Fluid-filled fractures in Earth's lithosphere: Gravitational loading, interpenetration, and stable height of dikes and veins

We present two analytical solutions for the boundary-value problem of a homogeneous, isotropic, and linear elastic solid with a vertical fluid-filled crack, and use these to investigate the effects of gravitationally induced stress and pressure gradients on the opening distribution, fracture-tip stress intensity, and stable heights of veins and dikes. Stable fractures are not propagating or closing anywhere along their tipline, so the vein or dike is in its final arrested configuration as preserved in the rock record. The solutions do not preclude interpenetration of crack surfaces, so we propose special constraints to avoid such ill-posed problems. Veins and dikes both grow as fluid-filled, opening fractures in Earth's brittle lithosphere, but attain markedly different heights. We show how gravity-induced pressure gradients in water and magma, and stress gradients in the host rock, lead to different stable heights. Dikes, but not veins, can attain stable heights near the level of neutral buoyancy in a density stratified rock mass, even if the host rock fracture toughness is zero. Depth to center and opening distributions for dikes are different if host rock density is stratified or continuously variable. These solutions offer a new set of analytical tools for interpreting field data from veins and dikes.