R.ABABOU and G.TREGAROT 2002 Proc. XIVth Internat. Conf. CMWR 2002, Delft: reformatted version of the original 8 page article.
Proceedings CMWR 2002, XIV International Conference on Computational Methods in Water Resources, 23-28 June 2002, Delft, The Netherlands.
Coupled modeling of partially saturated flow : macro-porous media, interfaces, variability. R.Ababou, G.Trégarot Institut de Mécanique des Fluides de Toulouse (
[email protected] ), Allée du Professeur Camille Soula, 31400 Toulouse, France.
ABSTRACT
This paper presents a novel approach for modeling hydrologic flows involving a variety of features : multiple free surfaces, flooding, partial saturation at interfaces, seepage faces, groundwater-streamflow interactions, heterogeneous and anisotropic structures, etc. The main issue is the efficient coupling of different types of flows. Our mathematical and numerical approach uses a spatially distributed, generic, single equation model, with or without vertical integration. The fully 3D model is based on a mixed formulation of variably saturated flow and the concept of macro-porous media, with a generalized nonlinear anisotropic form of Darcy's flux-gradient law which enables coupling. Several test cases are presented and analyzed. KEYWORDS q q q q q
Groundwater. Unsaturated flow. Macro-porous media. Stream-aquifer coupling. Finite volumes.
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Coupled modeling of partially saturated flow : macro-porous media, interfaces, variability. R.Ababou, G.Trégarot Institut de Mécanique des Fluides de Toulouse (
[email protected] ), Allée du Professeur Camille Soula, 31400 Toulouse, France. This paper presents a novel approach for modeling hydrologic flows involving a variety of features : multiple free surfaces, flooding, partial saturation at interfaces, seepage faces, groundwater-streamflow interactions, heterogeneous and anisotropic structures, etc. The main issue is the efficient coupling of different types of flows. Our mathematical and numerical approach uses a spatially distributed, generic, single equation model, with or without vertical integration. The fully 3D model is based on a mixed formulation of variably saturated flow and the concept of macro-porous media, with a generalized nonlinear anisotropic form of Darcy's flux-gradient law which enables coupling. Several test cases are presented and analyzed. Keywords : Groundwater. Unsaturated flow. Macro-porous media. Stream-aquifer coupling. Finite volumes.
1 GOVERNING EQUATIONS AND NUMERICS The foundation of our hydrologic flow model is a single, spatially distributed equation comprising a mass conservation principle and a generic flux-gradient law [cf. abstract]. The model equation is embedded in a 3D finite volume code, BF2000, derived from the previous BIGFLOW code. Both codes use the same numerics: implicit 3D finite volume discretization, conjugate gradient matrix solvers, and fixed point (modified Picard) nonlinear solver. Details on model equations and numerics can be found in [1]&[6], for BIGFLOW, and [7] for BF2000. In particular : The interfacial conductivities are evaluated by a geometric mean weighting scheme. At each Picard iteration, a symmetric linearized system obtains. The matrix system is then solved iteratively by Preconditioned Conjugate Gradients (PCG) - usually with Diagonal Scaling as preconditioner (DSCG). A diagonal storage scheme is used to take advantage of the sparse hepta-diagonal symmetric matrix structure [6]. The former numerical code BIGFLOW ([1],[6]) has been used previously to efficiently solve a mixed form of Richards' nonlinear unsaturated flow equation (Plate 1) for highly heterogeneous 3D soils : a sample output is depicted in Plate 2. The discrete structure, solver capability and efficiency of the former BIGFLOW code are preserved in the current, more general BF2000 code.
PLATE 1
BIGFLOWcode (Ababou et al. 1993) Numerical method : Fully implicit 3D finite volumes, with sparse Preconditioned Conjugate Gradients solvers, and a modified Picard iteration scheme (fixed point). Scope : Variably saturated flow in 3D heterogeneous anisotropic porous media. Model equation : Richards equation (1931) in mixed form (θ,h) ...
