EXCERPTS ON THE BIGFLOW NUMERICAL MODEL Main source (excerpts): SWIMED Deliverables D6 & D7: “D6 : Groundwater Flow Models” “D7: Seawater Intrusion Models”
R. Ababou & A. Al-Bitar January 2005
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TABLE OF CONTENTS Table of Contents ______________________________________________________________2 Acronyms and symbols__________________________________________________________3 1.
Groundwater flow modeling _________________________________________________4 1.1. 1.1.1. 1.1.2. 1.1.3. 1.1.4. 1.1.5. 1.1.6. 1.1.7. 1.1.8. 1.1.9.
2.
2.1.1. 2.1.2.
4.
Code abstract. ____________________________________________________________________ 4 Code history & status ______________________________________________________________ 4 BIGFLOW - Introduction and overview (2D & 3D)_______________________________________ 4 Remarks on 2D vs. 3D options _______________________________________________________ 5 Remarks on Sea Water Intrusion Modeling (SWIM) ______________________________________ 5 BIGFLOW - Model equations and physical basis (2D & 3D) _______________________________ 7 BIGFLOW - Inputs / Parameters / Limitations ___________________________________________ 9 General inputs ____________________________________________________________________ 9 Domain geometry _________________________________________________________________ 9 Boundary conditions and initial conditions______________________________________________ 9 Forcing Terms ____________________________________________________________________ 9 Physical Properties ________________________________________________________________ 9 Numerical solver parameters_________________________________________________________ 9 Limitations _____________________________________________________________________ 10 BIGFLOW - Numerical methods ____________________________________________________ 10 BIGFLOW - Operating systems _____________________________________________________ 10 BIGFLOW - Applications under third party software_____________________________________ 11 PBF 1.0 - Python BigFlow version 1.0 ________________________________________________ 11 Interactive 3D Visualization with VRML ______________________________________________ 12 Data processing : the DataFlow fortran program ________________________________________ 13 BIGFLOW - Benchmarks, performance tests, validation __________________________________ 14 BF-3D: ________________________________________________________________________ 14 BF-2D _________________________________________________________________________ 15
Surface processes and groundwater interactions ________________________________16 2.1.
3.
Groundwater flow modeling with BIGFLOW (2D/3D) ____________________________ 4
BIGFLOW integrated coupling of surface/GW flows ____________________________ 17 BIGFLOW coupling capabilities in 3D (“open” media) ___________________________________ 17 BIGFLOW coupling capabilities in 2D (diffusive wave) __________________________________ 19
BIGFLOW for 2D SeaWater Intrusion Modeling (BF-SWIM2D) __________________21 3.1.
Formulation of BF-SWIM2D (overview) _______________________________________ 21
3.2.
Formulation of BF-SWIM2D (equations) ______________________________________ 21
3.3.
Case of heterogeneous aquifers with K(x,y) or T(x,y) ____________________________ 22
3.4.
Implementation of source terms ______________________________________________ 23
References_______________________________________________________________26 4.1.
Bibliographic references ____________________________________________________ 26
4.2.
Internet web sites __________________________________________________________ 27
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ACRONYMS AND SYMBOLS BC: GIS: GW: IC: SWI: SWIM: SWIMED:
Boundary Condition Geographical Information System GroundWater Initial Condition SeaWater Intrusion SeaWater Intrusion Management or Modeling SeaWater Intrusion Management in Mediterranean Countries (this project)
1D: 2D: 3D:
One-dimensional Two-dimensional (plane view or cross-section, depending on context) Three-dimensional
K: T: H: ε: Q: q: V: θ: θS:
Hydraulic conductivity (m/s) Hydraulic transmissivity (m2/s) Total hydraulic head (m) Relative density contrast between seawater and freshwater Discharge rate (m3/s for a well) or specific discharge rate (m2/s for an aquifer) Flux density vector or “Darcy velocity”, in (m3/s/m2 ) or equivalently (m/s)1. Water velocity vector (or Darcy velocity depending on context) (m/s)2 Volumetric water content (m3/m3) Saturated water content, or porosity (m3/m3)
1 2
NOTE: bold type such as “q” or “V” indicates a vector. NOTE: V is the velocity of a tracer in water; Darcy velocity q differs from V; in general we have q = θV.
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1. GROUNDWATER FLOW MODELING 1.1.
Groundwater flow modeling with BIGFLOW (2D/3D) 1.1.1. Code abstract.
The spatially distributed flow model BIGFLOW (BF2000) is a generalized multidimensional 2D & 3D model for variably saturated flow processes in heterogeneous, anisotropic, macroporous, and partially saturated media. For instance, the model can efficiently represent the 3D dynamics of multiple interacting free surfaces. In addition, a vertically integrated 2D plane flow flow option is available, with several specialized capabilities including surface hydraulics coupling, and seawater intrusion. The general model is based on a single generalized 2D/3D nonlinear conservative flux-divergence equation (mixed form), discretized as a very sparse implicit finite volume system and solved using preconditioned conjugate gradient and modified Picard (fixed point) iterations. The material properties can be distributed arbitrarily in all directions. The code has been useful for direct simulations of highly heterogeneous GW flow problems on large grids (10 million nodes), but also, more recently, for simulating nonlinear/coupled phenomena such as perched water tables, streamaquifer-soil interactions, fast flow in single fractures and macroporous systems.
