DISCLAIMER: This document does not meet current format guidelines Graduate School at the The University of Texas at Austin. of the It has been published for informational use only. Copyright by Eric Cushman Bryant 2016 The Thesis Committee for Eric Cushman Bryant Certifies that this is the approved version of the following thesis: Hydraulic Fracture Modeling with Finite Volumes and Areas APPROVED BY SUPERVISING COMMITTEE: Supervisor: Mukul M. Sharma John T. Foster Hydraulic Fracture Modeling with Finite Volumes and Areas by Eric Cushman Bryant, B.A.; B.S. Thesis Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering The University of Texas at Austin August 2016 Dedication This work is dedicated to my dad George, who passed away last year. Acknowledgements The guidance and support of my supervisor Dr. Mukul M. Sharma is gratefully acknowledged. Parts of this work were accomplished in collaboration with co-workers. For example, the homogeneous residual stress formulation was developed by Dr. Jongsoo Hwang. Boundary (and internal continuity) conditions were developed under tutelage of Dr. Philip Cardiff, including work on finite areas. Dr. Krishnaswamy Ravi-Chandar provided me the context (in his fracture mechanics class) to describe residually stressed, poroelastic inhomogeneity with systematicity. Dr. Ripudaman Manchanda has provided excellent conceptual and code bugfixes to numerous projects, and in particular authored the implicit Carter leak-off formulation (which I like). Generally, I would like to thank Hisanao Ouchi for being a terrific consigliere and late-night programming companion. On that note, I would like to thank those contributing to the open-sourcing of the finite volume/area sold mechanics toolset used to undertake this work. These persons include Drs. Philip Cardiff, Hrvoje Jasak, Željko Tuković, Tian Tang, and Alojz Ivanković. I thank the UT Austin Hydraulic Fracturing and Sand Control JIP for its support. v Abstract Hydraulic Fracture Modeling with Finite Volumes and Areas Eric Cushman Bryant, M.S.E. The University of Texas at Austin, 2016 Supervisor: Mukul M. Sharma In Chapter 1, a finite volume-based arbitrary fracture propagation model is used to simulate fracture growth and geomechanical stresses during hydraulic fracture treatments. Single-phase flow, poroelastic displacement, and in situ stress tensor equations are coupled within a poroelastic reservoir domain. Stress analysis is used to identify failure initiation that proceeds by failure along Finite Volume (FV) cell faces in excess of a threshold effective stress. Fracture propagation proceeds by the cohesive zone (CZ) model, to simulate propagation of non-planar fractures in heterogeneous porous media under anisotropic far-field stress. In Chapter 2, we are concerned with stress analysis of both elastic and poroelastic solids on the same domain, using a FV-based numerical discretization. As such our main purposes are twofold: introduce a hydromechanical coupling term into the linear elastic displacement field equation, using the standard model of linearized poroelasticity; and, maintain the continuity of total traction over any multi-material interfaces (meaning an interface over which residual stresses, Biot’s coefficient, Young’s modulus, or Poisson’s ratio vary). vi In Chapter 3, we are concerned with modeling fluid flow in cracks bounded by deforming rock, and specifically, inside those initial discontinuities, softening regions and failed zones which constitute the solid interfaces of propagating hydraulic fractures. To accomplish this task the Finite Area (FA) method is an ideal candidate, given its proven facility for the discretization and solution of 2D coupled partial differential equations along the boundaries of 3D domains. In Chapter 4, rock formations’ response to a propagating, pressurized hydraulic fracture is examined. In order to initiate CZ applied traction-separation processes, an effective stress tensor is constructed by additively combining the total stress with pore pressures multiplied into a scalar factor. In effect, this scalar factor constitutes the Biot’s coefficient as acts inside the CZ. Integral analysis at the cohesive tip is used to show that this factor must be equal to the Biot’s coefficient in the bounding solid (for a small-strain constitutive relation). Also, effects of an initial residual stress state are accounted for. vii Table of Contents List of Tables ....................................................................................................... xiii List of Figures ...................................................................................................... xiv Chapter 1: Arbitrary fracture propagation in heterogeneous poroelastic formations using a finite volume-based cohesive zone model ........................................19 Field Equations .............................................................................................22 Verification of geomechanical model ..................................................24 Fractures and failure criteria ................................................................32 Benchmarking ......................................................................................37 Cohesive material parameters ..............................................................39 Numerical Results .........................................................................................42 Planes-of-weakness ..............................................................................43 Reversal of in-situ stress direction .......................................................45 Vertical propagation.............................................................................48 High angle intersection ........................................................................50 Stress shadow and poroelastic effects ..................................................54 Meshing-related error...........................................................................55 Conclusions ...................................................................................................57 Chapter 2: Application of finite volume-based bimaterial method to variously coupled poroelasticity ...................................................................................59 Interfaces .......................................................................................................61 Open pore interfaces ............................................................................61 Closed pore and impermeable interfaces .............................................62 Pore Fluid Flow.............................................................................................63 Linearized Poroelasticity ..............................................................................64 Traction decompositions ......................................................................65 Resolved effective stress .............................................................68 Interface resolved effective stress ...............................................68 Porosity ................................................................................................69 viii Pore Flow Discretization...............................................................................70 Coupling splits .....................................................................................73 Fixed-strain split .........................................................................73 Fixed-stress split .........................................................................74 Explicit fixed-stress ....................................................................75 Interface pressures ...............................................................................76 Open pore interfaces ...................................................................77 Closed pore and impermeable interfaces ....................................78 Momentum Balance Discretization...............................................................79 Bimaterial interfaces ............................................................................80 Normal traction ...........................................................................80 Tangential traction ......................................................................82 Skeletal tractions and boundary gradients ...........................................84 Skeletal tractions and boundary gradients ...........................................85 Numerical Results .........................................................................................86 Terzaghi’s problem ..............................................................................86 Mandel’s problem ................................................................................87 Inhomogeneous inclusion ....................................................................88 Pore pressure .............................................................................100 Residual stress ...........................................................................101 Metal casing .......................................................................................101 Conclusion ..................................................................................................102 Chapter 3: A finite area method for the solution of variously coupled hydraulic fracture