A Relaxation Projection Analytical–Numerical Approach in Hysteretic Two-Phase Flows in Porous Media

Abstract

Hysteresis phenomenon plays an important role in fluid flow through porous media and exhibits convoluted behavior that are often poorly understood and that is lacking of rigorous mathematical analysis. We propose a twofold approach, by analysis and computing to deal with hysteretic, two-phase flows in porous media. First, we introduce a new analytical projection method for construction of the wave sequence in the Riemann problem for the system of equations for a prototype two-phase flow model via relaxation. Second, a new computational method is formally developed to corroborate our analysis along with a representative set of numerical experiments to improve the understanding of the fundamental relaxation modeling of hysteresis for two-phase flows. Using the projection method we show the existence by analytical construction of the solution. The proposed computational method is based on combining locally conservative hybrid finite element method and finite volume discretizations within an operator splitting formulation to address effectively the stiff relaxation hysteretic system modeling fundamental two-phase flows in porous media.

Appendices

The Riemann Solution

In this Appendix, we obtain some Riemann solutions for three different regimes. These three regimes are the most representatives between the eight described in Sect. 2. To simplify our notation, we represent the rarefaction ${\mathcal {R}}_s$ through a state A as ${\mathcal {R}}_s(A)$; the shocks ${\mathcal {S}}_s$ and ${\mathcal {S}}_c$ from two states A to B as ${\mathcal {S}}_s(A,B)$ and ${\mathcal {S}}_c(A,B)$. Whenever necessary, we represent the shock ${\mathcal {S}}_c$ from a state (A) only as (${\mathcal {S}}_c(A)$). The last case is possible since the shocks ${\mathcal {S}}_c$ are straight lines parallels to the axis x. To represent the wave sequences, we use the notation introduced in Eq. (33). In Figs. 7, 8, 9, 10, 11, 12, 13, 14, 15, and 16, the solid curves with arrows indicate rarefaction direction (indicating the increasing of wave speed), the dashed curves are shocks without interaction with imbibition or drainage curves and the dashed dot curves are shocks that appear from the interaction between waves (shock or rarefactions) and the imbibition or drainage curves. A shock between two states A to B that interacts with imbibition curve is denoted as ${\mathcal {S}}_I(A,B)$, the shock interacting with the drainage curve is denoted as ${\mathcal {S}}_I(A,B)$ and the shock interacting with both imbibition and drainage is denoted as ${\mathcal {S}}_{ID}(A,B)$ or ${\mathcal {S}}_{DI}(A,B)$, depending on the order of interaction.

Fig. 7

(Left frame): Imbibition, drainage and fractional flows for the first regime. The arrow indicates the direction that $\lambda _s$ is an increasing function. ${\mathcal {I}}_s$ is the inflection of family s. (Right frame): For a fixed left state L, figure shows all 7 right regions, $R_1$ to $R_7$, for which the Riemann sequence is the same in each region. The solid curve represents rarefaction $({\mathcal {R}}_s)$ and dashed curves represent shocks $({\mathcal {S}}_c$ or ${\mathcal {S}}_s)$. States $P_1$, $P_2$ and line ${\mathcal {S}}_L$ are explained along of the text

Fig. 8

(Left frame): The Riemann solution for states on region $R_1$. Here we illustrate the projection method. For state L we have a shock ${\mathcal {S}}_s(L,N)$ followed by ${\mathcal {S}}_c(N,R)$. The projection of these waves on drainage curve gives origin to a wave with positive velocity. The interaction between waves faster with slower gives origin to an interaction shock ${\mathcal {S}}_D$. The wave sequence of Riemann solution is in Eq. (64). (Right frame): The Riemann solution for states on the regions $R_2$ or $R_3$. Here, there is no interaction of waves with the drainage or imbibition curves. If $R \in R_2$ the wave sequence is described in Eq. (65). If ${\tilde{R}}\in R_3$ the wave sequence is described in Eq. (66)

To obtain the solution, we use the Algorithm described in Sect. 3.1.4. To study the interaction between waves in the projection method we use the theory developed in Sects. 3.1.5 and 3.1.6. We also use Sect. 3.1.5 for the shock admissibility and the condition (42).

Fig. 9

(Left frame): The Riemann solution for right states in the region $R_4$. There is an interaction between ${\mathcal {R}}_s(L)$ and the imbibition curve. For right states R, the Riemann solution is described in Eq. (67). For state ${\tilde{R}}\in R_4$, for which there is an interaction with both imbibition and drainage curves, the wave sequence is described in Eq. (68). (Right frame): The Riemann solution for a right state ${\tilde{R}}\in R_5$ with y coordinates in the same region that $R_2$ is described in Eq. (69). Similarly, the wave sequence for a right state ${\tilde{R}} \in R_5$ with y coordinates in the same region that $R_3$ is described in Eq. (70)

1.1 First Regime: Case Described in Figs. 7, 8, 9, and 10

This case is described in Sect. 2 satisfying $(\sigma _\mathbf{g}>0,u=0)$ (Fig. 3f). In Fig. 7 (left), we draw the fractional flux for imbibition, drainage and for different hysteresis parameters. The fractional flow curves inside the equilibrium region are represented by solid curves and the arrow in each curve indicates the direction for which $\lambda _s$ increases. Notice that the speed is negative for each curve. The structure ${\mathcal {I}}_s$ represents the inflection of family ${\mathcal {R}}_s$. In Fig. 7-right, we fix a left state for the Riemann problem, represented by L. For this state, we divide the equilibrium region ${\mathcal {E}}{\mathcal {Q}}_{\mathcal {R}}$ into seven regions ($R_1$ to $R_7)$. In each region the Riemann solution exhibits the same sequence of waves. For this state L we also have two important states denoted as $P_1$ and $P_2$. The state $P_1$ lies at the intersection between the shock $S_L={\mathcal {S}}_s(L,P_1)$ and the drainage curve; the state $P_2$ is at the rarefaction ${\mathcal {R}}_s(L)$ and the drainage curve. The curve $S_L$ also defines a boundary of one region.

Fig. 10

(Left frame): The Riemann solution for state in $R_6$. Given a right state R, we obtain $M_1$ at the intersection between ${\mathcal {S}}_c(R)$ and the drainage curve, the wave sequence is described in Eq. (71). (Right frame): The Riemann solution for $R\in R_7$. Here, there is state $M_2$ for which the shock speed between L and $M_2$ is equal to the tangent (characteristic speed) at $M_2$ on the drainage curve. Given a right state R above the state $M_2$ (with respect to y-coordinate), we obtain $M_1$ from the intersection between ${\mathcal {S}}_c(R)$ and the drainage curve, in this case the wave sequence is described in Eq. (72). For a state ${\tilde{R}}$ above the state $M_2$ (with respect to y coordinates), we obtain $M_3$ as intersection between ${\mathcal {S}}_c(R)$ and the drainage curve. The wave sequence is described in Eq. (73)

Fig. 11

(Left frame): Imbibition, drainage and fractional flows for second regime. The arrow indicates the direction that $\lambda _s$ is increasing. ${\mathcal {I}}_s$ is the inflection of family s. (Right frame): For a fixed left state L, figure shows all 6 right regions, $R_1$ to $R_6$, for which the Riemann sequence is the same in each region. The solid curve represents rarefaction $({\mathcal {R}}_s)$ and dashed curves represent shocks $({\mathcal {S}}_c$ or ${\mathcal {S}}_s)$. States $P_1$, $P_2$, $P_3$ and line ${\mathcal {S}}_L$ are explained along of the text

Fig. 12

(Left frame): Imbibition, drainage and fractional flows for third regime. ${\mathcal {C}}_{sc}$ is the coincidence locus between $\lambda _s$ and $\lambda _c$. (Right frame): For a fixed left state L, figure shows all 8 right regions, $R_1$ to $R_8$, for which the Riemann sequence is the same in each region. The solid curve represents rarefaction $({\mathcal {R}}_s)$ and dashed curves represent shocks $({\mathcal {S}}_c$ or ${\mathcal {S}}_s)$. States $P_1$ to $P_4$ and curves ${\mathcal {C}}_3$ and ${\mathcal {C}}_4$ are explained along of the text

The region $R_1$ is defined as the states that are above the curves $S_c(P_1)$ and ${\mathcal {R}}_s(L)$. The Region $R_2$ is defined as the states that lay between the curves $S_c(P_1)$ and $S_c(L)$ crossing ${\mathcal {R}}_s(L)$. The region $R_3$ is defined as the states that lay between the curves $S_c(L)$ and $S_c(P_2)$ crossing ${\mathcal {R}}_s(L)$. The region $R_4$ are the states that lay in the equilibrium region below the curve $S_c(P_2)$. The region $R_5$ is formed by the states that lay between the curves ${\mathcal {S}}_c(P_1)$ and ${\mathcal {S}}_c(P_2)$ below the curve ${\mathcal {R}}_s(L)$. The region $R_6$ is formed by states that lay above the curve $S_c(P_1)$ and below the curve ${\mathcal {S}}_L(L, P_1)$, and finally, the region $R_7$ is formed by states that lay above the curve ${\mathcal {S}}_L(L, P_1)$ on left to state $P_1$. In Fig. 7-right, we show all regions.

