Building a simple delta numerical model: Part III

In this part of “Building a simple delta numerical model”, we’ll simply develop a module for calculating sediment transport at all locations within our model domain. This is naturally a critical piece of the model to get right, because the magnitude of sediment transport that occurs within the delta system will have a direct correlation with the magnitude of geomorphic change to the delta system. Just as before, we’ll be working our module for the Mississippi River, and so we will use the Engelund and Hansen, 1967 formulation for bed-material load sediment transport.

It is worth noting here that there are countless sediment transport equations published within the literature, with applicability for different scenarios, covering the entire range of alluvial rivers, the shelf environment, to turbidity currents. A nice review of some relations is given by Garcia, 2008, who produces the plot below with six selected relations, where τ* is shear stress (Shields number) and q* is a dimensionless sediment transport.

The only points I hope to make with this figure are that stress and transport are nonlinearly related, and that many of the relations are quite similar. In fact, the majority of sediment transport equations are based on an excess-shear stress formulation, where, as shown above, the sediment transport prediction scales with the shear stress. The Engelund and Hansen, 1967 equation is given in its dimensional form by (Equation 1)

where qs is the sediment transport per unit width (m2/s), β is an adjustment coefficient discussed below, R = 1.65 is submerged specific gravity of the bed sediment, = 9.81 m/s2 is gravitational acceleration, is the grain size in question, Cf is the coefficient of friction, ρ = 1000 kg/m3 is the fluid density, and τ = ρ Cf U2 is the fluid shear stress, where is depth-averaged fluid velocity.

The solution to this equation is obtained by simple algebra, and so I will display the code calculations below without any explanation.

``` U = Qw ./ (H .* B0); % velocity```

```function [qs] = get_transport(U, Cf, d50, Beta) % return sediment transport at capacity per unit width % 1000 = rho = fluid density % 1.65 = R = submerged specific gravity % 9.81 = g = gravitation acceleration u_a = sqrt(Cf .* (U.^2)); tau = 1000 .* (u_a.^2); qs = Beta .* sqrt(1.65 * 9.81 * d50) * d50 * (0.05 / Cf) .* ...```
```        (tau ./ (1000 * 1.65 * 9.81 * d50)) .^ 2.5; end```

Let’s first simply examine the behavior of Equation 1, in its dimensionalized form, for the Mississippi River. We need to define the grain size and adjustment coefficient to use in our model; we’ll follow from the literature and use 300 μm for the grain diameter, assuming this makes up the entirety of the bed, and β = 0.64 an empirical adjustment (Nittrouer et al., 2012; Nittrouer and Viparelli, 2014).

Evaluating the equation over a range of velocities from 0 to 3 m/s, we find that the nonlinear behavior shown in the above figure is also true for the Engelund and Hansen formulation.

Now, let us implement this into our delta model. Following from earlier, the flow depth and velocity are related by the cross sectional area and discharge, where for a rectangular cross section, the velocity = Qw / H B. We therefore can take our calculation of flow depth from the previous lesson, and solve for velocity at each location. We then simply plug this into the relation for τ given above and solve Equation 1.

The bottom plot reveals the behavior of the flow velocity calculated through the backwater region (shown in red), and the resulting sediment transport calculated (shown in green). The nonlinearity between stress (i.e., velocity) and transport is again revealed by the change in slope of the lines through the backwater region. Following from left to right along the green curve, it becomes obvious that there is a decrease in transport downstream. If there is more sediment being transported upstream than downstream, then there must be sediment that is deposited to the bed somewhere along the channel bed between ‘upstream’ and ‘downstream’.

This thought experiment represents one half of the famous Exner equation, which we will explore in the next part, and will provide the last module to complete our delta model. My complete code for the sediment transport calculation in our backwater delta model can be found here.

References

1. Engelund, F. & Hansen, E. A monograph on sediment transport in alluvial streams. (Technisk Vorlag, Copenhagen, Denmark, 1967).
2. García, M. H. Sediment Transport and Morphodynamics in Sedimentation engineering processes, measurements, modeling, and practice 21–163 (American Society of Civil Engineers, 2008).
3. Nittrouer et al., Spatial and temporal trends for water-flow velocity and bed-material sediment transport in the lower Mississippi River. 2012. Geological Society of America Bulletin, 5 (Table 1).
4. Nittrouer, J. A. & Viparelli, E. Sand as a stable and sustainable resource for nourishing the Mississippi River delta. Nature Geoscience, 7, 350–354 (2014).

This material is based upon work supported by the National Science Foundation (NSF) Graduate Research Fellowship under Grant No.145068 and NSF EAR-1427177. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Building a simple delta numerical model: Part II

Taking the simpler form of the backwater equation, that is, assuming that width does not vary over the channel reach, we find an expression for changing flow depth along the channel streamwise coordinate x: (Equation 1)where S is the channel bed slope, Cf  is the dimensionless coefficient of friction, and Fr is the Froude number: (Equation 2)

A couple quick notes about the overall delta model we are developing herein:

1. The code for this entire model will be developed in Matlab syntax, however, the math is all just math, and the code could easily be translated to another programming language.
2. The model will be set up for the conditions of the Mississippi River, to what I feel are the best constraints published in the literature. Surely not everyone will agree with some of my choices.

We set up our model domain by defining some parameters:

```L = 1200e3; % length of domain (m) nx = 400; % number of nodes (+1) dx = L/nx; % length of each cell (m) x = 0:dx:L; % define x-coordinates start = 63; % pin-point to start eta from (m) S0 = 7e-5; % bed slope eta = linspace(start, start - S0*(nx*dx), nx+1); % channel bed values H0 = 0; % fixed base level (m) Cf = 0.0047; % friction coeff ```

Each of these terms are necessary for defining the boundary conditions the backwater calculation requires. We will additionally need to define the water discharge (Qw), and it’s width averaged derivative qw = Qw / B, where B is the channel width. We’ll use a fixed width of 1100 m (Nittrouer et al., 2012) and a discharge of 10,000 m3/s; since width is fixed in our model, qw = 31.8182 m2/s.

Finally, to solve the above backwater equation, we’ll need to find the local slope at all nodes — our bed holds a constant slope so we could just skip this step and send S0 directly into the calculation, but we’ll be wise beyond our present experience, and prepare our calculation now for the future when our bed slope is constantly changing in time and space. A simple slope calculation is completed by a central difference calculation, applying forward difference and backward difference calculations at the domain upstream and downstream boundaries, respectively.

```function [S] = get_slope(eta, nx, dx) % return slope of input bed (eta) S = zeros(1,nx+1); S(1) = (eta(1) - eta(2)) / dx; S(2:nx) = (eta(1:nx-1) - eta(3:nx+1)) ./ (2*dx); S(nx+1) = (eta(nx) - eta(nx+1)) / dx; end```

As described in the previous section, we will use a predictor-corrector scheme to find a solution to Equation 1. This is not meant to be a comprehensive guide on solving ODEs, and so I will only summarize the method here.

1. We first can evaluate the flow depth H at the downstream boundary, where elevation is fixed (H = H0 – η = 21 m).
2. Then we predict the value at the next upstream node by solving Equation 1 for dH/dx at our known downstream-most node, using the inputs described above and H and S at the known node. This yields dH/dxp = 6.5784e-05.
3. Multiplying dH/dx by dx = 3000 m and subtracting this change from the known H at the downstream-most node, we find our predicted value Hp for the next upstream node.
4. We now repeat the evaluation of Equation 1 with the same inputs except using our new Hp value in Equation 2. This yields our corrected change in flow depth over x:
d
H/dxc = 6.5663e-05.
5. We then combine the results of these two predictions in flow depth change to give our best solution to the ODE Equation 1 at the downstream node. We combine the values using the trapezoidal rule for Hi = H(i+1) – ((0.5) * (dH/dxp + dH/dxc) * dx), giving a predicted change in flow depth of 0.1972 m, for a depth at Hi of 20.8028 m (it was 21 m at the downstream node).
6. repeat steps 2-5 solving for each node moving up the model domain until the final node is determined.

This method is fairly stable for most discharges, but may fail when discharge is very low. In thsi case, the backwater region becomes condensed, and the rate of change in H becomes very large over a very small distance. My code for the predictor corrector implementation can be found as the get_backwater_fixed() function within the complete backwater_fixed model. Below is a gif showing the results of a range of input discharges to our model.

and a still multi-surface example to demonstrate the variability.

The low and moderate discharge water surface curves in the above figure (solid and dashed lines) demonstrate the classical convex-up (aka M1, e.g., Chow, 1959) condition that characterizes lowland delta systems during most of the time. The third, high discharge, water surface profile exemplifies a convex-down water surface profile. This condition, called the M2 condition has been recently demonstrated to have importance to the cyclic processes of avulsion and delta-lobe growth (e.g., Chatanantavet et al, 2012; Ganti et al., 2016).

Building a simple delta numerical model: Part III

References:
1. Nittrouer et al., Spatial and temporal trends for water-flow velocity and bed-material sediment transport in the lower Mississippi River. 2012. Geological Society of America Bulletin (Table 1).
2. Chow, Ven Te. Open Channel Hydaulics. N.p., 1959. Print. McGraw-Hill Civil Engineering Series.
