We present analytical solutions for capillary-controlled displacement in one dimension by use of fractional-flow theory. We show how to construct solutions with a spreadsheet that can be used for the analysis of experiments as well as matrix-block-scale recovery in field settings. The solutions can be understood as the capillary analog to the classical Buckley-Leverett solution (Buckley and Lev...

To derive the fractional flow equation oil displacement will be assumed to take place under the so-called diffusive flow condition. The constraints are that fluid saturations at any point in the linear displacement path are uniformly distributed with respect to thickness; this allows the displacement to be described mathematically in one dimension. The simultaneous flow of oil and water can be ...

For the hyperbolic conservation laws with discontinuous flux function there may exist several consistent notions of entropy solutions; the difference between them lies in the choice of the coupling across the flux discontinuity interface. In the context of Buckley-Leverett equations, each notion of solution is uniquely determined by the choice of a “connection”, which is the unique stationary s...

In this study, traveling wave soliton solutions of hyperbolic and trigonometric functions are successfully obtained by using the modified exponential function method Buckley-Leverett equation. addition to these, there also rational solutions. Two three-dimensional graphs real imaginary parts included with contour simulations physically analysis solution equation analyzed as a mathematical model...

We discuss an extension of the Buckley-Leverett (BL) equation describing twophase flow in porous media. This extension includes a third order mixed derivatives term and models the dynamic effects in the pressure difference between the two phases. We derive existence conditions for traveling wave solutions of the extended model. This leads to admissible shocks for the original BL equation, which...

In this short course we will start by deriving the basic conservation laws for mass, momentum and energy. We will extend the results to mixtures in general and then to multiphase flow in porous media. We will describe the black-oil models and the Buckley-Leverett problem. Second, we will review the basic numerical methods for solving the differential equations that model a fluid in a porous med...

Several experiments have evidenced the occurrence of saturation overshoots for flows in homogeneous porous media. Such phenomena are ruled out by standard mathematical models, which are based on equilibrium assumptions. In this presentation we discuss nonequilibrium models, in particular including dynamic effects in the capillary pressure. This leads to extensions of the classical Buckley-Lever...

When the reservoir parameters are stochastic, then the flow in a reservoir is described by stochastic partial differential equations. Spatial stochastic relative permeability in one spatial dimension is modeled by the stochastic Buckley–Leverett equation s(x, t)t + f(s(x, t), x)x = 0 for x > 0 and t > 0. f is the stochastic flux function and s is the saturation. This equation is analyzed, and i...