# A Hyperbolic–Elliptic Model Problem for Coupled Surface–Subsurface Flow

- 671 Downloads
- 2 Citations

## Abstract

We consider a model problem for coupled surface–subsurface flow. The model consists of a nonlinear kinematic wave equation for the surface fluid’s height and a Brinkman model that governs fluid velocity and pressure for subsurface dynamics. For this coupled hyperbolic–elliptic model we establish the existence of weak solutions. The proof is based on a viscous approximation and the method of compensated compactness by virtue of appropriate energy estimates. To solve the coupled problem numerically, a finite volume method is applied. The numerical scheme is used to illustrate the influence of the Brinkman parameter on the coupled flow pattern for infiltration scenarios.

## Keywords

Free flow Porous medium Coupling Kinematic wave equation Brinkman’s equations Vanishing viscosity limit## 1 Introduction

Coupled surface and subsurface flows appear in a wide range of environmental settings such as infiltration of overland flows during rainfalls and the interactions of rivers, lakes or wetlands with the vadose zone. The coupled flow system contains different sets of entities: fluid on the surface, and fluid and solid in the porous medium region. This requires a separate model for each flow domain and accurate coupling of these models at the fluid–porous interface. The choice of the subdomain model depends on the application and the flow regime.

There are several possible models for the surface flow, ranging from the (compressible or incompressible) Navier–Stokes/Euler equations (Temam 2001) to source terms at the fluid–porous interface that represent, e.g., the rainfall rate and act as boundary conditions for the subsurface model (Berninger et al. 2014). For creeping flows, the advective inertial forces are small in comparison with the viscous forces, and therefore, the nonlinear inertial terms in the incompressible Navier–Stokes equations can be neglected leading to the Stokes equations. For many applications, such as dynamics of rivers and oceans, the horizontal length scale is much larger than the vertical one, and in this case, the surface equations can be averaged over the depth, leading from the incompressible Euler equations to the shallow water equations (Vreugdenhil 2010). The dynamic wave equation describes one-dimensional shallow water waves, and the kinematic wave equation is an approximation of the dynamic wave model (Takahashi 2014). In this work, we will consider the kinematic wave equation to model the surface flow.

To describe fluid flows through porous media, Darcy’s law (Bear 1972; Darcy 1856; Helmig 1997) is usually applied. For the description of flows in porous medium systems with high porosity, the Brinkman (1947) extension of Darcy’s law is typically used.

The governing equations describing physical processes in the free flow and porous medium flow domains have been widely investigated, but a challenge arises in describing the transition between the two flow systems at the fluid–porous interface. It can be a sharp interface void of thermodynamic properties (Discacciati et al. 2002; Layton et al. 2003; Mosthaf et al. 2011), or a transition region of a positive thickness which can store and transport mass, momentum, and energy (Jackson et al. 2012). Correct specification of coupling conditions at the fluid–porous interface is essential for a complete and accurate mathematical description of flow and transport processes in compositional systems.

To couple the Stokes equations and Darcy’s law that describe fluid flows in single-phase coupled systems, the Beavers–Joseph velocity jump condition (1967) is typically applied in addition to the mass conservation and the balance of normal forces across the fluid–porous interface. Mathematical models and numerical algorithms for solving such coupled flow problems have been developed and analyzed during the last decade (Discacciati et al. 2002; Discacciati and Quarteroni 2009; Layton et al. 2003; Rivière and Yotov 2005; Cao et al. 2010; Layton et al. 2013; Rybak and Magiera 2014). Many applications, such as overland flow interactions with unsaturated groundwater aquifers, require multiphase physics in the subsurface. In this case, the porous medium model typically includes multiphase Darcy’s law (Helmig 1997) that represents flows of several fluids or Richards’ equation (1931) that describes movement of only water through saturated/unsaturated porous media. Coupling of subsurface flows described by the Richards equation and overland flows has been studied recently (Dawson 2008; Rybak et al. 2015; Kollet and Maxwell 2006; Sulis et al. 2010; Sochala et al. 2009; Berninger et al. 2014; Mosthaf et al. 2011).

Despite the wide variety of models for coupled surface–subsurface flow, there are comparably less rigorous results on the existence, uniqueness and regularity of solutions. This applies in particular for surface flows which are governed by hyperbolic systems of balance laws (shallow water or kinematic wave approximation). In this case, one has to take into account that the surface model allows for discontinuous weak solutions.