Fig. 13

(Left frame): The Riemann solution for a right state in $R_1$ (third case). Given a right state R, we obtain $M_1$ at the intersection between ${\mathcal {S}}_c(R)$ and the imbibition curve and ${\tilde{M}}_1$ lies at the intersection of fractional flow passing through R and the imbibition curve. Here, we have a projection of ${\mathcal {S}}_s(L)$ on the imbibition curve. The state $M_1$ is reached first from this projection. The wave sequence is described in Eq. (74). (Right frame): The Riemann solution for a right state in $R_2$. There is no interaction between ${\mathcal {S}}_s(L)$ and imbibition curve. We obtain a state $M_1$ at the intersection between ${\mathcal {S}}_s(L,M_1)$ and ${\mathcal {S}}_c(M_1,R)$. The wave sequence is described in Eq. (75)

Fig. 14

(Left frame): The Riemann solution for a right state in $R_3$. There is no interaction between ${\mathcal {R}}_s(L)$ and drainage curve. We obtain a state $M_1$ at the intersection between ${\mathcal {R}}_s(L)$ and ${\mathcal {S}}_c(M_1,R)$. The wave sequence is in Eq. (76). (Right frame): The Riemann solution for state in $R_4$. The interaction between the ${\mathcal {R}}_s(L)$ and the drainage curve gives origin to a shock ${\mathcal {S}}_D(L,M_1)$, here the state $M_1$ explained along of the text. The wave sequence is described in Eq. (77)

Fig. 15

(Left frame): The Riemann solution for a right state in $R_5$. Given a right state R, we obtain ${\tilde{M}}_1$ at the intersection between ${\mathcal {S}}_c(R)$ and the drainage curves and ${M}_1$ at the intersection of fractional flow passing through R and the drainage curves. Here, we have a projection of ${\mathcal {R}}_s(L)$ on the imbibition curve. The state $M_1$ is reached first from this projection. The wave sequence is described in Eq. (78). (Right frame): The Riemann solution for a right state in $R_6$. Here, there is state ${\tilde{P}}$ at the intersection of fractional flow passing through R and the drainage curve and $P_3$ that is the state in drainage curve for which the tangent slope is equal to the shock slope. Here there is a rarefaction appearing as the interaction between waves, which we denote as ${\mathcal {R}}_D$. The wave sequence is described in Eq. (79)

Fig. 16

(Left frame): The Riemann solution for right state in $R_7$ (third case). At the state $P_4$ the tangent slope is parallel to ${\mathcal {S}}_c(P_4)$ and the solution bifurcates. We construct the extension from $P_4$, denoted as ${\mathcal {S}}_c(P_4)$. From R we draw the fractional flow crossing ${\mathcal {S}}_c(P_4)$ in ${\tilde{P}}$. The wave sequence is described in Eq. (80). (Right frame): The Riemann solution for right state in $R_8$. The difference of this solution and the previous one is that ${\tilde{P}}$ is connect to R with a rarefaction ${\mathcal {R}}_s({\tilde{P}})$. The wave sequence is described in Eq. (81)

The right state${{\varvec{R}}}\in {{\varvec{R}}}_1$. We start our Riemann solution taking a right state $R\in R_1$. From the geometrical compatibility, the first wave (slowest) is a shock ${\mathcal {S}}_s$ starting in L. However, this wave crosses the drainage curve, see Fig. 8-left. Thus, we need to apply our projection algorithm, see Sect. 3.1.4.

From the algorithm, we first consider the wave sequence without considering the interaction between source terms and wave sequence. For this case, the possible wave sequence is a shock ${\mathcal {S}}_s(L,N)$ (from L to state N). This state N lies at the intersection between ${\mathcal {S}}_s(L,N)$ and ${\mathcal {S}}_c(R)$. From N we have a shock ${\mathcal {S}}_c(N,R)$, with speed zero.

After we apply the first step of projection algorithm, we study the interaction between waves. For this case, we see that the projection of the sequence of waves (${\mathcal {S}}_s$ and ${\mathcal {S}}_c$) on the drainage curve gives origin to an interaction shock with viscous profile, satisfying the inequality (42) in Sect. 3.1.5. This shock connects an intermediate state $M_1$ to R. We represent this interaction shock as ${\mathcal {S}}_D$. This state $M_1$ is the same state $P_1$ at the intersection between ${\mathcal {S}}_s(L,M_1)$ and the drainage curve. Thus the Riemann solution, using the projection method, corresponds to:

$$\begin{aligned} L\xrightarrow {{\mathcal {S}}_s(L,M_1)}M_1\xrightarrow {{\mathcal {S}}_D(M_1,R)} R. \end{aligned}$$

(64)

The right state${{\varvec{R}}}\in {{\varvec{R}}}_2$ or ${{\varvec{R}}}_3$. For right states in regions $R_2$ and $R_3$, there are no interaction between the wave ${\mathcal {S}}_s$ or ${\mathcal {R}}_s$ with imbibition or drainage curves.

For a state R in $R_2$, our first wave (slowest) is a ${\mathcal {S}}_s$ from L to state $M_1$. This state lies at the intersection between ${\mathcal {S}}_s$ from L and ${\mathcal {S}}_c$ from R. From the state $M_1$ we have a shock ${\mathcal {S}}_c(M_1,R)$ with zero speed, and the wave sequence is:

$$\begin{aligned} L\xrightarrow {{\mathcal {S}}_s(L,M_1)}M_1 \xrightarrow {{\mathcal {S}}_c(M_1,R)}R. \end{aligned}$$

(65)

For a state ${\tilde{R}}$ in $R_3$, the first wave is a rarefaction ${\mathcal {R}}_s(L)$ to a state ${\tilde{M}}_1$ that lies at the intersection between ${\mathcal {R}}_s$ and ${\mathcal {S}}_c$; from ${\tilde{M}}_1$, we have a shock ${\mathcal {S}}_c({\tilde{M}}_1,{\tilde{R}})$, the wave sequence is:

$$\begin{aligned} L\xrightarrow {{\mathcal {R}}_s(L,{\tilde{M}}_1)} {\tilde{M}}_1\xrightarrow {{\mathcal {S}}_c({\tilde{M}}_1,{\tilde{R}})} {\tilde{R}}. \end{aligned}$$

(66)

Both solutions are shown in Fig. 8-right.

The right state${{\varvec{R}}}\in {{\varvec{R}}}_4$. For the right state in regions $R_4$, there are interaction between waves from the equilibrium region with the imbibition or drainage curves. Notice that, from L, to reach region $R_4$, we need first a ${\mathcal {R}}_s$. Using the projection method, there is an interaction between ${\mathcal {R}}_s$ and ${\mathcal {S}}_c$ with the imbibition curve. This interaction wave gives origin to a shock, satisfying (42). After projection method, the wave sequence from L to state R is a rarefaction ${\mathcal {R}}_s(L)$ to $P_2$ on the imbibition curve. From $P_2$ there is an interaction shock (${\mathcal {R}}_s$ and ${\mathcal {S}}_c$ with the imbibition curve) that we denote as ${\mathcal {S}}_I$ with positive speed to the state R. The wave sequence in this case is:

$$\begin{aligned} L\xrightarrow {{\mathcal {R}}_s(L,P_2)}P_2 \xrightarrow {{\mathcal {S}}_I(P_2,{R})}{R}. \end{aligned}$$

(67)

Moreover, notice it is possible that ${\mathcal {S}}_c$ interacts with the drainage curve. For a state ${\tilde{R}}$ in the region $R_4$, see Fig. 9-left, the wave sequence from state L, consists of a ${\mathcal {R}}_s(L)$ to $P_2$. From $P_2$ there is a shock to ${\tilde{R}}$, however, this shock interacts with the imbibition and drainage curve. We denote this shock as ${\mathcal {S}}_{ID}$. The wave sequence in this case is:

$$\begin{aligned} L\xrightarrow {{\mathcal {R}}_s(L,P_2)}P_2\xrightarrow {{\mathcal {S}}_{ID} (P_2,{\tilde{R}})}{\tilde{R}}. \end{aligned}$$

(68)

In Fig. 9-left we summarize these solutions.

