A model for hindered transport of water and ions is used to predict transient flow through a clay membrane caused by an initial difference in the concentration of salt solutions in reservoirs on the two sides of the membrane. Transport is assumed to be controlled by three coefficients, which are analogous to the permeability, salt diffusivity, and salt reflection coefficient of the membrane. Initial fluid motion is caused by osmosis, leading to a buildup of pressure on one side of the membrane. However, the clay forms an imperfect ion exclusion membrane and the final steady state is one of equal concentrations and pressures on the two sides of the membrane. The time-dependent differences in pressure and salt concentration across the membrane are predicted to vary as the sum of two decaying exponentials. When the salt reflection coefficient is small, one time scale governs Darcy how through the membrane and another the diffusion of salt. Experimental results confirm the analysis. Although the salt concentration in the reservoirs was monitored in the experiments, estimates of the transport coefficients can be obtained by measuring only the pressure change across the membrane.