Examples of Nonstrict Hyperbolic Systems

S.12.9. Examples of Nonstrict Hyperbolic Systems

We now consider several examples of nonstrict hyperbolic systems of the form (7), for which the matrix A has coincident eigenvalues, λ i (u) = λ j (u) for i ≠ j, in some domain.

Example 17. Let us discuss the Riemann problem for the 2 × 2 hyperbolic system

∂x ∂(cs) + ∂

c[1 − (s − c − 2) 2 ] =0

∂t

∂x ∂x

7 c=0

Figure 18. Graphical solution for the Riemann problem (45)–(47).

with initial conditions

This is a model system for a two-phase multicomponent flow through porous media in the gravitational field. We are going first to classify the elementary waves for system (45), (46) and then to construct the solution of the problem (45)–(47) from these elements.

1 ◦ . Differentiating both sides of system (45), (46), we find the 2 × 2 matrix A of (7) in the form

The eigenvalues of A are:

Figure 18 shows the graphs of the function f (s, c) = 1 − (s − c − 2) 2 for two fixed values of c: c = 0 and c = 1. From (48) it follows that the first eigenvalue is equal to the slope of a curve

f = f (s, c = const). The second eigenvalue is equal to the slope of the line segment linking the point (s,

f ) with the origin of coordinates. Points 5 and 6 are the points of tangency of the curves

c = 1 and c = 0 and the straight lines through the origin of coordinates, respectively. The eigenvalues λ 1 and λ 2 of (48) are equal at points 5 and 6. The locus of points with equal eigenvalues (48) for 0 <

c < 1 is shown in Fig. 18 by the dashed line linking points 5 and 6. The first eigenvalue is higher than the second one in the area below the dashed curve,

while in the area above the dashed curve, the inequality λ 1 < λ 2 holds.

From the Rankine–Hugoniot conditions (25) it follows that system (45), (46) allows for two types of shocks: shocks without jumps of

c (so-called s-shocks),

and shocks with jumps of

c (so-called c-shocks),

The calculation of the right eigenvector (19) for the first eigenvalue (48) shows that c is constant along the first-family rarefactions. These rarefactions are called s-waves. The calculation of (19) for the second-family rarefactions shows that they degenerate into c-shocks.

Hence, system (45), (46) allows for three elementary waves: an s-shock, a c-shock, and a rarefaction s-wave. The solution of the problem (45)–(47) is self-similar, i.e., can be found in the form (37). The initial conditions (47) for

the self-similar coordinate ξ = x/t become

2 ◦ . Let us calculate several values that will be helpful for solving problem (45), (46), (51). The values of λ 1 at points 1 and 4 (denote them by D 1 and D 4 ) can be calculated from (48): D 1 = 2 and D 4 = −2. The coordinate s 5 of point 5 follows from the condition of equality of the two eigenvalues of (48) on the curve

c = 1: s 5 =2 √ 2. The slope D 5 of the curve

c = 1 at point 5 is equal to λ 1 at this point: D 5 =6−4 √

2. Let us plot the intersection point (point 7) of the straight line 0–5 and the curve √

√ 2 − 3, and the slope D 7 of the

c = 0. The coordinate of point 7 is: s 7 =3

straight line 7–1 is: D 7 =8−5 2.

The solution of the problem (51) must connect point 4 with point 1. Both the s-shock and the c-shock from point 4 are unstable, so it is possible to exit from point 4 just by the s-wave. The point that would be reached by s-wave from point 4 could be located before or after point 5. In the former case, the c-shock from the curve c = 1 to the curve c = 0 is unstable for any point behind the shock located between points 4 and 5. In the latter case, the c-shock from the curve c = 1 to the curve

O c 0 s wi

xt /

Figure 19. Construction of a graphical solution for the displacement of oil by a chemical additive.

c = 0 is stable if the point behind the shock lies between points 5 and 2; nevertheless, the value ξ in this interval exceeds the shock speed, so the sequence of an s-wave and a forthcoming c-shock is not allowed. The only possibility left is a c-shock from tangent point 5.

The solution consists of an s-wave, a c-shock, and an s-wave; the structural formula is: 4 — 5 → 7 — 1. Finally, we can write out the solution in the form

5 x, t) = , D x/t < D c(x, t) = ∞ < x/t < D

where s 1 ( ξ) = 3 − 1

2 ξ, s 2 (

2 ξ, and ξ = x/t.

System (45), (46) is not strictly hyperbolic, and consequently both an s-wave and an s-shock are present in the solution of the Riemann problem (47).

Example 18. A two-phase immiscible flow of oil and water with a chemical additive in water is governed by a 2 × 2 system

∂ cs + a(c) + ∂ cf (s, c) = 0.

∂t

∂x

Here, s is the water saturation, c is the additive concentration, f (s, c) is the water flux, and a(c) is the adsorbed chemical concentration, the so-called sorption isotherm.

The function f (s, c) satisfies the following conditions: f (s, c) = 0 0 for 0<s<s i ; f

f (s, c) = 1 for s 0 ( c) < s < 1. The graphs of

c) < 0 for s i <s<s ( c);

f (s, c) at c = 1 and c = 0 are presented in Fig. 19 . The dependence

f = f (s, c) allows us to choose either (s, c) or (s, f ) to be the unknown functions in system (53). The problem of oil displacement by an aqueous solution of a chemical admixture is described by system (53) and the following initial and boundary conditions:

s =s i ,

c=0 at t = 0, s =s 0 ( 54 (1), ) c=1 at x = 0.