In this paper, we consider as a first step the coupling of the scalar kinematic wave equation with Brinkman’s equations. To couple these models, a sharp interface approach is applied, and coupling conditions based on the conservation of mass and the balance of forces at the interface are formulated. The main result is Theorem 6 that establishes the existence of a weak solution of the kinematic–Brinkman system. For the proof, we follow the idea of viscous regularization which is classical in the theory of hyperbolic conservation laws. The final existence proof relies then on the vanishing viscosity limit which is mastered by the theory of compensated compactness (Murat 1981). Finally, we introduce a finite volume scheme for the coupled system and present several numerical studies to validate the overall approach and to test different flow regimes.

The paper is organized as follows. In Sect. 2, the flow system of interest and the mathematical models including the corresponding interface conditions are described. In Sect. 3, the coupled model with viscous regularization is presented, the existence of a weak solution for such problem is proved (Theorem 5) and a priori estimates are obtained. The existence of a weak solution for the nonlinear case is proved in Sect. 4. The computational algorithm and numerical simulation results for the proposed model as well as comparison study of the considered model instance with some simplified cases are presented in Sect. 5. Finally, a conclusion is given, and possible extensions of this work are discussed.

## 2 The Mathematical Model

### 2.1 Geometry for the Coupled Model

### 2.2 The Surface Model: Kinematic Wave Equation

The flow on the surface \({\varOmega }_{\mathrm {ff}}\) is described by the kinematic wave equation, which describes one-dimensional, hydrostatic water flow through an open channel, and can be seen as a further simplification of the shallow water equations.

*Derivation of the Kinematic Wave Equation* We assume that we have a one-dimensional flow of an incompressible fluid with constant density through an open channel with a fixed channel bed and a small bottom slope. The flow is assumed to vary gradually, such that hydrostatic pressure prevails and vertical acceleration can be neglected.

*h*denotes the relative water height,

*v*is the velocity, and \(z_b\) is the height of the bottom (with respect to a fixed datum). The gravitational acceleration is denoted by

*g*, the bottom stress by \(\varvec{\tau }_b\) and the density of the water by \(\rho \). A source term

*f*is included, which describes the water flow into and out of the ground. Next, we assume that the bottom stress \(\varvec{\tau }_b\) is proportional to the slope of hydraulic friction \(S_f\), i.e.,

*R*is proportional to the water height

*h*, that is \(R^{2/3} = C_{\mathrm {Ch}} \, h^{2/3}\), with \(C_{\mathrm {Ch}} >0\) depending on the channel. If we substitute the proportionality coefficient into Eq. (3), we can describe the velocity

*v*in dependence of

*h*, by

*Remark 1*

*v*, i.e., \(S_f = c_f v\) for \(c_f > 0\), then the kinematic wave equation becomes the linear transport equation

*Surface Model Equation*The kinematic wave Eq. (4) has (for \(f = 0\)) the form of a scalar conservation law. Hereafter, we ignore variations in the bottom topology and let the flow on the surface \({\varOmega }_{\mathrm {ff}} \cong \mathbb {R}\) be described by the initial value problem

**Assumption 1**

For ease of notation, we further assume that \(\mathbf {n} = (0,1)^\top \). In this case, the subsurface coordinates are given by \(\mathbf {x}= (x_1, x_2) \in {\varOmega }_{\mathrm {pm}}\), with \(x_1 \in \mathbb {R}\), \(x_2 \in [0,a]\), and the surface coordinates are defined as \(x = x_1 \in \mathbb {R}\). Other domains with a different normal vector \(\mathbf {n}\) can be considered by adapting \(\phi \) according to the slope.

### 2.3 Subsurface Model: Brinkman’s Equations

*M*is set to zero, one obtains the Stokes equations.

Brinkman’s equations are derived by considering the friction of the fluid on the particles, and sets of particles are combined to create a porous medium. Brinkman’s equations can describe fluid flow through a porous medium, which was done successfully in, for example, Iliev and Laptev (2004). Nonetheless, one has to keep in mind that, by the nature of the derivation, Brinkman’s equations yield reliable results only for porous media with a very high porosity, rigid swarms of particles or fibers in low concentration. There are different opinions about the porosity range in which Brinkman’s equations are applicable. Most commonly, it is assumed that Brinkman’s equations only hold for porous media with a porosity higher than 0.8 (or even 0.9) (Kim and Russel 1985; Lundgren 1972; Auriault 2009). However, for example in Martys et al. (1994) it is pointed out that they may be applicable in cases where the porosity goes down to 0.5.