1.1.2. Code history & status 1987: 1993: 2000: 2004 :
BIGFLO (3D) is a university research code (Ababou, Gelhar, McLaughlin), MIT, Cambridge, Massachusetts, USA BIGFLOW 1.1 (3D) is a public domain code (Ababou, Bagtzoglou) US Nuclear Reg. Comm. (Washington D.C.) & SwRI/CNWRA (San Antonio TX) BF2000 (2D/3D) is a university research code (Ababou, Trégarot) Institut de Mécanique des Fluides de Toulouse, France BF-Python (2D/3D) is the current research code stemming from earlier versions; it contains a new seawater intrusion option & Python interface under construction: (Ababou, Al-Bitar) Institut de Mécanique des Fluides de Toulouse, France.
1.1.3. BIGFLOW - Introduction and overview (2D & 3D) The BIGFLOW code, named “BF” for short, is the research code used by the french partner (Partner 3 of SWIMED project) to develop analyses on porous media hydrodynamics and hydrogeologic processes, particularly in the presence of heterogeneities and couplings. Historically, BIGFLOW was initiated as a numerical tool for modeling 3D flow systems in randomly heterogeneous geologic formations, considering first only saturated GroundWater flow (Ababou et al. 1985) then unsaturated flow in heterogeneous soils (Ababou et al. 1987 & 1988) porous media with high resolution. The code was extensively tested between 1988 and 1992 resulting in a published manual of BGFLOW version 1.1 was published by the U.S. Nuclear Regulatory Commission (Ababou & Bagtzoglou 1993). The BIGFLOW code was later reshaped and enhanced, leading to BF-2000, a computer code for modeling diverse flow processes in heterogeneous variably saturated hydrologic media, including some surface/subsurface coupling. The BF2000 code is well documented in the Ph.D. thesis of Trégarot (2000).
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The spatially distributed flow model BF2000 generalizes variably saturated flow processes for heterogeneous, anisotropic, macroporous, and partially saturated media in 3D. The model can also efficiently track the dynamics of multiple interacting free surfaces in 3D. The model can accommodate nonlinear, anisotropic and heterogeneous material properties, in (x,y,z) for the 3D case, or in (x,y) for the 2D vertically integrated case. A new 2D module BFSWIM was introduced more recently in BF2000 to take into account saltwater intrusion modeling, and also, internal sink/source terms (see Deliverable 7 - “D7: Seawater intrusion models”). The BF-2000 code has two rather different options, 2D/3D, each one leading to various possible flow regimes as illustrated in the flow-charts below option (BF-3D) and 2D option (BF2D): • 3D option (BF-3D): finite volume model for fully heterogeneous, three-dimensional, variably saturated and nonlinear (porous or macroporous) soil-aquifer systems. • 2D option (BF-2D): finite volume model for vertically integrated quasi-plane flows, including Boussinesq-Dupuit aquifer flows, free surface hydraulics based on kinematic wave, Darcy-Forchheimer flow in rough fractures, and the seawater intrusion module. Remarks on 2D vs. 3D options The 3D option is fully three-dimensional, in the sense that all material properties being arbitrarily distributed in 3D space. On the other hand, the 2D option is vertically integrated and assumes quasi-plane flow (vertically hydrostatic). It should be emphasized, however, that the internal computational structure of the code is 3D. Indeed, the 2D option was developed by specializing the general algorithms of the code without modifying its internal 3D structure. It should be emphasized that the 3D algorithms of BIGFLOW have been tested quite extensively since the late 1980’s. Many features of the 2D plane flow option have been tested more recently, at least since the late 1990’s. For more details, see list of references (Ababou et al.; Trégarot et al.).
Remarks on Sea Water Intrusion Modeling (SWIM) In this report, we choose to emphasize the general capabilities of the 2D/3D code as they existed at the start of the project, rather than the new seawater intrusion modeling capabilities. Indeed, for the present project on coastal aquifers (SWIMED), a new algorithm for seawater intrusion was developed into the 2D version of BF, called BF-SWIM2D. This will be described in deliverable “D7: Seawater Intrusion Modeling”.
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Flow Chart 1 of BIGFLOW code : the 3D option (BF-3D)
Flow Chart 2 of BIGFLOW code : the 2D option (BF-2D)
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1.1.4. BIGFLOW - Model equations and physical basis (2D & 3D) BIGFLOW is an integrated model for groundwater and hydrologic flow processes based on a unified, generalized, conservative flux-divergence equation obtained from Darcy-Buckingham and mass conservation (see equations in Figure 3.2.A below).
Figure 3.2.A. BF single generic equation governing flow in several hydrologic configurations. Descriptions of variables and parameters are given in the next figures below.
