12.4 Mass Transport Equations
We represent mass transport in groundwater mathematically as a combination of:
- advective transport,
- molecular diffusion,
- mechanical dispersion,
along with retardation processes such as
- solutes exchanging places with ions on the surface of minerals that make up the subsurface material,
- sorption of mass to surface material,
- chemical reaction including dissolution of subsurface material, precipitation of mass to form solids, and
- decay (both radioactive and biological) of mass which can decrease concentration and introduce additional mass that are daughter products of a decay process.
Similar to the discussions in Sections 4, 5, and 6 of this book, it is possible to develop governing equations for transport of solutes and particles in groundwater systems by considering an elemental volume and deriving a general three-dimensional differential equation that can be solved for various practical applications. A detailed derivation of this equation is beyond the scope of this publication; however, a basic introduction to the concepts and mathematics is presented here.
Development of the mass transport equation in porous media starts with Fick’s Law, which describes the diffusive flux of solutes in response to a concentration gradient. Fick’s Law is presented in Equation (94).
| [latex]J = - D_{mol}\frac{\partial\varphi}{\partial x}[/latex] | (94) |
|---|
where (parameter dimensions are dark green font with mass as M, length as L, time as T):
| [latex]J[/latex] | [latex]=[/latex] | diffusion flux, which is the amount of substance diffusing per unit area per unit time (molesL-3T-1) |
|---|---|---|
| [latex]D_{mol}[/latex] | [latex]=[/latex] | molecular diffusion coefficient (L2T-1) |
| [latex]\frac{\partial\varphi}{\partial x}[/latex] | [latex]=[/latex] | concentration gradient, which has units of concentration per length (:) |
Like hydraulic conductivity, molecular diffusion can be measured in the laboratory.
Fick’s Law is then combined with the equation for advective transport, Equation (90), and a mechanical dispersion term that represents the dispersion described in Section 12.2.2 to develop Equation (95). Unlike the groundwater flow equation in which all the terms are second partial derivatives, this equation is a hyperbolic equation because the last term is not a second derivative. This variation makes it more difficult to solve the transport equation as compared with the flow equation.
| [latex]\frac{\partial C}{\partial t} = D_{l}\frac{\partial^{2}C}{\partial{l_{l}}^{2}} + D_{t}\frac{\partial^{2}C}{\partial{l_{t}}^{2}} + D_{v}\frac{\partial^{2}C}{\partial{l_{v}}^{2}} - \overline{v_{l}}\frac{\partial C}{\partial l}[/latex] | (95) |
|---|
where (parameter dimensions are dark green font with mass as M, length as L, time as T):
| [latex]C[/latex] | [latex]=[/latex] | concentration in fluid (ML-3) |
|---|---|---|
| [latex]t[/latex] | [latex]=[/latex] | time (T) |
| [latex]l[/latex] | [latex]=[/latex] | concentration gradient (ML-4) |
| [latex]D[/latex] | [latex]=[/latex] | hydrodynamic dispersion coefficient (L2T-1) |
| [latex]\overline{v}[/latex] | [latex]=[/latex] | interstitial velocity (LT-1) |
| [latex]l_{l}[/latex] | [latex]=[/latex] | in the direction of flow (:) |
| [latex]l_{t}[/latex] | [latex]=[/latex] | in the direction laterally transverse to flow (:) |
| [latex]l_{v}[/latex] | [latex]=[/latex] | in the direction vertically transverse to flow (:) |
Unlike the diffusion coefficient, the hydrodynamic dispersion coefficient cannot be measured in a laboratory experiment. Rather, the hydrodynamic dispersion coefficient is determined by adjusting its value until the distribution of calculated concentrations in space and time—as determined by Equation (95)—match the observed concentrations in a laboratory or field flow system. The hydrodynamic dispersion coefficient is a bulk parameter that combines the molecular diffusion coefficient of Equation (94), flow velocity, and a term called dispersivity, which is intended to represent the heterogeneous tortuosity of the medium through which the groundwater flows. Like hydraulic conductivity, dispersivity is viewed as anisotropic, that is, it has a different value in the longitudinal direction (i.e., in the direction of groundwater flow) and in the transverse lateral and vertical directions (i.e., perpendicular to the direction of fluid flow) thus there is a dispersion coefficient for each of the major directions of flow. In fact, the dispersion coefficient may vary in non-orthogonal directions, but using three directions as presented in Equation (96) simplifies the mathematics and captures concentration variations in the direction of flow as well as to the sides which is generally less and then above and below which is typically even more restricted due to the typical layering of geologic materials.