∂θe (h, x) = div[K(h, x) ∇h] − div[K(h, x) gB] ∂t h
: Water pressure or capillary pressure (m)
θe (h)
: Volumetric water content (m3/m3)
K(h)
: Hydraulic conductivity (m/s)
gB = -g / g : Normalized gravity vector in 3D frame (X1,X2,X3) R.Ababou et al. 2001
1
PLATE 2
3D moisture plume in a random soil Ababou et al. 1992 BIGFLOW code
R.Ababou et al. 2001
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The new generic equation used in BF2000 is a highly generalized form of Richards' equation (Plate 3). The coupling of un-saturated, sub-surface, stream-aquifer flows is implemented implicitly, without subdomains or front tracking. This is achieved by attributing adequate kinetic and dynamic properties to each cell in space. Thus, the dynamic "head loss law" can be anisotropic and can vary from cell to cell. More precisely, the BF2000 code comprises two distinct modules : volumetric 3D (described in Plate 4a) and vertically integrated planar 2D (Plate 4b -- not shown in the original paper for lack of space). In each module, the flow equation can vary in space. In this paper, unless stated otherwise, we present only some results obtained with the non-integrated, 3D volumetric module of BF2000. Simplified tests in 1D (column) or 2D (slice) are simulated as "3D volumetric" problems, by shrinking some of their dimensions to a single mesh size.
PLATE 3
The new BigFlow code (BF2000) An integrated model of groundwater and hydrologic flows based on a generalization of Darcy-Richards Hydraulic head H
Hˆ = hˆ + gˆ ( x ) ⋅ x
(
ˆ (hˆ, ∇Hˆ , x )∇ hˆ + gˆ (x ) ⋅ x qˆ = −T ∂θˆ e ( hˆ, x ) = −div[qˆ ] ∂t
[
)
Generalized Darcy
Mass conservation
]
[
∂θˆ e ( hˆ, x ) ˆ ( hˆ, ∇H ˆ , x ) ∇hˆ + div Tˆ ( hˆ, ∇H ˆ , x ) gˆ (x ) = div T ∂t R.Ababou et al. 2001
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]
PLATE 4A (3D MODULES)
BIGFLOW 2000 (3D Modules) The 3D Flow Modules
hˆ
θˆ e
Tˆ
gˆ
h
θ e (h )
K (h )
g B = −g / g
PRESSURE HEAD
W ATER CONTENT
HYDRAULIC CONDUCTIVITY
GRAVITY TERM
h
θ e (h )
2 K ii ( h )
g B = −g / g
(
δ + δ 2 + 4γK ii ( h )( ∇H ⋅ K ( h )∇H )1 / 2
)1 / 2
PRESSURE HEAD
W ATER CONTENT
MODIFIED HYDRAULIC CONDUCTIVITY (for non-darcian head loss)
GRAVITY TERM
h
θ e (h )
K (h )
g B = −g / g
PRESSURE HEAD
W ATER CONTENT (Step function)
HYDRAULIC CONDUCTIVITY (Step function)
GRAVITY TERM
H
Ss H
Ks
0
HYDRAULIC HEAD
W ATER STOCK
SATURATED HYDRAULIC CONDUCTIVITY
R.Ababou et al. 2001
3D FLOW MODEL Variably saturated flow in classical porous media (DARCY-RICHARDs ) Variably saturated non-darcian flow in macro-porous media (W ARD-RICHARDS)
Variably saturated or free surface flows in macro-porous media (FAST KINETICS) Fully saturated porous media (CONFINED AQUIFERS).