Several types of flows can be derived from this single generic equation. BF can also implements strong implicit coupling of different types of hydrologic flows via it’s single generic equation (e.g. coupled stream-aquifer flow, macroporous media with quadratic head loss law, perched groundwater with multiple free surfaces, etc). Table 3.2.(I) and Table 3.2.(II) give the description of the variables and parameters in the generic equation for the different configurations given in Figure 3.2.B and Figure 3.2.C. The tables list the flow regimes that can occur in different cells of the flow domain, depending on the coefficients. Each line corresponds to a type of flow regime. Different flow regimes can co-exist in the same flow domain (however, the 2D/3D options are mutually exclusive).
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Table 3.2.(I): Variables and parameters of the generic flow equation BF (3D option)
Table 3.2.(II) Variables and parameters of the generic flow equation BF (2D option)
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1.1.5. BIGFLOW - Inputs / Parameters / Limitations General inputs These are to define the main aspect of the simulation like type of simulation, transient/steady state simulation, and linear/nonlinear prob. There are also the less important identification code and date of simulation. Domain geometry BF assumes that the 3D domain has the shape of a rectangular box (parallelepiped rectangle). However the box can be slanted at any angle with respect to the vertical. The finite difference mesh is a fixed regular network of orthogonal links and nodes. Never the less physical boundaries like substratum bathymetry and surface topography, used in the coupled surfacesubsurface flow, can be distributed uniformly or spatially over the domain. Boundary conditions and initial conditions BF enables the use of three types of boundary conditions (BC) Dirichlet (fixed head), Neumann (fixed flux) and zero head gradient BC. All types can be fixed, uniformly or spatially distributed. Time dependence is also available. Mixed type BC can be defined separately for each BC. Mixed type means that the type of BC can vary within the boundary plane. Initial head pressures can be defined as fixed, uniformly distributed or spatially distributed. Forcing Terms Internal forcing terms are given in two different ways either locally for source/sink terms (e.i. pumping/recharge wells), or uniformly distributed over the domain. In the latter case these represent recharge or infiltration rates. Forcing terms are also time-varying. Physical Properties • For all cases: hydraulic properties are specified cell-by-cell (pixels in 2D, voxels in 3D) • For subsurface flow: hydraulics properties for saturated media are the hydraulic conductivity (Ks) and the specific storativity (Ss) both are spatially variable if needed. For spatially saturated or unsaturated media the hydraulic conductivity (K) is a function of pressure head h. The soil moisture retention curve can be chosen among several functions with multiple parameters, two of which can be spatially distributed. • For surface flow: where the diffusive-kinematic wave equations are applied, the hydraulic properties are the friction coefficients (C) derived from the Chezy, Manning or DarcyWeisbach formulas. • For macroporous media flow: another coefficient (γ) of the quadratic term in the generalized Ward’s law is defined. Numerical solver parameters These parameters allow the user to specify the choice of matrix solver and preconditioner, as well as its convergence criteria, like the maximum number of iterations, the minimum error criteria, and the type of norm used to compute those. These parameters also include the nonlinear Picard
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iteration convergence criteria. In addition, machine dependent criteria can be specified, such as machine precision, smallest and largest real floating point numbers, etc.
Limitations The geometry of the 3D domain is rectangular (however, in 2D, the bedrock is specified as ZINF(x,y)) The finite volume grid is rectangular, uniform, cartesian (similarly in 2D and in 3D) Some useful features such as leaky layers in the 2D case are not yet implemented The time step adaptivity and the nonlinear solution process are not always succesfull (a problem common to most computer models for highly nonlinear and transient problems)
1.1.6. BIGFLOW - Numerical methods BF is based on a fully implicit finite volume formulation of flux divergence equations in conservative form (mixed formulation), and it can solve transient as well as steady state problems (either by time marching or in a single infinite time step). It implements strong implicit coupling of different types of hydrologic flows via a single generic equation (e.g. coupled stream-aquifer flow, macroporous media with quadratic head loss law, perched groundwater with multiple free surfaces, etc). The «computational kernel» of the code has inherited features of previous versions and has been extensively tested, particularly for 3D saturated/unsaturated flow problems with randomly heterogeneous properties. The «kernel» is characterized by extremely sparse, specialized linear algebra based on preconditioned conjugate gradient solvers, interlooped with a robust modified Picard (fixed point) iteration scheme for nonlinear problems.
1.1.7. BIGFLOW - Operating systems BF has been developed in ANSI Standard Fortran77, free of machine dependent directives thus all versions of BF can be compiled for any specific platforms requirements. In the past, BIGFLOW has been tested on DOS, Windows 3.1, CRAY UNICOS (Unix), IBM AIX (Unix), Silicone Graphics machines, etc. For the time being executables are available for Windows® 98 / XP and for UNIX operating systems. BF has been programmed with very sparse data structures in order to achieve high resolution simulations of flow in heterogeneous multidimensional media. Estimations of CPU time performance and storage requirement (live-memory RAM) with respect to problem specifications are available. For example Figure 3.2.D’ below in the case of 2D Seawater Intrusion Modeling : the storage requirement is 200 Mbytes for a grid of 4 million cells with BF-SWIM2D single precision (or twice as much in double precision). In addition, the detailed numerical performance analyses reported in Ababou et al. (1992) describe solver convergence, storage requirements and CPU time estimates, for large 3D problems on coarse-grained parallel and vector machines (timings for more recent machines are also available).