It is possible to obtain Darcy’s law from Brinkman’s equations by taking the limit \(\mu \rightarrow 0\). However, this case is not covered by the analytical framework which is applied in this work, see Remark 5.

### 2.4 Coupling Conditions

*p*enters the subsurface model (7) as the Neumann boundary condition

### 2.5 The Coupled Kinematic–Brinkman Model

**Definition 1**

*weak solution of the Kinematic–Brinkman model*(11) iff

## 3 A Regularized Kinematic–Brinkman Model

In this section, we consider a regularization of the coupled model (11). The regularization relies on mollification operators and a viscosity approximation for the surface model. We show in this section the existence and uniqueness of solutions for the regularized model and derive important a priori estimates. These will allow us in Sect. 4 to prove the existence of a weak solution of the Kinematic–Brinkman model (11) by sending the regularization parameter to zero.

### 3.1 Description of the Regularization

**Definition 2**

*weak solution of the coupled model*(14) iff \(h^\varepsilon (\cdot ,0) = h^\varepsilon _0\) in \(\mathbb {R}\) and

*Remark 2*

*not*entropy solutions. The last choice might be even physically relevant because then (19) describes the dynamics of a thin film on a plane where concentration peaks might occur at the film’s head. This requires a nonentropic solution concept in the limit \(\varepsilon \rightarrow 0\). Up to our knowledge, such thin film dynamics on porous beds (subsurfaces) has not been mathematically analyzed yet.

### 3.2 A Priori Estimates for Weak Solutions of the Regularized Model (14)

For the coupled model (14) it is possible to prove the following a priori estimate. The result is essential to provide global solvability of (14) for fixed \(\varepsilon >0\) and to get an \(\varepsilon \)-independent estimate.

**Lemma 1**

*Proof*

*h*and Assumption 1 imply \(Q(h(\cdot ,t)) \in L^1(\mathbb {R}) \cap L^\infty (\mathbb {R})\) a.e. Next, we consider the subsurface model. In the weak formulation (17) of the subsurface problem, we set \(\mathbf {w}= \mathbf {v}^\varepsilon \in \mathrm {V}\) and \(q = p^\varepsilon \in \mathrm {P}\). This yields

### 3.3 Existence and Continuous-Dependence Estimates for Classical Solutions of the Uncoupled Kinematic Wave Equation

*Remark 3*

For some given function \(f\in L^2(0,T;L^2(\mathbb {R})) \) there exists a unique weak solution of (23) (see, e.g., Serre 1999, Theorem 6.2.5).

We now assume slightly more regularity on *f* and get the following result.

**Theorem 2**

*Proof*

The proof of Theorem 2 is based on the following Gronwall type inequality (from Dafermos 1979, Lemma 4.1).

**Lemma 2**

### 3.4 Existence and Regularity Estimates for Weak Solutions of the Uncoupled Brinkman System

*Remark 4*

Weak solutions satisfy the following regularity statement.

**Theorem 3**

*Proof*

A proof of the shift regularity can be done following the regularity proofs in Galdi (2011), Chapter IV, which extend to the unbounded domain \({\varOmega }_{\mathrm {pm}}\).\(\square \)

*Remark 5*

Because \(C_{\mathrm {Br}} = {\text {O}}(\mu ^{-1})\), the a priori estimate (39) no longer holds in the limit \(\mu \rightarrow 0\). Note that Darcy’s law results as the formal limit problem of Brinkman’s model for \(\mu \rightarrow 0\). Consequently, the subsequent analysis is not applicable when Darcy’s law is used as the subsurface flow model.

### 3.5 The Coupling Scheme

To verify the existence of a solution of the regularized fully coupled problem (14), we apply an alternating, iterative approximation method. That means, we solve the subsurface system and thereafter use the newly obtained solution as a source for the surface model, which then serves again as input for the subsurface equations.