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2 NUMERICAL EXPERIMENTS WITH PARTIALLY SATURATED SYSTEMS The capabilities of the model have been tested through a number of numerical experiments involving partially saturated or free surface flows. Some of the simulated flow problems were especially designed for analytical versus numerical validation tests, while others were too complex to be amenable to analytical solution and were only studied by numerical simulation (provided correct mass balance and convergence criteria, not described here for lack of space). 2.1 Numerical Experiments with Partially Saturated Systems One of the first numerical experiments analyzed in this framework involved variably saturated flow with abrupt porous interfaces and partial saturation phenomena : case of ponded infiltration with perched water tables (multiple and coalescing free surfaces). Another test involved soil and groundwater recharge and discharge into a fixed level stream (recession hydrograph; aquifer head drop vs. discharge rate). Another test problem was used to assess the macroporous medium approach : case of transverse imbibition from a canal into a dry porous bank with Polubarinova's hypotheses and quasi-analytical solution [3]. These test problems or similar variants of them were presented earlier, e.g. in reference [5]. The most salient results are summarized below, along with new tests not reported previously.
Perched groundwater during ponded infiltration : Plates 6-7 illustrate the perched groundwater problem in a vertical slice. Constant flooding (ponding) on a sandy soil with a subsurface clay lens, leads to multiple free surfaces and perched water phenomena. Similar tests with different data were presented in [4], [5], [7]. It was found that perched water only occurs under some conditions. The results demonstrate the ability of the model to represent partially saturated flows with multiple free surfaces, without front tracking. They illustrate the ephemeral nature of perched water and its sensitivity to soil conditions.
PLATE 6 :
Partially saturated flows in «3D» porous media
Schematic representation of the perched water test.
Flooding, infiltration, perched water CL : h = hsurf = 15 cm
Lame d'eau
Sable Argile
L = 20 m CL : q = 0
R.Ababou et al. 2001
l = 20 m
1
The domain is a 20mx20m vertical cross-section of sandy soil embedding a rectangular clay "lens". A constant water depth of 15cm is maintained on top. The bottom boundary is an impervious bedrock. The lateral boundaries are "no flow" (e.g., a periodic array of clay lenses).
PLATE 7 : Simulation of perched groundwater with BF2000 code.
Partially saturated flow: flooding and perched water
t2 = 0.22 jour
Initially "wet" soil (83% sand saturation).
R.Ababou et al. 2001
5
The arrows indicate fluxes. Colors or grey shades are water contents. Fully saturated regions are white, the initial state is black. The system is shown at time t=0.22 day. It becomes fully saturated hydrostatic at time : tFINAL≈0.65 day.
R.ABABOU and G.TREGAROT 2002 Proc. XIVth Internat. Conf. CMWR 2002, Delft: reformatted version of the original 8 page article.
PERCHED GROUNDWATER ANIMATION…
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Aquifer Discharge in a Stream: In [5] and [7] we studied soil-aquifer recharge by rainfall, and its discharge into a fixedlevel "stream", without modeling streamflow, but taking into account unsaturated flow. During discharge, 3 regimes were distinguished : 1) fast drainage of wet soil (hours); 2) fast drainage of watertable with exponentially decreasing discharge rate (days); 3) slow drainage of both water table and unsaturated soil (weeks). At aquifer scale, an equivalent head loss law was obtained by analyzing recession discharge rate versus head drop: Q(t) = ao (HL(t)2-H02) is a good fit for the entire transient recession phase. Interestingly, "ao" is close to the steady-state Dupuit-Forchheimer coefficient (ao=Ks/L ≈ 0.25 day -1 ). Slow imbibition from a canal in a dry macro-porous bank: This is Polubarinova's [3] classical problem of transverse infiltration from a fixed level canal into a dry porous bank, underlain by the same horizontal bed as the canal. Polubarinova's analytical solution for a Darcian medium gives the imbibition rate Q(t) ∝ t -1/2 [m2/s], and the position of the front upon the impervious bed, X(t) ∝ t +1/2 [m]. Our approach was to simulate partially saturated flow with highly nonlinear, kinetically macroporous anisotropic medium (with moderate horizontal Ks=1 m/day) in order to implement Dupuit’s hypotheses. A good fit was obtained : Q(t) ∝ t -0.516 , X(t) ∝ t +0.506 ([5],[7]).