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Storage Requirements for BFSWIM v1.1 500
2000 ABIG = 54.054 Nt + 0.0793 with ABIG : Array size (million elements) Nt: million nodes R2 = 1
1600
M = 211.42 Nt + 5.648 with M : memory (Mb) Nt: million nodes R2 = 1
300
1200
200
800 Totlal array (ABIG) E-06 Required memory Mb
100
Required Memory in Mb
Totla Array in millions
400
400
0
0 0
1
2
3
4
5
6
7
8
9
10
Nt (N1xN2x1) million nodes
Figure 3.2.D’: Memory requirements in Mb (right) or Total array size (left) with respect to total number of nodes Nt and for simple precision integer (4 bytes)
1.1.8. BIGFLOW - Applications under third party software PBF 1.0 - Python BigFlow version 1.0 A graphical user interface under Python®3 and using the wxPython® library has been developed for pre and post processing (see Figure 3.2.E’). The interface exists so far under the Windows OS. However, since the Python language is cross-platform, the interface is available on virtually all operating systems (it is currentlt being tested under the Linux OS). It is emphasized that the objective of the GUI (Graphical User Interface) is not to supply a commercial software package for BIGFLOW, but to provide BIGFLOW users with a convenient tool to manage the model inputs and outputs. This is mostly important for problems with complex geometry, but also, for optimal management problems and multiple scenario analyses (optimization; uncertainty). The Python interface manages directories, executables and data files with minimal user intervention. It also enables users to input simulation parameters like BC’s, grid size, time step, convergence criteria, initial head distribution, etc. Simulations are launched and tracked in realtime by numerical and graphical outputs. Outputs are also managed by the Python application. They can be plotted and exported in different graphical formats. Multiple simulations can be launched and managed interactively, for example Monte-Carlo simulations for stochastic uncertainty analyses, or iterative re-starts as occured in the sigma-homotopy method. (Note: the homotopy method is implemented as an iterative “continuation method” with respect to the σ parameter, where σ represents the degree of 3
Python is a Freeware, interpreted, Object Oriented Programming Language. wxPython is a Freeware, GUI toolkit for the Python programming language based on wxWindows.
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heterogeneity (standard deviation of lnK); the method is used to improve numerical convergence for seawater intrusion modeling in highly heterogeneous stochastic aquifers.)
Figure 3.2.E’ Illustration of the Python menu-driven and graphical interface of BigFlow.
Interactive 3D Visualization with VRML
VRML®4 is the object oriented Virtual Reality Modeling Language. VRML programs are interpreted (rather than compiled) by internet browsers such as IE5 or Netscape, provided a VRML plug in software such as Cosmo or Cortona. We developed some VRML programs, in particular, to enable interactive 3D visualization of coastal aquifers, like the simulated freshwater free surface and the freshwater/seawater interface.
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VRML stands for Virtual Reality Modeling Language
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Data processing : the DataFlow fortran program
A command-line interactive 3D data processing code (DATAFLOW) is also available. The Data processor is written in ANSI Standard Fortran77 (author: R.Ababou, circa 1988). It allows manipulation and analysis of BigFlow’s 3D datasets. DATAFLOW contains also a statistical analysis module for BIGFLOW inputs and outputs. The interactivity with the user exists in a rudimentory form (DOS-type window: screen text questions and answers). This auxiliary code is useful for generating and editing fully 3D datasets, but it will become obsolete, as it is currently being replaced by menu driven dialog boxes under Python.
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1.1.9. BIGFLOW - Benchmarks, performance tests, validation BF-3D: The reader is referred to Cray article 1992 (CPU performance tests) and to Ababou and Bagtzoglou (1993) for a series of completely documented validation tests in 3D and 2D crosssections. The computational kernel of the current BF code is analogous to that used in these publications (solvers and data structures are essentially the same). BIGFLOW has been extensively used for simulating highly heterogenous 3D flow systems, on large finite volume grids comprising on the order of 1 million nodes and, in some cases, up to 10 million nodes (3D cells, voxels). For such large grids, analyses of computational performance were conducted using Cray-2 and Cray-Y/MP8 supercomputers: this was described first in Ababou et al. (1992)’s article in Cray Research, and also in Ababou (1996). Figure 3.2.F (right). Visualization of a nonlinear variably saturated moisture plume in a 3D random soil with 0.3 million cells, from Ababou et al. (1992).
