This study develops approximate solutions for the problems of a plane strain and radial hydraulic fractures that are generated by a time-varying injection source. The mathematical model considers coupling between the effects of fracture toughness, fluid viscosity (Newtonian fluid), and leak-off (Carter’s model). The solution is constructed by approximating spatial variation of the fracture width by a function, whose near-tip asymptotic behavior is captured precisely while the solution in the vicinity of the wellbore is approximated. Such an assumption together with the closed form expression for the asymptotic solution allows one to reformulate the problem in terms of algebraic equations, which leads to construction of the closed form approximate solutions. To examine the accuracy of the developed solutions, the latter are tested against the corresponding numerical solutions for a wide range of parameters. Results indicate a good agreement for both the plane strain and radial fracture geometries, thus enabling one to use the developed solutions to address problems where rapid computations of fracture parameters are necessary.

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