Solving the Nonstationary Richards Equation With Adaptive hp-FEM Pavel Solin1;2, Ondrej Certik1;3, Michal Kuraz4 1Department of Mathematics and Statistics, University of Nevada, Reno, USA 2Institute of Themomechanics, Prague, Czech Republic 3Institute of Physics, Academy of Sciences of the Czech Republic, Czech Republic 4Department of Water Resources and Environmental Modeling, Czech University of Life Sciences, Prague, Czech Republic Abstract This paper introduces briefly the adaptive hp-FEM method and examines its potential for the numerical solution of time-dependent variably saturated groundwater flow problems described by the Richards equation. For this purpose we use a benchmark for the nonstationary Richards equation that has a known exact solution given in terms of Fourier series, and an example of groundwater seepage into a dry lysimeter box with time-dependent boundary conditions. In the second part of the paper we discuss in more detail several advanced approximation techniques related to adaptive hp-FEM and introduce the open source adaptive hp-FEM library HERMES. Keywords: Richards equation, Adaptive hp-FEM 1. Introduction oscillation [9]. It should be noted that these oscillations are an attempt of the numerical method to preserve mass The problem of predicting flow in an unsatu- within large elements where the solution is not captured rated/saturated zone is important in many fields rang- with sufficient accuracy. ing from agriculture and hydrology to technical appli- One of the goals of this paper is to show that to cer- cations such as contaminant transport, disposal of dan- tain extent, these problems can be avoided by resolving gerous waste in deep rock formations, and others. areas with steep solution gradients with adaptive finite The Richards equation is obtained by combining the element methods, so that the underlying reason for the Darcy-Buckingham’s law with the law of conservation nonphysical oscillations ceases to exist. In particular, of mass. Its numerical solution is the subject of an we focus on the application of adaptive hp-FEM, a mod- intense research since the early 1970s, beginning with ern version of the finite element method that is capable Neuman’s finite element solution of unsteady dam seep- of extremely fast, exponential convergence [36]. age [20], Celia’s mass conservative modified Picard it- The outline of the paper is as follows: To make the eration [2] method in the early 1990s, and the recent paper self-contained and to establish notation, in Sec- Kees’s work with focus on stabilized finite element ap- tion 2 we introduce the Richards equation and relevant proximations [16, 18, 5]. Numerical methods for the constitutive relations. In Section 3 we introduce briefly Richards equation can be tested on several simplified adaptive hp-FEM and demonstrate its potential on a examples with known exact solutions such as [38, 39]. simple elliptic benchmark problem with known exact It is well known that a straightforward application solution. In Section 4, an analogous comparison with of the Galerkin approximation to the Richards equation low-order finite element methods is done for the time- yields poor results. In certain types of media, wetting dependent Richards equation, in the context of a bench- fronts generate steep gradients with a dominant con- mark problem with known exact solution. Section 5 is vective term [16]. Classical finite element approxima- dedicated to solving with adaptive hp-FEM a nonsta- tions are therefore often modified using low-order mass- tionary groundwater seepage into a dry lysimeter box, lumping and upwinding schemes to prevent nonphysical with time-dependent boundary conditions. In the sec- Preprint submitted to Advanced Water Resources August 2, 2010 ond part of the paper we describe in more detail some pressure, and n and m [-] are a pore-size distribution particular aspects on adaptive hp-FEM that are relevant parameters affecting the slope of the retention function. for computations performed in this paper, and we intro- The relation between the hydraulic conductivity and duce the open source project HERMES [12]. the negative pressure head, an unsaturated conductivity function, might be defined by the Mualem’s model [19]: 2. Richards Equation 8 2 0 Rθ 1 > B 1 dθ C > q B h(θ) C > θ B θr C The mathematical model of unsaturated flow was > Ks B C > θs B RθS C < B 1 C originally published in [25]. This formula, usually iden- K(h) = @ h(θ) dθ A (5) > θ tified as the mixed form of the Richards equation, states > r > 8θ 2 (θr; θS ) ) h 2 (−∞; 0); that > : Ks 8θ = θS ) h 2 h0; +1): ∂θ @K(θ) − r · (K(θ)rh) − = 0: (1) @t @z