WGravitational Instability of Buoyant Layers Using Scaling Analysis, Long-wave Analysis and Linear Stability Analysis

University of Ibadan
Gravitational Instability of Buoyant Layers Using
Scaling Analysis, Long-wave Analysis and Linear
Stability Analysis
Presentation By
DONATIEN ISHIMWE (PAU-UI-0830)
PPG811 - Techniques In Structural Geology
Petroleum Geoscience, PhD
Monday 11th September 2023

Gravitational Instability of Buoyant Layers Using Scaling Analysis, Long-wave Analysis and Linear Stability Analysis
PPG811 (Techniques In Structural Geology) - Petroleum Geoscience, PhD
Donatien ISHIMWE (PAU-UI-0830)
Gravitational Instability of Buoyant Layers Using Scaling
Analysis, Long-wave Analysis and Linear Stability Analysis
1. Introduction
•Gravitational
Instability Theory
•Key Definition
2. Gravitational
Instability Analysis
•Scaling Analysis
•Long-wave Analysis
•Linear Stability
3. Application &
Case Study
•Applications
•Case Study
4. Reference
•Key authors

• The gravitational instability of buoyant layers is a fundamental concept in fluid
dynamics and astrophysics, which describes the process by which a layer of fluid
becomes unstable and forms structures under the influence of gravity.
• This phenomenon is important in various natural systems, including the formation
of galaxies, stars, and planetary rings, as well as in industrial applications like
combustion and crystal growth.
• This phenomenon can be explained or analyzed using different methods such as:
o Scaling analysis
o Long-wave analysis
o Linear stability analysis.
• These methods help us understand when and how instability occurs in fluid layers
under the influence of gravity (Chandrasekhar, 1961).
1.0. Analysis of Gravitational Instability of Buoyant Layers
1. Introduction

1.1. Gravitational Instability
• Gravitational instability occurs when a fluid layer becomes unstable due to
density differences.
• This instability leads to the formation of buoyant plumes or fingers, commonly
observed in various natural and industrial processes.
1.2. Gravitational instability is ubiquitous in nature and industry.
Its applications include:
o Astrophysics: Star formation and supernovae.
o Geophysics: Ocean circulation and mantle convection.
o Engineering: Mixing in chemical reactors and oil reservoirs.
1. Introduction

1.1. Gravitational Instability
• Gravitational instability occurs when a fluid layer becomes unstable due to
density differences.
• This instability leads to the formation of buoyant plumes or fingers, commonly
observed in various natural and industrial processes.
1.2. Gravitational instability is ubiquitous in nature and industry.
Its applications include:
-> Gravitational instability of buoyant layers is a complex phenomenon
1. Introduction

1.3. Gravitational instability of buoyant layers is a complex phenomenon:
• Analytical tools like scaling analysis, long-wave analysis, and linear stability
analysis provide valuable insights.
• Understanding these methods enhances our ability to predict and control
instabilities in various systems.
1. Introduction

1.4. Scaling Analysis
• Scaling analysis is a fundamental tool for understanding the dominant physical processes.
• In gravitational instability, it helps identify key parameters influencing the system.
• The key dimensionless numbers often include:
o The Rayleigh number (Ra)
o Prandtl number (Pr)
o Aspect ratio (Γ).
1. Introduction

1.5. Rayleigh-Taylor Instability
• A specific case of gravitational instability.
• Occurs at the interface between two fluids of different densities.
• Scales with the square root of the Rayleigh number (Ra^(1/2)).
• Commonly seen in supernova explosions, cloud formation, and ocean dynamics
1.6. Long-Wave Analysis
• Long-wave analysis is a mathematical approach to describe the evolution of
disturbances in the system.
• Assumes that the disturbances have long wavelengths compared to other
length scales.
• Leads to important results like the dispersion relation for the instability.
1. Introduction

1.7. Dispersion Relation
• The dispersion relation describes the growth rate and wavelength of
disturbances.
• In the context of gravitational instability, it relates the growth rate (σ) to wave
number (k).
• σ = f(k) - Important for understanding the stability of the system.
1.8. Linear Stability Analysis
• Linear stability analysis studies infinitesimally small disturbances to the
system.
• Provides insights into the initial stages of instability.
• Helps determine critical conditions for instability onset, often characterized by
a critical Rayleigh number.
1. Introduction

