Dynamic River Network Simulation at Large Scale
∗
Frank Liu
IBM Research Austin
Austin, TX
Ben R. Hodges
Center for Research for Water Resources
University of Texas at Austin
May 17, 2012
Abstract
Fully dynamic modeling of large scale river networks is still a challenge. In this paper we
describe SPRINT, an inter-disciplinary collaborative effort between computer engineering and
hydroscience to address the computational aspect of this challenge. Although algorithmic details
differ, SPRINT draws many design considerations from SPICE, one of the most fundamental
EDA tools. Experimental results demonstrate that SPRINT is capable of simulating large river
basins at over 100× faster than real time.
1
Introduction
EDA tools are essential for designing today’s VLSI semiconductor products. One of the earliest
and most fundamental EDA tools is SPICE[1][2]. By working in conjunction with compact device
models[3][4], SPICE provides a virtual environment so that designers can rapidly assess the function
correctness, performance as well as power consumption. This “virtual prototyping” environment
not only ensures the correctness of the complex design before they are committed to the lengthy
and expensive manufacturing processes, but also provides the designers directions to optimize the
design in order to meet the specifications.
Mathematically the function of SPICE can be described as a DAE (differential algebraic equation)
solver for electronic circuits. The circuit consists of nonlinear components (e.g., BJT or MOS
transistors) and linear components (e.g., resistors, capacitors or inductors). The circuit behavior
is described by governing laws: KCL (Kirchhoff’s Current Law), KVL (Kirchhoff’s Voltage Law)
and BCR (Branch Constituent Relations). The two Kirchhoff’s circuit laws (KCL and KVL) are
derived from Maxwell’s equations. KCL states that the algebraic sum of currents flowing into any
∗ Author’s preprint of invited paper to be presented in Special Session 32 (4 p.m. Wednesday, June 6, 2012) at
the Design Automation Conference 2012. CITE CONFERENCE PAPER AS: Liu, F. and B.R. Hodges (2012), “Dynamic River Network Simulation at Large Scale,” in Proceedings of the 49th ACM/EDAC/IEEE Design Automation
Conference, June 3-7, 2012, San Francisco, USA, 6 pgs.
1
node within a circuit is zero. KVL states that the algebraic sum of voltage drops along any branch
loop is zero. BCR describes the behavior of a circuit element with respect to its branch voltages and
currents. For example, the BCR of a resistor is basically the Ohm’s Law. The BCR of multi-terminal
MOSFET devices is described by a set of complex equations within the compact model. From a
given circuit specification, by applying KCL, KVL and BCR, SPICE constructs a system of nonlinear
differential-algebraic equations by using either sparse tableau[5], or modified nodal analysis[6]. The
voltages of the circuit nodes as well as branch currents of certain devices (e.g., of inductors) are the
canonical unknowns which need to be solved. For linear circuits (with only passive devices), a linear
matrix solver is all required to solve the circuits. When nonlinear devices (e.g., MOS transistors)
are present in the circuit, modified Newton-Raphson’s method and numerical integration methods
(e.g., Gear’s methods[7]) are applied to solve the associated nonlinear DAE. Three types of analyses
are available in the original version of SPICE: steady-state (DC), small-signal AC, and transient
analysis (TR) with time-varying input excitations.
The first version of SPICE was released to public in 1971, well before Open Source Software
became a big movement. Over the years, many revisions were made, including SPICE3, which is
the first version written in C, instead of FORTRAN as in earlier versions[8]. Today SPICE is still
part of the core curriculum on circuit simulation in universities across the globe[9][10][11]. Even
over forty years after its inception, modern versions of SPICE are still an important part of the
product portfolio of many EDA software companies and are generating multi-million dollar annual
revenues[12][13][14].
The success of SPICE has inspired many other simulation projects; some of them are well beyond
the traditional boundary of electrical engineering. For example, SUGAR[15] is an open source
simulation tool for micro-electromechanical systems (MEMS). Another example is Bio-SPICE, which
is an open source framework and software toolset for Systems Biology[16]. In this paper, we describe
a simulation software package which is intended to perform dynamic simulation of large scale river
networks.
