Exploring grain settling with Python

Grain settling is one of the most important problems in sedimentology (and therefore sedimentary geology), as neither sediment transport nor deposition can be understood and modeled without knowing what is the settling velocity of a particle of a certain grain size. Very small grains, when submerged in water, have a mass small enough that they reach a terminal velocity before any turbulence develops. This is true for clay- and silt-sized particles settling in water, and for these grain size classes Stokes’ Law can be used to calculate the settling velocity:

Screen Shot 2013-08-09 at 5.54.33 PM where R = specific submerged gravity (the density difference between the particle and fluid, normalized by fluid density), g = gravitational acceleration, D is the particle diameter, C1 is a constant with a theoretical value of 18, and the greek letter nu is the kinematic viscosity.

For grain sizes coarser than silt, a category that clearly includes a lot of sediment and rock types of great interest to geologists, things get more complicated. The reason for this is the development of a separation wake behind the falling grain; the appearance of this wake results in turbulence and large pressure differences between the front and back of the particle. For large grains – pebbles, cobbles – this effect is so strong that viscous forces become small compared to pressure forces and turbulent drag dominates; the settling velocity can be estimated using the empirical equation

Screen Shot 2013-08-09 at 5.54.40 PMThe important point is that, for larger grains, the settling velocity increases more slowly, with the square root of the grain size, as opposed to the square of particle diameter, as in Stokes’ Law.

Sand grains are small enough that viscous forces still play an important role in their subaqueous settling behavior, but large enough that the departure from Stokes’ Law is significant and wake turbulence cannot be ignored. There are several empirical – and fairly complicated – equations that try to bridge this gap; here I focus on the simplest one, published in 2004 in the Journal of Sedimentary Research (Ferguson and Church, 2004):

Screen Shot 2013-08-09 at 5.54.47 PM

At small values of D, the left term in the denominator is much larger than the one containing the third power of D, and the equation is equivalent of Stokes’ Law. At large values of D, the second term dominates and the settling velocity converges to the solution of the turbulent drag equation.

But the point of this blog post is not to give a summary of the Ferguson and Church paper; what I am interested in is to write some simple code and plot settling velocity against grain size to better understand these relationships through exploring them graphically. So what follows is a series of Python code snippets, directly followed by the plots that you can generate if you run the code yourself. I have done this using the IPyhton notebook, a very nice tool that allows and promotes note taking, coding, and plotting within one document. I am not going to get into details of Python programming and the usage of IPyhton notebook, but you can check them out here.

First we have to implement the three equations as Python functions:

import numpy as np
import matplotlib.pyplot as plt
rop = 2650.0 # density of particle in kg/m3
rof = 1000.0 # density of water in kg/m3
visc = 1.002*1E-3 # dynamic viscosity in Pa*s at 20 C
C1 = 18 # constant in Ferguson-Church equation
C2 = 1 # constant in Ferguson-Church equation
def v_stokes(rop,rof,d,visc,C1):
        R = (rop-rof)/rof # submerged specific gravity
        w = R*9.81*(d**2)/(C1*visc/rof)
        return w
def v_turbulent(rop,rof,d,visc,C2):
        R = (rop-rof)/rof
        w = (4*R*9.81*d/(3*C2))**0.5
        return w
def v_ferg(rop,rof,d,visc,C1,C2):
        R = (rop-rof)/rof
        w = ((R*9.81*d**2)/(C1*visc/rof+
        return w

Let’s plot these equations for a range of particle diameters:

d = np.arange(0,0.0005,0.000001)
ws = v_stokes(rop,rof,d,visc,C1)
wt = v_turbulent(rop,rof,d,visc,C2)
wf = v_ferg(rop,rof,d,visc,C1,C2)
plot([0.25, 0.25],[0, 0.15],'k--')
plot([0.25/2, 0.25/2],[0, 0.15],'k--')
plot([0.25/4, 0.25/4],[0, 0.15],'k--')
text(0.36, 0.11, 'medium sand', fontsize=13)
text(0.16, 0.11, 'fine sand', fontsize=13)
text(0.075, 0.11, 'v. fine', fontsize=13)
text(0.08, 0.105, 'sand', fontsize=13)
text(0.01, 0.11, 'silt and', fontsize=13)
text(0.019, 0.105, 'clay', fontsize=13)
xlabel('grain diameter (mm)',fontsize=15)
ylabel('settling velocity (m/s)',fontsize=15)
D = [0.068, 0.081, 0.096, 0.115, 0.136, 0.273,
    0.386, 0.55, 0.77, 1.09, 2.18, 4.36]
w = [0.00425, 0.0060, 0.0075, 0.0110, 0.0139, 0.0388,
    0.0551, 0.0729, 0.0930, 0.141, 0.209, 0.307]


The black dots are data points from settling experiments performed with natural river sands (Table 2 in Ferguson and Church, 2004). It is obvious that the departure from Stokes’ Law is already significant for very fine sand and Stokes settling is completely inadequate for describing the settling of medium sand.

