2-D Darcian flow in vicinity of permeable fracture perturbing unidirectional flow in homogeneous formation

A. R. Kacimov*, N. D. Yakimov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


A thin planar fracture with a highly permeable porous filling disturbs an incident unidirectional one-phase flow in a formation/aquifer and makes it essentially 2-D. The main objective of this paper is to get an analytical evaluation of the quantity of fluid intercepted, conveyed, and released back by the fracture and assessment of the near-fracture zone where 2-D effects are essential. The impact of the fracture is modelled by the Robin (third-type) boundary condition imposed along a mathematical cut. Flow in the fracture is 1-D and along the fracture axis. For a fracture of a rectangular shape, we use a conformal mapping of a physical plane with a cut onto the exterior of a unit disc in a reference plane and transform the Robin boundary condition into an infinite system of linear equations. Using the Kantorovich–Krylov theory, we rigorously prove that the system is entirely regular, i.e. it can be solved by the method of reduction (truncation to a finite system). A method of iterations is used to solve a reduced system. Convergence with respect to the truncation order is investigated. For non-rectangular (curvilinear) planar fractures with a straight axis, we use an inverse approach of Pilatovskii, viz., specifying the flow rate function along the fracture axis, i.e. the quantity of fluid which is intercepted by the fracture from the ambient rock. The fracture geometry is obtained as a part of solution. Pilatovskii’s function is involved as a weight of a singular integral with the Cauchy kernel. The corresponding integro-differential equation is a special case of the Prandtl equation for a thin airfoil in an ideal fluid flow with the circulation along the airfoil cord as a weight. The derivative of the Pilatovskii function is expanded into a series of Chebyshev’s orthonormal functions. This series is truncated, giving, in particular, the Pilatovskii fracture as a limit. Flow nets for both rectangular and curvilinear fractures are plotted. The dependences of the capture zone (quantity of intercepted–released fluid) on the ratios of permeabilities of the bulk rock-soil/fracture’s filling and fracture’s length/thickness, as well as the flow nets, are of interest to reservoir engineers and groundwater hydrologists working with Darcian flows in heterogeneous porous media.

Original languageEnglish
Pages (from-to)15-28
Number of pages14
JournalJournal of Engineering Mathematics
Issue number1
Publication statusPublished - Oct 1 2019


  • Boundary value problem
  • Complex potential
  • Flow net
  • Hydrofracking
  • Kantorovich-Krylov systems of linear equations
  • Robin’s boundary condition

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)


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