The right state${{\varvec{R}}}\in {{\varvec{R}}}_5$. In Fig. 9-right, we describe the Riemann solution for this case. Notice the right states in $R_5$ have the same y coordinates as the states in the regions $R_2$ and $R_3$. However, from L to reach a state in $R_5$ the wave ${\mathcal {S}}_c$ interacts with drainage curve. For a state ${\tilde{R}}$ in $R_5$ with y coordinates in the same region as $R_2$, the Riemann solution is similar the one obtained for states in the region $R_2$, i.e., from L there is a shock ${\mathcal {S}}_s$ to state ${\tilde{M}}_1$ lying at the intersection of ${\mathcal {S}}_s(L,{\tilde{M}}_1)$ and ${\mathcal {S}}_c({\tilde{M}}_1,{\tilde{R}})$. From ${\tilde{M}}_1$ to ${\tilde{R}}$ there is a shock with speed zero. Notice this shock represents the interaction between ${\mathcal {S}}_c$ and the drainage curve, satisfying (42). The interesting fact here is that the speed of the shock is also zero. We denote this shock as ${\mathcal {S}}_D({\tilde{M}}_1,{\tilde{R}})$, the wave sequence is:

$$\begin{aligned} L\xrightarrow {{\mathcal {S}}_s(L,{\tilde{M}}_1)} {\tilde{M}}_1\xrightarrow {{\mathcal {S}}_D({\tilde{M}}_1,{\tilde{R}})} {\tilde{R}}. \end{aligned}$$

(69)

For a state a ${\tilde{R}}\in R_5$ with y coordinates in the same region that $R_3$, the Riemann solution is similar that obtained for states in $R_3$, i.e., from L there is a rarefaction ${\mathcal {R}}_s$ to state ${M}_1$ that lies at the intersection of ${\mathcal {R}}_s(L)$ and ${\mathcal {S}}_c({M}_1,{R})$. From ${M}_1$ to R there is an interaction $S_D({M}_1,{R})$. The wave sequence is:

$$\begin{aligned} L\xrightarrow {{\mathcal {R}}_s(L,{M}_1)}{M}_1 \xrightarrow {{\mathcal {S}}_D({M}_1,{R})}{R}. \end{aligned}$$

(70)

The right state${{\varvec{R}}}\in {{\varvec{R}}}_6$. For this case, notice that any wave from L, using our projection method, will interact with the drainage curve. This interaction gives origin to a shock ${\mathcal {S}}_D$, satisfying (42). To facilitate our analysis, consider $R \in R_6$ and let $M_1$ be the state obtained from the intersection between the shock ${\mathcal {S}}_c(R)$ and the drainage curve, see Fig. 10-left. Since the $R_6$ is below the curve $S_L$, the shock speed of ${\mathcal {S}}_D$ from $P_1$ to $M_1$ is faster than the shock speed between L to $P_1$. Thus, the sequence ${\mathcal {S}}_s(L,P_1)$ followed by ${\mathcal {S}}_D(P_1,M_1)$ respects the geometrical compatibility. From $M_1$, there is a stationary shock ${\mathcal {S}}_s(M_1,R)$ from $M_1$ to the left state R. We summarize the wave sequence as:

$$\begin{aligned} L\xrightarrow {{\mathcal {S}}_s(L,{P}_1)}{P}_1 \xrightarrow {{\mathcal {S}}_D({L}_1,{M_1})}{M_1} \xrightarrow {{\mathcal {S}}_c(M_1,R)}{R}. \end{aligned}$$

(71)

In Fig. 10-left, we describe this sequence. We stress here, since drainage and imbibition curve are very close in $R_6$, we deformed the Figure to draw the states and wave sequence.

The right state${{\varvec{R}}}\in {{\varvec{R}}}_7$. In this case, any wave from L interacts with the drainage curve, given origin to ${\mathcal {S}}_D$, satisfying (42).

In $R_7$ there is a state $M_2$ with the property that the shock speed between L and $M_2$ (interaction shock ${\mathcal {S}}_D$) is equal to derivative of drainage curve, i.e., the shock speed is resonant with the characteristic speed.

Consider $R\in R_7$ below the state $M_2$ (with respect to coordinate y) and let $M_1$ be the state obtained from the intersection between the shock ${\mathcal {S}}_c(R)$ and the drainage curve, see Fig. 10-right. Since the $R_7$ is above the curve $S_L$, the shock speed of ${\mathcal {S}}_D$ from $P_1$ to $M_1$ is slower than the shock speed between L to $P_1$, from the geometrical compatibility principle, there is a direct shock between L and $M_1$, which we also denoted as ${\mathcal {S}}_D(L,M_1)$. From $M_1$ there is a shock ${\mathcal {S}}_c(M_1,R)$ to R, the wave sequence is:

$$\begin{aligned} L\xrightarrow {{\mathcal {S}}_D(L,M_1)}{M}_1 \xrightarrow {{\mathcal {S}}_c(M_1,R)}{R} \end{aligned}$$

(72)

In other hand, if we have a state ${\tilde{R}}$ above the state $M_2$ (with respect to the axis y) the Riemann solution from L to ${\tilde{R}}$ consists of a ${\mathcal {S}}_D(L,M_2)$ from L to $M_2$. From $M_2$, there is a rarefaction ${\mathcal {R}}_D(M_2)$ to state $M_3$. The state $M_3$ lies at the intersection between ${\mathcal {S}}_c({\tilde{R}})$ and the drainage curve. From state $M_3$ there is a ${\mathcal {S}}_c(M_3,{\tilde{R}})$. We summarize the wave sequence as:

$$\begin{aligned} L\xrightarrow {{\mathcal {S}}_D(L,M_2)}{M}_2\xrightarrow {{\mathcal {R}}_s(M_2)} {M_3}\xrightarrow {{\mathcal {S}}_c(M_3,R)}{R} \end{aligned}$$

(73)

In Fig. 10-right we summarize the solution.

1.2 Second Regime: Case Described in Fig. 11

This case is described in Sect. 2 satisfying $(\sigma _\mathbf{g}<0,u<0)$ (Fig. 3a) . In Fig. 11-left, we draw the imbibition, drainage and fractional flow for different values of hysteresis parameter. The curve ${\mathcal {I}}_s$ represents the inflection of family ${\mathcal {R}}_s$. In Fig. 11-right, for a left Riemann state given, we obtain all right states. Here, we can identify three important states: $P_1$-$P_3$. Here, the states $P_1$ and $P_2$ are obtained as in the previous regime; the state $P_3$ represents the state on the drainage curve with minimum value in the y-coordinate. As in the previous regime, we have a straight line ${\mathcal {S}}_L$ that is the extension of shock between the state L to a state $P_1$.

Using these important points and structures, we can divide the equilibrium region ${\mathcal {E}}{\mathcal {Q}}_{\mathcal {R}}$ in six regions $R_1$ to $R_6$.

The region $R_1$ is defined as the states that lay above the curves $S_c(P_1)$ and ${\mathcal {R}}_s(L)$. The Region $R_2$ is composed by the states that lay below the curves ${\mathcal {S}}_c(P_3)$, drainage and ${\mathcal {S}}_c(P_1)$ and above the imbibition and ${\mathcal {S}}_c(P_2)$. The region $R_3$ is composed by the states that lay below the curve ${\mathcal {S}}_c(L)$ and above the curve ${\mathcal {S}}_c(P_2)$ and imbibition curves. The region $R_4$ are composed by the states that lay below the curve $S_c(P_2)$ and above the imbibition curve. The region $R_5$ are composed by the states that are below the curve ${\mathcal {S}}_L(L,P_1)$ and above (i) the curve ${\mathcal {S}}_c(P_3)$ for states to left of $P_3$ and (ii) above the drainage curve for states between $P_3$ and P1. The region $R_6$ is formed by states that are above the ${\mathcal {S}}_L(L,P_1)$ to left of $P_1$. In Fig. 11-right, we show all regions.

