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Related Concept Videos

Design Example: Creating a Hydraulic Model of a Dam Spillway01:21

Design Example: Creating a Hydraulic Model of a Dam Spillway

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Scaled hydraulic models of dam spillways provide a practical way to replicate and study the intricate flow dynamics of these structures. Often built to a 1:15 ratio, these models allow for observing critical water behavior, such as velocity distribution, flow patterns, and energy dissipation.
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Typical Model Studies01:30

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Fluid mechanics model studies often utilize scaled-down systems to predict fluid behavior in full-scale environments, such as river flows, dam spillways, and structures interacting with open surfaces. Maintaining Froude number similarity in river models is crucial, as it replicates surface flow features like wave patterns and velocities.
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Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
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A hydraulic jump is a sudden rise in fluid depth in open channels, occurring when high-velocity (supercritical) flow transitions to low-velocity (subcritical) flow. This phenomenon requires an upstream Froude number greater than 1, as flows with Fr1<1 remain subcritical, making a hydraulic jump impossible due to the need for negative head loss, which violates thermodynamic principles.The characteristics of a hydraulic jump depend on the upstream Froude number and are classified as...
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Hydraulic Jump: Problem Solving01:16

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To analyze a hydraulic jump in a rectangular channel with a flow speed of 6 meters per second, follow these steps:Calculate Effective Upstream Velocity:When the downstream gate closes, a hydraulic jump forms, traveling upstream at 2 meters per second. This wave speed combines with the initial channel flow velocity, creating an effective upstream velocity.Identify Flow Velocities Before and After the Hydraulic Jump:Upstream of the hydraulic jump, the effective flow velocity includes both the...
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Rapidly Varying Flow01:24

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Rapidly varying flow (RVF) in open channels is characterized by abrupt changes in flow depth over a short distance, with the rate of depth change relative to distance often approaching unity. These flows are inherently complex due to their transient and multi-dimensional nature, making exact analysis difficult. However, approximate solutions using simplified models provide valuable insights into their behavior.Key Features of Rapidly Varying FlowRVF is commonly observed in scenarios involving...
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Implicit level set algorithms for modelling hydraulic fracture propagation.

A Peirce1

  • 1Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2 peirce@math.ubc.ca.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|September 7, 2016
PubMed
Summary
This summary is machine-generated.

This review details numerical methods for modeling hydraulic fracture propagation, focusing on asymptotic behavior near the fracture tip. These advanced techniques improve simulations of fluid injection in pre-stressed media.

Keywords:
displacement discontinuity methodextended finite-element methodhydraulic fracture propagationimplicit level set methodsmulti-scale tip asymptoticssingular free boundary

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Area of Science:

  • Geophysics
  • Computational Mechanics
  • Fluid Dynamics

Background:

  • Hydraulic fractures are tensile cracks formed by fluid injection in stressed rock.
  • Modeling their propagation is complex due to singular integro-partial differential equations and free boundary dynamics.
  • Existing numerical schemes struggle with the degenerate and hypersingular nature of these equations.

Purpose of the Study:

  • To present a class of numerical schemes for modeling hydraulic fracture propagation.
  • To utilize multiscale asymptotic behavior near the fracture tip to determine evolution.
  • To illustrate fundamental concepts in two distinct numerical formulations.

Main Methods:

  • Developing numerical schemes based on multiscale asymptotic behavior at the fracture tip.
  • Employing the displacement discontinuity boundary integral method.
  • Utilizing the extended finite-element method.
  • Incorporating new models for proppant transport and efficient solvers for coupled nonlinear systems.

Main Results:

  • Demonstration of numerical schemes that leverage tip asymptotics for fracture modeling.
  • Illustration of free boundary location and asymptotic behavior imposition in weak forms.
  • Presentation of practical considerations including proppant transport and solver efficiency.
  • Numerical examples validating the performance of the developed schemes.

Conclusions:

  • The reviewed numerical schemes offer a robust approach to modeling complex hydraulic fracture propagation.
  • Further research is needed on open questions related to fracture modeling and simulation.
  • These methods are crucial for understanding subsurface energy processes.