*iterative solution*\(((\mathbf {v}^{\varepsilon ,i}, p^{\varepsilon ,i}), h^{\varepsilon ,i})\)

*for*(41), (42), \( i \in \mathbb {N}\), we mean that \(h^{\varepsilon ,i}\) is a weak solution of (41) and that \((v^{\varepsilon ,i}, p^{\varepsilon ,i})\) is a weak solution of (42), i.e., it holds

*Remark 6*

- (i)
The iteration is well posed in terms of weak solutions. Starting with \(h^{\varepsilon ,0}\equiv h^\varepsilon _0 \in L^2(0,T; H^1(\mathbb {R}))\) we get from Remark 4 and Theorem 3 with \(\mathbf {g}_N :=-{h}^{\varepsilon ,0} \, \mathbf {n} \) the unique existence of a weak solution \((\mathbf {v}^{\varepsilon ,1}, p^{\varepsilon ,1}) \in L^2(0,T; (H^2({\varOmega }_{\mathrm {pm}} ))^2) \times L^2(0,T; H^1({\varOmega }_{\mathrm {pm}}))\) of (41). In turn Remark 3 ensures the unique existence of a weak solution \(h^{\varepsilon ,1} \in L^2(0,T; H^1(\mathbb {R}))\) of (42) since the trace \(f :={\mathbf {v}}^{\varepsilon ,1} \cdot \mathbf {n}\) on \({\varGamma }_{\mathrm {N}}\) is even in \(L^2(0,T; H^1(\mathbb {R})) \). This closes the loop.

- (ii)To satisfy higher regularity of the solution suppose Assumption 1 is valid. Then we have \(h^{\varepsilon ,0} \equiv h^\varepsilon _0 \in L^2(0,T; H^2(\mathbb {R}))\) and trivially \( \partial _t h^{\varepsilon ,0} \in L^2(0,T; L^2(\mathbb {R}))\). We apply Theorem 3 for (38) with \(\mathbf {g}_{\mathrm {N}} :=-{h}^{\varepsilon ,0} \, \mathbf {n} \) and \(\partial _t \mathbf {g}_{\mathrm {N}} :=-\partial _t {h}^{\varepsilon ,0} \, \mathbf {n} \). This gives for the traces \( f :={\mathbf {v}}^{\varepsilon ,1} \cdot \mathbf {n}\in L^2(0,T;H^1(\mathbb {R}))\) and \(\partial _t f \in L^2(0,T;L^2(\mathbb {R}))\). We conclude with Theorem 2 that a weak solution \(h^{\varepsilon ,1} \in L^2(0,T; H^1(\mathbb {R}))\) of (42) exists which satisfiesThis is also a classical solution of (42).$$\begin{aligned} h^{\varepsilon ,1} \in L^2(0,T;H^4(\mathbb {R})) \cap C([0,T];H^3(\mathbb {R})), ~ \partial _t h^{\varepsilon ,1} \in L^2(0,T;H^2). \end{aligned}$$

Our aim is to show that the sequence of iterative solutions \( \{((\mathbf {v}^i, p^i), h^i)\}_{i \in \mathbb {N}}\), obtained by the coupling scheme (41), (42), converges for \(i \rightarrow \infty \) to a weak solution of (14).

### 3.6 Local A Priori Estimates for the Iterative Solutions

The existence proof relies on certain a priori estimates for the iterative solutions. Note that these estimates hold only locally in time. Throughout this section, we assume that Assumption 1 holds such that all statements of Remark 6 apply.

**Lemma 3**

*Proof*

In the next step we consider higher-order derivatives of \(h^{\varepsilon , i}\).

**Lemma 4**

*Proof*

The proof is given for the case \(k=1\). The cases \(k>1\) follow exactly along the same lines. For the estimate on the time derivative the latter is expressed by Eq. (42), which is satisfied classically, and then the estimate follows directly from the higher-order space estimates. In the proof we will consider classical derivatives of the unknowns \( \mathbf {v}^{\varepsilon ,i}, p^{\varepsilon , i}, h^{\varepsilon , i}\). These exist since the regularity of the iterative solutions hinges only on the regularity of the initial datum \(h^\varepsilon _0 \in C^\infty (\mathbb {R}) \cap H^l(\mathbb {R})\) for any \(l\in \mathbb {N}\) (see (15)). Arguing as in Remark 6(ii) gives the classical differentiability by Sobolev embedding.

*x*yields

### 3.7 Existence of a Weak Solution for the Regularized Coupled Model (14)

The aim of this section is to show that the coupled model (14) has a weak solution in the sense of Definition 2. To this end we will use the coupling scheme (41), (42) to construct a sequence which will converge to a weak solution of (14). First, we will consider the properties of the coupling scheme.