3. Chatanantavet, Phairot, Michael P. Lamb, and Jeffrey A. Nittrouer. Backwater Controls of Avulsion Location on Deltas. 2012. Geophysical Research Letters 39.1
4. Ganti, V., A. J. Chadwick, H. J. Hassenruck-Gudipati, B. M. Fuller, and M. P. Lamb. Experimental River Delta Size Set by Multiple Floods and Backwater Hydrodynamics. 2016. Science Advances 2 (5).

This material is based upon work supported by the National Science Foundation (NSF) Graduate Research Fellowship under Grant No.145068 and NSF EAR-1427177. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Building a simple delta numerical model: Part I

This is the first in a series I’m going to write over an undetermined-number-of-parts series where we will develop all the pieces to a simple delta numerical model. I’ll outline all the pieces here while I work up an interactive version with a GUI in Matlab, and we write pieces of the code together within the posts.

——————————————————————————–

The water depth (H) over a channel bed with a fixed-elevation downstream boundary can be calculated by a mass-and-momentum conservative derivation following from the full Navier-Stokes equations (e.g., Chow 1959). The so-called backwater equation, in one form, is presented as (Equation 1)
where H is the flow depth, x is the streamwise coordinate of the flow, S is the channel bed slope, Cf  is the dimensionless coefficient of friction (corresponding to form drag in the channel bed and walls (i.e., roughness and bedforms)), and Fr is the Froude number, which scales with the ratio of inertia of the flow to the gravitational field driving the flow and can be given as (Equation 2)where qw = Qw is the width averaged water discharge, and g is the gravitational constant. Equations 1 and 2 are derived for a width-averaged, constant flow width scenario.

Equation 1 is useful to the research we do as scientists of lowland depositional systems because it predicts changes in flow depth through a delta system. The equation is solved with a boundary condition of a known flow depth (H) at the fixed-elevation downstream “receiving basin” boundary. For variations in river conditions (i.e., river water discharge), the surface elevation of receiving basins of large lowland rivers are effectively fixed, therefore the equation can be used to estimate a water surface over the lower reaches of the rivers we study for any input discharge conditions!

With a calculation of H, we can then use a conservation of water equation Qw = U A (where U is flow velocity and A = B H is channel cross sectional area for an approximately rectangular channel), to evaluate the flow velocity at all points along the channel. Calculation of U in turn allows for calculation of sediment transport, and ultimately sediment deposition — but more on these parts in future parts of this series!

Equation 1 is valid for a solution where width (B) is fixed over the downstream length of x. Where width is roughly constant, say within the lower reaches of a channel we can use this simple form of the backwater equation. However, where width is not constant over all x (i.e., dB/dx != 0), say in the river plume beyond the channel mouth, we must retain an addition term from the derivation from the Navier-Stokes equations to account for this change in width. Since we’re using computers to solve the equations, and the dB/dx term is basically only one extra gradient calculation, this will an easy addition to our model in the future. I’ll write a separate post and link it here with a solution for that version of the backwater equation as well.

So how do we solve Equation 1? Let’s see what we know: the elevation of the receiving basin is fixed at z = 0. Therefore, if we know the channel bed profile, we know at the downstream boundary for ALL possible values of Qw. We can evaluate Equation 1 from the known downstream boundary, and knowing all variables on the right hand side at this boundary, calculate the change in flow depth moving upstream over the channel bed (dH/dx)!

The solution to Equation 1 (or in its slightly more complex form for a varying width domain) can be numerically estimated pretty easily by a predictor-corrector scheme. We’ll work up a calculation for a fixed-width form of the backwater equation (Equation 1) in the next post, and then proceed to use this as one module of our simple delta model.

Building a simple delta numerical model: Part II