Here, we do not detail the Riemann solution, because the wave sequence is very similar to the one obtained in the case described in Sect. A.1. For right states in $R_1$, the wave sequence is the same as described in Eq. (64) and in Fig. 8-left. For right states on $R_2$ and $R_3$, the wave sequence is the same as described in Eqs. (65)–(66) and in Fig. 8-right. For right states on $R_4$, the wave sequence is the same as described in Eqs. (67)–(68) and in Fig. 9-left. For right states in $R_5$, the wave sequence is the same as described in Eq. (71) and in Fig. 10-left and, for right states on $R_6$, the wave sequence is the same as described in Eqs. (72)–(73) and in Fig. 10-right. Notice that the sequences (69) and (70) do not appear in this case.

1.3 Third Regime: Case Described in Figs. 12, 13, 14, 15, and 16

This case is described in Sect. 2 satisfying $(\sigma _\mathbf{g}>0,u=0)$ (Fig. 3f). In Fig. 12-left, we draw the imbibition, drainage and fractional flow for different values of hysteresis parameter. The curve ${\mathcal {C}}_{sc}$ is the coincidence between $\lambda _s$ and $\lambda _c$. In this model there is no inflection of family ${\mathcal {R}}_s$, the wave speed of fractional flow for different hysteresis parameters has increasing speed for decreasing water saturation values. In Fig. 12-right, for a left Riemann state given, we obtain all right states. Here, we can identify four important states denoted from $P_1$ to $P_4$. The state $P_1$ lies at the intersection between the rarefaction ${\mathcal {R}}_s(L)$ with drainage curve and the state $P_2$ lies at the intersection between rarefaction and imbibition curves. The state $P_3$ lies at on the drainage curve such that the slope of the shock between $P_1$ and L is equals to the slope of tangent of drainage curve at $P_1$. The state $P_2$ lies at the drainage curve for which the tangent is parallel to the x-axis. We also have two important curves, $C_3$ and $C_4$, that are the fractional flow passing through $P_3$ and $P_4$, respectively.

The region $R_1$ is composed by the states that are below the curves ${\mathcal {S}}_c(P_2)$ and imbibition and above the drainage curve. The region $R_2$ is composed by the states below the ${\mathcal {S}}_c(L)$ and imbibition curves and above the ${\mathcal {S}}_c(P_2)$ and drainage curves. The region $R_3$ formed by the states below the ${\mathcal {S}}_c(P_1)$ and imbibition curves and above the ${\mathcal {S}}_c(L)$ and drainage curves. The region $R_4$ is formed by the states that lay above the ${\mathcal {S}}_c(P_1)$ and drainage curves and below the ${\mathcal {C}}_{sc}$ and imbibition curves. The region $R_5$ composed by the states above the ${\mathcal {C}}_{sc}$ and drainage curves and below the ${\mathcal {C}}_3$ and imbibition curves. The region $R_6$ is formed by the states above the ${\mathcal {C}}_3$ and drainage curves and below the ${\mathcal {C}}_4$ and imbibition curves. The region $R_7$ is formed by the states above the ${\mathcal {C}}_4$ and ${\mathcal {S}}_c(P_1)$ and below the imbibition curve. The region $R_8$ is formed by the states below the ${\mathcal {S}}_c(P_4)$ and imbibition curves and above the drainage curve, see Fig. 12-right. The Riemann solution for this regime using another formulation was solved in [111] (for all cases). Our results are similar that obtained in [111], however, from the formulation using relaxation we do not need to make additional hypothesis on the model to obtain the Riemann solutions. The $L_1$ continuity discussed in [111] is obtained directly from the projection method.

The right state${{\varvec{R}}}\in {{\varvec{R}}}_1$. The right state defines two states on the imbibition curve: $M_1$, that lies at the intersection between ${\mathcal {S}}_c(R)$ and imbibition curve; and ${\tilde{M}}_1$ that lies at the intersection of fractional flow from R and imbibition curve.

From left state L, the first wave is a ${\mathcal {S}}_s$, this shock interacts with the imbibition curve and it is projected on this curve since the characteristic speed on the imbibition curve is smaller than the shock speed ${\mathcal {S}}_s$, this interaction gives origin to the shock ${\mathcal {S}}_I$, satisfying (42). Notice that, from the projected wave on the imbibition curve there are two possible sequence of waves connecting a projected state on the imbibition curve and the right state R (respecting the geometrical compatibility principle). One wave ${\mathcal {S}}_c(M_1,R)$ connecting $M_1$ and R, and another wave ${\mathcal {R}}_s({\tilde{M}}_1)$ connecting ${\tilde{M}}_1$ and R. This apparent loss of uniqueness was discussed in [111]. In that paper, authors proved that the correct sequence, preserving the $L_1$ norm, is L, the shock ${\mathcal {S}}_I(L,M_1)$ and ${\mathcal {S}}_c(M_1,R)$. Here, it is easy to see that the projection method gives the same wave sequence obtained in [111]. The projection of wave ${\mathcal {S}}_s$, starting at L, reaches first the state $M_1$ on the imbibition curve, thus the wave follows from $M_1$ to R. We summarize the wave sequence as:

$$\begin{aligned} L\xrightarrow {{\mathcal {S}}_I(L,M_1)}M_1 \xrightarrow {{\mathcal {S}}_c({L}_1,R)}{R}. \end{aligned}$$

(74)

In Fig. 13-left, we summarize this behavior.

The right state${{\varvec{R}}}\in {{\varvec{R}}}_2$. For right state in $R_2$, the wave sequence does not interact with imbibition or drainage curves. From L, there is a shock ${\mathcal {S}}_s(L,M_1)$, this state $M_1$ is obtained from the intersection of ${\mathcal {S}}_s(L)$ (starting at L) and ${\mathcal {S}}_c(R)$ (starting at R). From $M_1$, there is a shock ${\mathcal {S}}_c(M_1,R)$ to R, see Fig. 13-right. We summarize the wave sequence as:

$$\begin{aligned} L\xrightarrow {{\mathcal {S}}_s(L,M_1)}{M}_1 \xrightarrow {{\mathcal {S}}_c(M_1,R)}{R}. \end{aligned}$$

(75)

The right state${{\varvec{R}}}\in {{\varvec{R}}}_3$. Similar behavior appears for right state $R\in R_3$, the difference is that we substitute the shock ${\mathcal {S}}_s$ by the rarefaction ${\mathcal {R}}_s$. The solution from L is ${\mathcal {R}}_s(L)$ to state $M_1$ (here, $M_1$ is obtained between the intersection of ${\mathcal {R}}_s(L)$ and ${\mathcal {S}}_c(R))$ and then a shock ${\mathcal {S}}_c(M_1,R)$. The wave sequence for this case is:

$$\begin{aligned} L\xrightarrow {{\mathcal {R}}_s(L)}{M}_1 \xrightarrow {{\mathcal {S}}_c(M_1,R)}{R}. \end{aligned}$$

(76)

The solution is summarized in Fig. 14-left.

The right state${{\varvec{R}}}\in {{\varvec{R}}}_4$. The derivative of all fractional flow is negative for all states in $R_4$. On the other hand, notice that from state L, the interaction between ${\mathcal {R}}_s$ and the drainage curve gives origin to an interaction shock ${\mathcal {S}}_D$ from L to the drainage curve. Similarly the case discussed for states in $R_1$, from the drainage curve, there are two states for which the wave sequence respect the geometrical compatibility criterion.

The first state is $M_1$, that is obtained from the intersection between ${\mathcal {S}}_c(R)$ and the drainage curves; the second state ${\tilde{M}}_1$ lies at the fractional flow from R crossing ${\tilde{M}}_1$ on the drainage curve. Notice, however, using the projection method, the first state to be reached is $M_1$ (it is in according to [111]) and thus from $M_1$ there is a shock ${\mathcal {S}}_s(M_1,R)$ connecting $M_1$ to R. The wave sequence is:

$$\begin{aligned} L\xrightarrow {{\mathcal {S}}_D(L,M_1)}{M}_1 \xrightarrow {{\mathcal {S}}_c(M_1,R)}{R}. \end{aligned}$$

(77)

The solution is summarized in Fig. 14-right.

The right state${{\varvec{R}}}\in {{\varvec{R}}}_5$. Here, we define two states on the drainage curve: $M_1$ that is at the intersection of the fractional flow from R and crossing the drainage curve at $M_1$; ${\tilde{M}}_1$ that is at the intersection of ${\mathcal {S}}_c(R)$ with the drainage curve. To obtain the wave sequence, we notice that the waves leaving the left state interact with the drainage curve, from the projection method this wave is projected on the drainage curve.