**Proposition 1**

*Proof*

*H*. Therefore we can conclude (see Theorem 2 for the definition of \(\theta \)) that there is a \(\bar{\theta }=\bar{\theta }(H) >0\) such that

By applying the contractive property of Proposition 1 we are now able to prove the existence of a solution of the coupled model (14) on a short time interval.

**Theorem 4**

*Proof*

*h*we see that \(((\mathbf {v},p), h)\) is a weak solution of (14).\(\square \)

Actually, the argument of Theorem 4 can be extended to the time interval [0, *T*].

**Theorem 5**

(Global existence) Let Assumption 1 hold. For each \(\varepsilon \in (0,1]\) there exists a weak solution \(((\mathbf {v}^\varepsilon , p^\varepsilon ), h^\varepsilon ) \in (\mathrm {V}\times \mathrm {P}) \times \mathrm {H}\) of the coupled model (14) in [0, *T*] which satisfies in particular the estimate from Lemma 1.

*Proof*

From Theorem 4 we have the existence of a weak solution \(((\mathbf {v}^\varepsilon , p^\varepsilon ), h^\varepsilon )\) in the time interval \([0,t_\mathrm {max}]\). The number \(\bar{\theta }(H)\) in the proof of Proposition 1 depends only on \(H= {||}{h_0}{||}_{L^2(\mathbb {R})}\) (and the fixed values of \(\varepsilon ,\mu ,M,T,\beta ,\alpha \)) such that the existence interval \([0,t_\mathrm {max}]\) depends only on *H*. As the limit of iterative solutions we can bound \({||}{h^\varepsilon (\cdot ,t_\mathrm {max})}{||}_{L^2(\mathbb {R})}\) only by \(1 +(1-\beta )^{-1} H\) due to Lemma 3. However, Lemma 1 gives the improved bound \({||}{h^\varepsilon (\cdot ,t_\mathrm {max})}{||}_{L^2(\mathbb {R})}\le H\) for a weak solution component \(h^\varepsilon \). As a consequence we can repeat all arguments of Proposition 1 and Theorem 4 to extend the weak solution to \([0,2t_\mathrm {max}]\). Iteratively we achieve the existence on [0, *T*].\(\square \)

## 4 Existence of a Weak Solution for the Kinematic–Brinkman Model

**Theorem 6**

(Existence) Let Assumption 1 hold and additionally let the measure of the set \(\{ s \in \mathbb {R}~ : ~ \phi ''(s) = 0 \}\) be zero.

To prove Theorem 6 we will rely on the a priori estimates from Lemma 1 and—what concerns the limit procedure for \(h^{\varepsilon }\)—on the Lemma of Murat (1981) in the \(L^p\)-framework (Schonbek 1982), and in particular we shall refer to the arguments used in Corli and Rohde (2012) and Lu (1989). Note that the additional regularity condition on \(\phi \) in Theorem 6 is needed in these papers.

**Lemma 5**

*Q*, such that

*Proof*

With the compactness result of Lemma 5 we finalize the proof of Theorem 6.

*Proof (of Theorem 6)*

*h*if we take into account the Lipschitz continuity of \(\phi \), the uniform \(L^2\)-boundedness of \(h^\varepsilon \), (15) and the weak convergence (59) in the linear boundary term. For the weak formulations for velocity \(\mathbf {v}\) and pressure

*p*we note again that the Brinkman part is linear. Thus the weak convergence as stated in (58) suffices to pass to the limit \(\varepsilon \rightarrow 0\) and to obtain (12)\(_{2}\). \(\square \)

## 5 Numerical Experiments

In this section, results from numerical simulations for the coupled surface–subsurface model are presented and discussed. First, a discretization for the coupled model is introduced. Next, we make a qualitative comparison of the model for different parameters and fluxes, and examine the behavior of the discrete coupling algorithm for some test cases.

### 5.1 Discretization

*Coupled Discretization*To combine the surface and the subsurface flow on a discrete level, we couple the models at each time-step \(t_n = n \, \Delta t\), \(n \in \mathbb {N}\), \(\Delta t > 0\), and, if necessary, iterate for each time-step \(K_{\mathrm {It}}\)-times between the flow domains. Thus, a general, discrete coupling algorithm can be schematically written in the following form:

In spite of the robust performance of the numerical method for the presented experiments, the convergence analysis remains an open issue. In principle, the sequence of numerical approximations can be understood as the discrete analogon of the viscous approximations from Sect. 3. An \(L^2\)-contraction estimate for finite volume schemes has been proven in Jovanović and Rohde (2006), and the use of compensated compactness methods (Murat 1981) for numerical approximations is well established. This would be the basis to transfer the analysis of the continuous model to the discrete case.