This material is based upon work supported by the National Science Foundation (NSF) Graduate Research Fellowship under Grant No.145068 and NSF EAR-1427177. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Dietrich Settling Velocity Matlab code

Just about two years ago, I published a code for calculating the settling velocity of a particle in water by the Dietrich, 1982 method. I later realized an error in the code that made it produce erroneous estimates. I replaced it with a corrected version of the code written in R, that you can still find on my Github page, but won’t link here. I’ve taken down the old Matlab code and unpublished the original post.

Since the first publication two years ago, I’ve dramatically improved my technical coding skills, and, just as importantly, learned a ton about how to share code with others. A new version of the code can be found linked a the end of this post, but first some philosophy about coding.

As scientists, we want to be able to reliably reproduce results and we have an attraction to openness and access to knowledge, as such, we inevitably share code with one another. I’ve shared a good bit of my code thus far in my career, and I’ve made a few notes about the matter below:

1. It is better to organize your code in a way that may not be the most succinct (when speed is not an issue) while improving the readability of the code. In essence, making your code readable to those unfamiliar with it, so they can follow along with your logic.
2. Comment everything. It is so easy to get into the groove of writing something and blow through it until it’s finished and then never return to clean up and document the code. When others (or even you) go back to look at it, simple comments explaining what a line does go miles further than a detailed explanation in the documentation.
3. Write your code to withstand more that you are currently using it for. What I mean by this is to not write code that works only when you are manipulating this particular dataset or working this particular problem, but code that will work over a broader range of data or can be applied to another problem. Along the same line, use explicit values as sparingly as possible, e.g., if X = [1:12] is a 12 element vector and you want the middle value, do not use X(6), but instead use X(length(X)/2); or slightly better yet, X(ceil(length(X)/2)) to at least not throw an error and give you the first of the two middle rows if X is an odd number of elements in length.

Anyway, the code is uploaded to my Github and you can find it there. The documentation output on a help get_DSV call in Matlab is produced below.

References:
Dietrich, W. E. Settling velocity of natural particles. Water Resources Research 18, 1615–1626 (1982).

This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No.1450681. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors(s) and do not necessarily reflect the views of the National Science Foundation.

Yellow River shoreline traces

For some ongoing research I needed a set of shoreline traces of the Yellow River delta that I could measure some properties of to compare modeling results to. My model predicts delta lobe growth rates, for example the growth of the Qingshuiguo lobe on the eastern side of the delta from 1976 to 1996.

early growth stage of Qingshuiguo lobe of the Yellow River delta, figure from van Gelder et al., 1994.

It’s a little early for me to release the data I’ve collected from the shoreline traces (n=296), or the code I used to generate the traces — but below is a figure mapping all the shoreline traces I have from 1855 to present. My model can also predict rates of delta system growth (i.e., growth over many lobe cycles) and I have generated data from these traces that I can compare to as well. I like the below figure, because I think it nicely demonstrates the rapid reworking of delta lobes (yes even on “river dominated” deltas!); evidenced most clearly by the retreat of the Diakou lobe to the north side of the delta.

This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No.1450681. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors(s) and do not necessarily reflect the views of the National Science Foundation.

Yellow River (Huanghe) 2016 survey plans