Here θr is the so called residual water content [-] – a bot- tom limit of water content for Darcian flow to occur, Ks Here θ is the water content of a porous material [-], h is is the saturated hydraulic conductivity (the maximum the pressure head [L], K(θ) is the unsaturated hydraulic value). − conductivity function [L:T 1], z denotes the vertical di- Due to the great complexity of the van Genuchten’s mension [L], assumed positive upwards, and the porous relations (4) and (5), an older exponential Gardner’s medium is assumed to be isotropic. Appropriate con- model [10] is sometimes used in works treating analyti- stitutive relationships among θ and h, K and h, are also cal aspects such as, e.g., [38, 39]. The Gardner’s model assumed. has the form This PDE can be generalized as a quasilinear elliptic- ( θ + (θ − θ )eαh: 8 h 2 (−∞; 0); parabolic differential equation and as a degenerate θ(h) = r s r (6) convection-diffusion problem. A proof of the existence θS ; 8 h 2 h0; +1); of a solution is given in [1]. This model is applicable to unsteady unsaturated and flow, thus for h 2 (−∞; 0i. An extension into saturated ( αh Kse ; 8 h 2 (−∞; 0); zone can be obtained by modifying the time derivation K(h) = (7) Ks; 8 h 2 h0; +1): term as considered in Neumann’s and Huyakorn’s works (see, e.g., [20, 21, 22, 13, 14]). The modification has the form 3. Adaptive hp-FEM ∂θ θ @h @K(θ) + S − r · (K(θ)rh) − = 0; (2) The finite element method (FEM) is the most widely @t θS @t @z used numerical technique to solve partial differential equations (PDE). By adaptive hp-FEM we mean a mod- where θS is the saturated water content [-] and S the specific storativity [L−1]. Such an approach is suitable, ern version fo FEM that combines adaptively finite ele- h e.g., for modeling an unsteady dam seepage. Equation ments of different spatial diameters ( ) and polynomial p (2) can also be assumed in the standard h-based form, degrees ( ) in order to maximize the convergence rate. For theory and technical details of higher-order FEM ! θ @h @K(h) and automatic hp-adaptivity we refer, e.g., to [36] and C(h) + S −r·(K(h)rh)− = 0: (3) θS @t @z many references therein. In the following let us present a simple example that illustrates the huge computational 2.1. Constitutive Relations potential of the method. A relation between the water content and the pressure We consider the Poisson equation −∆u = f with head is called retention curve. Its most commonly used zero Dirichlet boundary conditions, in a square domain 2 analytical formula is the van Genuchten’s equation [40]: Ω = (0; π) . Exact solution to this problem is chosen to be u(x; y) = sin(x) sin(y) and the right-hand side func- ( − θs θr + θ ; 8 h 2 (−∞; 0) tion f is calculated accordingly. Note that having the θ(h) = (1+(−αjhj)n)m r (4) θS ; 8 h 2 h0; +1): exact solution means that we are able to calculate the approximation error exactly, and thus compare fairly the Here α [L−1] is an empirical parameter whose inverse performance of different methods. The exact solution u is often referred to as the air entry value or bubbling is shown in Fig. 1. 2 Figure 1: Exact solution. The problem is solved using adaptive hp-FEM, adap- tive h-FEM with quadratic elements, and adaptive h- FEM with linear elements. To solve the problem we have implemented a simple C++ program that uses the open source adaptive hp-FEM library HERMES that is Figure 2: Convergence in terms of DOF. described in more detail in Section 7. Each computation starts from a one-element mesh and it is stopped when the number of unknowns in the discrete problem (DOF) exceeds 40000, or when the ap- proximation error drops below 10−5%. Due to these lim- itations, each method stops with a different number of DOF and accuracy, as shown in Table 1: Method DOF error [%] h-FEM (p = 1) 27469 0:3917379 h-FEM (p = 2) 39185 0:0022127 hp-FEM 49 0:0000427 Table 1: Performance comparison of the three methods. The table shows that adaptive hp-FEM attained an er- ror below 10−4% with 49 DOF, which is an accuracy that neither of the two remaining methods was able to Figure 3: Convergence in terms of CPU time. attain with much larger numbers of DOF. Fig. 2 presents a more complete convergence com- Final meshes for all three computations are shown in parison of the three methods, and it also shows that each Figs. 4 – 6. Note that the hp-FEM mesh (Fig. 6) only method attained its theoreticaly predicted asymptotic contains one element of polynomial degree 8. convergence rate – h-FEM with linear and quadratic ele- ments have on a log-log convergence graph slopes −1=2 and −1, respectively, and the convergence of hp-FEM is exponential.