The focus of the thesis is on constructing the solution of both a semi-infinite and a finite fluid-driven cracks propagating in an impermeable linear elastic medium of arbitrary toughness. The tip region of a hydraulic fracture propagating in a permeable saturated medium is also considered under certain limiting conditions. First, we consider a sem i-in fin it e fluid-driven fracture propagating steadily in an impermeable linear elastic medium and seek a solution which accounts for the presence of a lag of a priori unknown length between the fluid front and the crack tip. The asymptotic forms of the solution near and away from the tip are discussed. It is shown that the solution is not only consistent with the square root singularity of linear elastic fracture mechanics, but that its asymptotic behavior at infinity is actually given by the singular solution of a semi-infinite hydraulic fracture constructed under the assumption that the fluid flows to the tip of the fracture and that the solid has zero toughness. Further, the asymptotic solution for large dimensionless toughness is derived, including the explicit dependance of the solution on the toughness. The intermediate part of the solution (in the region where the solution evolves from the near tip to the far from the tip asymptote) of the problem in the general case is obtained numerically and relevant results are discussed, including the universal relation between the fluid lag and the toughness.

Then we use this solution of the semi-infinite crack to build a consistent solution of a finite two-dimensional fluid-driven fracture propagating in an impermeable solid of non-zero toughness. The solution is constructed in the spirit of a singular perturbation technique based on the presence of three different lengthscales in the problem of finite fracture. Namely, the lengthscale Lh pertaining to the near tip processes over which the “inner” solution (given by the appropriately rescaled solution of a semi-infinite crack) is applicable;​ the lengthscale of the finite crack L, L≫L_h, over which the “outer” solution (provided by the zero-toughness singular solution of the finite crack) is applicable;​ and the lengthscale over which the “inner” and the “outer” solutions match and which is given by the intermediate asymptote (which is the common asymptote of “inner” and “outer” solutions). It is shown that the condition for matching of the “inner” and the “outer” solutions for large enough time is satisfied for arbitrary toughness if the pumping rate increases with time, and for toughness less than certain critical value if the pumping rate is constant. In this solution the fracture length, fluid pressure and crack opening at the inlet of the fracture are insensitive to the details of the inner solution and, in particular, are independent of toughness.

Finally, we have analyzed the problem of the tip region of a hydraulic fracture propagating steadily in a permeable elastic region. In this analysis, the length of the lag zone or tip cavity which is filled by pore fluid is assumed known. This solution is sought to provide the appropriate tip conditions when considering either the finite or semi-infinite crack. We have shown that a consistent solution for the near-tip cavity can be constructed that simultaneously satisfies the diffusion equation in the permeable rock, the equations of linear elastic fracture mechanics, and the lubrication theory for the flow of pore fluid in the cavity. The solution indicates that circulation of pore fluid takes place between the rock and the tip cavity:​ the pore fluid is sucked in at the tip of the crack and is returned to the porous medium near the interface between the fracturing fluid and the pore fluid. The condition for which cavitation (corresponding to the situation where the absolute fluid pressure becomes zero) takes place in the tip cavity and, consequently, the solution is no longer valid, is predicted.