Thrust-related fluid flow coupled with isotopic exchange between fluid and rock is simulated in a three-dimensional finite-element model of the Glarus nappe, eastern Swiss Alps. Numerical simulations are matched against well-established oxygen isotope gradients on the kilometer scale along the thrust. At internal southern locations, strongly channelized thrust-parallel fluid flow requires a high permeability contrast of >100:1 between the mylonite zone and country rocks and a high hydraulic head in the hinterland and footwall. In contrast, isotopic patterns ∼5–10 km farther north indicate a predominantly vertical, upward drainage of fluids. We propose a situation in which the Glarus thrust evolved northward across the boundary between the lithostatic and hydrostatic fluid- pressure regimes—the “impermeable cap.” A cyclic behavior of fluid-pressure buildup, fracturing, channelized fluid escape, and sealing explains structural and geochemical observations and the best-fit three-dimensional fluid-flow model parameters.