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Keywords: asymptote

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Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 53rd U.S. Rock Mechanics/Geomechanics Symposium, June 23–26, 2019

Paper Number: ARMA-2019-1617

... of the stage and, more importantly, spatial variations of the entry friction due to near-wellbore fracture tortuosity have a more pronounced adverse effect on the fluid partitioning. Engineering variation MPa injection Upstream Oil & Gas stress interaction

**asymptote**hydraulic fracture...
Abstract

ABSTRACT: In this contribution, we explore different aspects of the multi-stage fracturing process such as stress interaction between growing hydraulic fractures, perforation and near-wellbore tortuosity effects as well as the wellbore flow dynamics using a fit-for-purpose numerical model, which accounts for the full fluid-solid coupling nature of hydraulic fracture propagation, stress interactions between multiple growing fractures, and the coupling with the wellbore flow via entry friction. We restrict the hydraulic fractures to be fully contained in the reservoir. After presenting several verifications of this model, we investigate the effect of spatial variation of the entry friction associated with the near-wellbore fracture tortuosity. We show that, although large perforation friction helps to equalize the fluid partitioning between fractures, the pressure drop along the length of the stage and, more importantly, spatial variations of the entry friction due to near-wellbore fracture tortuosity have a more pronounced adverse effect on the fluid partitioning.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 53rd U.S. Rock Mechanics/Geomechanics Symposium, June 23–26, 2019

Paper Number: ARMA-2019-1689

... of poroelasticity on the contact stress and the transient indentation force response is then discussed. The

**asymptotic**behaviors for the normalized transient force at early and late times are derived. Though derivation of these fully coupled solutions requires the aid of a variety of mathematical techniques...
Abstract

ABSTRACT: Indentation of a poroelastic solid by a smooth rigid sphere is analyzed within the framework of Biot's theory. The particular cases when the spherical indenter is loaded instantaneously to a fixed depth with the surface of the semi-infinite domain being either fully undrained or in a mixed drainage condition are solved. Constituents of the poroelastic medium are assumed to be slightly compressible. The solution procedure based on the McNamee-Gibson displacement function method is adopted in this work. Problem formulation and the solution procedure are first introduced. Effect of poroelasticity on the contact stress and the transient indentation force response is then discussed. The asymptotic behaviors for the normalized transient force at early and late times are derived. Though derivation of these fully coupled solutions requires the aid of a variety of mathematical techniques, the normalized transient force responses are remarkably simple and show only weak dependence on one derived material constant, which lend itself to convenient use for poroelastic characterization of geomaterials such as rocks in the laboratory.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 52nd U.S. Rock Mechanics/Geomechanics Symposium, June 17–20, 2018

Paper Number: ARMA-2018-076

... initiation equation indentation numerical integration reservoir geomechanics radial stress

**asymptote**expression material constant indenter deformation contact axis 1. INTRODUCTION The process of indentation by a rigid tool has been widely studied for its versatility as an experimental technique...
Abstract

ABSTRACT: Indentation of a poroelastic solid by a spherical-tip tool is analyzed within the framework of Biot’s theory. We seek the response of the indentation force as well as the field variables as functions of time when the rigid indenter is loaded instantaneously to a fixed depth. We consider the particular case when the surface of the semi-infinite domain is permeable and under a drained condition. Compressibility of both the fluid and solid phases is taken into account. The solution procedure based on the McNamee- Gibson displacement function method is adopted. One of the difficulties in solving this class of problems analytically is in evaluating integrals with oscillatory kernels over an unbounded interval. We show that such issues can be overcome by the use of a series of special functions. Problem formulation and the solution procedure are first introduced. Implications of the poroelastic solution for incipient failure in form of tensile crack initiation and plastic deformation are then discussed. An interesting outcome from this analysis is that if the indentation forces at the undrained and drained limits are known, relaxation of the indentation force with time can be used to determine the diffusion coefficient of a porous medium.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 51st U.S. Rock Mechanics/Geomechanics Symposium, June 25–28, 2017

Paper Number: ARMA-2017-0763

... (? 2/3 ) solutions for the fracture aperture co-exist near the fracture tip. Depending on the ratio of viscous dissipation and fracture resistance, the region for the validity of each tip

**asymptote**varies. In this paper, we developed a 2D displacement discontinuity based hydraulic fracture propagation...
Abstract

ABSTRACT: In fluid driven fractures, energy is dissipated by creating new fracture surfaces and overcoming viscous resistance to fluid flow. It is known that when fluid driven fractures propagate in a rock with a finite amount of resistance to fracture, the toughness (? 1/2 ) and viscosity (? 2/3 ) solutions for the fracture aperture co-exist near the fracture tip. Depending on the ratio of viscous dissipation and fracture resistance, the region for the validity of each tip asymptote varies. In this paper, we developed a 2D displacement discontinuity based hydraulic fracture propagation model for multiple large-scale hydraulic fracture propagation while taking into account the multi-scale nature of the fracture tip. The numerical model is capable of distinguishing between viscosity and toughness dominated propagation regimes, and takes into account their corresponding solutions as boundary conditions near the fracture tips. We provide numerical examples of large-scale multiple fracture propagation in a rock with finite fracture toughness using injection fluid viscosities range from 1-1000 cP. The results include fracture geometry, aperture distribution and regime of fracture propagation. To study the effect of fracture tip asymptote on the resulting fracture cluster, we approximated multi-scale fracture tip asymptotes with toughness asymptote showing that for a rock with finite toughness and for the range of fluid viscosities considered, the toughness solution provides sufficiently accurate results irrespective of the regime of fracture propagation.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 49th U.S. Rock Mechanics/Geomechanics Symposium, June 28–July 1, 2015