Another interesting issue concerns the choice of the iteration number \(K_{\mathrm {It}}\) in Algorithm 1, which should be chosen such that the coupling error is of the same order as the discretization error. We are not aware of any rigorous analysis that attempts to answer this question (even for simplified situations). Our specific choice of \(K_{\mathrm {It}}\) in the numerical experiments is in fact just an ad-hoc decision. However, in Sect. 5.3 we investigate the influence of \(K_{\mathrm {It}}\) numerically for a test case.

### 5.2 Qualitative Comparison of the Models

In this section we qualitatively investigate the effects of choosing different surface and subsurface models. We make numerical computations using a finite volume discretization and Algorithm 1, and compare the results. To this end, we consider the coupled model (11). It is possible to achieve different models by setting the flux \(\phi \) and \(M \ge 0\). If we have \(\phi (h) = Vh\), as in Remark 1, we get the linear transport equation as the surface model with a velocity field \(V:{\varOmega }_{\mathrm {ff}} \rightarrow \mathbb {R}\). For \(\phi (h) = V{|}{h}{|}^{5/3}\) we obtain the kinematic wave equation as the surface model (Sect. 2.2). Likewise, if we set \(M = 0\) the Stokes equations are considered as the subsurface model, and in case \(M > 0\) one considers Brinkman’s equations.

*The Linear Model*Figure 3 shows the numerical solution at \(t = 80\) of the coupled model consisting of the linear transport equation and Stokes equations. It can be seen that the fluid on the surface flows through the subsurface from places with a higher water height to places with a lower water height. In comparison, it can be observed in Fig. 4 that the flow through a subsurface which is modeled by Brinkman’s equations is much slower. Evidently, this is due to the lower permeability of the porous medium, since the parameter

*M*plays the role of the inverse permeability.

*The Nonlinear Model*The same behavior can be seen, if instead of the transport equation, the kinematic wave equation is considered—see Figs. 5 and 6. In contrast to the coupled models with the transport equation, we observe a much steeper slope of the water front. This is due to the strict convexity of the flux function \(\phi (h) = V{|}{h}{|}^{5/3}\) in the kinematic wave equation.

*M*on the surface flow, modeled by the kinematic wave equation, is examined. The numerical experiments indicate that the coupling with Brinkman’s equations has a certain smoothing effect on the fluid height at the interface, which becomes more distinctive for lower

*M*, i.e., higher permeability of the subsurface. In that case, more water moves through the subsurface to form an equilibrium, driven by pressure differences coming from the surface. However, outside of the interface the smoothing effect vanishes, which is why new shock waves are formed. Nonetheless, the flow exchange with the subsurface seems to reduce the occurrence of discontinuities in the fluid height at the interface.

### 5.3 The Influence of the Iteration Number

*T*, i.e., the difference

### 5.4 Examination of the Mass Conservation

For the numerical computations, we consider the domains \({\varOmega }_{\mathrm {ff}} = [0,10]\) and \({\varOmega }_{\mathrm {pm}} = [0,10] \times [0,5]\), discretized with mesh width \(\Delta x = \Delta x_1 = \Delta x_2 = 0.05\). The time-step size is \(\Delta t = 0.25\), and the iteration number is \(K_{\mathrm {It}} = 1\). For convenience we set \(V= 0\). We consider the initial data \(h_0(x) = \chi _{x \in [4.5, 5.5]}(x)\) and consequently have the initial mass \(\Sigma (0) = 0.1\).

## 6 Conclusion

In the paper, a coupled surface–subsurface model is considered. The subsurface is modeled by Brinkman’s equations in a two-dimensional unbounded domain. For the surface model a one-dimensional scalar conservation law, as a generalization of the kinematic wave equation, is applied. For this surface–subsurface model the existence of a weak solution is proven. This is done by introducing a regularized coupled model, and, via a fixed-point argument, proving the existence of a weak solution for it. Then, by applying the method of compensated compactness it is possible to show the existence for the original coupled model without regularization. In Sect. 5, an alternating, iterative numerical algorithm for the coupled model is presented and tested for different models.

