Recharge, leakage and transient storage¶
Like in any flow model, the flow equation in Anaqsim is based on Darcy's Law and conservation of mass (and volume, with constant density). The conservation equation, in its simplest form, is
where \(\nabla Q\) is the divergence of the two-dimensional aquifer discharge vector field and \(\gamma\) is the net extraction per area (sink term, units of L/T). The sink term \(\gamma\) may have contributions from leakage out the top of the subdomain (\(L_t\)), leakage out the bottom of the subdomain (\(L_b\)), and transient discharge/area into storage (\(S\,\partial h/\partial t\)). See equations 4–6 of Fitts (2010).
Vertical leakage¶
The vertical leakages \(L_t\) and \(L_b\) are specific discharges proportional to the head difference between the domain and the head above or below (could be specified head or another domain at a different level), and proportional to the equivalent vertical hydraulic conductivity \(K_e\) that is based on the vertical conductivities and the saturated thicknesses at the average heads specified for the domains involved. If there is leakage between two domains at different levels, the equation used for computing \(K_e\) is
where \(b_1\) and \(b_2\) are the average saturated thicknesses of the two layers and \(K_1\) and \(K_2\) are the vertical hydraulic conductivities of the two layers. In Anaqsim, \(K_e\) is assumed to be constant and independent of head and actual saturated thickness.
Uniform vs spatially-variable extraction¶
Under restricted circumstances (uniform recharge, steady flow, and a single-level model), \(\gamma\) is constant and independent of location. In such cases, the uniform \(\gamma\) distribution may be modeled exactly using a uniform area sink.
In many practical cases, the model needs spatially-variable extraction (\(\gamma\) varies with \(x\), \(y\)) due to spatially-variable vertical leakage and/or spatially-variable storage changes. When that is the case, the model needs spatially-variable area sinks to approximate the proper distribution of \(\gamma\). The spatially-variable area sink functions in Anaqsim create a smooth, continuous, irregular \(\gamma\) surface within a subdomain. The model is using equation 13 of Fitts (2010) as a model of the distribution of \(\gamma\), with \(\gamma\) equal to the perfect extraction/area (computed by equation 6 of Fitts (2010)) at each basis point, but approximating equation 6 of Fitts (2010) between basis points. The modeled distribution of \(\gamma\) satisfies the flow equation perfectly at basis points and approximates the flow equation between basis points. This approximation is more accurate if the basis point spacing is smaller.
Accuracy and computational cost¶
To check the accuracy of this approximation, Anaqsim provides an analysis tool (Check a solve). If checks with this tool reveal a poor approximation, the basis point spacing is too large. If the approximation is satisfactory, you may be able to increase the basis point spacing and reduce computation time.
Spatially-variable area sinks can rack up a large number of equations and computational burden on the system, so use them sparingly. Where possible (often the far-field), use a single level in the model, which if it is steady means you can use the very efficient uniform area sinks instead of spatially-variable area sinks. Use the special well basis point spacings around wells to gain accuracy with minimal computation.