Paper Number: ARMA-2015-297

... or viscous regimes of propagation by using the corresponding

**asymptotic**solution at the tip element. Since either the viscous or toughness**asymptote**is used, the intermediate regime is not described accurately. To deal with this problem, this study aims to implement the intermediate**asymptotic**solution...
Abstract

Abstract The design of a hydraulic fracturing treatment typically requires using a computational model that provides rapid results. One such possibility is to use the so-called classical pseudo-3D (P3D) model with symmetric stress barriers. Unfortunately, the original P3D model is unable to capture effects associated with fracture toughness in the lateral direction due to the fact that the assumption of plane-strain (or local) elasticity is used. On the other hand, a recently developed enhanced P3D model utilizes full elastic interactions and is capable of incorporating either toughness or viscous regimes of propagation by using the corresponding asymptotic solution at the tip element. Since either the viscous or toughness asymptote is used, the intermediate regime is not described accurately. To deal with this problem, this study aims to implement the intermediate asymptotic solution into the enhanced P3D model. To assess the level of accuracy, the results are compared to a reference solution. The latter reference solution is calculated numerically using a fully planar hydraulic fracturing simulator (Implicit Level Set Algorithm (ILSA)), which also incorporates the asymptotic solution for tip elements that captures the transition from viscous to toughness regime. 1. INTRODUCTION Hydraulic fracturing (HF) plays a crucial role in the petroleum industry, as it allows one to perform reservoir stimulation and intensify hydrocarbon production [1]. To design a HF treatment, an appropriate HF model needs to be utilized. The simplest model is the onedimensional Khristianovich-Zheltov-Geertsma-De Klerk (KGD) model [2], in which the fracture propagates in a plane, the elastic interactions are modelled assuming that plane strain conditions prevail, and the coupling between viscous fluid flow and elasticity is included. To represent the fracture geometry more realistically, the Perkins-Kern- Nordgren (PKN) model [3, 4] was developed to predict fracture propagation in a horizontally layered medium. The PKN model assumes that the fracture height is always equal to the thickness of the reservoir layer, the fracture opening in each vertical cross-section is taken to be elliptic, while the fluid pressure is calculated assuming that a plane strain condition holds in each cross-section. Given the fact that the PKN model does not allow for the height growth, the pseudo-3D (P3D) model, which permits height growth, has been developed [5]. Later, with the increase of the computational power, more accurate planar 3D models (PL3D) were developed [7, 8]. As follows from the name, the fracture is contained in one plane, where the fracture geometry within this plane is discretized using a two-dimensional grid. Since the KGD, PKN and P3D are essentially one-dimensional models, while all varieties of PL3D are two-dimensional, the CPU time increases dramatically. The PL3D models improve accuracy and open the possibility of capturing different fracture geometries. Recently, researchers have shifted their effort to investigate the interaction between multiple hydraulic fractures that are growing simultaneously [9], and to describe non-planar fracture propagation [10].

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the DC Rocks 2001, The 38th U.S. Symposium on Rock Mechanics (USRMS), July 7–10, 2001

Paper Number: ARMA-01-0243

...) is expressed as 1 1 - lf 20II (17) 3' The global equation of fluid volume balance (4) re- duces to r = 23' f d (18) and the continuity equation (5) to Of _ lST (19) The boundary condition (6) is given by 1 =5' ate=0. (20) 2.4 Crack Propagation and Tip

**Asymptotic**Behav- ior According to linear elastic...
Abstract

ABSTRACT: This paper describes a numerical model to simulate the propagation of a plane-strain (KGD) hydraulic fracture in an elastic, impermeable medium with zero toughness. The fracture is driven by injection of an incompressible fluid with power-law rheology. The numerical model, which is formulated in terms of a moving coordinates system, is based on the displacement discontinuity method and on an explicit finite difference scheme. The accuracy of the algorithm is validated against the available self-similar solution for a Newtonian fluid. INTRODUCTION Hydraulic fracturing (HF) is a technique widely used to enhance the flow of oil or natural gas from the reservoir formations towards the extracting wells. The uncertainty about the in situ conditions, the complexity of the mechanisms taking place and the difficulty in obtaining precise measurements of the fracture geometry, make necessary the use of idealized models (e.g., the KGD or "plane-strain" model, the "pennyshaped" or radial model and the PKN model) for studying this process. Even with these simplified models, the mathematical formulation for the propagation of hydraulic fractures is given by a relatively complicated system of integral and non-linear differential equations. Some analytical solutions of these mathematical models are already available (Spence & Sharp, 1985; Savitski & Detournay, 1999; Garagash, 2000; Savitski, 2000; Adachi, 2001; Adachi et al., 2001). However, these solutions are constructed on the basis of various restrictive assumptions (e.g., constant injection rate; very small or very large material toughness, Newtonian theology for the fracturing fluid, no fluid leakoff, etc.). In order to extend the applicability of these models, it is necessary to release some of these assumptions and consequently, the solution of the governing equations demands the use of numerical techniques. Commonly, the numerical solution of non-linear problems is restricted to the use of implicit schemes. Explicit schemes are not often applied in this type of problems due to the difficulty in reaching numerical stability, even though the latter are simpler to implement. In this paper, we introduce an explicit finite difference scheme with a moving mesh that can be used to simulate the propagation of a KGD hydraulic fracture. This scheme is shown to be numerically stable, accurate and "flexible", in the sense that additional features (e.g., fluid leak-off, poroelastic effects, etc.) can be easily incorporated into the model (Detournay et al., 1990).