Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 84 / 3

Take $Q$ as the fluid volume flux per unit source length into one side of the aquifer. For a symmetric release with total rate $Q$, use $Q/2$ on each side. Introduce a constant porosity $\varphi$ to distinguish Darcy velocity from the advecting interstitial velocity; if the given flux is already normalized to pore volume, this factor is absorbed in its definition. With a pressure gradient $G=-p_x>0$, Darcy law givesThus, with $\xi =z/h$, the pore velocity and its depth average areThe tracer concentration initially obeys the two-dimensional advection-diffusion equationReflecting transverse boundaries and constant effective molecular diffusion are assumed. No independent radioactive decay rate is supplied; it can be added separately if it is significant over the experiment.

For times long compared with $h^2/D$, transverse diffusion exchanges tracer between fast and slow streamlines. Write $c=C(x,t)+\chi (z)C_x+\cdots$, where $C=\overline{c}$ and $\overline{\chi }=0$. At the leading transverse-correction order,Integration yieldsIts mean and boundary derivatives are zero. Averaging the concentration equation gives $C_t+U{C}_x=[D-\overline{(u-U)\chi }]C_{xx}$. Integration by parts in the cell equation shows that the enhancement is positive:Consequently Taylor dispersion in a parabolic porous-layer velocity profile givesInjection fixes the inlet concentration or solute-flux boundary condition. This depth-averaged equation requires transverse equilibration and longitudinal concentration variations slow enough for the cell expansion. The supplied PDF hint has both a different polynomial and a false numerical value: its integral evaluates exactly to $-83/2520$, not $0.3525$. The cell problem above supplies the coefficient independently.

After transverse equilibration, a localized release on a sufficiently long aquifer has the advection-diffusion heat-kernel solutionwhere $\mathcal{M}=\int Cdx$ is conserved. The actual tracer amount per unit source length is $\varphi h\mathcal{M}$ in this normalization. Its center moves at the mean interstitial velocity $U$, its variance is $2D_{\mathrm{eff}}t$, and its peak decreases as $t^{-1/2}$. A finite initial packet is convolved with this kernel.

Before times of order $h^2/D$, this effective Gaussian description need not hold. The parabolic velocity profile initially shears a packet, and particles initially at different depths acquire different along-flow displacements. Transverse diffusion subsequently erases depth memory and produces the long-time Taylor dispersion law. A short pulse can therefore have an initially asymmetric or strongly distorted longitudinal shape before entering the averaged regime.

Interpret absorption as irreversible capture of radioactive tracer by the formation at the stated first-order rate. With stored amount $S$ measured per the same reference pore-storage volume as $C$, irreversible capture of an advecting tracer givesAdding the equations conserves dissolved plus captured tracer, apart from advective and diffusive transport. A different solid-volume normalization inserts the corresponding porosity factor in $S_t$. The averaged equation has the same transverse-equilibration limitation as in part (a).

For a maintained inlet concentration $C(0,t)=C_0$, a bounded downstream stationary water profile satisfies $D_{\mathrm{eff}}{C}^{\prime \prime }-U{C}^{\prime }-\lambda C=0$. Its characteristic roots are $(U\pm \Delta )/(2D_{\mathrm{eff}})$, where $\Delta =\sqrt{U^2+4\lambda D_{\mathrm{eff}}}$. Rejecting the growing root givesIf the inlet instead prescribes solute flux $J_0=UC-D_{\mathrm{eff}}{C}_x$, replace $C_0$ by $J_0/(U+D_{\mathrm{eff}}r)$. In the advection-dominated limit, $r\simeq \lambda /U$, giving an attenuation length $U/\lambda$.

The corresponding capture-rate profile is $\lambda C_{\infty }(x)$. There is an important qualification to the requested adsorbed profile: with a continuous source, irreversible adsorption and no solid decay or saturation, the water concentration can be stationary but the accumulated solid concentration cannot. At every point with positive $C_{\infty }$,Thus the deposited amount grows asymptotically linearly with a spatially exponential rate profile. A time-independent deposit would require additional desorption, finite capacity or radioactive decay in the solid; for example $S_t=\lambda C-{\gamma }_{s}S$ would give $S_{\infty }=\lambda C_{\infty }/{\gamma }_s$ if ${\gamma }_s>0$.

If the source is the localized release in part (b), there is instead a finite final deposit and no nonzero steady water concentration. On the full line, multiply the moving Gaussian by $e^{-\lambda t}$. Integrating its governing equation over all time gives $D_{\mathrm{eff}}{J}^{\prime \prime }-U{J}^{\prime }-\lambda J=-\mathcal{M}\delta (x-x_0)$ for $J={\int }_0^{\infty }Cdt$. Continuity and the derivative jump at $x_0$ then giveIts integral is $\mathcal{M}$, so all the released tracer is eventually captured in this infinite-domain model. This distinguishes the finite-pulse deposited profile from the continuously accumulating deposit under sustained injection.