Water is essential for the life forms as we know today. Only 3% percent of earth’s water is in
the form of fresh water, mainly locked in polar icecaps and continental glaciers. River networks are
the fastest distribution conduits through the landscape for fresh water, and hence play vital roles
in municipal water supply, low-cost bulk transport of grain, agricultural irrigation, wildlife habitat,
and recreation. River networks also play a key role in flooding, either carrying away potential flood
waters or causing inundation when the carrying capacity is exceeded. Floods cause devastating loss
of lives and economic harm throughout the world. For example the Thailand floods in 2011 not only
caused hundreds of casualties and billion dollars of direct property damage to the local communities,
but also severely disrupted the electronics supply chain with implications across the globe.
Perhaps the most commonly used software for river modeling is HEC-RAS, developed by US
Army Corps of Engineers[17]. It primarily serves as a desktop tool for hydraulic engineers conducting
flow analysis and flood-plain assessments over main stem rivers or local catchments. What has not
been previously available is an efficient scalable river network simulator, which capable of simulating
not only the main river stems, but also thousands of kilometers of upstream tributaries. This paper
presents the initial development of just such a simulator.
The present work is the result of close inter-disciplinary collaboration between computer engineering and civil engineering hydroscience. Although the target applications and detailed algorithms
differ, the design philosophy and many design considerations in our simulator are strongly influenced
by SPICE. The result is an efficient river network simulator which is capable of simulating large
river basins at over 100× of real time. With the recognition to the influence of SPICE, we name
2
our software SPRINT (Simulation Program for RIver NeTworks). For the remaining portion of this
paper, we present the first-principle physics model of river networks in Section 2; the simulation and
implementation of SPRINT in Section 3, followed by some experimental results in Section 4.
2
Modeling River Networks
To fully describe the intricacies of water movement within a river channel requires the 3D NavierStokes and continuity equations. Much as in electrical circuit analyses where Maxell’s equations
(the fundamental physical laws with high computational costs) are replaced with simpler models
with lower computational costs, in river networks a simpler model is commonly used in place of the
Navier-Stokes equations. This model is named after a French mathematician, Adhémar Jean Claude
Barré de Saint-Venant, and is usually called the Saint-Venant equations (although is sometimes seen
as the St. Venant equations).
Several approximations are made to derive and apply a model based on the Saint-Venant equations. Some of these are:
• River flow is approximately 1-dimensional;
• River bed slope is relatively small;
• Effects of river bed friction and turbulence can be modeled as resistance terms.
The 1D approximation allows us to integrate 3D Navier-Stokes and continuity equations over a
river cross-section without knowledge of the exact velocity distribution. The second approximation is
part of a “hydrostatic approximation” that removes smaller-scale dynamic pressure effects. The third
approximation allows simplified empirical equations to model resistance, which we will discuss later.
Although these approximations are generally valid for large-scale rivers analyses, there are places
(such as steep mountain streams) where departures from these idealized flows may be significant.
2.1
Continuity Equation
The continuity equation describes the mass conservation in river flow. To illustrate this, consider the
cross section of a river segment in Fig. 1. For a control volume highlighted in the diagram (between
A2
A1
x2
x1
Q
Figure 1: Illustration of a river channel cross section.
x1 and x2 ), the mass of water flowing into the control volume between time instances t1 and t2 can
be written as:
Z t2
[(ρvA)x1 − (ρvA)x2 ] dt
(1)
t1
3
where ρ is the water density, v is the flow velocity. x1 and x2 defines the control volume along the
channel, and the cross-sectional area A is a variable that is a function of the water surface elevation
and riverbed geometry.
In the same time window, the change of storage within the control volume can be written as:
Z x2
[(ρA)t2 − (ρA)t1 ] dx
(2)
x1
Mass conservation dictates that the algebraic sum of the water flowing into the control volume
equals the accumulation of the water in the control volume:
Z t2
Z x2
[(Q)x1 − (Q)x2 ] dt = 0
(3)
[(A)t2 − (A)t1 ] dx +
t1
x1
where Q = vA is the flow rate. For fresh water in a river, variations of density ρ with temperature
have negligible effects on flow dynamics, hence we can drop it from both sides of the equation.
