In fluid dynamics

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

, the Buckley–Leverett equation is a transport equation

Generic scalar transport equation

The generic scalar transport equation is a general partial differential equation that describes transport phenomena such as heat transfer, mass transfer, fluid dynamics , etc. A general form of the equation is \big...

 used to model two-phase flow in porous media. The Buckley–Leverett equation or the Buckley–Leverett displacement can be interpreted as a way of incorporating the microscopic effects to due capillary pressure in two-phase flow into Darcy's law

Darcy's law

Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand...

.

In a 1D sample (control volume

Control volume

In fluid mechanics and thermodynamics, a control volume is a mathematical abstraction employed in the process of creating mathematical models of physical processes. In an inertial frame of reference, it is a volume fixed in space or moving with constant velocity through which the fluid flows...

), let be the water saturation, then the Buckley–Leverett equation is


where


is the fractional flow rate, is the total flow, is porosity

Porosity

Porosity or void fraction is a measure of the void spaces in a material, and is a fraction of the volume of voids over the total volume, between 0–1, or as a percentage between 0–100%...

 and is area of the cross-section in the sample volume.

Assumptions for validity

The Buckley–Leverett equation is derived for a 1D sample given

• mass conservation
  Conservation of mass
  The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...
  
  
• capillary pressure  is a function of water saturation  only
• causing the pressure gradients of the two phases to be equal.
• Flow is Linear
• Flow is Steady-State
• Formation is one Layer

General solution

The solution of the Buckley–Leverett equation has the form which means that is the front velocity

Front velocity

In physics, front velocity is the speed at which the first rise of a pulse above zero moves forward.In mathematics, it is also used to describe the velocity of a possibly propagating front in the solution of hyperbolic partial differential equation....

 of the fluids at saturation .

See also

• Capillary pressure
• Permeability (fluid)
  Permeability (fluid)
  Permeability in fluid mechanics and the earth sciences is a measure of the ability of a porous material to allow fluids to pass through it.- Units :...
  
  
• Relative permeability
  Relative permeability
  In multiphase flow in porous media, the relative permeability of a phase is a dimensionless measure of the effective permeability of that phase. It is the ratio of the effective permeability of that phase to the absolute permeability...
  
  
• Darcy's law
  Darcy's law
  Darcy's law is a phenomenologically derived constitutive equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand...