Solution Tutorial — 2D Steady Flow & Particle Tracking

The implementation in this applet is based on the RESSQ program of Javandal, Doughty, and Tsang (1984). A more rigorous treatment using complex variable (potential flow) methods is given by Strack (1989).

For 2D steady flow in a homogeneous aquifer, the velocity potential $\Phi$ is defined as:

where $K$ is hydraulic conductivity and $h$ is hydraulic head. Darcy's law gives specific discharge components $V_x = -\partial\Phi/\partial x$ and $V_y = -\partial\Phi/\partial y$. The steady-state governing equation is the Laplace equation:

Regional Flow

For uniform regional flow with specific discharge magnitude $V_\text{reg}$ directed along $+x$, the potential is:

The arbitrary constant is set to zero since only relative values of $\Phi$ are needed (the applet plots equally spaced contours between minimum and maximum values).

Single Pumping Well

For a single pumping well of flow rate $Q_w$ centered at the origin of a confined aquifer of thickness $b$, with no regional flow, the radially symmetric potential is:

where $r = \sqrt{x^2 + y^2}$ is the radial distance from the well. Positive $Q_w$ corresponds to injection (flow away from the well); negative $Q_w$ corresponds to extraction (flow toward the well).

Because the Laplace equation is linear, solutions may be superimposed. For an extraction well in a uniform regional flow field, combining Eqs. (1) and (2) gives:

This result generalizes directly to $N$ wells at arbitrary locations $(x_i, y_i)$ with rates $Q_{w,i}$:

where $r_i = \sqrt{(x-x_i)^2 + (y-y_i)^2}$ is the distance from point $(x,y)$ to well $i$.

Taking the negative gradient of $\Phi$ in Eq. (4) gives the Darcy specific discharge components at any point $(x, y)$ due to the combination of regional flow and all wells:

The pore-water (seepage) velocity components, which drive particle advection, are obtained by dividing by porosity $n$:

Once the seepage velocity field is known analytically everywhere, a particle placed at position $(X_p, Y_p)$ is tracked by integrating the trajectory equations:

The applet uses a forward Euler (first-order) time-stepping scheme. At each step $k$:

Because $\bar{V}_x$ and $\bar{V}_y$ are known exactly at any point (no grid interpolation needed), particle accuracy is limited only by the size of $\Delta t$ and not by grid resolution.

Forward vs. Backward Tracking

Forward tracking integrates Eq. (7) with $\Delta t > 0$, tracing the path a particle takes from its starting point downstream to where it exits the domain or converges on an extraction well. Backward tracking uses $\Delta t < 0$ (reversing flow), tracing a particle back to its source — useful for determining capture zones and source areas.

Travel Time Marking

With the Fixed Incremental or Specific Times contouring modes, the applet marks particle positions at specified elapsed times, effectively drawing isochrones (lines of equal travel time) on the domain.