Linear hydraulic fracture with tortuosity: Conservation laws and fluid extraction. (March 2016) - Kgatle, M.R.R.; Mason, D.P.
Abstract: Fluid extraction from a pre-existing two-dimensional hydraulic fracture with tortuosity is investigated. The tortuous fracture is replaced by a symmetric open fracture without asperities (deformations) on opposite crack walls but with a modified Reynolds flow law and a modified crack law (the linear crack law). The Perkins–Kern–Nordgren approximation is made in which the normal stress at the fracture walls is proportional to the half-width of the symmetric model fracture. By using the multiplier method two conservation laws for the non-linear diffusion equation for the half-width are derived. Two analytical solutions generated by the Lie point symmetries associated with the conserved vectors are obtained. One is the known solution for a fracture with constant volume. The other is new and is the limiting solution for fluid extraction. A jet of fluid escapes from the fracture entry and the volume of the fracture decreases. There is a dividing cross-section between fluid flowing towards the fracture entry and fluid flowing towards the fracture tip which explains why the length of the fracture continues to grow as fluid is extracted. As tortuosity increases the position of the dividing cross-section moves closer to the entry. A numerical solution is presented for the other cases of fluid extraction. Comparison of the fluid flux for different operating conditions within the fluid extraction region shows that the limiting solution yields the maximum rate of fluid extraction from the fracture. As the fracture becomes more tortuous its length becomes less dependent on the operating conditions at the fracture entry. For fluid extraction working conditions close to the constant volume operating condition the width averaged fluid velocity increases approximately linearly along the whole length of the fracture. For these working conditions, an approximate analytical solution for the half-width for fluid extraction, which agrees well with the numerical solution, is derived by assuming that the width averaged fluid velocity increases exactly linearly along the fracture. Abstract : Highlights: Analysis of a hydraulic fracture with tortuosity is extended to fluid extraction. Two conservation laws are derived for a linear hydraulic fracture with tortuosity. A solution for fluid extraction is derived using an associated Lie point symmetry. A dividing cross-section separates flow to the entry from flow to the tip. Extraction of fluid from a fracture becomes more difficult as tortuosity increases.