CE 574
Reactive transport in porous media

0.2 Reactive transport equations and key concepts

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Most reaction transport codes solve equations of mass, momentum, and energy conservation (Steefel et al., 2005). For mass conservation, reactive transport models usually partition aqueous species into primary and secondary species (Lichtner, 1985). The primary species are the building blocks of chemical systems of interest, upon which concentrations of secondary species are written through laws of mass action for reactions at thermodynamic equilibrium. The partition between primary and secondary species allows the reduction of computational cost by only solving for mass conservation equations for primary species and then calculating secondary species through thermodynamics. Detailed discussion on primary and secondary species will be in lesson 1 on Aqueous Complexation. 

Please watch the following video: Reactive Transport Reactions (7:25)

Reactive Transport Reactions
Click for a transcript of the Reactive Transport Reaction video

Reactive Transport Reactions

PRESENTER: What I'm going to do today is really to introduce the general idea of reactive transport equation and the overview of it. This equation that I put here is we call mass conservation equation for chemical species in aqueous phase, for one of the representative species. I don't expect you to know every term, or the details of every term in this equation. Because we will be talking more about each term later. But this is really to give you a general idea and overview before we start about everything.

And the importance of this is that we run these reactive transport codes. And there are a lot of built up architecture behind the code, in terms of what they solve. And it's always a good idea to know some of these, like what they solve, and what are the things behind these codes. So what I will do today is really talk through each of these terms and talk about the [? fake ?] meaning of each term, so that you get a general idea what are they really for.

So the first term we call the mass conservation term. It's called mass accumulation rate. And it should have the units of mol per length cubed, which is volume, of porous media per time, whatever time you pick. But all the terms have to be consistent, have the same length and time units. So this equation really says mass accumulation rate depends on several different processes, right?

So the first term is the rate itself. And the overall rate, the second term, what we call dispersive and diffusive transport. So you think about how a chemical species in water has a change over time, which is this. So some of these rates coming from, for example, the chemical species gold to get different concentrations in different locations. For example, you think about a dye put in a cup of water. And over time, they tend to have the same color everywhere. So this is one of the driving forces, in terms of what we call diffusive or dispersive transport. And they should have the same unit as the first term. Every term should have the same unit. So that's a second term.

And then, the third term is what we call the advective transport. And this is a process where, for example, you think about rivers, right? And the chemical species will flow together with the water. And so essentially, the water brings the chemical species to different places. So this advective transport, in this term, you have the hue, which is we call Darcy velocity. And then, the concentration actually I probably should explain here, the Ci will be the concentration of one chemical species of species i, a representative species, i. And the Ci everywhere is the same. So this is advective transport.

But also in a lot of systems, you have reactions, right? So the last term, four, is for the total reaction rates. And again overall, it has the same units as the first term. But essentially, this could it be a summation of several different-- let's call this equal to summation of i, which is for the chemical species, i. But this i could be involved in, let's say, ik, different numbers of reactions. So essentially, you will be adding all these reactions that this can chemical species, i, is participating in. And this ik would be a total number of reactions.

OK, so this is one representative equation. We write this equation form for the species, i. But I put the i from 1 to n here, meaning you can have any arbitrary number of what we call a primary species. So if you have, let's say, 10 different chemical species, then the primary species you have n equal to 10. And you will be writing 10 of these equations.

And these equations, essentially, if we solve these equations, you get the concentration or different chemical species as a function of time and space. So essentially, the outcome of this is the temporal and spatial distribution of chemical species. So essentially, you can tracing after each chemical species and look at how they change as a function of time and space and how, in different parts, they have different rates, and all that. Species i and i from 1 to n.

OK, so that's what you hope you guys will be exposed to in using this code. You will be solving this equations for particular, specific questions, problem applications. And then, you're usually given a set of initial conditions, like where is the concentration of different species at time 0 at different locations, and then over time, how these countries have different species evolve over time. And you will see you will learn a lot about these, using this code. And we'll be talking more about this in each of these terms, what are they, over time in different lessons that will follow.

Source: The Pennsylvania State University

The following is a representative mass conservation equation for a primary species I in the aqueous phase:

∂(Ci)∂t+∇·(−D∇Ci+vCi)+Nr∑r=1virRr+Nm∑m=1vimRm=0,i=1,n
(1)

Here Ci is the total concentration of species i (mol/m3 pore volume), t is the time (s), n is the number of primary species, D is the combined dispersion–diffusion tensor (m2/s), u (m/s) is the Darcy flow velocity vector and can be decomposed into ux and uz in the directions parallel and transverse to the main flow direction. Nr is the total number of kinetic aqueous reactions that involve species i, vir is the stoichiometric coefficient of the species i associated with the reaction r, Rr is the rate of aqueous reaction r.

Equation (1) implies that the mass change rate of species i depends on physical and chemical processes: the diffusion/dispersion processes that are accounted for by the first term of the right hand side of the equation, the advection process that is taken into account by the second term of the right hand side, and reactions that are represented by the last term of the equation. The last term is the summation of multiple reaction rates, the form of which depend on the number and type of kinetic reactions that species i is involved in. The reaction terms include the rates of kinetically controlled reactions including microbe-mediated bioreduction reactions, mineral dissolution and precipitation reactions, and redox reactions. The reactions also include fast reactions that are considered at thermodynamics equilibrium, including aqueous complexation, ion exchange, and surface complexation. These fast reactions that are at equilibriums however do not show up in the above governing equation (1). Instead they exhibit themselves in the non-linear coupling of primary and secondary species through the expression of equilibrium constants (laws of mass action), as will be detailed later.

The dispersion-diffusion tensor D is defined as the sum of the mechanical dispersion coefficient and the effective diffusion coefficient in porous media D*(m2/s). At any particular location (grid block) with flow velocities in longitudinal (L) and transverse (T) directions, their corresponding diffusion / dispersion coefficients DL (m2/s) and DT (m2/s) are calculated as follows:

DL=D*+LavZ
(2)
DT=D*+aTvT
(3)

Here αL and αT are the longitudinal and transverse dispersivity (m). The dispersion coefficients vary spatially due to the non-uniform distribution of the permeability values.

As will be discussed in lesson 1, in a system with N total number of species and m fast reactions, the total number of primary species is n = N – m. With specific initial and boundary conditions, reactive transport codes solve a suite of n equation (1) with explicit coupling of the physical processes (diffusive/dispersive + advective transport) together with m algebraic equations defined by the laws of mass action of fast reactions. The output is the spatial and temporal distribution of all N species. This type of process-based modeling allows the integration of different processes as a whole while at the same time differentiation of individual process contribution in determining overall system behavior.