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An aquifer drainage hydrograph plots the time-dependent discharge from an aquifer. In the shallow constant-recharge model, early discharge grows linearly according to the rain-fed aquifer similarity solution. Sustained recharge $R$ on a finite length $L$ gives eventual steady discharge $RL$ when the other end is impermeable. After recharge stops, a separable aquifer drawdown has height proportional to $1/(t+t_0)$ and discharge proportional to $1/(t+t_0{)}^2$; this is generally a late-time description rather than an exact solution for arbitrary inherited height profiles.

Darcy law is a linear relation between the bulk Darcy flux and the net driving force per unit volume. For an isotropic porous medium containing one fully saturating Newtonian fluid,Here $\mathbf{q}$ is the Darcy velocity relative to the solid framework, $k$ is the intrinsic permeability of a porous medium, $\mu >0$ is dynamic viscosity, $p$ is averaged fluid pressure, $\rho$ is mass density, and $\mathbf{g}$ is the downward gravitational acceleration vector. The pressure gradient exerts force density $-\nabla p$, while gravity contributes $\rho \mathbf{g}$; the minus sign makes the discharge follow their sum. The symbols $\mathbf{u}$ and $\mathbf{q}$ are both commonly used for Darcy velocity; neither denotes pore velocity here.

With upward coordinate $z$ and positive gravitational acceleration magnitude $g$, $\mathbf{g}=-g{\mathbf{e}}_z$. Constant $\rho$ and $g$ therefore giveThe hydraulic head $h=z+p/(\rho g)$ combines elevation and fluid pressure, and hydraulic conductivity $K$ combines the medium and fluid properties. Hydrostatic pressure satisfies $\nabla p=\rho \mathbf{g}$, so it produces zero Darcy flux even though fluid pressure increases downward. Conversely, equal pressures at the two ends of a vertical column do not imply zero flow: their elevations give different hydraulic heads.

The appeal of Darcy law is that a complicated connected pore geometry enters a macroscopic calculation through a measurable coefficient. It predicts filter discharge and aquifer flow; combined with mass conservation, it gives an equation for the evolving fluid pressure. Its simple linear form requires a small pore-scale Reynolds number, slow adjustment compared with pore-scale inertial relaxation, and a valid continuum approximation. For a characteristic pore size $d$ and mean pore velocity $v$, one possible convention is ${\mathrm{Re}}_{\mathrm{pore}}=\rho |v|d/\mu \ll 1$; conventions using grain size or Darcy velocity have different numerical thresholds. Merely calling flow laminar does not guarantee negligible inertia. Unsaturated pores, several fluid phases, appreciable inertia, or a fluid whose stress is not Newtonian require further modeling. Woessner and Poeter discuss these limits in their authored hydrology text.^[ref]

Henry Darcy's 1856 book Les fontaines publiques de la ville de Dijon was published by Victor Dalmont; the University Grenoble Alpes digital collection verifies this bibliographic record and supplies a digitized copy.^[ref] Woessner and Poeter's account connects Darcy law to experiments on water flowing through sand filters, relating discharge to cross-sectional area, hydraulic head difference and sample length.^[ref] The modern intrinsic permeability of a porous medium notation above should not be read as a quotation of Darcy's original notation.

Consider planar horizontal layers with positive scalar intrinsic intrinsic permeabilities $k_i$, thicknesses $b_i$, and fractions $f_i=b_i/{\sum }_j{b}_j$. Assume each layer is uniform and isotropic, fully saturated by the same Newtonian fluid of constant mass density and dynamic viscosity, with perfect hydraulic contact and steady incompressible flow. Use boundaries that impose pure flow normal to the layers or a common gradient of hydraulic head along them; additional interface resistance and finite-edge effects are excluded.

For vertical flow normal to the layers, every layer carries the same normal Darcy flux by mass conservation. The hydraulic head drops add. Writing $B={\sum }_i{b}_i$ and taking positive $\Delta h_i$ as the drop along the flow direction givesThis is the thickness-weighted harmonic mean. It is hydraulic head, rather than fluid pressure alone, that accounts correctly for gravity in the vertical calculation. A thin layer of very small permeability of a porous medium can control the whole discharge because all flow must pass through it.

For horizontal flow parallel to the layers, each layer experiences the same gradient of hydraulic head. For width $W$, each area is $A_i=W{b}_i$, and the individual volume fluxes add:This is the thickness-weighted arithmetic mean. A highly permeable layer can carry most of the horizontal flow because it supplies a continuous route from inlet to outlet. Woessner and Poeter derive both cases for hydraulic conductivity; identical fluid properties allow the common factor $\rho g/\mu$ to cancel.^[ref]

For two equally thick layers, take $k_1=k_0$ and $k_2=9k_0$, where $k_0=10^{-12}{\mathrm{m}}^2$. The values 1 and 9 are therefore permeabilities in units of $k_0$, rather than dimensionless intrinsic intrinsic permeabilities. ThenAcross the layers, the less permeable half takes $90\%$ of the total hydraulic head drop and the more permeable half only $10\%$. Along the layers, the more permeable half carries $90\%$ of the total volume flux. At the same magnitude of imposed gradient of hydraulic head, horizontal Darcy flux is $5/1.8=25/9\simeq 2.78$ times the vertical Darcy flux.

The arrangement thus creates physical anisotropy from locally isotropic ingredients. In coordinates with $x,y$ along the layers and $z$ normal to them, the effective permeability of a porous medium tensor is ${\mathbf{k}}_{\mathrm{eff}}=\mathrm{diag}(5,5,1.8)k_0$. A driving gradient of hydraulic head with both horizontal and vertical components generally produces a discharge in a different direction. This distinction matters when deciding whether a geological bed conducts aquifer flow along itself or acts as a barrier to flow across it.

Regenerate the diagram by running python ../layered-porous-permeability.py from _media; it writes its PNG into the current directory.

The condition $f(0)=0$ means the river maintains the water table at the impermeable base level: $h(0,t)=0$. The liquid is exposed to atmospheric pressure there. It does not imply zero drainage, because $h_x$ may be singular at the outlet.

Far from the river, lateral drainage has not significantly affected storage. Rainfall simply fills pore volume, so the dimensional height is $h\sim Rt/\varphi$. In the dimensionless variables this is $h\sim t$, giving $f(\infty )=1$.

Although the physical aquifer is finite, these boundary conditions describe its early similarity solution before the drainage-affected region reaches $x=L$. In dimensional variables its horizontal extent is of order $\sqrt{KR}t/\varphi$, so the semi-infinite approximation requires $t\ll T=\varphi L/\sqrt{KR}$.

For the dimensionless rainfall-driven Boussinesq equation for an unconfined aquifer, $h_t=(h{h}_x{)}_x+1$, balancing every time power in $h=t^{a}f(x/t^b)$ gives $a=b=1$. The similarity profile obeys $(f{f}^{\prime }{)}^{\prime }+\eta f^{\prime }-f+1=0$. A river drained to base level gives $f(0)=0$, while undrained storage far away gives $f(\infty )=1$. The solution approximates a finite aquifer before its drainage-affected length reaches the far boundary. If $J={\lim }_{\eta \downarrow 0}f{f}^{\prime }>0$, the dimensional drainage flux is $\sqrt{K}{R}^{3/2}Jt/\varphi$.

An aquifer containing saline rather than fresh water. A deep saline aquifer can provide pore space for carbon capture and storage, but safe retention depends on reservoir connectivity, continuous caprock and pressure management.