Beyond Poiseuille flow: A transient energy-conserving model for flow through fractures of varying aperture

Gee, Bruce; Gracie, Robert

doi:10.1016/j.advwatres.2022.104192

Cited by 8 publications

(23 citation statements)

References 33 publications

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“…The momentum correction factor depends on the shape of the fluid velocity profile across the aperture and is a measure how the velocity across the profile differs from the mean velocity. 14 The momentum correction factor is defined as…”

Section: Governing Equationsmentioning

confidence: 99%

“…The fundamental frequency of a fracture with constant aperture is given by 𝜙 = 12𝜈∕w 2 , in which 𝜈 is the kinematic fluid viscosity, and w is the fracture aperture. 14 When the loading frequency, Ω, is sufficiently small (Ω ≪ 𝜙), the quasi-steady cubic law solution is recovered. When loading rates are faster (Ω ⪆ 𝜙), transient effects dominate the response.…”

Section: Introductionmentioning

confidence: 99%

“…21,22 Seismic activity can also induce similar loading rates and has been shown to create long-lasting changes in the fracture permeability. 23 The reduced dimension fracture flow model, 14 which will be referred to as the GG22 flow model in this article, is a recently derived model for fracture flow that can capture nonlinear transient and convective inertial effects. The GG22 model is derived by integrating the higher-dimension incompressible Navier-Stokes equations over the fracture aperture, 14 creating a reduced-dimension model that captures more complex fluid behavior without the computational burden that accompanies the full Navier-Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

“…23 The reduced dimension fracture flow model, 14 which will be referred to as the GG22 flow model in this article, is a recently derived model for fracture flow that can capture nonlinear transient and convective inertial effects. The GG22 model is derived by integrating the higher-dimension incompressible Navier-Stokes equations over the fracture aperture, 14 creating a reduced-dimension model that captures more complex fluid behavior without the computational burden that accompanies the full Navier-Stokes equations. Unlike Poiseuille flow, the GG22 model does not assume parallel fracture faces and is therefore suitable for fractures of variable aperture.…”

Section: Introductionmentioning

confidence: 99%

“…While the GG22 model was developed to simulate flows at any Reynold's number, it has been shown that: both the GG22 and the cubic law flow models perform equally well at Re < 1, the GG22 model performs significantly better than the cubic law in the range 1 < Re < 100; and the GG22 model predictions may deviate from actual flow behavior at Re > 100. 14 While the GG22 model is superior to the cubic law in terms of the physical behaviors it can capture, the increased physical accuracy comes at the cost of increased complexity. While the cubic law is a one-field model (fluid pressure) governed only by the conservation of fluid mass, the GG22 model is a two-field model (fluid flux and fluid pressure) governed by the conservation of fluid mass and momentum.…”

Section: Introductionmentioning

confidence: 99%

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Beyond the cubic law: A finite volume method for convective and transient fracture flow

Gee

Gracie

2022

Numerical Methods in Fluids

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The reduced dimension fracture flow model (referred to as the GG22 model) is a recently derived extension of the cubic law model for flow through fractures of variable aperture with fluid inertia effects. Novel numerical methods are required to solve the nonlinear partial differential equations governing the GG22 model, as it is more complex than the cubic law. The GG22 model is derived from Navier–Stokes, which allows the adoption of similar numerical methods to the Navier–Stokes equations, but the model contains its own idiosyncrasies which must be addressed. This article presents the first numerical methods to solve the GG22 model. An explicit multi‐step finite volume method is developed and verified. The method is based on deriving a Poisson equation for pressure with an additional continuity correction to overcome numerical instabilities. The critical timestep is derived and shown to be a function of the fundamental frequency of the fracture‐fluid system and the maximum fluid velocity. The results show excellent agreement with analytical solutions, and the method demonstrates a first‐order rate of fluid flux convergence in time and a second‐order rate of pressure convergence in space. The model is applied to a traveling aperture wave which shows that higher pressures are required to generate lower average fluxes than predicted by the cubic law.

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Section: Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Beyond the cubic law: A finite volume method for convective and transient fracture flow

Gee

Gracie

2022

Numerical Methods in Fluids

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Inertial and turbulent flow in hydro‐mechanically coupled KGD‐like fractures

Gee

Gracie

2023

Num Anal Meth Geomechanics

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Fluid flow in deformable fractures is important to many applications including hydraulic stimulation. Fracture flow simulations typically rely on the Poiseuille flow model which presumes laminar quasi‐steady‐state flow conditions with negligible inertia. The high flow rates involved in industrial applications bring these assumptions into question. The GG22 flow model is a new method to capture inertial effects and turbulent flow phenomena with a moderate increase in computational complexity. Here we develop and verify the first hydro‐mechanically coupled finite element – finite volume model for fracture flow which uses the GG22 model. We use the model to examine fracture flow between oscillating elastic plates and show that inertia may elicit phase‐shifts in the fluid response, larger fluid pressures and rock mass stresses, and induce wave‐like behaviour even in a quasi‐static rock mass. We then apply the model to hydraulic stimulation of a cemented fracture in both the viscous and toughness dominant propagation regimes and examine the influence of inertia and turbulence compared to Poiseuille flow. We demonstrate that inertial effects due to the variation of aperture are negligible in two‐dimensional KGD‐like fractures with constant injection rates. If the flow rates are large enough to invoke turbulent flow behaviour, then significantly different fracture propagation behaviour is observed. The modelling established in this article will serve as the basis to examine the role of inertia and turbulence in axisymmetrical radial fractures and dynamic stimulation with pressure pulsing where we expect the role of inertia to be more significant.

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Utilization mechanism of foam flooding and distribution situation of residual oil in fractured-vuggy carbonate reservoirs

Wen¹,

Hou²,

Xiao³

et al. 2023

Petroleum Science

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Beyond Poiseuille flow: A transient energy-conserving model for flow through fractures of varying aperture

Cited by 8 publications

References 33 publications

Beyond the cubic law: A finite volume method for convective and transient fracture flow

Beyond the cubic law: A finite volume method for convective and transient fracture flow

Inertial and turbulent flow in hydro‐mechanically coupled KGD‐like fractures

Utilization mechanism of foam flooding and distribution situation of residual oil in fractured-vuggy carbonate reservoirs

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