideal gas constant , T is the absolute temperature (K). If we take the logarithm of this equation, we obtain:
This means if we draw lnk versus , we get a straight line, as shown in Figure 2. For a particular reaction, if and at the temperature T1 is known, at another temperature T2 can be calculated by
Figure 3 shows the dependence of calcite rate constants on temperature under different CO2 partial pressure.
To model the mineral dissolution/precipitation, we need to inform the code both reaction thermodynamics and kinetics. Here we introduce how to set up simulations for mineral dissolution/precipitation in a well-mixed batch reactor, where the concentrations of all species and temperature are assumed to be spatially uniform. That is, the rate of mixing is faster than the rate of dissolution or precipitation, which is in general true for most minerals except carbonates under very low pH conditions. No transport processes and boundary conditions need to be defined because of the assumption of uniform concentration. The example 6 included in the CrunchFlow package is also a good reference for kinetic reaction setup.
Calcite dissolution proceeds through three parallel pathways as shown in reactions (1)-(3). In addition, a series of fast aqueous complextion / speciation reactions occur simultaneously. These fast reactions are similar to those in the examples in the lesson on Aqueous Complexation.
In a batch reactor of 200 ml, the initial solution is at a pH of 5.0 with close to zero salinity. The initial total carbonate concentration is . The kinetic rate constant for three parallel reactions are , and respectively. The volume fraction of the calcite grains in the batch reactor is 0.5%, with a specific surface area (SSA) of 0.24 m2/g. Please set up the simulation and plot the following quantities as a function of time:
Here is a light board video (12:44) of what equations the code solves for this system.
PRESENTER: So in this video we're going to talk about mineral dissolution and precipitation, the equation involved and interaction involved. This is different from the previous lesson, aqueous [INAUDIBLE], because the previous reaction only involved reaction [INAUDIBLE] dynamics. Meaning we care about the end point of the reaction.
When we have mineral dissolution precipitation, because reaction usually occur at the interface of solid phase and water, reaction occur much slower. So we often need to consider the kinetics of the reaction. Meaning we care about how long it takes for the reaction to occur to reach the end point.
So what I'm going to do today here is using the carbonate dissolution system in a battery reactor. So battery reactor, meaning it's closed while mixed, so that it does not have constitution gradient in different parts of reactor. So essentially you're solving for one constitution for each species for the whole reactor. So that's our systems.
If you think about if you want to draw something relevant to what you do in the lab, but you think about these water, and then you have these calcite grains, and you have these mixers that will keep the system well-mixed. And again, the reaction involving this system is similar to what we talked about last time for the aqueous speciation. The carbon and water-- for example, all these three reactions are the same as what I wrote before. Invest in one. But the key thing is we are adding this calcite dissolution, which is calcite dissolving out to become calcium and carbonate.
Calcite or carbonate in general is a very common type of rock or mineral fissure or surface. So it's a reaction. Compared to the aqueous speciation or [INAUDIBLE], it's much slower. But compared to other type of mineral dissolution, it's actually reactive fast.
So we have these four reactions here. You can think about the calcite dissolution essentially kind of releasing out the elements from the solid phase the water phase. So it dissolve as an add to the water with calcium species and carbonate species. But then once carbonate is released into water, it quickly goes through the speciation reaction to become either bicarbonate or carbonic acid, depending on the pH a system, as we talked about last time.
So if you think about the species in this system, now that the species we have, in addition to the five species, carbonic acid, bicarbonate, carbonate-- this is wrong. So it should be carbonate. And this should be two.
So you have these three carbonate species. You have hydrogen ion, which minus acid five, as before. But on top of that, you are adding calcium.
So essentially, now you have six species. You would have Ca2 plus carbonic acid carbonate, bicarbonate carbonate. And then you also have H plus or H minus. So that's six species.
So what does that mean, is that we essentially have six unknowns to solve for. And we will need six equations to solve for this, to solve for the species. But notice that one things that we need to pay attention to is that we have these three reactions that occur fast.
So you have these three equations already there. We can think about as one, two, three. That is as a three relationship. But we cannot use this one, because this is not a faster reaction. These are fast reaction.