Since the fractional flow have positive slope for any state in $R_5$, the state $M_1$ is reached first (than ${\tilde{M}}_1$) by this projection, see Fig. 15-left. From $M_1$ there is a shock ${\mathcal {S}}_s(M_1,R)$ to R. The wave sequence is:

$$\begin{aligned} L\xrightarrow {{\mathcal {S}}_D(L,M_1)}{M}_1 \xrightarrow {{\mathcal {S}}_c(M_1,R)}{R}. \end{aligned}$$

(78)

The solution is summarized in Fig. 15-left.

The right state${{\varvec{R}}}\in {{\varvec{R}}}_6$. Consider ${\tilde{P}}$ the state that lies at the intersection between the fractional flow from R crossing drainage curve, see Fig. 15-right. The projection of ${\mathcal {R}}_s(L)$ on the drainage curve leads first to an interaction shock ${\mathcal {S}}_D(L,P_3)$ at the state $P_3$. This state has the property that the tangent slope at $P_3$ is equal to the slope of shock ${\mathcal {S}}_D(L,P_3)$. From $P_3$, the projected wave (interaction) has increasing speed, which gives origin to a rarefaction, which we denote as ${\mathcal {R}}_D(P_3)$. We draw this rarefaction to ${\tilde{P}}$, which we connect to R state with a shock ${\mathcal {S}}_s({\tilde{P}},R)$. The wave sequence is given by:

$$\begin{aligned} L\xrightarrow {{\mathcal {S}}_D(L,P_3)}{P}_3 \xrightarrow {{\mathcal {R}}_D(P_3)}{{\tilde{P}}} \xrightarrow {{\mathcal {S}}_s({\tilde{P}},R)}{R}. \end{aligned}$$

(79)

The solution is summarized in Fig. 15-right.

The right state${{\varvec{R}}}\in {{\varvec{R}}}_7$. In Fig. 16, we can notice the existence of a state $P_4$ on the drainage curve. This state has the property that the slope of tangent in drainage curve is parallel to ${\mathcal {S}}_c$, i.e., the solution bifurcates when crosses this point. We define the extension of this point at left of $P_4$, denoted as ${\mathcal {S}}_c(P_4)$. In Fig. 16, we draw all solutions to cases ${{\varvec{R}}}\in {{\varvec{R}}}_7$ and ${{\varvec{R}}}\in {{\varvec{R}}}_8$. For a right state R in $R_7$, the fractional flow from R crosses the extension ${\mathcal {S}}_c(P_4)$ at state ${\tilde{P}}$. The wave sequence of the Riemann solution in this case, from discussion of previous solutions, is summarized as:

$$\begin{aligned} L\xrightarrow {{\mathcal {S}}_D(L,P_3)}{P}_3 \xrightarrow {{\mathcal {R}}_D(P_3)}{P_4} \xrightarrow {{\mathcal {S}}_c(P_4,{\tilde{P}})}{\tilde{P}} \xrightarrow {{\mathcal {S}}_s({\tilde{P}},R)}{R}. \end{aligned}$$

(80)

The right state${{\varvec{R}}}\in {{\varvec{R}}}_8$. For right states in this region, the difference with the previous solution is the existence of a ${\mathcal {R}}_s({\tilde{P}})$ connecting ${\tilde{P}}$ to R. We summarize the solution as:

$$\begin{aligned} L\xrightarrow {{\mathcal {S}}_D(L,P_3)}{P}_3 \xrightarrow {{\mathcal {R}}_D(P_3)}{P_4} \xrightarrow {{\mathcal {S}}_c(P_4,{\tilde{P}})}{\tilde{P}} \xrightarrow {{\mathcal {R}}_s({\tilde{P}})}{R}. \end{aligned}$$

(81)

Numerical Procedures for Hyperbolic and Diffusion Models

1.1 Domain Decomposition Iteration and Mixed Finite Element Approximations for the Diffusion Equation

In this section, we overview the numerical procedure that we employ for the solution of the diffusion equation (60)–(61). Hybridized mixed finite elements are used for the spatial discretization along with a domain decomposition strategy towards the solution of the resulting algebraic problem. Indeed, hybridized mixed finite elements is locally conservative by construction and is quite adequate for accurate velocity field computation in the case of porous media transport problems [29, 39, 41, 53, 54]. It is completely feasible and can be computationally advantageous to define the finite element method for the saturation equation over different partitions of the domain. Moreover, our choice for the HMFEM as such is related to the use of Robin boundary conditions as natural interfacial transmission conditions in the domain decomposition iterative procedure as discussed by [29, 53, 96] and its application [1] to the case of numerical simulation of wave propagation in nonlinear three-phase flows in porous media with spatially varying flux functions. Moreover, the implementation of the code is simple because the domain decomposition of elements of the finite element method localizes the computations and the scheme is also naturally parallelizable [29, 39, 53, 96]. We also point out the issue of discontinuous flux for convection–diffusion models with convection-dominated problems is not well understood, see, e.g., [17, 19, 21, 35].

Consider the diffusion equation to be solved as seen in Sect. 4,

$$\begin{aligned} \frac{\partial S}{\partial t} - \frac{\partial w(S,x,\pi )}{\partial x} = 0, \quad w(S,x,\pi ) = D(S,x,\pi ) \frac{\partial S}{\partial x}, \end{aligned}$$

(82)

along with $\forall x \in \varOmega = [0,X] \subset {\mathbb {R}}$, $t > 0$, and proper initial and boundary conditions. We shall take the computational domain to be a uniformly partitioned interval and employ the simplest RT space. We notice that the same partition will be used to the discretization of the convective transport system. Domain decomposition methods have been studied for the most part as algebraic tools for solving problems on parallel machines (see, e.g., [54] and references therein). This is not what we do here. Instead, we employ a spatial decomposition of the diffusion problem (82) to construct a simple and efficient iterative method for the numerical solution.

To discretize Eq. (82) we consider the spaces $V = \{v \in H^{1}(\varOmega )\}$ and $W = L^{2}(\varOmega )$. The global weak form for the diffusion equation (82) for a time $t_n<t<t_{n+1}$, is given by finding $\{w, S\} \in V \times W$ such that,

$$\begin{aligned} \left( D^{-1} w,{\bar{w}}\right) - \left( \frac{\partial S}{\partial x},{\bar{w}} \right)&= 0, \qquad \forall {{\bar{w}}} \in V, \end{aligned}$$

(83)

$$\begin{aligned} \left( \frac{\partial S}{\partial t},{\bar{S}} \right) - \left( \frac{\partial w}{\partial x},{\bar{S}}\right)&= 0, \qquad \forall {\bar{S}} \in W, \end{aligned}$$

(84)

along with the initial condition from the convective step,

$$\begin{aligned} S(x,t_n) = s^{i_c-1}(x). \end{aligned}$$

(85)

Without loss of generality, let $\varOmega = [0,X]$ and $\{\varOmega _j,\ j=1,\dots ,M\}$ be a regular partition of $\varOmega$, with size $\varDelta x = h$, such that: ${\bar{\varOmega }} = \cup ^M_{j=1} {\bar{\varOmega }}_j$, $\varOmega _j \cap \varOmega _k = \emptyset ,\, j\ne k$, and $\varGamma = \partial \varOmega$, $\varGamma _{j} = \partial \varOmega _{j}$, $\varGamma _{jk} = \varGamma _{kj} = \partial \varOmega _{j} \cap \partial \varOmega _{k}$. Since we are considering the one dimensional case, we have a partition $\{ x_{0}, x_{1},\dots ,x_{M},x_{M+1} \}$, where $x_{0} = 0$, $x_{M+1} = X$ and $h = x_{j+1} - x_{j}$, $j = 0,\dots ,M$. Also,

$$\begin{aligned} \varOmega _{j} = [x_{j},x_{j+1}], \quad \varGamma = \{ x_{0},x_{M+1} \}, \quad \varGamma _{j,j+1} = \varGamma _{j+1,j} = \{x_{j+1}\}. \end{aligned}$$

(86)

Consider decomposing the problem (83)–(85) over the partition $\{\varOmega _j\}$, for $j=1,\dots ,M$, and consider the spaces $V_{j} = H^{1}(\varOmega _{j})$ and $W_{j} = L^{2}(\varOmega _{j})$. The localized weak form corresponding to the domain decomposition above are given by seeking “global solutions” (i.e., approximate solutions) $\{w_{j}, S_{j}\} \in V_{j} \times W_{j}$, $j=1,\dots ,M$, such that, $\forall {{\bar{w}}} \in V$ and $\forall {\bar{S}} \in W$ we have

$$\begin{aligned}&\left( D^{-1} w,{\bar{w}}_{j} \right) _{\varOmega _{j}} - \left( S_{j}, \frac{\partial {\bar{w}}}{\partial x}\right) _{\varOmega _{j}} \!\!\! - S_{j}(x_{j}){{\bar{w}}}(x_{j}) + S_{j}(x_{j+1}){{\bar{w}}}(x_{j+1}) = 0, \end{aligned}$$

