The Use of for Integrated Hydrological Modeling Ting Fong May Chui *, David L. Freyerg Department of Civil and Environmental Engineering, Stanford University *Corresponding author: 380 Panama Mall, Terman Building, Room M-13, Stanford, CA 94305-4020, maychui@stanford.edu Astract: Hydrological processes and components are intrinsically coupled, and thus often must e modeled as an integrated system. Unfortunately, although a few modeling codes are availale, integrated hydrological modeling remains a challenge. The ojective of this paper is to explore the feasiility of using Multiphysics for integrated hydrological modeling. In particular, using the generic oundary conditions and Richards Equation in the Earth Science Module, we set up two important oundary conditions: rainfall infiltration and seepage faces, and implement them in example simulations. Given s versatility and user-friendly modeling environment, shows promise as a tool for integrated hydrological modeling. Keywords: Hydrology, Groundwater, Rainfall, Evapotranspiration, Face 1. Introduction Hydrological processes (e.g., rainfall and evapotranspiration) and components (e.g., lakes and groundwater) are intrinsically coupled, and thus often must e conceptualized as an integrated system. However, ecause of the complexity of their interactions they have traditionally een studied independently or with only loose coupling. As our understanding of and concern for water resources increase, it is ecoming evident that in many contexts hydrologic processes and components need to e modeled together as an integrated system (Winter, et al., 1998). Fortunately, with the dramatic growth in computing power, it is ecoming more feasile to analyze and predict hydrologic systems through truly integrated modeling. Because of their complexity, there are only a few integrated hydrological modeling codes currently availale. One example is MODHMS, proprietary software developed y HydroGeoLogic, Inc. (HydroGeoLogic, Inc., 1996). MODHMS is ased on a popular U.S. Geological Survey groundwater modeling code, MODFLOW (HydroGeoLogic, Inc., 1996), ut has additional capailities, such as varialysaturated susurface conditions and surface flow modeling. Another example is HydroGeoSphere, which was developed y the Groundwater Simulations Group at the University of Waterloo and HydroGeoLogic, Inc. (Therrien, et al., 2006). HydroGeoSphere is very comprehensive, covering various surface and susurface flows and transport modeling. In this paper we explore the feasiility of using Multiphysics ( AB, 2005a and 2005) for integrated hydrological modeling. provides generic partial differential equation (PDE) solvers that are roust in handling coupled equations. In addition, its unified graphical modeling environment is oth flexile and very easy to use. Therefore, appears to have the potential to e oth a versatile and an effective integrated hydrological simulator. has developed several modules helpful for hydrologic applications, e.g., the Earth Science Module ( AB, 2005a), ut implementations of oundary conditions important for integrated hydrological modeling are not availale. Also, no module has yet een uilt particularly for surface-water modeling. Therefore, in this paper, we examine the feasiility of extending to integrated hydrological modeling through the use of generic oundary conditions and PDE solvers. 2. Methods and Results We ase all of our studies on the Richards Equation solver within the Earth Science Module for varialy-saturated susurface flow modeling (conservation of mass comined with Darcy s Law). We examine the implementation of two oundary conditions important for integrated hydrological modeling: rainfall infiltration and seepage faces through one- and two-dimensional, steady and transient simulations.