Concerning the model, there are still some extensions possible. First, higher space dimensions should be considered, and instead of the steady-state Brinkman’s equations, a time-dependent subsurface model could be studied. Another issue of the model is that it cannot be expected that the water height *h* stays nonnegative. The model would have to be adapted to have an admissible solution set \(h \ge 0\), for example as it was done in Sochala et al. (2009), for a model that couples the kinematic wave equation and Richards’ equation. Finally, nontrivial bottom topology should be regarded, which would be a further step toward a realistic coupled surface–subsurface model.

## References

- Auriault, J.L.: On the domain of validity of Brinkman’s equation. Transp. Porous Media
**79**(2), 215–223 (2009)CrossRefGoogle Scholar - Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)Google Scholar
- Beavers, G.S., Joseph, D.D.: Boundary conditions at a naturally permeable wall. J. Fluid Mech.
**30**, 197–207 (1967)CrossRefGoogle Scholar - Berninger, H., Ohlberger, M., Sander, O., Smetana, K.: Unsaturated subsurface flow with surface water and nonlinear in- and outflow conditions. Math. Models Methods Appl. Sci.
**24**, 901–936 (2014)CrossRefGoogle Scholar - Bertozzi, A., Münch, A., Shearer, M.: Undercompressive shocks in thin film flows. Phys. D
**134**(4), 431–464 (1999)CrossRefGoogle Scholar - Brinkman, H.C.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res.
**1**, 27–34 (1947)CrossRefGoogle Scholar - Cao, Y., Gunzburger, M., Hua, F., Wang, X.: Coupled Stokes–Darcy model with Beavers–Joseph interface boundary condition. Commun. Math. Sci.
**8**(1), 1–25 (2010)CrossRefGoogle Scholar - Corli, A., Rohde, C.: Singular limits for a parabolic-elliptic regularization of scalar conservation laws. J. Differ. Equ.
**253**(5), 1399–1421 (2012)CrossRefGoogle Scholar - Dafermos, C.M.: The second law of thermodynamics and stability. Arch. Ration. Mech. Anal.
**70**(2), 167–179 (1979)CrossRefGoogle Scholar - Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 3rd edn. Springer, Berlin (2010)CrossRefGoogle Scholar
- Darcy, H.: Les Fontaines Publiques de la Ville de Dijon. Dalmont, Paris (1856)Google Scholar
- Dawson, C.: A continuous/discontinuous Galerkin framework for modeling coupled subsurface and surface water flow. Comput. Geosci.
**12**, 451–472 (2008)CrossRefGoogle Scholar - Discacciati, M., Miglio, E., Quarteroni, A.: Mathematical and numerical models for coupling surface and groundwater flows. Appl. Numer. Math.
**43**, 57–74 (2002)CrossRefGoogle Scholar - Discacciati, M., Quarteroni, A.: Navier–Stokes/Darcy coupling: modeling, analysis, and numerical approximation. Rev. Mat. Complut.
**22**(2), 315–426 (2009)CrossRefGoogle Scholar - Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence (2010)Google Scholar
- Galdi, G.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Springer, Berlin (2011)CrossRefGoogle Scholar
- Helmig, R.: Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems. Springer, Berlin (1997)CrossRefGoogle Scholar
- Iliev, O., Laptev, V.: On numerical simulation of flow through oil filters. Comput. Vis. Sci.
**6**(2–3), 139–146 (2004)CrossRefGoogle Scholar - Jackson, A.S., Rybak, I., Helmig, R., Gray, W.G., Miller, C.T.: Thermodynamically constrained averaging theory approach for modeling flow and transport phenomena in porous medium systems: 9. Transition region models. Adv. Water Res.
**42**, 71–90 (2012)CrossRefGoogle Scholar - Jovanović, V., Rohde, C.: Error estimates for finite volume approximations of classical solutions for nonlinear systems of hyperbolic balance laws. SIAM J. Numer. Anal.
**43**(6), 2423–2449 (2006)CrossRefGoogle Scholar - Kim, S., Russel, W.B.: Modelling of porous media by renormalization of the Stokes equations. J. Fluid Mech.