To derive the differential form of the mass equation, we follow the Taylor expansion:
(A)t1 = (A)t2 +
∂A
∂ 2 A ∆t2
∆t + 2
+ ···
∂t
∂t 2
(4)
(Q)t1 = (Q)t2 +
∂Q
∂ 2 Q ∆t2
∆t +
+ ···
∂t
∂t2 2
(5)
and
where ∆t = t1 − t2 .
Retaining only the first two terms in the approximations in Eqn. (4) and (5), then in the limit
as ∆t → 0, we have:
Z x2
Z x2 Z t2
∂A
[(A)t2 − (A)t1 ] dx =
lim
dt dx
(6)
t2 →t1 x
x1
t1 ∂t
1
and similarly:
Z
t2
x2 →x1
Z
t2
Z
x2
[(Q)x2 − (Q)x1 ] dt =
lim
t1
t1
x1
∂Q
dx dt
∂x
Hence the mass conservation in Eqn. (3) becomes:
Z x 2 Z t2 ∂A ∂Q
+
dx dt = 0
∂t
∂x
x1
t1
(7)
(8)
or the commonly known continuity equation in differential form:
∂A ∂Q
+
=0
∂t
∂x
(9)
When there is lateral inflow to the channel, the continuity equation becomes:
∂A ∂Q
+
= ql
∂t
∂x
where ql is the lateral inflow per unit length along the channel.
4
(10)
2.2
Dynamic Equation
Although continuity can be used with empirical models for simplified representation of river networks,
the correct dynamical solution of the Saint-Venant equations requires momentum conservation. The
flows into/out of a control volume carry momentum, which is also affected by four forces on the
control volume: the water pressure upstream and downstream of the control volume, the pressure
from the river banks, gravity, and frictional force developed from the river bed and internal fluid
shear. The relationship between forces and momentum is governed by Newton’s second law of
motion, which reduces to the dynamic Saint-Venant equation for 1D flow. It can be presented as:
∂ Q2
∂h
∂Q
+
+ gA
= gA(S0 − Sf )
(11)
∂t
∂x A
∂x
where g is gravity, h is the depth. S0 is the slope of the river bottom. Sf is the friction slope which
we will discuss in the next subsection. Note that unlike the continuity equation in Eqn. (10), the
dynamic equation has multiple nonlinearities, with variables Q, A Sf and h. The derivation for Eqn.
(11) can be found in many hydraulics books, e.g. [18].
2.3
Friction Term
The frictional forces that extract momentum and energy from the flow are represented by empirical
resistance models. A commonly-used model (adopted herein) is the Chézy-Manning formula, which
was introduced by an Irish engineer Robert Manning[19]:
v=
Q
1 2/3 1/2
= Rh Sf
A
n
(12)
where Rh is the hydraulic radius, which is the ratio of the wetted cross section area (A) and the
“wetted perimeter” (i.e. the distance measured along the river bottom across the river). n is
known as “Manning’s n,” serving as an empirical friction coefficient that may have a wide range of
uncertainty, so serves as a calibration parameter.
2.4
Saint-Venant Equation
To summarize the behavior of a river segment is modeled by nonlinear Saint-Venant equations:
(
∂Q
∂A
= ql
∂t + ∂x
(13)
2
∂Q
∂ Q
∂h
= gA(S0 − Sf )
∂t + ∂x ( A ) + gA ∂x
where the nonlinear friction slope is described by the Chézy-Manning equation:
Sf = n2
Q2 1
A2 R4/3
(14)
h
There are two independent variables in Saint-Venant equations, the flow rate Q and the wetted
area A. Once these two quantities are known, the other quantities, e.g., water depth h, flow velocity
v, are dependent functions of Q, A and river cross-section geometry (which may have substantial
variability throughout a network).
5
A key difference between electric circuits and river networks is in the upstream propagation of
information in the latter. Although mass flows downstream in a river, information of downstream
events (e.g. a collapsed bridge that obstructs the flow) propagates upstream as a rising/falling
water depth (changing A). Thus, the directionality of information propagation is both upstream and
downstream through the network. Thus, the Saint-Venant equations must be solved as a distributed
model and cannot be replaced with an equivalent lumped model as used in electronic circuits.