2.1 Rainfall Infiltration Boundary Condition To create a oundary condition to simulate infiltration from rainfall, we implement an algorithm similar to that in another popular varialy-saturated flow model, VS2D (Lappala, et al., 1983). The oundary condition is set as a specified flux at the rainfall rate at the start of the rainfall. If the calculated pressure head at the oundary ecomes greater than zero gage (atmospheric), meaning that the soil surface has saturated, we switch the oundary condition to a specified pressure head of zero. This assumes that all excess water on the surface runs off immediately and there is no ponding (i.e., incipient ponding). If the calculated flux crossing the specified pressure oundary exceeds the rainfall rate, we switch the oundary condition ack to a Neumann one at the rainfall rate. The rainfall infiltration oundary condition thus involves switching etween Dirichlet and Neumann oundary conditions depending on the solution itself. The timing and the location of these switches are not known eforehand and typically have to e implemented using an iterative technique (e.g., Mein and Larson, 1973; Lappala, et al., 1983; Hsu, et al., 2002). An algorithm governing such switches is not explicitly availale in. However, we use s general mixed (Cauchy) oundary condition, together with conditional statements, to identify the switches etween the two oundary conditions. defines a general mixed oundary condition as ( AB, 2005a): ( p + ρ gz) = N + R ( H H ) κ s n k r f 0 η n normal to the oundary κ s intrinsic permeaility [m 2 ] η fluid viscosity [Pa s] k r relative permeaility [-] p pressure [Pa] ρ f fluid density [kg/m 3 ] g gravitational acceleration [m/s 2 ] z vertical coordinate [m] N 0 non-head dependent flux [m/s] R external resistance [s -1 ] (1) p H = z + external total head [m] ρ f g z external elevation [m] p external pressure [Pa] H p = z + total head [m] g ρ f The aove general mixed (Cauchy) oundary condition reduces to a Neumann condition if R = 0 and reduces to a Dirichlet condition if R is infinite (very large) (Forsyth, 1988). In other words, we can use the mixed oundary condition, along with conditional statements, to represent either Neumann or Dirichlet conditions through the specification of R. In short, we use the following parameters for the two parts of the oundary condition: p = 0 z = z (2a-) For p < 0 or R ( H H ) N, N = N 0. Otherwise, N 0 = 0. (2c) For p 0 and R ( H H ) N, R = < large numer. Otherwise, R = 0. (2d) The switching etween the Neumann and the Dirichlet condition is conceptually a step function. The oundary condition switches at the moment when the pressure at the land surface is precisely zero. However, to minimize the numer of switches during iterative solution and to aid convergence when pressures are near zero, we smooth the switching etween the two oundary conditions. In other words, instead of performing a complete switch when pressure is right at zero, we turn on/off the oundary conditions gradually over a small range of pressure. More specifically, we turn on/off the specified flux and the resistance, R, gradually with a multiplier, as shown in Figure 1. With less smoothing (i.e., the oundary conditions are turned on/off within a smaller range of pressure), the switch is closer to the ideal discontinuous step. However, it may also ecome more difficult to solve. Therefore, there is a trade-off etween accuracy and computational challenges
Multiplier and the optimal level of smoothening varies case y case. 1 0.8 0.6 0.4 0.2 Specified Head Resistance Specified Flux sets of results, which are likely ecause of the fitting of the van Genuchten model to the soil moisture retention and relative permeaility curves used y Mein and Larson, and some distortion in the scanned image from the original paper. 0-1.5-1 -0.5 0 0.5 Pressure Head ( x -6 m) Figure 1. Example multiplier smoothening oundary condition switches. Smoothed Heaviside Functions with continuous second derivatives generated y ( AB, 2005). To demonstrate the rainfall infiltration oundary condition implementation descried aove, we reproduce some of the simulations from Mein and Larson (1973). Mein and Larson computed numerical solutions of a Richards Equation model of infiltration under a constant intensity rainfall into a homogeneous soil with uniform initial moisture content in order to test a simplified two-stage model. The medium is a sandy loam with a saturated hydraulic conductivity, K sat, of 1.39 x -3 cm/s and a porosity of 0.518. We visually fit a van Genuchten model to the soil moisture retention and relative permeaility curves provided y Mein and Larson to otain a residual moisture content of 0.119, an α of 0.0268 cm -1 and an n of 4.36. Mein and Larson do not specify the depth of the simulated soil column. We chose a depth of 2 m, which is deep enough not to interact with the wetting front (i.e., any significant increase in moisture content ecause of the rainfall event) y the end of the simulation period of 24 min. The rainfall rates applied to the top of the column are uniform and always greater than the saturated hydraulic conductivity. Once the surface of the column reaches saturation, the top of the column remains at incipient ponding (zero ponding depth). Figures 2 compare our infiltration curves with the numerical results of Mein and Larson. The infiltration curves from agree very well with those from Mein and Larson. We can oserve minor disparities etween the two Figure 2. Infiltration curves from Mein and Larson (1973) (lack solid lines) and (red dotted lines). Results of different rainfall rates (+ = 4 K sat, o = 6 K sat and = 8 K sat ). The initial moisture content is 0.125 for all three cases. 