**154**, 269–286 (1985)CrossRefGoogle Scholar - Kollet, S.J., Maxwell, R.M.: Integrated surface-groundwater flow modeling: a free-surface overland flow boundary condition in a parallel groundwater flow model. Adv. Water Res.
**29**, 945–958 (2006)CrossRefGoogle Scholar - Layton, W., Schieweck, F., Yotov, I.: Coupling fluid flow with porous media flow. SIAM J. Numer. Anal.
**40**, 2195–2218 (2003)CrossRefGoogle Scholar - Layton, W., Tran, H., Trenchea, C.: Analysis of long time stability and errors of two partitioned methods for uncoupling evolutionary groundwater - surface water flows. SIAM J. Numer. Anal.
**51**, 248–272 (2013)CrossRefGoogle Scholar - LeFloch, P., Rohde, C.: High-order schemes, entropy inequalities, and nonclassical shocks. SIAM J. Numer. Anal.
**37**(6), 2023–2060 (2000)CrossRefGoogle Scholar - Lu, Y.G.: Convergence of solutions to nonlinear dispersive equations without convexity conditions. Appl. Anal.
**31**(4), 239–246 (1989)CrossRefGoogle Scholar - Lundgren, T.S.: Slow flow through stationary random beds and suspensions of spheres. J. Fluid Mech.
**51**, 273–299 (1972)CrossRefGoogle Scholar - Manning, R.: On the flow of water in open channels and pipes. Trans. Inst. Civ. Eng. Irel.
**20**, 161–207 (1891)Google Scholar - Martys, N., Bentz, D.P., Garboczi, E.J.: Computer simulation study of the effective viscosity in Brinkman’s equation. Phys. Fluids (1994-present)
**6**(4), 1434–1439 (1994)CrossRefGoogle Scholar - Mosthaf, K., Baber, K., Flemisch, B., Helmig, R., Leijnse, A., Rybak, I., Wohlmuth, B.: A coupling concept for two-phase compositional porous-medium and single-phase compositional free flow. Water Resour. Res.
**47**, W10,522 (2011)CrossRefGoogle Scholar - Murat, F.: L’injection du cône positif de \(H^1\) dans \(W^{1,q}\) est compacte pour tout \(q<2\). J. Math. Pures Appl. (9)
**60**(3), 309–322 (1981)Google Scholar - Richards, L.A.: Capillary conduction of liquids through porous mediums. J. Appl. Phys.
**1**, 318–333 (1931)Google Scholar - Rivière, B., Yotov, I.: Locally conservative coupling of Stokes and Darcy flow. SIAM J. Numer. Anal.
**42**, 1959–1977 (2005)CrossRefGoogle Scholar - Rybak, I., Magiera, J., Helmig, R., Rohde, C.: Multirate time integration for coupled saturated/unsaturated porous medium and free flow systems. Comput. Geosci.
**19**(2), 299–309 (2015)Google Scholar - Rybak, I., Magiera, J.: A multiple-time-step technique for coupled free flow and porous medium systems. J. Comput. Phys.
**272**, 327–342 (2014)CrossRefGoogle Scholar - Schonbek, M.E.: Convergence of solutions to nonlinear dispersive equations. Commun. Partial Differ. Equ.
**7**(8), 959–1000 (1982)CrossRefGoogle Scholar - Serre, D.: Systems of Conservation Laws 1: Hyperbolicity, Entropies. Cambridge University Press, Shock Waves (1999)CrossRefGoogle Scholar
- Sochala, P., Ern, A., Piperno, S.: Mass conservative BDF-discontinuous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows. Comput. Methods Appl. Mech. Eng.
**198**, 2122–2136 (2009)CrossRefGoogle Scholar - Sulis, M., Meyerhoff, S.B., Paniconi, C., Maxwell, R.M., Putti, M., Kollet, S.J.: A comparison of two physics-based numerical models for simulating surface water-groundwater interactions. Adv. Water Res.
**33**, 456–467 (2010)CrossRefGoogle Scholar - Takahashi, T.: Debris Flow: Mechanics, Prediction and Countermeasures. CRC Press, Boca Raton (2014)CrossRefGoogle Scholar
- Temam, R.: Navier–Stokes Equations: Theory and Numerical Analysis. AMS (2001)Google Scholar
- van Duijn, C.J., Peletier, L.A., Pop, I.S.: A new class of entropy solutions of the buckleyleverett equation. SIAM J. Math. Anal.
**39**(2), 507–536 (2007)Google Scholar - Vreugdenhil, C.B.: Numerical Methods for Shallow-Water Flow. Kluwer, Dordrecht (2010)Google Scholar
- Yen, B.: Open channel flow resistance. J. Hydraul. Eng.
**128**(1), 20–39 (2002)CrossRefGoogle Scholar