(87)

$$\begin{aligned}&\left( \frac{\partial S_{j}}{\partial t},{\bar{S}} \right) _{\varOmega _{j}} - \left( \frac{\partial w_{j}}{\partial x},{\bar{S}}\right) _{\varOmega _{j}} = 0. \end{aligned}$$

(88)

To require $\{w_j,S_{j}\}$ to be a solution of (83)–(84) for $x \in \varOmega _{j}$, $j=1,\dots ,M$, it is also necessary to impose consistency conditions,

$$\begin{aligned} S_{j}=S_{k}, \quad w_{j} = w_{k}, \quad x \in \varGamma _{jk}, \end{aligned}$$

(89)

1.1.1 Hybridized Mixed Finite Element Approximation

Let $W^h_j \times V^h_j$ be the mixed finite element spaces over $\{\varOmega _{j}\}$. Each of these spaces is defined through local spaces $W_j \times V_j= W(\varOmega _j) \times V(\varOmega _j)$ (see [1, 29, 39, 41, 54] for more details). In each spaces of mixed elements referenced above, the generic functions $S \in W^{h}_j$,   and $w \in V^{h}_j$ are allowed to be discontinuous across each $\varGamma _{j\,k}$. As a consequence, attempting to impose the consistency conditions (89) would force continuity of S on the discrete level, an uninteresting case. Thus, let us introduce the Lagrange multipliers [1, 29, 39, 41, 54] $\ell _{jk}$ on the edges $\varGamma _{j\,k}$, defined as the value of the water saturation S on $\varGamma _{j\,k}$. Assume that $w_{j}= w|_{\,\varOmega _j}$, $w \in V^h_{j}$, and $w_{j} \in \varGamma _{j\,k}$ is a polynomial of some fixed degree $\varrho = \text{ degree } w^h_{j}$, where for simplicity we shall assume $\varrho$ to be independent of $\varGamma _{j\,k}$ [1, 39, 41, 54]. Moreover, it can be quite intricate to design correct coupling condition at the interfaces (93)–(94) for nonlinear parabolic convection–diffusion transport problems in porous media (see [1, 17, 35]), in particular for the case where we have discontinuous quantities involved as such hysteresis. Therefore, Robin transmission boundary conditions seems to be a natural choice. Let,

$$\begin{aligned} \varLambda ^{h} = \{\ell : \ell |_{\,\varGamma _{j\,k}} \in P_{\varrho }(\varGamma _{j\,k}) = \varLambda _{j\,k},\ \varGamma _{j\,k} \ne \emptyset \}. \end{aligned}$$

Notice that there are two copies of $P_{\varrho }$ assigned to the set $\varGamma _{j\,k}$: $\varLambda _{j\,k}$ and $\varLambda _{k\,j}$.

The hybridized mixed finite element method is given by (dropping the superscript h) seeking $\{S_{j}, w_j, \ell _{jk}\} \in W_j \times V_j \times \varLambda$, $\{j=1,\dots ,M\}$, such that:

$$\begin{aligned}&\left( D^{-1} w,{\bar{w}}_{j} \right) _{\varOmega _{j}} - \left( S_{j}, \frac{\partial {\bar{w}}}{\partial x}\right) _{\varOmega _{j}} - \ell _{j,j-1}(x_{j}){{\bar{w}}}(x_{j}) + \ell _{j,j+1}(x_{j+1}){{\bar{w}}}(x_{j+1}) = 0, \nonumber \\ \end{aligned}$$

(90)

$$\begin{aligned}&\left( \frac{\partial S_{j}}{\partial t},{\bar{S}} \right) _{\varOmega _{j}} - \left( \frac{\partial w_{j}}{\partial x},{\bar{S}}\right) _{\varOmega _{j}} = 0, \end{aligned}$$

(91)

$\forall {{\bar{w}}} \in V_{j}$ and $\forall {\bar{S}} \in W_{j}$ along with consistency conditions,

$$\begin{aligned} \ell _{j\,k} = \ell _{k\,j}, \quad w_{j} = w_{k}, \quad x \in \varGamma _{j\,k}. \end{aligned}$$

(92)

1.1.2 Algebraic Method for Solving the Discrete Parabolic Problem

To define an iterative method for solving the diffusion problem (82), it will be convenient to replace (92) by equivalent Robin transmission boundary conditions [29, 53, 96]. Recently in [1], Robin transmission boundary conditions were used as coupling condition in the case of three-phase flow in porous media with discontinuous (in space) capillary pressures functions. Consider the Lagrange multipliers as seen from $\varOmega _j$ and its adjacent elements $\varOmega _{j-1}$ and $\varOmega _{j+1}$. The Robin conditions for an element $\varOmega _{j}$ can be written as,

$$\begin{aligned} \chi _{j,j-1} w_{j} + \ell _{j,j-1} = \chi _{j,j-1} w_{j-1} + \ell _{j-1,j},&\quad x \in \varGamma _{j,j-1} = x_{j}, \end{aligned}$$

(93)

$$\begin{aligned} -\chi _{j,j+1} w_{j} + \ell _{j,j+1} = -\chi _{j,j+1} w_{j+1} + \ell _{j+1,j},&\quad x \in \varGamma _{j,j+1} = x_{j+1}, \end{aligned}$$

(94)

where $\chi _{j \, k}$ is a positive function on $\varGamma _{j\,k} \subset \partial \varOmega _j$ and $\varGamma _{k\,j} \subset \partial \varOmega _k$. We shall make the choice of the mixed finite element space $W_j^h \times V_j^h$ to be the lowest index RT space. The degrees of freedom on $\varOmega _{j}$, are given by the local spaces,

$$\begin{aligned} V_{j}^{h} = \{v: v \text{ is } \text{ linear } \text{ in } \varOmega _{j} \}, \quad W_{j}^{h} = \{w: w \text{ is } \text{ constant } \text{ in } \varOmega _{j}\}. \end{aligned}$$

The Lagrange multipliers $\ell _{jk}$ and $\ell _{kj}$ on $\varGamma _{j\,k}$ are also constants. We also shall assume the variable D to be constant on each element, and the relative permeability functions to be independent of position. Lets consider in more details the calculation of w. Suppress for the moment the subscript j. Now, consider $\{\phi _{L},\, \phi _{R} \}$ as basis functions for $V_{j}^{h}$, given by,

$$\begin{aligned} \phi _{R}(x) = \displaystyle \frac{x - x_{j}}{h}, \quad \phi _{L}(x) = \displaystyle \frac{x - x_{j+1}}{h}, \quad x \in \varOmega _{j}. \end{aligned}$$

(95)

The basis considered here slightly differ from the usual Lagrange basis; here we have $\phi _{R}(x_{j}) = 0$, $\phi _{R}(x_{j+1}) = 1$, $\phi _{L}(x_{j}) = -1$, $\phi _{L}(x_{j+1}) = 0$. Standard Lagrange basis can be used. Writing $w_{j}(x) \in V_{j}^{h}$ as a linear combination of the basis functions, $w_{j}(x) = w_{R}\,\phi _{R} + w_{L}\,\phi _{L}$, we have $w_{j}(x_{j}) = -w_{L}$ and $w_{j}(x_{j+1}) = w_{R}$, defining a diffusive flux w in then the interface of $\varOmega _{j}$. The saturation function $S_{j}(x)$ on $\varOmega _{j}$ is approximated by a constant, $S_{j}(x) = S_{j}$. In order to write the iterative procedure, consider a element $\varOmega _{j}$ and its adjacent elements. The superscript  $\, \widetilde{\,}$  indicates the variables of the adjacent elements and the subscripts indicates the variables on the right (R) and on the left (L) edges of an element. Using the trapezoidal integration rule [1, 53, 54], we discretize Eqs. (90)–(91) obtaining,

$$\begin{aligned} w_{R} + w_{L} = h\,\displaystyle \frac{\partial S}{\partial t}, \quad w_{R} = \frac{2D}{h}\left( S - \ell _{R} \right) , \quad w_{L} = \frac{2D}{h}\left( S - \ell _{L} \right) , \end{aligned}$$

