Zones of anomalous pressure, higher and lower than hydrostatic pressure, have been observed in many sedimentary basins around the world. These normally consist of groups of pressure compartments: volumes of higher-permeability rock surrounded on all sides by lower-permeability barriers. Knowledge of the timescales over which these abnormal pressures are maintained and the mechanisms by which they dissipate is critical for understanding how fluids, such as oil and gas, move in the subsurface. Existing analytic solutions investigate pressure dissipation through low-permeability barriers on top of or underneath an isolated pressure compartment. There are no analytic solutions describing pressure dissipation through lateral barriers, such as faults, or investigating the impact of groups of pressure compartments on the rate of pressure dissipation. This paper presents simple analytic models to investigate pressure dissipation through barriers, such as faults, forming the sides of pressure compartments. The timescales are compared with a solution for pressure dissipation through barriers on top of and underneath the compartment. It also investigates analytically the rate of pressure dissipation from groups of pressure compartments. Lateral seal permeabilities of 10-19 m2 may delay pressure equilibration for millions of years provided the compartment has a sufficiently high fluid storage capacity. Factors contributing toward a high fluid storage capacity include a high fluid compressibility (as is the case in hydrocarbon reservoirs) and a high porosity. The grouping of abnormally pressured compartments into megacompartment complexes may delay pressure dissipation for hundreds of millions of years.