3
Simulation and Implementation
The Saint-Venant equations in Eqn. (13) are a coupled set of nonlinear, time-varying, partial differential equations that are solvable by either explicit or implicit numerical methods. Over the years,
there have been plethora of explicit methods proposed. For example, the leap-frog method in [20],
the popular Lax-Wendroff method[21] in [22], and an modified Lax-Friedrichs method in [23]. The
biggest advantage of explicit methods is that there is no need to construct and solve the nonlinear
matrix otherwise required for implicit solution of the Saint Venant equations. However, this benefit
comes at the cost of limited step size. Explicit methods have to meet the Courant-Friedrichs-Lewy
(CFL) condition[24] in every element of a network; a constraint that can result in time steps on the
order of seconds in a river simulation.
On the other hand, implicit methods are not stability limited by the CFL condition. A moderate
step size (CFL ¡ 10) is still preferred for accuracy, but localized high CFL conditions are tolerable.
Many practical hydrology software packages use implicit methods[18] due to their robustness. The
most commonly used implicit methods to solve Saint Venant equations are the method on staggered
grid in [25], and the method on collocated grid in [26]. We focus on the four-points scheme from
[26].
3.1
Four-Point Scheme
The four point scheme approximates spatial and temporal differences by using the average of four
neighboring points in the x − t plane[26]. It is generally over-damped and works very well for slow
moving flow conditions.
In a simplified version of the four-point scheme, a time difference is approximated by:
1
∂
n
f (x, t) '
(f n+1 − fj+1
+ fjn+1 − fjn )
∂t
2∆t j+1
(15)
where the subscript j represents spatial indices and the superscript n represents temporal indices.
The spatial difference is approximated by:
1
∂
f (x, t) '
(f n+1 − fjn+1 )
∂x
∆x j+1
(16)
and we take the average of two adjacent points to approximate the function value itself.
f (x, t) '
1 n+1
(f
+ fjn+1 )
2 j+1
(17)
Applying these discretization formulae, we can discretize Saint-Venant equations in Eqn. (13).
At each time point n + 1, the discretized continuity and dynamic equation specify a nonlinear
relationship between unknowns (Q and A) at time point n + 1, as well as at time point n.
6
3.2
Nonlinear Equation Solver
One of the most popular method to solve nonlinear equations is the Newton-Raphson’s method. To
compute the solution to a nonlinear problem F(x) = 0, the method iteratively improve solution x
from a starting point x0 by using the local gradients:
xk+1 = xk − α · (
∂F −1
) ·F
∂x
(18)
where α is a damping factor.
Although Newton-Raphson’s method is easy to implement, some cautions need to be exercised.
One issue is the convergence. Since Newton-Raphson’s method is a local search algorithm, it may
fail to converge. In transient simulations this is rarely an issue because the solution at time n + 1
is close to the solution at time n, which can be used as the starting point. However, for DC
solve, convergence strongly depends on the selection of initial starting points; hence convergence
could fail. In SPICE, this problem is partially solved by using a bounding algorithm to limit
the operating range change of nonlinear devices. For difficult circuits, more advanced homotopy
methods are still required circuits[27]. For river flow simulation, again this issue mainly arises for a
steady-state solution with limited initialization information. We address this issue by first solving
the Saint-Venant equations approximately, so that we can place a reasonable starting point for
iteration. Secondly we implement a bounding algorithm based on fluid dynamics principles to limit
the solution range. The combination of the two approaches provides good results.
Another issue with Newton-Raphson’s method is that the functions have to be C 1 (i.e., the
functions should be continuous themselves and also have continuous first-order derivatives with
respect to unknowns). Depending on the complexity of the river cross-section shape, this is not
always the case. We address this issue by approximating the river channel data into a C 1 function
template. The approach not only solves the function continuity issues, but also help us to achieve
considerable speed-up in the simulation.
As shown in Eqn. (18), Newton-Raphson’s method requires the factorization of the Jacobian
matrix ∂F
∂x . The size of the matrix is 2m where m is the number of the computational nodes, which
can be quite large for a continental-scale river network. However, as we have observed in SPICE,
the runtime complexity of factorizing a sparse matrix is usually O(N 1.2 ). In other words, it is
only slightly super linear, versus the cubic complexity of factorizing dense matrix. It can be shown
that the stencil of the Jacobian matrix in Saint-Venant equations using the four-point discretization
scheme is only 4. Given the continuing rapid improvement of computer hardware and the efficiency
of today’s sparse linear matrix packages such as [28][29], the capacity required and the runtime for
the factorization of the Jacobian matrix is not a concern.