(96)

where $\displaystyle \frac{\partial S}{\partial t}$ in $\varOmega _{j}$ will be approximate by a backward Euler scheme. To complete the system of discrete equations, we rewrite Eqs. (93) and (94) as,

$$\begin{aligned} \ell _{j,j-1} = \chi _{j,j-1}\,\left[ w_{j-1}(x_j) - w_{j}(x_{j})\right] + \ell _{j-1,j},&\, x \in \varGamma _{j,j-1} = x_{j}, \end{aligned}$$

(97)

$$\begin{aligned} \ell _{j,j+1} = \chi _{j,j+1}\left[ w_{j}(x_{j+1}) - w_{j+1}(x_{j+1})\right] + \ell _{j+1,j},&\, x \in \varGamma _{j,j+1} = x_{j+1}, \end{aligned}$$

(98)

Since $w_{j}(x_{j}) = -{w}_{L}$, $w_{j}(x_{j+1}) = w_{R}$, $w_{j-1}(x_{j}) = {\tilde{w}}_{R}$ and $w_{j+1}(x_{j+1}) = -{\tilde{w}}_{L}$, the Eqs. (97)–(98) become,

$$\begin{aligned} \ell _{L} = \chi _{L}\left( {\tilde{w}}_{R} + w_{L}\right) + {\tilde{\ell }}_{R},&\quad x \in \varGamma _{j,j-1} = x_{j}, \end{aligned}$$

(99)

$$\begin{aligned} \ell _{R} = \chi _{R}\left( w_{R} + {\tilde{w}}_{L} \right) + {\tilde{\ell }}_{L}.&\quad x \in \varGamma _{j,j+1} = x_{j+1}. \end{aligned}$$

(100)

Combining Eqs. (96)–(100) we can easy get the following equations for the water diffusion saturation for an element $\varOmega _{j}$,

$$\begin{aligned}&\displaystyle \frac{\partial S}{\partial t} - \frac{1}{h}\left[ w_{R} + w_{L}\right] = 0, \end{aligned}$$

(101)

$$\begin{aligned}&(1 + \chi _{L}\, \xi _{L})\,w_{L} - \xi _{L}\,S = -\chi _{L}\, \xi _{L}\,{\tilde{w}}_{R} - \xi _{L}\,{\tilde{\ell }}_{R}, \end{aligned}$$

(102)

$$\begin{aligned}&(1 + \chi _{R}\, \xi _{R})\,w_{R} - \xi _{R}\,S = -\chi _{R}\, \xi _{R}\,{\tilde{w}}_{L} - \xi _{R}\,{\tilde{\ell }}_{L}, \end{aligned}$$

(103)

where $\xi _{\alpha } = 2D_{\alpha }(\ell _{\alpha },\, \pi )/h$,  $\alpha = L,R$. This completes the derivation of the continuous-time mixed finite element discrete equations.

1.1.3 Discretization in Time and a Domain-Decomposition Iterative Procedure for the Diffusion Subproblem

In the domain decomposition procedure, we apply the Robin transmission conditions (99)–(100) on the interfaces of each element. Let us solve for the pair (S, w) when this transmission condition is used. If the domain decomposition is to be performed at the level of individual elements, then the parabolic stage of the time step accounting for the diffusive effects is based on a backward Euler approximation as follows (written for a generic jth element over the partition $\{\varOmega _j\}$ and by inserting the initial data into ${\overline{S}_{j}}$):

$$\begin{aligned}&\left[ \displaystyle \frac{\partial S}{\partial t}\right] _{j,\,n+1} \approx \displaystyle \frac{S^{n+1}_{j}-{\overline{S}}_{j}}{\varDelta t_d}. \end{aligned}$$

(104)

Then, the diffusive-stage equation equivalent to that of (101)–(103) is given on the $j^{th}$-element by

$$\begin{aligned}&\displaystyle \frac{S^{n+1}_{j}-{\overline{S}}_{j}}{\varDelta t_d} - \displaystyle \frac{1}{h}\left[ w^{\,n+1}_{\,L}+w^{\,n+1}_{\,R}\right] =0, \end{aligned}$$

(105)

$$\begin{aligned}&(1+\chi _{L}\,\xi _{L})\,w^{\,n+1}_{L}-\xi _{L} \, S^{n+1}_{j} =-\chi _{L}\,\xi _{L}\,{\tilde{w}}_{R}-\xi _{L}\, {{{\tilde{\ell }}}}_{R}, \end{aligned}$$

(106)

$$\begin{aligned}&(1+\chi _{R}\,\xi _{R})\,w^{\,n+1}_{R}-\xi _{R} \, S^{n+1}_{j} =-\chi _{R}\,\xi _{R}\,{\tilde{w}}_{L}-\xi _{R}\, {{{\tilde{\ell }}}}_{L}, \end{aligned}$$

(107)

where now $\xi _{\alpha } = 2D_{\alpha }(\ell _{\alpha },\pi )/h$, and $D_{\alpha }(\ell _{\alpha },\pi ) \approx D_{\hbox {eff}\,\alpha } (S,\pi ,{\tilde{S}},{\tilde{\pi }})$ as will be described bellow. The superscript   $\, \widetilde{\,}$  have the same meaning as above. For simplicity of notation, the time index $n+1$ is suppressed on $\xi _{\alpha }$ and on the right-hand side of (106) and (107). As announced, modeling hysteresis in porous media flow via relaxation naturally leads to discontinuous hyperbolic–parabolic flux functions. Moreover, $S \in W^{h}_{j}$ and $w \in V^{h}_{j}$ are allowed to be discontinuous across each $\varGamma _{jk}$. As a consequence, it is needed to define an interface transmission condition for the diffusive flux w, which takes into account the new parameter $D_{\alpha }(\ell _{\alpha },\pi ) \approx D_{\hbox {eff}\,\alpha } (S,\pi ,{\tilde{S}},{\tilde{\pi }})$ defined on the interface of each element. It follows from (96) that (where $\alpha = L,R$),

$$\begin{aligned} w_{\alpha } = \displaystyle \frac{2}{h} D_{\alpha }(S-\ell _{\alpha }), \quad {\tilde{w}}_{\alpha ^{\prime }} = \displaystyle \frac{2}{h} {\tilde{D}}_{\alpha ^{\prime }} ({\tilde{S}}_{\alpha } - {\tilde{\ell }}_{\alpha ^{\prime }}), \quad D_{\hbox {eff}\,\alpha } = \frac{2\,D_{\alpha }\,{\tilde{D}}_{\alpha ^{\prime }}}{D_{\alpha } +{\tilde{D}}_{\alpha ^{\prime }}}, \end{aligned}$$

(108)

where $D_{\alpha }=D_{\alpha }(S,\pi )$ and ${\tilde{D}}_{\alpha ^{\prime }}={\tilde{D}}_{\alpha ^{\prime }}({\tilde{S}},{\tilde{\pi }})$. The subscript $\alpha ^{\prime }$ denotes the edges of the adjacent elements corresponding to the edge $\alpha = L, R$ of the element under consideration. Because the domain decomposition localizes the computations, it is natural as well as advantageous to let $D_{\hbox {eff}\,\alpha }$ be defined on each interface in terms of local properties of the medium, i.e., as saturation S and hysteresis parameter $\pi$. The novelty here is in defining a value for the hysteresis parameter $\pi$ in the interfaces of each element through $D_{\hbox {eff}\,\alpha }$. The parameter $\chi _{\alpha }$ might lead to faster convergence of the iterations if is it properly chosen; see [54] and reference therein for details. A dimension analysis of the diffusive Robin transmission condition (93)–(94), or (99)–(100), at the interfaces of each element leads to the choice:

$$\begin{aligned} \chi _{\alpha } = {h}/{D_{\hbox {eff}\,\alpha }}. \end{aligned}$$

(109)

Notice we are solving the differential Eqs. (1)–(2) by splitting diffusive and convective effects. The hyperbolic problem (56)–(57) is first solved and then next we solve the parabolic problem (60)–(61) in [$t_n,t_{n+1}$]; the nonlinear coefficient $\xi _{\alpha }$ make use the of solution for the convection equation at $t_{n+1}$. The error in this approximation is expected to be smaller than the error of a zero order linearisation at $t_{n}$ (i.e., local Picard iteration per time step). Moreover, the introduction of this approximation makes the diffusive system “locally” linear for each time step $\varDelta t_d$. The iteration for the diffusive effects over saturation quantities at times $t_n$, $n>0$, is based on an element-by-element domain decomposition and the Eqs. (105)–(106). Here, we use an ordinary checkboard ordering to separate the subdomains into “red” and “black” subsets. Then, with a fixed time $n+1$ (replace for a moment the time index $n+1$ for the iteration index j) and iterates as follows:

• For a given level j of iteration, set $S^{j} = {\overline{S}}$.

