Abstract: Based on Terzaghi’s one-dimensional consolidation theory applied to partially drained boundaries, conventional models often prescribe an exponential decay function (E-function) to represent boundary pore-water pressure. While this approach effectively reconciles Terzaghi’s boundary and initial conditions, it generates a physical paradox: at t = 0, the drainage boundary exhibits a nonzero rate of change of pore-water pressure, implying an instantaneous drainage velocity that contradicts the assumed initial undrained state. In this work, we revisit the physical basis of boundary conditions and derive a novel decay function for boundary pore-water pressure that simultaneously satisfies the initial condition of zero drainage velocity and the prescribed boundary constraint. This new formulation yields a mathematically consistent one-dimensional consolidation model for saturated soils with a partially permeable drainage boundary. Analytical solutions are obtained by applying Fourier integral transforms in the spatial coordinate and Laplace transforms in time. A key feature of the model is the introduction of a dimensionless parameter, α, which continuously controls drainage capacity. When α is set near 500, the boundary pore-water pressure decays almost instantaneously to zero, recovering the classical fully drained condition and Terzaghi’s original solution. Conversely, as α approaches 0.01, the boundary behaves as undrained, with pore-water pressure remaining effectively constant over time. Under identical α values and at any given dimensionless time, the pore-water pressures predicted by our model exceed those computed with the conventional E-function boundary condition. Correspondingly, the predicted consolidation time is longest for our solution, intermediate for the E-function condition, and shortest for Terzaghi’s fully drained case. Parametric studies further illustrate the sensitivity of consolidation rates to α and provide practical guidance for selecting boundary functions in geotechnical design. Results demonstrate that physically grounded boundary conditions are critical for accurately capturing consolidation mechanisms and timing. This work thus offers a robust reference for developing and calibrating one-dimensional consolidation models. Future research may extend the analysis to include soil relaxation and creep effects, which remain outside the scope of Terzaghi’s original theory.