1.9. Critical Rayleigh Number
• The critical Rayleigh number is the threshold value above which instability
occurs.
• The Rayleigh number depends on ΔT, gravity, the height of the container, the
viscosity of the fluid, and the thermal diffusivity of the fluid.
• Low Rayleigh numbers indicate weaker convection, and higher Rayleigh
numbers indicate stronger, more intense convection.
• Below , the system is stable, and above it, instability sets in.
• It can be determined through linear stability analysis.
1. Introduction

• The gravitational instability of buoyant layers can be
explained using scaling analysis, long-wave analysis,
and linear stability analysis.
• These methods help us understand when and how
instability occurs in fluid layers under the influence of
gravity.
2. Analysis of Gravitational Instability of Buoyant Layers

• Scaling analysis is a method that allows us to estimate the order of magnitude of
different terms in the governing equations of fluid dynamics.
• In the context of gravitational instability in buoyant layers, we consider a layer of
fluid with a density difference Δρ across it, under the influence of gravity.
• The key parameters involved in scaling analysis are the thickness of the layer (h),
the gravitational acceleration (g), and the kinematic viscosity (ν).
(Chandrasekhar, 1961).
2.1. Scaling Analysis

• This Richardson number quantifies the relative importance of buoyancy (gravity)
and viscosity in the system.
• When Ri is less than a critical value (typically around 1/4 to 1/16), it indicates that
the layer is susceptible to gravitational instability.
• This means that buoyancy dominates over viscosity, and the layer can develop
instabilities and form structures due to gravity.
(Chandrasekhar, 1961).
2.1. Scaling Analysis ( Cont’d. )

• In the long-wave analysis, we examine the behavior of small perturbations (waves)
with long wavelengths in the fluid layer. This is particularly relevant in cases where
the layer's thickness is much larger than the wavelength of perturbations.
• The key equation used in long-wave analysis is the Rayleigh-Taylor instability
equation:
• Here, ξ represents the perturbation amplitude, g is the gravitational acceleration, Δρ is
the density difference, σ is the surface tension, and ρ is the fluid density.
• The long-wave analysis focuses on the growth rate of perturbations and stability
criteria. It typically leads to a dispersion relation that relates the growth rate of
perturbations to the wavenumber. If the growth rate is positive for some range of
wavenumbers, the layer is unstable to long-wavelength perturbations, and
gravitational instability occurs.
(Drazin & Reid, 1981).
2.2. Long-Wave Analysis:

• Linear stability analysis is a mathematical method used to determine the stability of a
given equilibrium state in a fluid system. In the context of gravitational instability, we
start with a base state (e.g., a flat fluid layer) and analyze how small perturbations
grow or decay over time.
• The linearized equations are derived by linearizing the governing equations (e.g.,
Navier-Stokes equations) around the base state and solving for the perturbations'
evolution. The critical parameter in linear stability analysis is the growth rate of
perturbations. If the growth rate is positive for any perturbation mode, the system is
unstable.
• The critical condition for instability is often expressed in terms of a dimensionless
parameter, such as the Reynolds number or Froude number, depending on the specific
problem.
2.3. Linear Stability Analysis:

• It has broad applications in various scientific and engineering disciplines.
3. Applications & Case Study

• Gravitational instability analysis is a crucial aspect of understanding mixing processes in oil
reservoirs.
• Predicting and managing the movement of fluids (Enhanced Oil Recovery and Reservoir
management and Engineering).
• There exist various applications of gravitational instability analysis in the context of mixing
in oil reservoirs:
3. 1. Applications

3.1.1. Reservoir Fluid Mixing:
• Enhanced Oil Recovery (EOR): Gravitational instability analysis can help optimize EOR
techniques such as water flooding, gas injection, or chemical flooding.
• Understanding how different fluids mix in a reservoir allows engineers to design efficient
injection strategies for displacing trapped oil ( Lake, 1989).
3.1.2. Fluid Saturation Modeling:
• Reservoir Simulation (Gravitational instability analysis is essential for accurate reservoir
simulation models).
• It helps in predicting the distribution of oil, water, and gas within the reservoir over time,
which is crucial for production forecasting and reservoir management (Aziz & Settari ,
1979).
3. 1. Applications

3.1.3. Fractured Reservoirs:
• Fracture Network Analysis ( In fractured reservoirs, gravitational instability analysis is
used to study how fluids move and mix within the fractures).
• This knowledge aids in optimizing well placement and injection strategies for efficient
production ( Warren & Root, 1963).
3.1.4. Gas-Oil Gravity Drainage:
• Gas Cap Reservoirs (Gravitational instability analysis is applied to gas cap reservoirs to
understand how gas and oil interact) .
• This knowledge helps in maintaining reservoir pressure, maximizing oil production, and
managing gas-oil ratios (Fanchi, 2006).
3. 1. Applications