Fast imbibition (quadratic head loss) from a canal in a dry macro-porous bank : The same geometry was used again to simulate high rate imbibition in a high permeability, dynamically macro-porous bank (Ks=10+4m/day). Darcy and Ward-Forchheimer laws were compared. Briefly, we used a non-isotropic version of the quadratic law : (1 + C.(ρK/gµ)+1/2 q) q = -K.gradH , with C=0 for Darcy and C=0.55 for Ward. After about 10 seconds (t=10-4day) the saturation front reached only 2.75 m with Ward, compared to 5.50 m with Darcy. At high flow rates, we believe that the saturation front obtained with the Ward model is more realistic, as it accounts for inertial quadratic head losses. Other tests with composite hydrological systems : We have also studied other simplified composite systems, where a "macro-porous" domain represents a coarse medium or even a free water body, e.g. to simulate the drainage of a water column through a porous obstruction or layer. We have also tested the use of non-isotropic macro-porous media with simple boundary conditions to implement water extraction or injection to/from a water table without using volumetric sinks or sources such as pumping wells : this is shown in more detail below.
Pumping well test : This is a problem of steady-state well extraction from a water table. The numerical simulation, with two different non-isotropic macro-porous materials in the well and in the aquifer, fits precisely the Dupuit analytical solution. This is illustrated in Plates 8-9. PLATE 8 3D variably saturated groundwater flow with macroporous media Steady pumping from a water table located between two streams CL : Q = QsP > 0
z z=l
CL : q = 0 Aquifère (KsA )
Puit (KsP )
Rivière
Rivière Zs (x) Nappe
QsP /2 > 0
z = Zs0 = ZsL
z = ZsP -
QsP /2 < 0 z=0
CL : H = Zs0
x=0
Z s (x) =
CL : H = Zs0 2
Z sP +
x=L
Q sP ( x − L / 2) K sA
R.Ababou et al. 2001
x
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PLATE 9 :
E coulem ents 3D de nappes souterraines libres Steady pum ping from a w ater table located betw een tw o stream s
Z s0
Z s (m ) : num érique
Z sL
Z s (m ) : analytique
x (m ) R .A babou et al. 2001
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2.2 Numerical Simulation of Coupled Stream-Soil-Aquifer Flow Finally, a more complex case involving fully coupled stream-bank-aquifer interactions during a stream flood rise and recession, was studied extensively under various conditions (slow flood versus fast flood). The results for the "slow flood” are presented here in Plates 10 through 14. Several simplifications are made : The river stretch is U-shaped, so the river bank is vertical. The stream is reduced to a thin slice. Thus, axial streamflow is not fully modeled in this particular experiment. Nevertheless, there is a two-way coupling between surface and subsurface waters, as both the stream and the porous subsurface are part of the computational domain. This coupling is achieved without a need for boundary conditions at the stream-bank interface. The simulated “behavior” of the coupled aquifer-stream flow system cannot be compared to analytical solutions…, but the results are encouraging : (a) the stream water level correctly rises and falls; (b) the flood propagates into the porous bank during flood rise… and back into the stream during recession; (c) internal seepage naturally occurs at the stream-bank interface.
PLATE 10
PLATE 11
Coupling of « 3D » Stream-Aquifer Flow Systems
Coupling of « 3D » Stream-Aquifer Flow Systems Stream-Soil-Groundwater exchanges during a flood event:
Stream-Soil-Groundwater exchanges during a flood event
Storage
z
Vx (t)
Bank
GW
q=0
Stream
z 5m Zs (t)
Zs (t)
0.5 m qx (t) > 0 qz (t) > 0
qx (t) < 0 O
20 m
x 500 m
O
x
Stream flood : recession phase.
Stream flood : ascending phase R.Ababou et al. 2001
qz (t) < 0
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R.Ababou et al. 2001
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R.ABABOU and G.TREGAROT 2002 Proc. XIVth Internat. Conf. CMWR 2002, Delft: reformatted version of the original 8 page article.