Figure 3.2.G (above). Computational performance tests of BIGFLOW for 3D aquifers with randomly heterogeneous permeabilities K(x,y,z) on grids on the order of 1-10 million cells : parallel speed-up (left); convergence of Conjugate Gradient solver (right). (Ababou et al. 1992). 14
BF-2D The BF-2D code can perform GW flow simulations based on the classical Boussinesq equations, as with many other popular codes (both public and commercial), albeit with more rigid geometry (the BD grid is cartesian and the domain is rectangular). Nevertheless, it is possible to achieve realistic simulations of GW flow with BF 2D in irregularly shaped domains inscribed within the template rectangular domain : this is illustrated in Figure 3.2.H for the Tadla aquifer, a phreatic aquifer with salinity problems, located in an irrigated plain in central Morocco (Trégarot 2000).
Figure 3.2.H. Groundwater flow simulation of the unconfined Beni Amir aquifer with BF 2000 (2D plane flow option. This simulation takes into acount the actual, detailed domain geometry, and uses realistic boundary conditions infered from piezometric maps, as well as a substratum topography map, and a kriged transmissivity map (obtained with exponential variogram).
NB: Concerning BIGFLOW’s surface/subsurface coupling capabilities, the reader is referred to Sec.(5.2) further below.
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2. SURFACE PROCESSES AND GROUNDWATER INTERACTIONS GroundWater (GW) flow modeling of real site-specific problems requires taking into account other hydrologic processes such as surface flows, which interact with subsurface flows. Another example is unsaturated flow in the vadose zone extending from soil surface to the water table, Unsaturated flow can interact with the saturated GroundWater flow in the aquifer. Other interactions include the partition between rainfall, infiltration and runoff on sloping soil surfaces, and the relation between evaporation and surface soil moisture. In each of the cited cases, the interactions can run both ways, although most popular GW models neglect some interactions, often assuming “one-way coupling”, e.g. in a stream-aquifer system. Overall, the different types of interactions between surface waters and groundwater can be distinguished and classified as follows: 1. Atmospheric vapor flow (evaporation) interactions with surface soil moisture 2. Surface/subsurface flow interactions (particularly in coastal aquifers): i) Stream/Aquifer coupling in the presence of seawater intrusion, e.g. in the neighborhood of a river outlet to the sea, or in an alluvial floodplain, etc (it should be noted that most models implement only “one-way coupling”, e.g. stream Î aquifer); ii) Surface/Subsurface flow coupling due to highly transient, variably saturated processes: e.g. run-off, ponding, and flooding processes (it should noted that most models implement only “one-way coupling”, e.g. unsaturated soilÎexcess rainfall) 3. Spatially distributed and time-varying surface flow inputs and boundary inputs like rainfall, irrigation, leakage, pumping wells, etc : one issue is how to represent and assess the affect of spatially distributed recharge on seawater intrusion; another issue is how to build efficient interfacing of these variable inputs with GW flow models. 4. Salt concentration coupling due to natural and/or man-induced recharge: solute return flux due to extraction/irrigation cycle; hydraulic remediation by (re)-injection, etc… We do not attempt here a complete review of models that integrate such coupled processes. Instead, the reader is referred to Section 2 (above) for a review of general GW models (some of which include a fair amount of coupling), and to Section 5 (below) concerning the specific coupling and interfacing capabilities in this project.
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2.1.
BIGFLOW integrated coupling of surface/GW flows 2.1.1. BIGFLOW coupling capabilities in 3D (“open” media)
The existing coupling capabilities of the BIGFLOW 3D option (X-Y-Z flows) are exemplified through several representative examples not involving salt water density effects (the numerical experiments are restricted to X-Z vertical cross-sections but the model is intrinsically 3D) :
Transient formation of localized perched groundwater under flooding conditions (partially saturated flow with multiple free surfaces); Soil-aquifer-stream interactions during and after rainfall (accounts for one way coupling streamÎaquifer, and drainage of unsaturated soil into the free surface groundwater), Aquifer-stream interaction during the passage of a river flood (with two-way coupling !)
Only the latter phenomenon, stream-aquifer interaction, will be detailed here. Figure 5.2.A below is a schematic representation of the passage of a river flood in a coupled stream-aquifer system. q=0
z 5m Zs
Berge
Stream
0.5 m q = qz
O 20 m
x 500 m
Figure 5.2.A. Schematic representation of coupled stream-aquifer system (vertical crosssection). The stream, porous banks, and aquifer form a single computational domain (halfdomain by symmetry). The geometry is simplified : U-shaped stream with vertical banks. The next figures show the stream-aquifer simulation results obtained with the BIGFLOW code for the case of a “slow” river flood, whose main characteristics were :
Stream half-width X = 20 m (note : ∆x=1m); Initial water depth in stream : Zo=0.5 m; Maximum water depth in stream (“peak”) : Zmax=4.3 m (note: ∆z=0.10m); Time-to-peak of streamflood : tmax=5.0 days (“slow flood”).
