Cutoff walls are extensively applied as seepage and contaminant leakage obstruction structures. However, stochastic defects can exist within cutoff walls due to randomness in the construction of cutoff walls and faciliate the penetration of polluted groundwater through cutoff walls. To assess the contaminant leakage risk, definition and judge criterion for breakthrough time of cutoff walls with inhomogeneous Dirichlet boundary conditions are derived, and breakthrough time diagrams are plotted, which are based on the normalized analytical solution to one-dimensional linear steady seepage and transient contaminant advection-diffusion coupled process. The analytical solution also verifies its finite element method counterpart. For the purpose of assessing the contaminant leakage risk in a realistic way, a nonlinear model with nonconstant cross-sections along the penetration channel is obtained, which can be combined with planar and spacial models for cutoff walls with random defects, and a finite difference method algorithm for the nonlinear model is proposed, while an algorithm by Adomian decomposition method is also included for a trial.