PLATE 12 : Macroporous Coupling: «3D» Aquifer-Stream Flood Porous bank (clayey sand): θ e = 0.35 , K s= 1 m/d
Input streamflow hydrograph (« slow » flood): m
Zs ( t) = Zs0
t + Z s 1 e − t / t tc
c
dZ s Z s1 t q z (t) = = dt t c t c
m −1
t e − t / tc m − tc
Input parameters: m=2 , tc =2.5 j , Z s1=7 m
Zs (m)
Z smax = 4.30 m at tmax = 5.0 days :
t (jour) R.Ababou et al. 2001
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PLATE 13 : Macroporous Coupling: «3D» Aquifer-Stream Flood Stream
Porous bank Simulation of stream-bank flow during and after a flood event.
Z (m)
The free surface, flux vectors, and hydraulic head contours are depicted at time t=20 days (after the flood). Observe flow separation around x=32m, and internal seepage face about x=22m.
h=0
Z (m)
X (m) X (m) R.Ababou et al. 2001
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PLATE 12
Macroporous Coupling: «3D» Aquifer-Stream Flood Porous bank (clayey sand): θe= 0.35 , Ks= 1 m/d
Input streamflow hydrograph (« slow » flood):
t Z s ( t ) = Z s 0 + Z s1 tc
q z (t) =
m
−t / tc e
dZs Zs1 t = dt tc tc
m −1
t e − t / t c m − tc
Input parameters: m=2 , tc=2.5 j , Zs1=7 m
Zs (m)
Zsmax= 4.30 m at tmax= 5.0 days : t (jour) R.Ababou et al. 2001
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"3D"
PLATE 14 :
Macroporous Coupling: «3D» Aquifer-Stream Flood Net volume in the river stretch
Net volume infiltrated in the bank
V x (m 3 /m)
V z (m 3 /m)
t (day)
t (day)
V Xmax /V Zmax ≈ 7.4 / 75.5 ≈ 9.8 % R.Ababou et al. 2001
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2.3 Numerical Experiment with Vertically Integrated Aquifer-Stream Flows We have also developed a different, complementary approach for dealing with coupled flows in 2D plane view. The 2D planar module of BF2000 can model a broad class of vertically integrated flows. The equation type can vary cell-by-cell in the plane, with a generalized "transmissivity". This 2D planar module is built upon a single generic equation, which can model : (1) Subsurface water tables with one or several underlying connected aquifers, and (2) Surface streamflows with one or several underlying connected aquifers. Briefly, the models used are as follows Subsurface flows : Dupuit-Boussinesq model (vertical integration of Darcy), Surface flows : Kinematic-Diffusive Wave model, derived from Saint-Venant equations by vertical integration of Navier-Stokes plus other simplifying assumptions. The 2D plane flow simulation tests [7] are not shown here[in the original paper] for lack of space.
3 CONCLUSIONS The spatially distributed model BF2000 generalizes variably saturated flow processes for heterogeneous, anisotropic, macroporous, partially saturated media. The model can efficiently track the dynamics of multiple interacting free surfaces. It reproduces correctly some well known dynamics, such as recession hydrographs. Moreover, it is able to simulate a highly coupled stream-aquifer problem, involving the ebb and flux of water in a porous bank coupled to a flood wave in the stream, with twoway coupling and with internal seepage at the stream-bank interface. Other phenomena modeled with macroporous media include pumping wells. The numerical experiments presented here are aimed at efficiently representing, modeling, and analyzing environmental flows through different types of natural media, interfaces, and structures. Our model appears suitable as a tool for analyzing fissured and karstic hydrogeologic systems, stream-soil-aquifer flows and other coupled sub/surface flows. ACKNOWLEDGMENTS This research was funded by GIS ECOBAG (Prog.HYDROBAG). Support was also provided by PNRZH (Prog.Nat.Rech.Zon.Humid.), GIP Hydrosystèmes (DIES-IRAE), and PNRH (Prog.Nat.Rech.Hydrol.).
