Solute Transport in the Vadose Zone Quantification of oozing, spreading and smearing
Overview • Much of the attention in vadose zone and groundwater in general results from interest in contaminant transport. • We will review • basic formulations of sorption and degradation • Plug flow (piston flow) modeling approach • Convective/Dispersive approach • Remember: • Any errors in you solution to water flow will be propagated in your solute transport estimates
Partitioning between phases: Sorption • The total concentration C (in mass per volume) is the sum of sorbed and aqueous • b = bulk density of the porous media [mass dry media per total volume] • cs = concentration adsorbed to media [mass of solute adsorbed per mass of dry media] • = volumetric water content [volume of water per total volume], • cl = solute concentration liquid phase [mass of solute in liquid phase per volume of water].
A Brief Discussion of Sorption • An isotherm relates cs to cl in a mathematical form • Typical Assumptions: • Each chemical species acts independently • Rub: with limited number of adsorption sites, this doesn’t work • Desorption and adsorption follow the same isotherm • Rub: There is hysteresis between adsorption and desorption AND time dependence! Solid with adsorbed Concentration cs Liquid with concentration cl
Freundlich cs = kf cl1/n Linear cs = kd cl cs Langmuir cs = a cl Q/(1+acl) cl Isotherms Three most popular relationships
Linear Isotherm • cs = Kd cl • What is so great about the linear isotherm? • Two things: • For low concentration (i.e., when most sorption sites are unoccupied), the linear isotherm is a good description. • It makes the math easy! (allows us to find solutions that we can understand). • Problems: • If dealing with concentrated sources or limited sorption sites.
Langmuir Isotherm • cs = a cl Q/(1+acl) Q = adsorption sites/mass a = k1/k2 where k1 = rate of adsorption k2 =rate of desorption • Notes: • for acl <<1 this reduces to cs = aQcl (linear isotherm) • for acl >>1 this reduces to cs =Q Makes sense since Q is the sorption capacity of the soil (recall CEC)
What’s so great about Langmuir isotherm? • The high and low concentration behavior makes intuitive sense • We can “derive” the Langmuir relationship from a simplified model. • Consider a block of stuff with Q adsorption sites per unit mass • at equilibrium the rate of sitesbeing filled (ra) will equal therate of sites being vacated (rd) • Assuming that each site acts independently, the probabilityof sorption will be proportionalto the probability of a solutemolecule hitting that site
“Deriving” the Langmuir isotherm • So we estimate the adsorption rate asSimilarly, we may estimate the rate of desorption as being proportional to the number of sites filled Proportionality Constant Concentration in Liquid Fraction of sites filled Fraction of sites filled
Langmuir derivation... • At equilibrium, ra = rd. Equating theseletting k = k’/k” and multiplying each side by QSolving for the sorbed concentration • as desired.
Transport: Basic Processes • 3 basic mechanisms by which solutes move: • advection • diffusion • dispersion • Advection (A.K.A. convection) movement of the solute with the bulk water in a macroscopic sense. • Advective transport ignores the microscopic processes, but simply follows the bulk Darcian flow vectors. • The crowd metaphor: in a march with thousands, a small group will still stay together
Basic Processes 2: Diffusion • Diffusion: • the spreading of a compound through the effects of molecular motion • Governed by Fick’s law • tends to mix areas of high concentration with areas of lower concentration. • the rapidity of diffusive spreading linked to molecular velocities and path length between collisions.
Diffusion Cont. • For a given temperature, any given molecule has a particular energy, and thus velocity. • Since kinetic energy is related to the square of velocity, diffusion rates changes with the square-root of temperature (as measured in degrees K), • varies little over typical groundwater temperature ranges. • Summarized by the diffusion coefficients • are on the order of 0.2 cm2/sec in gases • 0.00002 cm2/sec in liquids • a factor of 10,000 higher in gases due to the lower rate of molecular collisions.
Diffusion, cont. • The Crowd Scene metaphor: • diffusion corresponds to the movement that happens when they put the dance music on as darkness falls at the end of the march. • People start bouncing around • Slowly you and your buddies spread out in the crowd, making your designated driver very anxious about how you will all ever be brought together again. • Right to worry in that she is working in direct opposition to the aggressive force of entropy, a tough foe.
Basic Processes 3: Dispersion • Dispersion is: • Mixing which occurs due to differences in velocities of neighboring parcels of fluid. • Occurs at many scales (compared to diffusion which is strictly a molecular-scale process). • The crowd scene metaphor: • they have turned the music off, and your chaperone has reassembled the group to leave. • Some members get stripped as the crowd moves past obstructions, others caught up quick moving groups • 2 problems: • (1) People hitting poles get left behind • (2) people in the center of the crowd exit too quickly.