3.1.5. Chemical Flooding:
• Surfactant and Polymer Flooding (Gravitational instability analysis is crucial in designing
chemical flooding processes).
• It helps in optimizing the distribution of injected chemicals within the reservoir, improving
sweep efficiency, and ultimately enhancing oil recovery (Farajzadeh et al. 2017).
3.1.6. Thermal Recovery Processes:
• Steam and Hot-Water Injection (In thermal recovery processes, gravitational instability
analysis helps in understanding the movement of heated fluids and their mixing with oil).
• This knowledge is vital for optimizing steam and hot-water injection strategies
(Butler,1991).
3. 1. Applications

3.1.7. Geological Heterogeneity:
• Reservoir Characterization (Gravitational instability analysis assists in characterizing
reservoir heterogeneity, including variations in rock properties and fluid mobility).
• This information is used to make informed decisions regarding well placement and
production techniques (Aguilera, 2014).
3. 1. Applications

3. 2. Case Study : Niger Delta
Fig.1-(a) Overview of offshore Niger Delta structural domains,
modied from Leduc et al. (2012).
(b) Crosssection PP' [Morgan, 2006] with the three structural
domains and the position of the assumed detachment
surface in the Akata formation, adopted from Butler [2010].
(Yuan, et al., 2017)

(a) Schematic illustration of upslope extensional (B), transitional and downslope compressive (A) provinces, composing the
gravity-driven collapse of offshore deltas (modified from King et al. [2010]).
(b) The stability domain in the (α , β ) plane according to the CCW theory [Dahlen, 1984], accounting for either extension
or compression, and the amendment due to the three province interaction (dashed curve C).

Fig.2- The general prototype and the collapse mechanism for gravitational instabilitie (Yuan, et al., 2017)

Fig.4- The velocity field for the extensional and the compressive parts of the collapse mechanism in (a) and (b), respectively, with the associated
hodographs of the velocity jumps across the axial surfaces JI and GF.

Eq.4- Stability
condition
Eq.1 – External Resisting Power
Eq.2 – External Resting Power
Eq.3 – Maximum Resisting Power

Eq.7
Eq.8

Eq.9

Eq.9- The maximum resisting power detailed in eq.3

Aguilera, R. (2014). Geology of Tight Gas Reservoirs. Gulf Professional Publishing.
Anyiam, U., Nduka, V., & Opara, A. (2010). 3-D seismic interpretation and reserve estimation of Ossu Field in OML 124,
onshore Niger Delta Basin, Nigeria. In SEG Technical Program Expanded Abstracts 2010 (pp. 1327-1331). Society of
Exploration Geophysicists.
Aziz, K., & Settari, A. (1979). Petroleum Reservoir Simulation. Applied Science Publishers.
Butler, R. M. (1991). Thermal Recovery. Prentice Hall.
Chandrasekhar, S. (1961). "Hydrodynamic and Hydromagnetic Stability." Dover Publications.
Drazin, P. G., & Reid, W. H. (1981). "Hydrodynamic Stability." Cambridge University Press.
Fanchi, J. R. (2006). Principles of Applied Reservoir Simulation. Gulf Professional Publishing
Haack, R. C., Sundararaman, P., Diedjomahor, J. O., Xiao, H., Gant, N. J., May, E. D., & Kelsch, K. (2000). Niger delta
petroleum systems, Nigeria. MEMOIRS-AMERICAN ASSOCIATION OF PETROLEUM GEOLOGISTS, 213-232.
Lacoste, A., B. C. Vendeville, R. Mourgues, L. Loncke, and M. Lebacq (2012), Gravitational instabilities triggered by fluid
overpressure and downslope incision - insights from analytical and analogue modelling, Journal of Structural Geology, 42,
151 - 162.
Lake, L. W. (1989). Enhanced Oil Recovery. Prentice Hall.
Warren, J. E., & Root, P. J. (1963). The Behavior of Naturally Fractured Reservoirs. Society of Petroleum Engineers Journal.
Yuan, X. P., Leroy, Y. M., & Maillot, B. (2017). Reappraisal of gravity instability conditions for offshore wedges: consequences
for fluid overpressures in the Niger Delta. Geophysical Journal International, 208(3), 1655-1671.
5. Reference

WGravitational Instability of Buoyant Layers Using Scaling Analysis, Long-wave Analysis and Linear Stability Analysis

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