• For all red elements, compute Eqs. (105)–(109) at level j by using adjacent black elements at level $j-1$ of iteration. Then, update the Lagrange multiplies associated to red elements by Eqs. (99)–(100).

• For all black elements, compute Eqs. (105)–(109) at level j by using adjacent red elements at level $j-1$ of iteration. Then, again, update the Lagrange multiplies associated to black elements by (99)–(100).

• Check convergence for values between iteration levels $k-1$ and k by

  $$\begin{aligned} \frac{1}{m}\sqrt{\sum _{j = 0}^{m+1}{\left| S_j^{n, \, k} - S_j^{n, \, k-1}\right| ^2}} < \varepsilon . \end{aligned}$$

  (110)

1.2 Numerical Solution of the Hyperbolic-Convection Equation

We make use of the robust Nessyahu and Tadmor [103] scheme along with the first convergence proof for the Lax–Friedrichs given by Karlsen and Towers [89] for the numerical approximation of the hyperbolic problem (56)–(57) exhibiting discontinuous nonlinear hyperbolic flux functions with respect to quantities S and $\pi$. Indeed, the Nessyahu and Tadmor scheme is fully based on the Lax–Friedrichs. This shed light on about our careful choice on the construction of a robust and high-resolution procedure to deal with hyperbolic problem with discontinuous flux. The theory of scalar hyperbolic conservation law with discontinuous flux is, no doubt, not yet well understood [19, 21]; see also [1, 17, 35].

The central idea here was to build a central type-scheme respecting the local equilibria linked to the nonlinear hyperbolic–parabolic flux functions without any special interface-coupling condition, which is heavily based Riemann solvers [17, 35] that do not extend in any obvious truly multidimensional way. Karlsen and Towers [89] gave the first convergence proof for the Lax–Friedrichs finite difference scheme for non-convex genuinely nonlinear scalar conservation laws of the form $u_t+f(u,k(x,t))_x = 0$, $u=u(x,t)$, where the coefficient k(x, t) is allowed to be discontinuous along curves in the (x, t) plane. Moreover, the authors prove stability and uniqueness for an extended Kruzkov entropy solution, provided that the flux function satisfies a so-called crossing condition, and that strong traces of the solution exist along the curves where k(x, t) is discontinuous. In this way they were also able to show that a convergent subsequence of approximations produced by the Lax–Friedrichs scheme to the above equation converges to such an entropy solution [89]. We notice that $\pi (x,t)$ induces discontinuities in the flux function associated to the hyperbolic conservation law (56). Thus, in this work the flux function ${f}({S},\pi )$ for the two-phase system (1)–(2) can be discretized with respect to x and consistent with a conservative finite-volume interpretation in a staggered grid in view of the same approach as performed by Karlsen and Towers [86, 89]; see [1] for more information applied to multi-D problems.

To explain the key idea, lets consider as a prototype to the hyperbolic problem (56) linked to problem (1)–(2) the 1D equation:

$$\begin{aligned} \frac{\partial S}{\partial t} +\frac{\partial }{\partial x} f(S,\pi ) = 0, \end{aligned}$$

(111)

where $S= S(x,t)$ is the water saturation and $\pi = \pi (x,t)$ is the hysteresis parameter as seen in Sect. 4. At each time level, a piecewise constant approximate solution over cells of width $\varDelta x=x_{j+\frac{1}{2}}-x_{j-\frac{1}{2}}$, namely,

$$\begin{aligned} {\overline{S}}(x,t)=S_{j}(t),\quad x_{j-\frac{1}{2}} \le x \le x_{j+\frac{1}{2}}, \end{aligned}$$

(112)

is first reconstructed [103],

$$\begin{aligned} L_j(x,t)=S_{j}(t)+(x-x_j){\frac{1}{\varDelta x}}{S^{\prime }_{j}(t)}, \quad x_{j-\frac{1}{2}}\le x\le x_{j+\frac{1}{2}}, \end{aligned}$$

(113)

where we use slope limiters for computation of $(1/\varDelta x)S^{\prime }_{j}(t)$ to prevent oscillations and to improve resolution. Equation (113) also retains local conservation ${{\bar{L}}}_j(x,t)={\bar{S}}(x,t) = S_{j}(t)$. Next, the piecewise linear interpolant (113) is evolved in time through the solution of successive noninteracting Riemann problems over the cells $x_j<x<x_{j+1}$. The solution (112) is then projected back into the space of staggered piecewise constant grid-functions to yield,

$$\begin{aligned} \begin{array}{c} S_{j+\frac{1}{2}}(t+\varDelta t_c) \, \equiv \, \displaystyle \frac{1}{\varDelta x}\displaystyle \int _{x_j}^{x_{j+1}} S(x,t+\varDelta t_c)\,dx. \end{array} \end{aligned}$$

(114)

Considering the conservation law (111) and the last integral in (114) reads,

$$\begin{aligned} \begin{array}{l} S_{j+\frac{1}{2}}(t+\varDelta t_c) = \displaystyle \frac{1}{\varDelta x}\bigg [\int _{x_j}^{x_{j+\frac{1}{2}}}L_{j}(x,t)\,dx + \displaystyle \int _{x_{j+\frac{1}{2}}}^{x_{j+1}}L_{j+1}(x,t)\,dx \bigg ]\\ \quad -\,\displaystyle \frac{1}{\varDelta x} \bigg [\displaystyle \int _{t}^{t+\varDelta t_c} \!\!\!\!\!\!\!\! f(S(x_{j+1},\tau ),\pi (x_{j+1},\tau ))\,d\tau - \int _{t}^{t+\varDelta t_c} \!\!\!\!\!\!\!\! f(S(x_{j},\tau ),\pi (x_{j},\tau ))\,d\tau \bigg ].\\ \end{array} \end{aligned}$$

(115)

In (115) the integrands involving $L_{j}(x,t)$ and $L_{j+1}(x,t)$ can be integrated exactly. Next, if the following Courant–Friedrichs–Lewy condition holds,

$$\begin{aligned} \frac{\varDelta t_{c}}{\varDelta x} \max _{x_j\le x\le x_{j+1}} \left\{ f^{\prime }(S(x,t),\pi (x,t)) \right\} <\frac{1}{2}, \end{aligned}$$

(116)

then we can use the midpoint rule on the RHS of (115), to yield the corrector step, where ($t=t^{n}\,$, $t+\varDelta t_c = t^{n+1}\,$ and $t+\frac{\varDelta t_c}{2} = t^{n+\frac{1}{2}}$),

$$\begin{aligned} {\begin{array}{l} S_{j+\frac{1}{2}}(t^{n+1}) = {\frac{1}{2}}[S_{j}(t^n)+ S_{j+1}(t^n)]+ {\frac{1}{8}}[S^{\prime }_{j}(t^n)- S^{\prime }_{j+1}(t^n)] \\ \\ \quad -\,\frac{\varDelta t_c}{\varDelta x}\left[ f\left( S(x_{j+1},t^{n+\frac{1}{2}}), \pi (x_{j+1},t^{n+\frac{1}{2}})\right) - f\left( S(x_j,t^{n+\frac{1}{2}}), \pi (x_j,t^{n+\frac{1}{2}})\right) \right] . \end{array}} \end{aligned}$$

(117)

We might use $S(x_j,t+{\varDelta t_c}/{2})=S_{j}(t)- \displaystyle \frac{\varDelta t_c}{\varDelta x} f^{\prime }_{j}(t)$, for the approximation of the saturation midpoint rule values of the numerical fluxes that appears in (117). Here, $\frac{1}{\varDelta x}f^{\prime }_j$ stands for an approximate numerical derivative of the numerical spatially varying flux function $f(S(x_j,t))$,

$$\begin{aligned} \frac{1}{\varDelta x}f^{\prime }_{j}(t)= \frac{\partial }{\partial x} \left[ f(S(x_j,t),\pi (x_{j},t))\right] +O(\varDelta x). \end{aligned}$$

(118)

Notice at this stage quantities S(x, t) and $\pi (x,t)$, involved in the computation of the discontinuous numerical flux, $\int _{t}^{t+\varDelta t_c}f(S(x_{j},\tau ),\pi (x_{j},\tau ))\,d\tau$, are all well defined away from the interfaces of adjacent generalized Riemann problems. The design of simple, robust and feasible scheme for flow in porous media to deal with nonsmooth flux functions is very challenging, in particular to the case of systems of equations [1,2,3, 8].