I’ll be departing soon for my second field campaign on the Yellow River, China. We traveled there around the same time last year to generate a dataset we could begin to explore and develop research ideas; now, a year later, we will return with a more focused plan to gather the data we need to address our research questions. We will bring with us a suite of sophisticated data collection equipment. Below is a terrible sketch I made to demonstrate most of the data we will collect at each of our stations located on the river.

Pretend that the grey thing floating on the water might look like a boat, viewed from the back. We aim to characterize the mechanisms for sediment transport within the Yellow River at various stages of a flood discharge curve. We use a point-integrating sediment sampler to collect numerous suspended sediment and water samples from various depths in the water column (yellow stars). In order to compare the compositions of suspended material to its source (i.e., the bed), we use a Shipek grab sampling device to collect sediment from the bed for our analysis. Deployed off the other side of the boat, we will can characterize the velocity structure of the water column through the use of an aDcp (acoustic velocity profiler) and a mechanical propeller driven velocimeter for near-bed measurements where the aDcp may lack resolution.

This setup represents only a portion of our survey plans, of course we’ll be there for six weeks that will generate a diverse and (hopefully!) comprehensive dataset for our future work.

GSA abstract acceptance!

A little while ago, I heard back from the annual conference planners for Geological Society of America that my abstract submission was accepted for presentation! I’ll be presenting the following poster at GSA 2015 in Baltimore in early November.

Paper No. 311-3
Presentation Time: 9:00 AM-6:30 PM

SPATIOTEMPORAL CHANNEL-BED EVOLUTION PATTERNS OBSERVED FOR THE HUANGHE (YELLOW RIVER), CHINA: IMPLICATIONS FOR EVALUATING SYSTEM RESPONSE AND COMPLEXITY TO EXTERNAL PERTURBATIONS

MOODIE, Andrew J.1, MA, Hongbo1, NITTROUER, Jeffrey A.2, CARLSON, Brandee N.3 and KINEKE, Gail C.4, (1)Earth Science, Rice University, 6100 Main Street, MS-126, Houston, TX 77005, (2)Dept of Earth Science, Rice University, 6100 Main Street, MS-126, Houston, TX 77005, (3)Department of Earth Science, Rice University, 6100 Main Street MS-126, Houston, TX 77005, (4)Earth and Environmental Science, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA 02467

The Huanghe (Yellow River), China, is an end-member fluvial system because it is characterized by very high sediment loads, variable water discharge over annual and decadal timescales, and substantial anthropogenic influences. Over the past several centuries, the channel bed of the Huanghe has aggraded significantly, resulting in several catastrophic avulsions. In order to mitigate channel-bed aggradation, in 2002, Chinese engineers implemented an annual Water and Sediment Discharge Regulation (WSDR) period when, over a two-week period, a controlled flood wave designed to erode the channel bed is triggered by releasing water from a network of upstream dams and reservoirs. Here, we present the results of an extensive time-series field survey that measured fluid and sediment properties to assess the effect of a WSDR flood wave on a portion of the river located ~80 km upstream of the ocean outlet. This controlled experiment represents a classic external perturbation to a fluvial-deltaic system; the morphodynamic responses, however, were anything but predictable. For example, multibeam bathymetric data illuminate the uniqueness of channel-bed properties, including a broad, flat, and shallow sand-bed devoid of a thalweg or bedforms in the straight-reach segments of the Huanghe. Alternatively, in bend segments, the channel-bed deepens significantly, and longitudinal dunes emerge as the dominant bed features. Interestingly, as the flood wave progresses over time, the channel bed in bend segments rapidly erodes upstream and downstream, thus propagating dunes in both directions. In order to further ascertain linkages between the fluid flow, sediment transport, and morphological evolution of the bed, we couple water velocity, suspended sediment concentration, and bedload transport datasets to standard physical models. The results are then used to constrain a numerical model that assesses stability of the channel bed over the duration of the flood wave. In effect, this study shows the inherent unpredictability that emerges between contemporaneously evolving “external” (allogenic) and “internal” (autogenic) processes. Finally, the results of this field and numerical investigation are used to constrain a prediction for the long-term stratigraphic development of the Huanghe fluvial-deltaic system.

