Drilling a cylindrical borehole is the first and important step in oil mining. Borehole design and strength check are big problems of utmost importance. Biot introduced a poroelastic constitutive theory for porous rock with freely moving fluid inside. In this paper, by using Biot poroelastic model, we analyze a borehole with drilling fluid in an infinite porous rock with three-dimensional in situ stresses and obtain whole domain solutions for instantaneous, short-time, and long-time stress distributions. Maximum and minimum allowable drilling pressures are given for tensile failure and shear failure criterions, and allowable drilling pressure regions are drawn in the space of in situ hydrostatic stress , deviatoric stress , and pore pressure . By comparing with classical elastic constitutive relations, or Hooke's model, the necessity of Biot poroelastic constitutive relations is shown.