| [latex]D_{l,t,v} = \alpha_{l,t,v}\overline{v_{l}} + D_{mol}[/latex] | (96) |
|---|
where (parameter dimensions are dark green font with mass as M, length as L, time as T):
| [latex]D_{l,t,v}[/latex] | [latex]=[/latex] | dispersion coefficient in the longitudinal, transverse lateral, or transverse vertical direction of flow (L2T-1) |
|---|---|---|
| [latex]\alpha_{l,t,v}[/latex] | = | dispersivity in the longitudinal, transverse lateral, or transverse vertical direction of flow (L) |
| [latex]\overline{v_{l}}[/latex] | = | average linear velocity in the longitudinal direction of flow (LT-1) |
| [latex]D_{mol}[/latex] | [latex]=[/latex] | molecular diffusion coefficient in porous medium (L2T-1), typically insignificant, being on the order of 1×10-9 m2/s |
Dispersivity, [latex]\alpha[/latex], is a function of heterogeneity at the pore-, unit-, and formational-scale. It is a very difficult term to characterize. Previous studies (e.g., Schulze-Makuch, 2005) have shown that dispersivity is a function of scale of measurement such that studies which incorporate larger volumes of aquifer (e.g., larger contaminant plume sizes) will likely experience behavior consistent with greater dispersivity coefficients.
Equation (95) is often reduced to its one-dimensional form as shown in Equation (97) and used for predicting movement and distribution of contaminants along a groundwater flow path.
| [latex]\frac{\partial C}{\partial t} = \ D_{x}\frac{\partial^{2}C}{\partial x^{2}} - \overline{v_{x}}\frac{\partial C}{\partial x}[/latex] | (97) |
|---|
There are many analytical solutions for the mass transport equation that can be used to calculate concentration at a specified location and time. These can be used to predict the spatial distribution of contaminant at a given time, or the shape and timing of contaminant breakthrough curves at a given location. Analytical expressions are available for one-, two‑, and three- dimensional spreading from both continuous and slug sources. Some are enhanced to reflect retardation and/or decay of contaminants. The equations may look daunting, but a solution can be obtained by systematically entering the values and following through each step of the calculation. A couple of example solutions are provided here to offer insight into the character of the equations. One is for one-dimensional spreading of a continuous source in a uniform, one-dimensional flow field. It is representative of contaminant moving through a column full of sand in a laboratory, Equation (98). This equation is discussed by Fetter (1999) in section 2.8.4 of his book. This is similar to the plume shown in Figure 93a but with no lateral spreading. The other equation is for three-dimensional spreading of a specified amount of mass introduced at a point in a uniform, one-dimensional flow field, Equation (99). This is similar to the plume shown in Figure 93b but with the source defined as an amount of mass at a point location intend of a block of fluid with a given concentration. lateral spreading.
| [latex]C = \frac{C_{o}}{2}\left\lbrack erfc\left( \frac{x - \overline{v_{x}}t}{2\sqrt{D_{x}t}} \right) + \exp\left( \frac{x\overline{v_{x}}}{D_{x}} \right)erfc\left( \frac{x + \overline{v_{x}}t}{2\sqrt{D_{x}t}} \right) \right\rbrack[/latex] | (98) | |
|---|---|---|
| [latex]C = \frac{M}{8{(\pi t)}^{\frac{3}{2}}\ \sqrt{D_{x}D_{y}D_{z}}}\exp\left( - \frac{X^{2}}{4D_{x}t} - \frac{Y^{2}}{4D_{y}t} - \frac{Z^{2}}{4D_{z}t} \right)[/latex] | (99) |
where (parameter dimensions are dark green font with mass as M, length as L, time as T):
| [latex]C[/latex] | [latex]=[/latex] | concentration in fluid (ML-3) |
|---|---|---|
| [latex]C_{o}[/latex] | [latex]=[/latex] | concentration of the source (ML-3) |
| [latex]erfc[/latex] | [latex]=[/latex] | complimentary error function, which is the same as 1-erf, if the value within the erf is negative then the erf is minus the erf of its absolute value, so erfc(x)=1-erf(x) and if x is negative then erfc(x)=1-(-erf(x)); erf can be looked up in a mathematical table or obtained using an online calculator such as https://www.medcalc.org/manual/erf-function.php↗ |
| [latex]x[/latex] | [latex]=[/latex] | distance from the source in the x direction (L) |
| t | [latex]=[/latex] | time since source began releasing contaminant (T) |
| [latex]\overline{v_{x}}[/latex] | [latex]=[/latex] | interstitial velocity in the x direction (LT-1) |
| [latex]D[/latex] | [latex]=[/latex] | hydrodynamic dispersion coefficient (L2T-1), subscript indicates the directionality of the coefficient |
| X | [latex]=[/latex] | distance from the center of mass of the plume in the x direction at time t (L), the center of mass is at x=[latex]\overline{v}t[/latex] |
| [latex]Y[/latex] | [latex]=[/latex] | distance from the center of mass of the plume in the y direction (L) |
| Z | [latex]=[/latex] | distance from the center of mass of the plume in the z direction (L) |
It is essential to determine the dimensionality of the setting and the character of the source when selecting an equation to calculate concentration. Exercise 24⮧ provides an opportunity to calculate a concentration from one-dimensional spreading of a continuous contaminant source using Equation (98). Exercise 25⮧ provides an opportunity to calculate three-dimensional spreading of a slug contaminant source Equation (99).