REFERENCES 1. R.Ababou and A.C.Bagtzoglou: BIGFLOW, a Numerical Code for Simulating Flow in Variably Saturated, Heterogeneous Geologic Media-Theory and User's Manual Ver.1.1. NUREG/CR-6028, U.S.NRC, 1993. 2. R.Ababou, B.Sagar, and G.Wittmeyer: Testing Procedures for Spatially Distributed Flow Models, Advances in Water Resources, Vol.15, pp.181-198, 1992. 3. P.Ya.Polubarinova-Kochina: Theory of Ground Water Movement [Ch.13]. Transl. from russian by J.M.R. de Wiest, Princeton University Press, NJ, 613 pp., 1962. 4. R.Ababou, G.Trégarot, and M.Bouzelboudjen: Variably Saturated Subsurface Flow with Layers & Interfaces: Perched Water Tables & Stream-Aquifer Connection. ModelCare96 Proc, GWMI96-OX, 10pp, 1996. 5. R.Ababou, G.Trégarot, A.Larabi: Partially Saturated Hydrological Flows : Numerical Experiments and Analyses. XII CMWR Proc., Crète, Greece, 8 pp., 15-19 June 1998. 6. R.Ababou, L.W.Gelhar, and Ch.Hempel: Serial and Parallel Performance on Large Matrix Systems. Cray Channels, Cray Research, 14(3), Summer Issue 1992. 7. G.Trégarot: Modélisation Couplée des Ecoulements à Saturation Variable avec Hétérogénéités, Forçages, et Interfaces Hydrologiques. Doct. thesis, Institut National Polytech., IMFT, Toulouse, France, May 2000. 8. J. Dupuit : Etudes Théoriques et Pratiques sur le Mouvement des Eaux dans les Canaux Découverts et à Travers les Terrains Perméables. Dunod, Paris, 1863. 9. P. Forchheimer : Hydraulik. Teubner Verlaggesellschaft, Stuttgart, 1930.
R.ABABOU and G.TREGAROT 2002 Proc. XIVth Internat. Conf. CMWR 2002, Delft: reformatted version of the original 8 page article.
REFERENCES 1. R.Ababou and A.C.Bagtzoglou: BIGFLOW, a Numerical Code for Simulating Flow in Variably Saturated, Heterogeneous Geologic Media-Theory and User's Manual Ver.1.1. NUREG/CR-6028, U.S.NRC, 1993. 2. R.Ababou, B.Sagar, and G.Wittmeyer: Testing Procedures for Spatially Distributed Flow Models, Advances in Water Resources, Vol.15, pp.181-198, 1992. 3. P.Ya.Polubarinova-Kochina: Theory of Ground Water Movement [Ch.13]. Transl. from russian by J.M.R. de Wiest, Princeton University Press, NJ, 613 pp., 1962. 4. R.Ababou, G.Trégarot, and M.Bouzelboudjen: Variably Saturated Subsurface Flow with Layers & Interfaces: Perched Water Tables & Stream-Aquifer Connection. ModelCare96 Proc, GWMI96-OX, 10pp, 1996. 5. R.Ababou, G.Trégarot, and A.Larabi: Partially Saturated Hydrological Flows : Numerical Experiments and Analyses. XII CMWR Proceedings, Crète, Greece, 8 pp., 15-19 June 1998. 6. R.Ababou, L.W.Gelhar, and Ch.Hempel: Serial and Parallel Performance on Large Matrix Systems. Cray Channels, Cray Research, 14(3), Summer Issue 1992. 7. G.Trégarot: Modélisation Couplée des Ecoulements à Saturation Variable avec Hétérogénéités, Forçages, et Interfaces Hydrologiques. Doct. thesis, Institut National Polytech., IMFT, Toulouse, France, May 2000. 8. J. Dupuit : Etudes Théoriques et Pratiques sur le Mouvement des Eaux dans les Canaux Découverts et à Travers les Terrains Perméables. Dunod, Paris, 1863. 9. P. Forchheimer : Hydraulik. Teubner Verlaggesellschaft, Stuttgart, 1930.
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