The simulations were performed with the BIGFLOW “BF-3D” option (Trégarot 2000; Ababou and Trégarot 2002). Only the right half-domain is simulated (stream at left, aquifer at right) in a thin vertical slice of the full 3D stream-aquifer system. In principle, it is possible to conduct similar coupled simulations with ful 3D geometry for a real stream-aquifer-flood plain system.
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Figure 5.2.B: BIGFLOW Stream-Aquifer Coupling (BF-3D / macroporous stream): “Slow Flood“ Experiment with time to peak tmax = 5 days; each plot displays the free surface (blue) and total head contours (red) at times t= 5, 8, 10, 20, 30, 50 days.
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Figure 5.2.C. BIGFLOW Stream-Aquifer Coupling (BF-3D / macroporous stream): detailed snapshot of the “Slow Flood“ experiment at time t = 20 days.
2.1.2. BIGFLOW coupling capabilities in 2D (diffusive wave) The existing coupling capabilities of the 2D plane flow version of BIGFLOW (vertically integrated flows in the X-Y plane) : 1. vertically integrated stream-aquifer system : full coupling capability for a flood plain 2. vertically integrated multilayer stream-aquifer-aquifer (flood plain with two aquifer layers) 3. salt water density effects (seawater intrusion in confined or unconfined aquifers) The first type of coupling is illustrated schematically in Figure 5.2.D below. It concerns the ability of the model to take into account fully coupled GroundWater and Surface Flow hydraulics in plane view, using the Boussinesq equation for 2D GW flow, and the kinematic-diffusive wave equation for 2D surface hydraulics. The latter is usually valid for slowly varying river flows and/or flood plain hydraulics. In BIGFLOW, these equations are fully coupled both ways (surfaceÎsurface & subsurfaceÎsurface). The model is valid provided the assumption of plane flow and the assumed continuity of hydraulic head at the contact between groundwater and open water (their respective free surfaces must remain connected at all times). 19
Figure 5.2.D Schematic representation of BIGLOW surface/subsurface coupling capability for 2D vertically averaged plane flow systems : stream, flood plain and alluvial aquifer. The second type of coupling (multilayered aquifer coupled to surface flow) can be considered as a relatively simple (but important) extension of the previous case. Finally, the third type of coupling - saltwater density effects - concerns the ability of the model to take into account variable water density, e.g. due to salinity variations, based on plane flow assumptions. A special aplication of this case is the modeling of seawater intrusion via the sharp interface - plane flow approximation, of major relevance for the SWIMED project. The BIGFLOW model is presently capable of modeling seawater intrusion in very heterogeneous coastal aquifers (vertically averaged heterogeneity based on hydraulic transmissivities). This particular feature of the BIGFLOW model (BF-SWIM2D) will be presented in the next deliverable D7: Seawater Intrusions Models Î (see excerpts inserted below)...
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3. BIGFLOW FOR 2D SEAWATER INTRUSION MODELING (BF-SWIM2D) 3.1.
Formulation of BF-SWIM2D (overview)
The BF-SWIM2D module is based on the vertically integrated GroundWater (GW) flow equation, along with “sharp interface” approximations for simplying seawater intrusion modeling while retaining the plane view heterogeneity of real coastal aquifers. The sharp interface approximation used in BF-SWIM2D assumes that seawater and freshwater are immiscible fluids, and relies on pressure equilibrium relations to close the resulting system of equations (Ghyben-Herzberg). Again, it is emphasized that these assumptions are implemented in the framework of heterogeneous aquifers in plane view. The sharp interface assumption may be acceptable if salt diffusion phenomena can be neglected. One possible justification is that, over time, salt diffusion may be overwhelmed by other hydrological processes. That is, the (neglected) diffusion width remains smaller than the length scale of hydrodynamic dispersion due to both aquifer heterogeneity and temporal fluctuations (rainfall events, pumping/irrigation cycles, tides, seasonal cycles, etc). Thus, if spatial distributions and temporal fluctuations of hydrologic inputs can be explicitly represented and simulated in sufficient detail, then (as a counterpart) the local mixing of salt by molecular diffusion and/or sub-grid dispersion can be neglected. The Dupuit-Boussinesq approximation in freshwater implies that freshwater flow is quasihorizontal with vertically hydrostatic pressure profiles at all points (x,y). That is, freshwater velocity is quasi-horizontal and vertically uniform. If the free surfaces and the substratum slopes are small enough (except perhaps locally), neglecting vertical velocities of freshwater may be acceptable. Finally, the seawater itself, i.e. the seawater wedge, is assumed totally hydrostatic (quasiequilibrium approximation for seawater). The pressure equilibrium relation used in BF-SWIM2D is derived from these assumptions, taking into account pressure continuity at the salt/fresh interface, and also, taking into account the existence of a freshwater outflow face at the coast, beneath the sea surface (Figure B).
3.2.
