The exact nature of the seals maintaining abnormally pressured compartments in sedimentary basins is not well understood, despite decades of research. We use the Lattice Boltzmann method for immiscible fluids to investigate a novel idea that capillary pressures alone are able to seal in the observed abnormal pressures. A capillary seal forms when a non-wetting fluid phase is generated within or introduced into grain sized layered sediment and the pressure drop across the coarse/fine interface is less than the capillary pressure. For such seals to maintain abnormal pressures, both phases must be blocked, the capillary pressure drops must be accumulative over many fine/coarse interfaces and the seals must be able to re-form after rupture. We show all three to be true, and hence lay the numerical foundation for the validity of capillary seals. Lattice Boltzmann Method showed itself to be applicable to other problems of interest. The morphology of the meniscus between wetting and non-wetting fluids in a capillary tube when the fluids are in motion is controversial, for example. We also encountered instabilities and limitations in code as originally implemented. We investigated and instituted methodologies which drastically reduce the instabilities and worked around the encountered limitations.