Abstract: One embodiment of the invention is a variable density drilling mud comprising compressible particulate material in the drilling mud wherein the density of the drilling mud changes in response to pressure changes at depth. A second embodiment is a method for varying drilling mud density. The method comprises estimating the pore pressure and fracture gradient, and choosing a drilling mud with compressible materials wherein the effective mud weight of the drilling mud remains between the pore pressure and the fracture gradient in at least one interval of the wellbore. A third embodiment is an apparatus for drilling a wellbore.