Formulation of BF-SWIM2D (equations)
The resulting closure relation is a modified version of the Ghyben-Herzberg equation with a nonzero outflow face (δZ): Z SALT = Z SEA −
(H
− Z SEA )
ε
− δ Z where ε = ρS − ρF ≈ 1 . 40 ρF
The vertical depth δZ of the freshwater outflow face at the shoreline is an external parameter, typically much smaller than aquifer thickness. It can be evaluated from an exact solution of seawater intrusion in a vertical slice in a homogeneous confined aquifer (without resorting to planar flow approximations).
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Figure B: Illustration of BF-SWIM2D freshwater/seawater configuration, showing a schematic view of a seawater wedge (sea level ZSEA shown at right). The sea intrudes into a free surface freshwater aquifer (shown at left), delimited by the seawater interface ZSALT(x,y) and the substratum ZINF(x,y). The magnifyer at right shows a zoom on the outflow face. Finally, the generalized Ghyben-Herzberg closure relation (as expressed above) is inserted in the vertically-averaged Boussinesq flow equations for freshwater, leading to the following nonlinear system of equations (for more details, see Ababou & Al-Bitar 2004): 1.
2.
Steady-state mass conservation (freshwater) : ∂Θ = −div(Q) ∂t Darcy’s law (vertically integrated) :
3.
Freshwater transmissivity :
Q = -T (H,Z SALT ,Z INF ,x,y ) grad(H )
T=
3.3.
K(x, y) × (H − ZINF(x, y)) if ZSALT < ZINF K(x, y) × (H − ZSALT(x, y)) if ZSALT ≥ ZINF
Case of heterogeneous aquifers with K(x,y) or T(x,y)
Figure C shows a 3D plot of a seawater/freshwater interface ZSALT(x,y) obtained from the BFSWIM2D module by solving the above nonlinear equations with random-type heterogeneity of K(x,y). Note that K(x,y) represents in our 2D model the vertically averaged permeability at each location (x,y) in plane view. The figure shows a perspective view of ZSALT(x,y), H(x,y), and
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log K(x,y) for a random field K(x,y) having gauss-shaped isotropic covariance, with correlation scale λ (L/λ = 30) and lnK standard deviation σ = ln10.
3.4.
Implementation of source terms
Two types of internal source terms were developed into the general BF code. Spatially distributed source terms are used to represent time-varying/constant recharge or infiltration rates. Local source terms are used to represent pumping wells (extraction and injection wells). This new capability enabled us in particular to develop, more recently, salt water intrusion tests in the presence of pumping wells in heterogeneous aquifers using the BF-SWIM2D module. Thus, Figure D shows an example of seawater intrusion modeling with a pumping in well in a heterogeneous aquifer. However, concerning possible usage and limitations of such source terms in the presence of saltwater, see the sub-section further below entitled “Limitations”.
Figure C: Seawater interface in a randomly heterogeneous aquifer (301x301 cells), with the freshwater free surface on top. The color map is permeability K (lows in blue, highs in red).
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Figure D: Seawater interface in a randomly heterogeneous aquifer with a pumping well, shown in plane view (here, the color map and contour lines both represent ZSALT elevation). Numerical methods The BF-SWIM2D module uses the same methods available for the other 3D/2D modules of the general BF code (see Deliverable D6). Briefly, BF is based on a fully implicit finite volume formulation of flux-divergence equations in conservative form (mixed formulation). It can solve transient as well as steady state problems (either by time marching or in a single infinite time step). The computational «kernel» is characterized by very sparse, specialized linear algebra based on preconditioned conjugate gradient solvers (CG, DSCG, SIP), inter-looped with a robust modified Picard (fixed point) iteration scheme for nonlinear problems. Some special considerations were given to the non-linear aspects of the SWIM problem. Indeed, Seawater Intrusion Modeling is “doubly non-linear” because of the non-linearity of the Dupuit Boussinesq equation (head-dependent transmissivity) and because of the nonlinearity due to the interception of the salt interface by the aquifer’s substratum. This non-linearity is even stronger when the domain is highly heterogeneous (K(x,y). For these cases, we used a special iterative “re-start method” (mathematically a sigma homotopy or continuation method). Also, double precision computations needed to be implemented, rather than single precision, for highly heterogeneous stochastic aquifers. The use of such methods will be detailed in the D8 deliverable of SWIMED, which emphasizes stochastic seawater intrusion analyses.