2.2 Face Simulation A seepage face develops when a discharging water tale intersects with a land surface (Figure 3). The pressure along a seepage face is atmospheric and is represented mathematically as a Dirichlet oundary with a pressure head of zero or a total head of its own elevation. The oundary condition along the ground surface aove the upper limit of the seepage face is typically a Neumann condition, e.g., no flow, specified evaporative or infiltration flux. It is challenging to model a seepage face, as its upper limit (the star in Figure 3) is usually unknown a priori. Traditionally, the extent of the seepage face has een determined y iterative approaches. A seepage face oundary condition with an iterative searching algorithm is not an explicit option in. However, adopting a similar approach to that used for the rainfall infiltration oundary condition, we implement a mixed oundary condition to split a oundary into a Dirichlet portion for the seepage face and a Neumann portion for the region aove the
seepage face. We accomplish this through the specification of R along with conditional statements. h = h 1 Figure 3. Example prolem with a seepage face, showing oundary conditions around the domain. The red star indicates the upper limit of the seepage face. More specifically, we use the mixed oundary condition, as stated in Equation (1), with the parameters shown elow on a oundary along which a seepage face could potentially form. p = 0 z = z (3a-) For infiltration or evapotranspiration along and aove the seepage face, N 0 = N. Otherwise, N 0 = 0. (3c) For p 0 and R ( H H ) < 0, R = large numer. Otherwise, R = 0. (3d) The non-head dependent flux, N 0, is zero if there is no infiltration and evapotranspiration along and aove the seepage face. If pressure is negative or if a zero pressure results in an inward flux, R is also set to e zero giving a Neumann oundary condition with a specified flux at N. Otherwise, R is set to e a large numer so that the resulting head-dependent flux gives a total head very close to the specified external total head (i.e., h h as R ). Vadose zone Face p = 0 h = h 2 < h 1 2.2.1 Two-Dimensional Steady State Simulations from Clement, et al. (1996) As mentioned previously, the actual oundary condition along a seepage face should Mixed Boundary Condition in e p = 0 (h = z). Therefore, the idea is to make the head-dependent flux oundary as close to a Dirichlet oundary of p = 0 (h = z) as possile. We can achieve this y setting R to e as large as possile. However, it ecomes more difficult to converge when R is large. Therefore, there is a trade-off etween accuracy and convergence. We express R in terms of the saturated hydraulic conductivity and a coupling length scale (Equation 4), parameters that are common in the context of groundwater modeling. K sat R L (4) K sat saturated hydraulic conductivity [m/s] L coupling length scale [m] We then perform a sensitivity analysis for the choice of coupling length scale using the simulations in Clement, et al. (1996) as a test case. Clement, et al. simulate steady-state flow through square domains of various dimensions, as shown in Figure 3. The top and the ase of the domains are no-flow oundaries. The water levels on the left and on the right are at the top of the domain and at a specified tailwater level, respectively. The saturated hydraulic conductivity is 1.0 m/d and the van Genuchten parameters are α = 0.64 m -1, n = 4.65. To locate the seepage face, Clement, et al. use an iterativesearch procedure proposed y Neuman (1973), later modified y Cooley (1983). The prolem is first solved with an initial guess of the seepage face location. Pressure is set to e zero along the seepage face and the flux is set to e zero aove it. Calculations are performed with these oundary conditions and the results are checked for consistency with the assumed conditions. If there is an inward flow into the domain at a node specified with a zero pressure, the node will then e changed to a zero-flux one in the next iteration. If a positive pressure is found at a node specified with a zero flux, the node will then e changed to a zero-pressure one for the next iteration. The nodes are adjusted one y one from the lower end of the seepage face until convergence occurs for each of the nodes. Tale 1 shows the computed results and Figure 4 shows the pressure along and aove the seepage face for a 1 m x 1 m model.
