Math 2.0

Fluid mechanics Definition

Buckley–Leverett equation

Builds on Hyperbolic conservation lawDarcy's lawMultiphase flow

Definition

Consider one-dimensional flow of two immiscible incompressible fluids, water and oil, through a homogeneous porous medium of porosity ϕ∈(0,1)\phi \in (0, 1), neglecting gravity and capillary pressure. Let S(x,t)∈[0,1]S(x, t) \in [0, 1] be the water saturation, and let u>0u > 0 be the total Darcy velocity of the two fluids, which is constant in xx. Let krw(S)k_{rw}(S) and kro(S)k_{ro}(S) be the relative permeabilities and μw\mu_w, μo\mu_o the viscosities of water and oil, and define the fractional flow function f(S)=krw(S)/μwkrw(S)/μw+kro(S)/μo.f(S) = \frac{k_{rw}(S) / \mu_w}{k_{rw}(S) / \mu_w + k_{ro}(S) / \mu_o}. The Buckley–Leverett equation is the scalar conservation law ∂S∂t+uϕ∂f(S)∂x=0.\frac{\partial S}{\partial t} + \frac{u}{\phi} \frac{\partial f(S)}{\partial x} = 0.

Remarks

It follows from conservation of water and Darcy's law for each phase: the water flux is uf(S)u f(S).

Example: with krw=S2k_{rw} = S^2, kro=(1−S)2k_{ro} = (1 - S)^2 and μw=μo\mu_w = \mu_o, f(S)=S2/(S2+(1−S)2)f(S) = S^2 / (S^2 + (1 - S)^2). This ff increases from 00 to 11 and has an inflection point at S=1/2S = 1/2, so it is neither convex nor concave. For injection of water into a medium with S=0S = 0, the entropy solution is a shock from 00 to a saturation S∗S^* followed by a rarefaction. The value S∗S^* is fixed by the tangent condition f(S∗)/S∗=f′(S∗)f(S^*) / S^* = f'(S^*), the Welge construction. Here S∗=1/2S^* = 1/\sqrt{2}.

Non-example: if capillary pressure is kept, an extra term of the form ∂x(D(S)∂xS)\partial_x (D(S) \partial_x S) appears, and the equation is parabolic, not a pure hyperbolic conservation law.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHyperbolic conservation law, concept, no evidence levelHyperbolic conservati…conceptMultiphase flow, concept, no evidence levelMultiphase flowconceptStokes flow, concept, no evidence levelStokes flowconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptDarcy's law, statement, no evidence levelDarcy's lawstatementBuckley–Leverett equation, definition, no evidence levelBuckley–Leverett equa…definition
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  1. Euclidean space object
  2. Function concept
  3. Hyperbolic conservation law concept
  4. Multiphase flow concept
  5. Stokes flow concept
  6. Total derivative concept
  7. Function of several variables concept
  8. Vector-valued function concept
  9. Differentiable function of several variables concept
  10. Partial derivative concept
  11. Vector field concept
  12. Gradient concept
  13. Divergence concept
  14. Darcy's law statement
  15. Buckley–Leverett equation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Buckley%E2%80%93Leverett_equation
About this page
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MSC76S05, 76T06, 35L65
DefinesBuckley–Leverett equation
UsesHyperbolic conservation law, Darcy's law, Multiphase flow
Changed3 November 2026

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