Dispersion in Groundwater • Start at the scale of the intergranular channels which the fluid moves through. • In these channels the fluid velocity is proportional to the square of the distance from the local surfaces, leading to separation of particles across these areas
Tortuosity and beyond • The tortuousity of the intergranular space also smears solutes. • At a larger scale (say the 1 m scale), there is typically heterogeneity between materials of differing permeability, which will again lead to areas of higher and lower flow velocity, and therefore dispersion. • Dispersion increases with increasing scale as each new dispersive process is added to those which occur at all of the lower scales.
Plug or Piston Flow models • Movement is taken to be only due to advection • Processes of sorption and degradation still may be included • How could this assumption be reasonable? • Typically don’t have data on the magnitude of dispersion for media. • May argue that it is better to be explicit with lack of knowledge rather than making a wild guess • If the solute is distributed relatively uniformly (as in nitrogen), then dispersion and diffusion are not big players • If we don’t care about position, but just about final loading
Plug Flow model • The notion is that all water molecules move in lock-step. • Visualize marbles moving down a rubber tube • Push one in the top, and one comes out the bottom • The order of the marbles never changes (no mixing) • Solutes move in proportion to the fraction in the liquid state • If non-adsorbed, solutes move with the water • For sorbed solutes it makes sense to use linear partition, which does not cause dispersion
Example: Plug Flow Transport (Mills et al., 1985) • 50,000 g/ha of naphthalene spilled • sandy loam soil with bulk density of 1.5 g/cm3 • = 0.22 cm3/cm3 • water table at 1.5 meters • mean annual percolation of 40 cm. • first order partition coefficient kd =11, • half life of 1,700 days (decay rate = 0.149 yr-1) • We want to know: the quantity of naphthalene that will reach the aquifer.
Plug flow example (cont.) • Computing the plug flow velocity is simply a matter of computing the ratio of the water to solute velocity (retardation factor) • The water velocity is the flux divided by the moisture content.
Plug flow example (completed) • At 2.3 cm/yr, it takes 65 yr. to go 1.5 m • The half life is 4.66yrs • From the definition of half life we find the decay rate c/co = 0.5 = exp(-t1/2) = exp(- 4.66) = 0.149 yr-1 • Thus the final mass is M = M0 exp(-0.149 x 65) =50,000gr/ha x exp(-0.149 x 65) = 3.1 gr/ha
What was so great about that? • Advantages of the plug flow approach: • No hidden steps or highly uncertain parameters • obtain expression which allows direct assessment of uncertainty in key transport parameters (sorption, percolation velocity, decay rate) • Disadvantages: • Not conservative in terms of the leading edge of the plume which will get to the aquifer perhaps years before the center of mass through diffusion/dispersion • Reinforces a false sense of deterministic knowledge of the outcome.
The Advective/Dispersive Equation (ADE) • Also called the Convection-Dispersion Equation (CDE) • Most widely used approach to describe solute transport in porous media. • Derived by imposing the conservation of mass upon transport which includes convection, diffusion, and dispersion. • Scale dependent dispersion! In general requires numerical methods for solution. • There are some very useful analytical solutions to the ADE for special cases which give insight into many real world problems.
Scope of Application of ADE • Applicable in contexts as varied as • riverine discharges • atmospheric plumes • groundwater transport • In the vadose zone, the equation may be used to describe any contaminant which does not move a as a free phase (e.g., not NAPLs) • ADE assumes the solutes are hydrodynamically inactive. • concentrations are so small that density induced flow is ignored. • Flow field must be known a priori. Any error in the flow modeling will cause errors in solute modeling
Derivation of the ADE • Road map of our approach • (1) use a mass balance on an REV to obtain solute mass conservation equation • (2) look at the flux term at a microscopic and macroscopic level to identify transport processes which are added to the solute conservation equation • (3) add in chemical reactions (decay and absorption) to obtain the ADE
Mass Balance about an REV • Take an arbitrary volume and compute the total solute flux into the volume, accounting for source/sink terms. rate of change of mass in the volume contribution of sources or sinks rate of delivery through surface
Recall derivation of Richards equation. Transform the surface integral into a volume integral using the Divergence TheoremWhich gives usgathering the integrals
Since the volume V is completely arbitrary, we could choose this to be any given point. The integrand must be zero everywhere. So we have • which can be summarized as Rate of Change = Fluxes in/out + sources/sinks storage • a.k.a.: conservation of mass
Now what about that Flux term? • We will now discuss in more detail • 1. Advection • joint movement of the water/solute ensemble • 2. Diffusion • Purely microscopic molecular solute movement • 3. Dispersion • Scale dependent • Intrinsically anisotropic: tensor property
u dA jc n Advection • The advected flux is computed through an area dA with unit normal vector n in a local flow with vector velocity u • total flux = (u•n c) dA [mass/time] • = jc•ndA • where jc = uc is the convective flux vector with units [mass/(area•time)]
Diffusive Transport • Fick’s Law states that the net rate of diffusive mass transport is proportional through the diffusion coefficient D to the negative gradient of concentration normal to the area, dA:In flux notation
Advective/Diffusive Transport • the diffusive mass flux, jdiff, is definedCombining this with the advective results we have the net local (micro-scale) flux
Macroscopic Phenomena: Dispersion • The rub: how to deal with the variability in velocity in a macroscopic sense? • Taylor’s approach of mean and deviations • Consider the local velocity to be composed of a sum of the average local velocity with a deviation term accounting for the departure of the local velocity from the average • We may do the same for the concentration
Macroscopic flux Now we may put these mean/deviation expressions into our flux equationCarrying out the products we obtain To obtain volume averaged flux, multiply by the fraction of the volume taking part in the flow () and take averages of all terms
the average of a deviation is zero, so any constant time the average of a deviation is also zero. Thusand so our total flux becomes
b • Dispersion is due to correlations between variations in solute concentration and fluid velocity
Great, how are we going to handle this? • For mathematical convenience we will take dispersion to follow a pseudo-Fickian form: • D3 is the dispersion coefficient (second rank tensor) • Watch out: D3 is always anisotropic even if flow is isotropic. • Dispersion in the longitudinal direction (in the direction of flow) is always much greater than in the transverse direction.
Back to the ADE ... • Putting this form of the dispersion into the fluxPutting this into the conservation of mass Eq.we obtain the governing equation for solute transport, the Advection Dispersion Equation!
y u x The dispersion tensor • The dispersion is a 3 x 3 tensor. If D3 is aligned with the velocity field the off-diagonal terms go to zero if the media is uniform lateral to the direction of flow, say in the y and z directions, then Dy = Dz, and we may write this as a 2 x 2 tensor • DL = longitudinal dispersion • DT = transverse dispersion • typically DL 10 DT z
About those dispersion Coefficients • We have two basic relationships to look at:We need to look particularly at the velocity deviation • Remember that the flow is laminar AND non-inertial • So if your double the flow rate, you double the velocity everywhere • This doubles the mean velocity as well as the deviation velocity • SO D3 is linear with velocity • If D3 dominates, then velocity*time and position of the position of the solutes are related in a 1-to-1 manner
More on velocity and dispersion • The upshot: • IF DISPERSION DOMINATES • plume spreading will yield the same shape of plume with consistent values of u * t 1 u = 4, t = 1 u = 1, t = 4 C/C0 0 Position
Dispersivity • D3 is a function of the porous media and the velocity of the flow field. • We need a parameter which is a function of the media alone • Define dispersivities: • As before, L 10 T • Intrinsic permeability and pore-scale dispersivity are properties of the porous media, they are related
Scale Dependence of Dispersion • Consider the various scales at which velocities will be regionally distributed • Micro: at no-slip boundaries compared to channel core • Meso: Along structural elements (fissures/cracks/ bedding planes. • Macro: between units of differing properties (e.g., soil horizons) • Field: Pinch-outs, low permeability lenses etc. • Point: Dispersion increases monotonically with scale due to additive processes
Combining dispersion and diffusion • Wouldn’t it be nice if we could simply add the D’s? • Define the “Hydrodynamic Dispersion” D3’ D3’ D + D3 • To assign vales to D3’ we need to assess the relative importance of diffusion and dispersion: the dimensionless Peclet number, Pe, the ratio of dispersion effects to diffusion effectswhere d is the mean grain size
Hydrodynamic Dispersion Zones • Zone I: 0< Pe <0.4 Diffusion dominates • Zone II: 0.4 < Pe < 5 Mixed dispersion/diffusion • Zone III: 5 < Pe < 10 Dispersion dominates in longitudinal, combined effects in transverse • Zone IV: Pe > 10 and Re < 1Dispersion dominates: laminar, non-inertial flow • Zone V: Pe > 10 and Re > 1Dispersion dominates,but now D3 is a function of v
Typical values of Pe • For dispersion to dominate we need Pe > 5 • For a sandy soil, with a mean grain diameter, d = 10-3 m and D = 10-11 m2/sec • Pore water velocity needs to be greater than about 5 x 10-8 m/sec (1.6 m/year) to neglect the effects of diffusion on the longitudinal spreading of the solute. • Had we considered a finer texture of soil this velocity would increase linearly. • In the vadose zone we are typically in the tough regions II and III. • Critical to correctly identify the values of d and u that apply to your problem to determine the relative importance of diffusion and dispersion.