Tale 1: Results for 1 m x 1 m seepage face simulations with different coupling length scales. Tailwater level = 0.2 m. Yellow row contains results from L = 1 which is the coupling length scale used for susequent simulations. L (m) Inflow Total Below Tailwater Elevation Face Length (m) Clement, et al. NA 0.55 NA NA 0.32 00 0.521 0.515 0.181 0.334 0.389 0.545 0.544 0.269 0.275 0.319 1 0.545 0.545 0.273 0.272 0.317 0.1 0.545 0.545 0.272 0.273 0.317 0.001 0.545 0.545 0.272 0.273 0.317 Elevation (m) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 L = 00 m L = m L = 1 m L = 0.1 m L = 0.001 m 0.2-0.4-0.3-0.2-0.1 0.0 0.1 Pressure Head (m) Figure 4. Pressure heads along the downstream oundary. Domain: 1 m x 1 m. All curves except L = 00 m plot on top of each other. From Figure 4, we see that the pressure along the seepage face with L = 00 m is quite far from zero, while all the other plots are nearly identical with p = 0 along the seepage face. Given the convergence challenges for smaller Ls (e.g., L = 0.1 and 0.001 m) and the reasonale accuracy achieved with L = 1 m in this sensitivity analysis, we set L to e 1m for all of the susequent simulations. Tale 2 compares computed results, oth discharges and seepage face lengths, for all of the domains used in Clement, et al. (1996). The results all used L = 1 m. Tale 2: Comparisons of the two-dimensional steadystate seepage face simulation results from Clement, et al. (1996) and. Prolem Tailwater (m x m) Level (m) Clement Discharge Discharge Clement Face (m) Face (m) 1 x 1 0.2 0.55 0.54 0.32 0.32 x 2 5.25 5.24 2.8 2.8 25 x 25 5 12.56 12.57 6.0 6.2 50 x 50 24.56 24.62 11.0 11.3 For prolems with dimensions of 1 m x 1 m and m x m, the discharges and the seepage face lengths from Clement, et al. and from are almost the same. However, as the domain ecomes larger, the two sets of results start to deviate. The differences in discharges for prolems with dimensions of 25 m x 25 m and 50 m x 50 m are oservale, ut very small. However, the seepage face lengths from are clearly larger than those from Clement, et al., y around 3 % for prolems with dimensions of 25 m x 25 m and 50 m x 50 m. These differences are acceptale in most contexts ut certainly are larger than those in smaller domains. Overall, oth sets of results match reasonaly well, and we can confirm the mixed oundary condition approach for twodimensional steady-state simulations. 2.2.2 Two-Dimensional Transient Simulation from Wise, et al. (1994) The transient simulation in Wise, et al. (1994) is very similar to the steady-state simulations in Clement, et al. (1996). The square domain is m x m with no-flow oundaries along the top and the ase. The initial total head throughout the domain is at m, i.e., water tale is at the top of the domain. Wise, et al. keep the water tale on the left at the top of the domain, while they lower the head on the right oundary instantaneously at t = 0 to an elevation of 3 m. The homogeneous and isotropic saturated hydraulic conductivity is 5.1 m/d and the specific storage is 5x -5 m -1. The residual moisture content is 0.01 and the porosity is 0.46. The van Genuchten parameters are α = 2.0 m -1 and n = 2.8. We use a coupling length scale, L, of 1 m. Figures 5 and 6 compare the
Elevation (m) Elevation (m) Wise, et al. and water tales and flows at several different times. 8 6 4 2 8 6 4 2 Wise, et al. (1994) Steady State t = 0.0002 d 0 2 4 6 8 x (m) t = 1.0 d 0 2 4 6 8 x (m) t = 0.02 d t = 0.14 d Figure 5. Water tales at different times from Wise, et al. (1994) and. Water tale at t = 1.0 d was plotted for instead of the steady-state. Comparing the solutions shown in the aove figures, we find reasonale agreement etween the two solutions. Close inspection reveals that the water tales at later times from are generally slightly higher than those in Wise, et al, which implies the seepage faces from are generally slightly longer than those in Wise, et al. t = 0.60 d t = 0.0002 d t = 0.02 d t = 0.14 d t = 0.60 d In Figure 6, the inflow, outflow, and pore drainage from the -ased model are all lower than those from Wise, et al. Given that the seepage faces are slightly longer and the system discharges are smaller in, reducing the coupling length scale in might produce etter matches etween the two solutions. However, oth sets of results match reasonaly well overall and the seepage face oundary condition formulation developed for use with appears successful when applied in two-dimensional transient simulations. Flow Rate (m 3 /d/m) Flow Rate (m 3 /d/m) 50 40 30 20 50 40 30 20 Wise, et al. (1994) Inflow Pore Drainage 0.0 0.2 0.4 0.6 0.8 1.0 Time (d) Inflow Pore Drainage 0.0 0.2 0.4 0.6 0.8 1.0 Time (d) Figure 6. Comparisons of flows at different times from Wise, et al. (1994) and.