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Data, parameters and limitations Parameters
In addition to the standard groundwater flow parameters (see Deliverable D6: GroundWater Flow Models), a small number of additional parameters needs to be defined for the BF-SWIM2D seawater intrusion module. These include (non exhaustive list) : • The saltwater/freshwater density contrast. • The height of the outflow face computed from analytical solutions. • The sea water level • Special numerical parameters used for the sigma homotopy method (to be detailed in D8). Data The data needed for running the BF-SWIM2D seawater intrusion module are boundary conditions (heads, fluxes), initial conditions (heads), permeability or transmissivity maps (generated stochastically in our case), and sink/source terms (wells and distributed recharge). Limitations • The sharp interface approximation is widely acceptable for modeling regional scale aquifers, as explained earlier. On the other hand, it is recognized that the Dupuit-Boussinesq plane flow approximation may not be applicable for some special aquifer configurations. • The sharp interface formulation with immiscible salt/freshwater and plane flow approximation, has the advantage of being much more efficient in terms of CPU time and memory allocation (dynamic allocation is used in BF-SWIM2D), in comparison with miscible salt transport models which involve solving coupled flow and transport equations. • As a consequence, the BF-SWIM2D module has the ability to address seawater intrusion problems in heterogeneous aquifers with high resolution simulations, compared to other methods. There remain, as can be expected with most methods, some numerical limitations when heterogeneity levels are extremely high (slower convergence or divergence). • Finally, a note of caution should be added concerning the use of localized sinks (such as extraction wells) in the presence of saltwater. No provision was made, thus far, to take into account the proportion of saltwater extracted by the well. Nevertheless, while the amount of salt taken up by the well remains unknwon with our model, the amount of freshwater extracted by the well can be correctly simulated provided an iterative update of the freshwater pumping rate at the well. Since this procedure has not yet been tested, the current version of the module BFSWIM2D can be considered valid in cases where the uptake of saltwater at the well is negligible. For instance, the code can be used to assess the critical pumping rate corresponding to salt breakthrough at the well.
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4. REFERENCES 4.1.
Bibliographic references
Ababou R. and A. Al-Bitar, 2004: “Random Field Approach to Seawater Intrusion in Heterogeneous Coastal Aquifers : Unconditional Simulations and Statistical Analysis”. GEOENV’04 Internat. Conf., Pre-Proceedings, 16 pp. [Book in preparation, Springer 2005]. Ababou R. and A. Al-Bitar, 2004 : Salt Water Intrusion with Heterogeneity and Uncertainty : Mathematical Modeling and Analyses. Proceedings CMWR 2004, Comput. Meth. in Water Resources, Special Session on Coastal Aquifers, 13-17 June 2004, Chapel Hill, North Carolina, USA, 12pp. Ababou R. and G. Trégarot, 2002 : Coupled Modeling of Partially Saturated Flows : MacroPorous Media, Interfaces, and Variability. Proceedings CMWR 2002, Comput. Meth. in Water Resources, 23-28 June 2002, Delft, The Netherlands, 8pp. Ababou R., Trégarot G., Bouzelboudjen M. Variably Saturated Subsurface Flow with Layers and Interfaces : Perched Water Tables and Stream-Aquifer Connection. ModelCare'96 Proc., International Groundwater Modeling Center (IGWC), GWMI Series No.96-OX, Colorado School of Mines, 10 pp., 25-27 Sept. 1996. Ababou R. and Bagtzoglou A.C. , 1993 : BIGFLOW : A Numerical Code for Simulating Flow in Variably Saturated, Heterogeneous Geologic Media (Theory and User's Manual, Version 1.1). Report NUREG/CR-6028, US Nuclear Regulatory Commission, Government Printing Office, Washington DC, USA, 139 pp., 1993. Ababou R., B. Sagar, and G. Wittmeyer, 1992 : Testing Procedures for Spatially Distributed Flow Models, Advances in Water Resources, Vol.15, pp. 181-198, 1992. Ababou R., D. McLaughlin, L.W. Gelhar, et A.F.B. Tompson, Numerical Simulation of Three Dimensional Saturated Flow in Randomly Heterogeneous Porous Media, Transport in Porous Media, 4, 549-565, 1989. Al-Bitar A. and R. Ababou, 2004 : Modeling of Seawater Intrusion in Mediterranean Coastal Aquifers. Internat. Conf. Thermal Engg. Theory & Appli., Beyrouth, Lebanon, 31 May-4 June 2004. Trégarot G. Modélisation Couplée des Ecoulements à Saturation Variable avec Hétérogénéités, Forçages, et Interfaces Hydrologiques. Thèse de Doctorat de l'Institut National Polytechnique de Toulouse (Sci. Terre et Environnement). Institut de Mécanique des Fluides. Toulouse, Mai 2000. Trégarot G., Ababou R., Larabi A. Inondations, Infiltrations, Hétérogénéités et Couplages d' Ecoulements Partiellement Saturés et Non-Saturés. 22èmes journées GFHN 25-26 Novembre 1997, Meudon, France. Bulletin du GFHN (Group. Francophone d’Hydrologie Non-saturée), N°42: Milieux Poreux et Transferts Hydriques, pp.101-109, Nov.1998. Trégarot G., Ababou R., Bouzelboudjen M. Hydrologic Modeling Under Rainfall Infiltration and Flooding : Wetting Fronts, Perched Water Tables, and Stream-Aquifer Relations. International Association of Hydrological Sciences, IAHS'97, Poster Proc., Symposium S1, Rabat, Morocco, 4 pp., 23 Avril - 3 Mai 1997.
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4.2. Internet web sites WEB SITE (URL)
WEB SITE OWNER
http://www.crs4.it/EIS/SWIMED/index.html
SWIMED project
http://rachid.ababou.free.fr
R.Ababou
http://www.imft.fr
IMFT laboratory
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