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We will discuss the solutions to a Riemann problem of a non-strictly hyperbolic system of conservation laws $s_t+f(s, c)_x= 0$, $(cs+a(c))_t+ (cf(s, c))_x= 0.$ Here $s=s(x, t)$ is the water phase saturation, $c=c(x, t)$ is the concentration of the chemical agent in the water phase; the function $f$ denotes the fractional flow of water; the function $a$ denotes the chemical’s adsorption on the rock. It is commonly assumed that $f$ is an S-shaped function of $s$ for every $c$, and $a$ is an increasing concave function. This system is often used to describe the displacement of oil by a hydrodynamically active chemical agent (polymer, surfactant, etc) and thus we will call it a chemical flooding model. We will focus on two situations:

\noindent $\bullet a≡0:$ (zero adsorption). This is the so called KKIT model (Keyfiz, Kranzer, Issacson, Temple, c.f. [1])). It has many interesting properties. We will focus on the Isaacson-Glimm admissibility criteria for shock waves and show how to deduce it naturally if one remembers the physics of the process (just add adsorption).

\noindent $\bullet a\neq0:$ (non-zero adsorption). This model was first considered by Johansen and Winther [2] for $f(s, c)$ being a monotone function of $c$. We will consider the case when $f(s, c)$ is non-monotone increasing. In this case the non-Lax shocks appear, which depend on the ratio of diffusion parameters. Such shocks are also known as under-compressive shocks or transitional shocks.

References

[1] Isaacson E. L. and Temple J. B. $Analysis\ of\ a\ singular\ hyperbolic\ system\ of\ conservation\ laws$, Journal of Differential Equations, 1986, Vol. 65, no. 2, P. 250–268.

[2] Johansen T. and Winther R. $The solution\ of\ the\ Riemann\ problem\ for\ a\ hyperbolic\ system\ of\ conservation\ laws\ modeling\ polymer\ flooding$, SIAM journal on Mathematical Analysis, 1988, Vol. 19, no. 3, P. 541–566.

[3] Bakharev F., Enin A., Petrova Y., and Rastegaev N. {\it Impact of dissipation ratio on vanishing viscosity solutions of the Riemann problem for chemical flooding model}, arxiv:2111.1500