This is a slow reaction, which means we need to care about its kinetics. Like how far it release as these chemical species over time. So when we solve these equations, we actually need to consider these time component.
In this case, we will need to think about the mass balance of system. So these calcite dissolving out and release out. For example, Ca2 plus, and releasing out CO3 bicarbonate. And then this will be exchanged into bicarbonate and exchange into carbonic acid. So these three specie are exchangeable.
So when we think about mass balance of system, you would think about the constitution of a single body calcium first. Volume-- this is a volume of the water in the system. And you have the constitution of calcium.
Ultimately, we are solving for contrition, not activity. So this will give you the mass. So contrition of the unit and mass per volume. And then we will need to solve for order and differential equation now because just time component here. So in this kind of system we don't really have fro transport so the only source of zinc of this calcium specie is through these reactions.
So let's say we have a rate constant of this reaction is Ksp. So that's the rate constant. And we talk about the tst rate, which is transition state rate theory in the online material. So you can go look this up.
So this Ksp and this is rate constant times a surface area of the mineral. And then times for example, how far away the reaction is from equilibrium, which will depends on the activity of Ca2 plus activity of carbonate, and then divided by Ksp, which is how far away this is from equilibrium. So this is Iap.
Now when we have dilute system-- if we have dilute system, then this activity would be equal to c. So you can replace every a is c for simplicity. So essentially, is this equation saying mass of calcium is added to system, to the water, by having this rate lower for calcium. But we also will have adding total carbon to the system. So similarly, you will have V, d, and then we call it total carbon cT, dt.
It will have the same expression here as Ksp A or minus activity ca. Because, essentially, it's through the same reaction. So this rack essentially adding calcium total carbonate-- this CT will be equals to again, constitution of carbonic acid plus concentration of bicarbonate and concentration of carbonate.
But then we need one more equation. One, two, three. And then this is four, this is five. We need a six equation, which would it be-- if you think about it, when this mineral dissolving out to form calcium and carbonate, it's actually changing the pH of the system. So the hydrogen ion-- or somehow we also call it's a measure of acidity of system will also change.
I'm not going through the detailed derivation of the whole process how we come up with. But let's call this dC, which is-- so this a constitutional cT-- concentration of hydrogen ion total dt. And by dissolving out calcite, actually decrease acidity of system. So it will actually have minus Ksp. And then activity one minus a Ca2 plus.
So again, it's the same rate law, but it do actually have the minus signs. That meaning it's decreasing the total acidity of a system. So the expression for the total acidity will be different if we define different to comnination primer specie.
But in general, if we define this carbonate as a primary species, we should have HT equal to hydrogen, concentration hydrogen ion. This should be CHt. Hydrogen ion plus concentration of H2CO3 minus concentration of bicarbonate and minus concentration of OH minus. You can actually derive this equation from the [INAUDIBLE] method. And these expressions is actually-- you can look at [INAUDIBLE] And by combining different terms, you come up with this. So essentially, we have six unknowns, and you have six equation to solve with that. But on top of that, one thing we need to pay attention is that these are OD equations.
So we call it ordinary differential equation. Differential equation, meaning we have one independent variable, which is time in this equation. So typically, the code will be solving these equation first using time stepping.
And then once we get the concentration calcium, concentration of CT, concentration of CHt, you have three kind of variable in particular time step. And then we solve for the algebraic and relationship with these numbers together with these three relationship. And by doing that, we essentially get the concentration of all the six species as a function time in these aqueous species. And what do you see? For example, your homework is essentially generated by solving these equations.
Here is a video demo in CrunchFlow on how to set up the simulation in CunchFlow (43:51 minutes).
Thermodynamics and kinetic parameters that affect mineral dissolution include specific surface area (SSA), kinetic rate constants (k), and equilibrium constants. In addition, geochemical conditions can also have a large impact on mineral dissolution rates, including salinity (e.g. NaCl) and pH. With the provided CrunchFlow template files in example 1 as a starting point, please do the following analysis comparing Ca(II) concentration evolution under different parameters and geochemical conditions. In each question, please only change the parameter that is discussed and keep all other parameters the same as those in Example 1.
1) Calcite specific surface area (SSA). In three different simulations, run the code using SSA being 0.024, 0.24, and 2.4 m2/g. Plot the total Ca(II) as a function of time under these surface area values in one figure. Discuss how SSA affects calcite dissolution kinetics.