3. Conclusions In this paper, we explore the use of for integrated hydrological modeling. With the generic oundary condition and Richards Equation in the Earth Science Module, we set up two important hydrologic oundary conditions: rainfall infiltration and seepage faces, and implement them in example simulations. Although not included here, we have also estalished an evapotranspiration oundary and performed simulations of groundwater-lake interactions y coupling the Richards Equation with an ordinary differential equation representing the lake mass alance. Given s versatility and user-friendly modeling environment, we elieve that can also model other areas of integrated hydrological modeling (e.g. stream and overland flow modeling) with reasonale effort. In a nutshell, shows promise as a tool for integrated hydrological modeling. 4. References 1. Clement, T. P., W. R. Wise, F. J. Molz, and M. Wen, "A comparison of modeling approaches for steady-state unconfined flow", Journal of Hydrology (Amsterdam), 181(1-4), 89-209 (1996) 2. AB, Multiphysics Earth Science Module User s Guide (Version 3.2), 120p (2005a) 3. AB, Multiphysics Modeling Guide (Version 3.2), 336p (2005) 4. Cooley, R. L., "Some New Procedures for Numerical Solution of Varialy Saturated Flow Prolems", Water Resources Research, 19(5), 1271-1285 (1983) 5. Forsyth, P. A., Comparison of the singlephase and two-phase numerical model formulation for saturated-unsaturated groundwater flow, Computer Methods in Applied Mechanics and Engineering, 69 (2), 243-259 (1988). 6. Hsu, S. M., C. F. Ni, and P. F. Hung, "Assessment of three infiltration formulas ased on model fitting on Richards equation", Journal of Hydrologic Engineering, 7(5), 373-379 (2002) 7. HydroGeoLogic, Inc., MODHMS Software (Version 2.0) Documentation, 426p, Herndon, VA (1996) 8. Lappala, E. G., R. W. Healy, and E.P. Weeks, Documentation of computer program VS2D to solve the equations of fluid flow in varialy saturated porous media, 184p, U.S. Geological Survey Water-Resources Investigation Report 83-4099, Denver, CO (1987) 9. Mein, R. G. and C. L. Larson, Modeling infiltration during a steady rain, Water Resources Research, 9(2), 384-394 (1874).. Neuman, S. P., "Saturated-Unsaturated y Finite Elements", Journal of the Hydraulics Division, American Society of Civil Engineers, 99(hy12), 2233-2250 (1973) 11. Panday, S. and P. S. Huyakorn, A fully coupled physically-ased spatially-distriuted model for evaluating surface/susurface flow, Advances in Water Resources, 27, 361-382 (2004) 12. Therrien, R., R. G. McLaren, and E. A. Sudicky, HydroGeoSphere-A Three-Dimensional Numerical Model Descriing Fully-integrated Susurface and Surface Flow and Solute Transport, 349p Groundwater Simulations Group (2006). 13. Winter, T. C., J. W. Harvey, O. L. Franke, and W. M. Alley, Ground Water and Surface Water. A Single Resource, 9p, U.S. Geological Survey Circular 1139, Denver, CO (1998) 14. Wise, W. R., T. P. Clement, and F. J. Molz, "Varialy saturated modeling of transient drainage: Sensitivity to soil properties", Journal of Hydrology (Amsterdam), 161(1-4), 91-8 (1994) 5. Acknowledgements This work was supported y a Stanford Graduate Fellowship awarded to the first author. The authors wish to thank Mike Cardiff, Peter Kitanidis, Michele Minihane and Jiahe Wang for sharing their knowledge of.