One year of graduate school travel

The 24th of August 2015 marks my one year anniversary of beginning graduate school. It has been an awesome year, filled with new science, great friends, and lots of travel. I kept a record of where I slept every night for the past year to see just how far I made it. I visited three continents (plus Central America), six states, and 32 different cities. I spent just over 30% of the year away from Houston (112 nights). I got to see some really amazing places on the planet. It puts into perspective for me how just how great of a gig being a grad student is — get paid to do cool science.

Below, I made some maps to show where I went. The color of a dot represents when I traveled there during the year (Aug 24 2014 to Aug 24 2015) according to traditional rainbow color order. The maps were made in GMT using this script.

Blow ups:
global map
US map
Gulf Coast map

Cheers to what next year brings!

Back from China!

I’ve finally returned from China after a 6-week field campaign to get a first round look at my field location. We worked hard every single day after our gear was released by China Customs officials on the Yellow River Delta doing Vibracoring for Brandee Carlson’s research and doing Multibeam and river survey work for my work. Below you can see me running our Multibeam setup aboard our research vessel. More pictures and descriptions to come!

River hysteresis and plotting hysteresis data in R

Hysteresis is the concept that a system (be it mechanical, numerical, or whatever else) is dependent of the history of the system, and not only the present conditions. This is the case for rivers. For example, consider the following theoretical flood curve and accompanied sediment discharge curve (Fig. 1a).

Figure 1. Theoretical plots to demonstrate the hysteresis of a river.

With the onset of the flood, the increased sediment transport capacity of the system entrains more sediment and the sediment discharge curve (red, Fig. 1a) rises. However, the system may soon run out of sediment to transport (really just a reduction in easily transportable sediment), and the sediment discharge curve decreases although the water discharge curve remains high in flood.

In Fig. 1b, the sediment discharge and water discharge are plotted through time, a typical way of observing the hysteresis of a system. Note that for the rising limb and falling limb of the river flood, the same water discharge produces two different sediment transport values.

Now, let’s imagine that we want to investigate how important the history of the system is to the present state of our study river. You can grab the data I’ll use, here. This is data from one year of flow on the Huanghe (Yellow River) in China, and it has been smoothed with a moving average function to make the hysteresis function more visible.

Making the plot

It is easy enough to plot a line with R (the lines function) but with a hysteresis plot, it is important to be able to determine which direction time is moving forward along the curve. For this reason we want to use arrows. So we plot the line data first, with:

```lines(df[1:366,"Qw"],df[1:366,"Qs"],
lwd=2,
)
```

and then using a constructed vector of every 22nd number, we plot an arrow over top of the lines using:

```s <- seq(from=1, to=length(df[,"Qw"])-1, by=22)
arrows(df[,"Qw"][s], df[,"Qs"][s], df[,"Qw"][s+1], df[,"Qs"][s+1],
length=0.1,
code=2,
lwd=2
)
```

Finally, with a few more modifications to the plot (see the full code here), we can produce Fig. 2 below. This plot is comparable to the theoretical one above.

Figure 2. Hysteresis plot from actual data.

Using the green lines and points, I have highlighted the observation that for the rising limb and falling limb of a flood, there can be substantially different sediment discharges for the same water discharge — this observation is not so easily made from the plot on the left, but it is immediately clear in the hysteresis plot on the right.