3.3
General Simulation Flow
To illustrate the simulation flow, we use the symbolic three-branch river network in Fig. 2. The river
network is first partitioned into computational nodes indicated by red dots. At each computational
node, the river cross section description is used as input, along with the friction coefficients (Manning’s n) and the bottom slope S0 . Next the computational nodes are connected from upstream to
downstream following discritized Saint-Venant equations.
At the junction point of the three branches, two linear relationships are specified. The first
equation is mass conversation: the sum of the flow rate of two tributaries should equal the flow rate
7
Qs
ql
Qs
ql
ql
h
Figure 2: Illustration of computational nodes on a river network
the downstream stem. Another linear relationship is used to specify the relative contributions of
two upstream branches to the downstream branch.
Boundary condition flow rates are the forcing input Qs at two upstream points, as well as the
possible lateral inflows along the channels. A downstream boundary condition in the form of the
depth at the most downstream node is applied. This value is typically the depth where a river enters
a lake or ocean.
These procedures are used to build an entire river network with any number of tributary branchings. Once the network is constructed, we first perform steady-state solution with fixed forcing
terms. We then apply the time-varying forcing terms to compute unsteady solutions. The solution provides the depth and flow velocity at every computational node for every time point in the
simulation.
3.4
SPINT Implementation
SPRINT is implemented in C++ in a modular design approach. Besides the nonlinear and matrix
solution methods described in the previous subsections, it also has many functionalities to enhance
runtime performance and robustness, as well as other facilities such as model pre-processing and
topological checking of the river networks. To facility the deployment in a cloud-based simulation
environment, SPRINT also has an http front-end which enables remote simulation.
4
Experimental Results
In this section, we present some experimental results of SPRINT.
8
4.1
Comparison with Analytical Solutions
In this experiment, a river segment with rectangular cross section is used. Due to its simplicity,
the steady-state solutions can be calculated analytically, which are compared with the steady-state
solutions computed by SPRINT. The depth at each node is plotted in Fig. 3. The difference is
negligible at the resolution of the graph.
20
Analytical Solution
Simulation
19
Flow Depth (m)
18
17
16
15
14
13
0
50
100
150
200
250
300
Distance from pour port (km)
350
400
450
500
Figure 3: Comparison between SPRINT output and analytical solutions. Y-axis represents depth.
4.2
A Small Creek in Central Texas
This experiment is a creek in central Texas. It has three branches with the total length of approximately 7.2 miles. The simulation takes merely a few seconds on a common desktop computer for a
5-day event. Fig. 4 shows the simulated results at a computational node where observed gauge data
are available. Note that we haven’t gone through detailed calibration process so the simulated results will not match the observed depth data exactly. However, all the key waveform characteristics
of the observed data are captured by the simulation results.
4
observed
model output
3.5
3
2.5
2
1.5
1
0.5
0
5
10
15
20
25
30
35
40
Figure 4: Comparison between SPRINT output and observed depth data. Note no calibration was
performed.
9
4.3
A River Basin in Central Texas
This experiment is a relatively large river basin in central Texas. It consists of over 3, 500 river
branches with the total length of over 9, 000 miles. The whole river network is modeled by over
110K nodes and the simulation is performed on a regular desktop computer. It takes about an hour
to simulate a 12-day event, which translates into about 100× speedup of real-time. Three snapshots
of the simulated results are presented in Fig. 5. From the top figure, one can see the approximated
diagram of the river networks. Note that many small tributaries are not included in the figure to
avoid over crowding the graph. The width of each segment approximates the flow rate, while the
color represents relative depths.
Figure 5: A river basin in central Texas. Three snapshots show the simulation output at different
time points. The width represents the relative flow rate in the river branches. The color represents
the relative depth.
10
5
Final Remarks
SPRINT represents an inter-discipline collaboration between computer engineering and civil engineering hydroscience. Although its algorithms cannot be directly applied, SPICE has been influential
on the design of SPRINT, particularly in terms of design principle and design philosophy. Experimental results show that our tool can achieve considerable capacity and performance. We hope our
effort will be beneficial to hydrology and hydraulic modeling community, just as SPICE has long
lasting impact on the semiconductor industry.
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