International Journal of Fracture 103: 243–258, 2000. © 2000 Kluwer Academic Publishers. Printed in the Netherlands. An analytical method for mixed-mode propagation of pressurized fractures in remotely compressed rocks M.K. RAHMAN, M.M. HOSSAIN and S.S. RAHMAN ∗ School of Petroleum Engineering, University of New South Wales, Sydney 2052, Australia (fax: +61 2 9385 5182; e-mail: sheik.rahman@unsw.edu.au) Received 14 April 1999; accepted in revised form 10 November 1999 Abstract. An analytical method for mixed-mode (mode I and mode II) propagation of pressurized fractures in remotely compressed rocks is presented in this paper. Stress intensity factors for such fractured rocks subjected to two-dimensional stress system are formulated approximately. A sequential crack tip propagation algorithm is developed in conjunction with the maximum tensile stress criterion for crack extension. For updating stress inten- sity factors during crack tip propagation, a dynamic fictitious fracture plane is used. Based on the displacement correlation technique, which is usually used in boundary element/finite element analyses, for computing stress intensity factors in terms of nodal displacements, further simplification in the estimation of crack opening and sliding displacements is suggested. The proposed method is verified comparing results (stress intensity factors, propagation paths and crack opening and sliding displacements) with that obtained from a boundary element based program and available in literatures. Results are found in good agreements for all the verification examples, while the proposed method requires a trivial computing time. Key words: Hydraulic fracture, fracture propagation, compressed rocks, stress intensity factors, crack opening displacement, crack sliding displacement. 1. Introduction Underground rocks are naturally fractured and subjected to in-situ compressive stresses. These natural fractures (or initiated new fractures) are further extended and inflated by injecting fluid to enhance its permeability in order to allow enhanced hydrocarbon recovery and geothermal energy extraction. This process is usually known as hydraulic fracturing. The hydraulic frac- ture stimulation is thus essentially based on fundamental principles of fracture mechanics. To date, the standard design tools in oil and gas industries are mainly based on idealized planar fractures. The role of these fractures on permeability enhancement is generally postulated based on fracture extension by opening mode (mode I) under injected fluid pressure. The existing approaches are not adequate to simulate the non-planar fracture extension in hydraulic fracture stimulation. This has resulted in the failure of many hydraulic stimulation operations, particularly under complex in-situ stress conditions. The non-planar fracture extension can be best studied by considering mixed-mode (opening and shearing) crack tip propagation. The final geometry (shape, size, orientation, etc.) of such non-planar fractures is then required to incorporate in the design of hydraulic fracture stimulation. The prediction of the final geometry of such non-planar fractures is a complex fracture mechanics problem which is often solved by computation intensive finite element or boundary element methods. The analysis by these methods is very time consuming and therefore these methods are not suitable for routine design tasks carried out in an industry environment. This