Analytical description of seepage from disconnected surface water: improved approximate solution and insights for disconnection assessment

Seepage across a streambed is typically expressed by Darcy's law, q = K_c · ΔH / w_c, where K_c and w_c are the hydraulic conductivity and thickness of the bed, respectively, and ΔH is the hydraulic head gradient. When the stream is “disconnected” – i.e., an unsaturated zone exists beneath it – the suction at the lower bed interface no longer follows the groundwater head. Capturing this effect requires solving a nonlinear equation that incorporates the unsaturated hydraulic conductivity (UHC) relationship of the underlying aquifer (most commonly described by van Genuchten–Mualem or Brooks‑Corey–Burdine parametrizations). Because of the added computational burden, many modelling packages (e.g., MODFLOW) simplify the problem by assuming zero suction, which leads to systematic underestimation of seepage.
In this study we analyse the governing nonlinear equation using a generic UHC function that exhibits a power‑law behaviour in the dry limit. First, we examine the zero‑stage (h = 0) case and demonstrate that three distinct asymptotic solutions arise, depending on the relative magnitude of the dimensionless groups (K_a/K_c)^1+1/b and B^1/b w_cK_a/h_gK_c, where K_a is the aquifer hydraulic conductivity. Here b and B are shape parameters that depend on the chosen parameterisation (for example, for Van Genuchten–Mualem, b = (5n − 1)/2 and B = (1 − 1/n)²) and h_g is the associated scale parameter. From these asymptotes we identify two clogging regimes. For hard clogging, seepage becomes independent of aquifer properties; in this regime the MODFLOW simplification is exact. For soft clogging, zero-stage seepage converges to q₀ = K_a · (w_cK_a/h_gK_c)^-b/(1+b).
We then derive an analytical approximation that smoothly bridges the asymptotic limits. Validation against the exact numerical solution shows that the new expression provides a rapid yet more accurate alternative to the conventional MODFLOW formulation. Finally, we argue that a positive stage simply adds a linear term to the flux, yielding q  ≈  q₀  +  w_c/K_c · h. Based on this insight, we propose a novel method for assessing surface water - groundwater disconnection.