2) Kinetic rate constant: increase and decrease the original three rate constant values in Example 1 by an order of magnitude. Compare the Ca(II) concentration evolution under the three k values in one figure. Also compare this figure here with the figure in 1) with the SSA of 0.24 m2/g. Do changing k and A have the same impact on dissolution rates?
3) Equilibrium constant Keq quantifies how much a mineral can dissolve in aqueous phase. Increase and decrease the equilibrium constant by an order of magnitude. Please plot the three curves (total Ca(II) ~ t) with different Keq values in one figure. What do the rate kinetics change with the total Ca(II) evolution figure?
4) Initial pH condition. Compare the base case with two more cases where you have the initial pH being 4.0 and 6.0.
5) The role of speciation. In this question you remove all secondary species such as CaOH+, CaCO3(aq), and CaHCO3+. Draw the same figures as those in Example 1 and solution from this question if the same figure. Compare and describe your observations. Does the reaction system evolve in the same way in these two systems? What do you think are the effects of speciation?
6) Salinity: in the base case scenario there is no NaCl in the solution. Simulate two more cases with NaCl at concentrations of 0.01 and 0.1 mol/kgw, respectively. Please compare total Ca(II) evolution under these three salinity conditions.
7) Go back to your answer to each question 1)-6). Summarize your observations in these questions. Which factors have the most significant control on calcite dissolution?
Feldspars dissolve following the TST rate law and have very different dissolution rates depending on their composition (figure 7 in [Blum and Stillings, 1995]). Pick a feldspar mineral from the paper and/or the cited reference that has a complete TST rate law from acidic to alkaline conditions. Set it up in CrunchFlow to simulate its dissolution with initial pH conditions varying from acidic (for example, pH =4.0), to neutral (pH = 7), and to alkaline condition (pH = 10.0). Compare the reaction production evolution under these three conditions. Please make sure that you use reasonable rate constant and specific surface area values from literature.
In this lesson, we learned about mineral dissolution and precipitation reactions. We discussed two aspects of the reactions: 1) reaction thermodynamics that control how much minerals can dissolve in aqueous solutions; 2) mineral reaction kinetics (TST rate law) and parameters that control the rates of mineral dissolution and precipitation. We also learned how to set up running simulations in CrunchFlow for mineral dissolution in well-mixed reactors.
You have reached the end of Lesson 2! Double-check the to-do list on the Lesson 2 Overview page to make sure you have completed all of the activities listed there before you begin Lesson 3.
Homework assignments for lesson 2 is due on Tuesday before midnight.
Appelo, C. A. J., and D. Pstma (2005), Geochemistry, groundwater, and pollution, 2nd ed., 649 pp., CRC Press, Taylor and Francis group, Boca Raton, FL.
Blum, A. E., and L. L. Stillings (1995), Feldspar dissolution kinetics, in Chemical Weathering Rates of Silicate Minerals, edited by A. F. White and S. L. Brantley, pp. 291-351, Mineralogical Soc America, Washington, D. C.
Brantley, S. L. (2008), Kinetics of mineral dissolution, in Kinetics of water-rock interaction, edited, pp. 151-210, Springer.
Chou, L., R. M. Garrels, and R. Wollast (1989), Comparative study of the kinetics and mechanisms of dissolution of carbonate minerals, Chem Geol, 78(3), 269-282.
Lasaga, A. C. (1984), Chemical kinetics of water-rock interactions, Journal of Geophysical Research, 89(B6), 4009-4025.
Plummer, L., T. Wigley, and D. Parkhurst (1978), The kinetics of calcite dissolution in CO 2-water systems at 5 degrees to 60 degrees C and 0.0 to 1.0 atm CO 2, Am J Sci, 278(2), 179-216.
Pokrovsky, O. S., S. V. Golubev, J. Schott, and A. Castillo (2009), Calcite, dolomite and magnesite dissolution kinetics in aqueous solutions at acid to ciarcumneutral pH, 25 to 150° C and 1 to 55 atm p CO2: New constraints on CO2 sequestration in sedimentary basins, Chem Geol, 265(1), 20-32.