The solution of problem (53), (54) is obtained by the same method as in Example 17. The initial-boundary value problem (54) can be transformed to the following boundary value problem for the self-similar coordinate:

s =s 0 (1),

The point at ξ = 0 lies on the curve c = 1, the point at x → ∞ is located on the curve c = 0 (Fig. 19). The self-similar path (s( ξ), f (ξ)) should connect the points (s 0 (1), 1) and (s i , 0) on the plane (s,

f ); see Fig. 19.

O c 0 s wi

Figure 20. Graphical construction of the solution for the displacement of oil by a chemical additive in the case of high sorption of the additive.

System (53) can be reduced to an equivalent system of the form (7) with matrix 

The eigenvalues of system (53) are evaluated as

Right eigenvectors corresponding to the first and second eigenvalues (57) are given by

s + ′ − f a s 58

Let us take the unknown s in the ordinary differential equations for rarefaction waves (19) to be the independent variable, so we look for a solution of (19) in the form

f = f (s), ξ = ξ(s). The equations for the s-waves and c-waves (first and second families of rarefactions) read

From (59) it follows that the first eigenvalue is equal to the slope of the curve f = f (s, c = const); see Fig. 19. The second eigenvalue (60) is equal to the slope of the line segment connecting the points (s,

f ) and (−a ′ ( c), 0). From the Rankine–Hugoniot conditions (25) it follows that system (53) admits two types of shocks: shocks without jumps of

c (so-called s-shocks),

and shocks with c-jumps (so-called c-shocks),

The case of a convex sorption isotherm is presented in Fig. 19. Point 2 is the tangent point of the curve c = 1 and the straight line through point O c with coordinates (−[ a]/[c], 0). The shock 2 → 3 is evolutionary. Let us plot the tangent point 4 of the curve

c = 0 and the straight line through s i . Figure 19 shows the case where point 4 is located above point 3, which corresponds to low sorption. The solution consists of an s-wave and two shocks; the structural formula is: s 0 (1) — 2 →3→s i . The speeds D 2 and D 3 of the shocks 2 → 3 and 3 → s i are calculated by formulas (62) and (61), respectively. The solution is given by

c(x, t) =

where the function s 1 ( ξ) is determined by the inversion of the relation ξ = f s ′ (s 1 ).

Figure 19 shows the correspondence between the solution image on the planes (s, curve s = s 1

f ) and (s, ξ), ξ = x/t. The continuous ( ξ) of (63) corresponds to the motion along the curve c = 1 from the point s 0 (1) to point 2; the slope of the curve

c = 1 at a point s is equal to the coordinate ξ that corresponds to the value s of the curve s = s 1 ( ξ). The shocks 2 → 3 and 3 →s i on the plane (s,

f ) correspond to discontinuities in the curve s = s(ξ) at the points ξ = D 2 and ξ=D 3 . If sorption is high, and point 4 is located below point 3, as in Fig. 20, the structural formula for the solution is: s 0 (1) — 2 →3—4→s i .

Example 19. If the sorption isotherm in (53) is concave, the transition from c = 1 to c = 0 occurs by a c-wave. The

structural formula is: s 0 (1) — 2 — 3 →s i .

If the sorption isotherm in (53) has inflection points, the transition from c = 1 to c = 0 occurs by a sequence of c-shocks and c-waves that correspond to a concave envelope of the sorption isotherm (see Fig. 21).

s 0 (1)

( -a c ¢ ( ), 0 1 )

( -a c ¢ ( ), 0 2 ) 0 s wi

Figure 21. Graphical solution for the displacement of oil by a chemical solution in the case of the sorption isotherm having inflection points.

Example 20. A two-phase (liquid–gas) three-component incompressible flow in porous media is governed by the system

Here, the following notation is adopted:

C=l 1 s + g 1 (1 − s), U=l 1 f (s, g 2 )+ g 1 [1 −

f (s, g 2 )], α(g 2 )= l 2 − g 1 2 , β(g 1 2 )= g 2 − αg 1 , l n = l n ( g 2 ), g 1 = g 1 ( g 2 ), ( 65 )

l − g where s = s( x, t) is the liquid saturation, l n and g n are the volume concentrations of the nth component in the liquid and

gas phases respectively, and f (s, g 2 ) is the liquid phase flux. The independent concentration in this system is g 2 , the other

concentrations are functions of g 2 . The unknowns in system (64) are s and g 2 or

C and g 2 .

The functions f,U,l 1 , and g 1 satisfy the following conditions:

2 ) < 0, C, g g ′ 1 ( g 2 ) > 0, l 1 ′ ( g 2 ) < 0. System (64) is analogous to system (53) analysed in Example 18.

The problem of the displacement of oil with composition A by gas with composition B corresponds to the initial and boundary conditions

The solution of problem (64)–(66) is expressed as

 0 if 0 < 2  , ξ<D

C(x, t) =

C(ξ) if D 2 < ξ<D 3 ,

where ξ = x/t, the function C(ξ) is the inverse of the function ξ = ∂U ∂C ( 2B ), and the constants C, g D 2 , D 3 , D 4 , C 4 (and also C 2 and C 3 ) are determined by the following transcendental equations:

s✂t The structural formula for solution (67) is: (0, g 2B ) → (C 2 , g 2B )—( C 3 , g 2B ) → (C 4 , g 2A ) → (C A , g 2A ). References for Subsection S.12.9: C. Wachmann (1964), L. W. Lake (1989), P. G. Bedrikovetsky and M. L. Chumak (1992a, 1992b).