This plot only captures particle sizes finer than medium sand; let’s see what happens as we move to coarser sediment. A log-log plot is much better for this purpose.

d = np.arange(0,0.01,0.00001)
ws = v_stokes(rop,rof,d,visc,C1)
wt = v_turbulent(rop,rof,d,visc,C2)
wf = v_ferg(rop,rof,d,visc,C1,C2)
plot([1.0/64, 1.0/64],[0.00001, 10],'k--')
text(0.012, 0.0007, 'fine silt', fontsize=13,
plot([1.0/32, 1.0/32],[0.00001, 10],'k--')
text(0.17/8, 0.0007, 'medium silt', fontsize=13,
plot([1.0/16, 1.0/16],[0.00001, 10],'k--')
text(0.17/4, 0.0007, 'coarse silt', fontsize=13,
plot([1.0/8, 1.0/8],[0.00001, 10],'k--')
text(0.17/2, 0.001, 'very fine sand', fontsize=13,
plot([0.25, 0.25],[0.00001, 10],'k--')
text(0.17, 0.001, 'fine sand', fontsize=13,
plot([0.5, 0.5],[0.00001, 10],'k--')
text(0.33, 0.001, 'medium sand', fontsize=13,
plot([1, 1],[0.00001, 10],'k--')
text(0.7, 0.001, 'coarse sand', fontsize=13,
plot([2, 2],[0.00001, 10],'k--')
text(1.3, 0.001, 'very coarse sand', fontsize=13,
plot([4, 4],[0.00001, 10],'k--')
text(2.7, 0.001, 'granules', fontsize=13,
text(6, 0.001, 'pebbles', fontsize=13,
xlabel('grain diameter (mm)', fontsize=15)
ylabel('settling velocity (m/s)', fontsize=15)


This plot shows how neither Stokes’ Law, nor the velocity based on turbulent drag are valid for calculating settling velocities of sand-size grains in water, whereas the Ferguson-Church equation provides a good fit for natural river sand.

Grain settling is a special case of the more general problem of flow past a sphere. The analysis and plots above are all dimensional, that is, you can quickly check by looking at the plots what is the approximate settling velocity of very fine sand. That is great, but you would have to generate a new plot – and potentially do a new experiment – if you wanted to look at the behavior of particles in some other fluid than water. A more general treatment of the problem involves dimensionless variables; in this case these variables are the Reynolds number and the drag coefficient. The classic diagram for flow past a sphere is a plot of the drag coefficient against the Reynolds number. I will try to reproduce this plot, using settling velocities that come from the three equations above.

At terminal settling velocity, the drag force equals the gravitational force acting on the grain:

Screen Shot 2013-08-09 at 5.49.31 PM

We also know that the gravitational force is given by the submerged weight of the grain:

Screen Shot 2013-08-09 at 5.34.42 PM

The drag coefficient is essentially a dimensionless version of the drag force:

Screen Shot 2013-08-09 at 5.49.08 PM

At terminal settling velocity, the particle Reynolds number is

Screen Shot 2013-08-09 at 5.59.14 PM

Using these relationships it is possible to generate the plot of drag coefficient vs. Reynolds number:

d = np.arange(0.000001,0.3,0.00001)
C2 = 0.4 # this constant is 0.4 for spheres, 1 for natural grains
ws = v_stokes(rop,rof,d,visc,C1)
wt = v_turbulent(rop,rof,d,visc,C2)
wf = v_ferg(rop,rof,d,visc,C1,C2)
Fd = (rop-rof)*4/3*pi*((d/2)**3)*9.81 # drag force
Cds = Fd/(rof*ws**2*pi*(d**2)/8) # drag coefficient
Cdt = Fd/(rof*wt**2*pi*(d**2)/8)
Cdf = Fd/(rof*wf**2*pi*(d**2)/8)
Res = rof*ws*d/visc # particle Reynolds number
Ret = rof*wt*d/visc
Ref = rof*wf*d/visc
loglog(Res,Cds,linewidth=3, label='Stokes')
loglog(Ret,Cdt,linewidth=3, label='Turbulent')
loglog(Ref,Cdf,linewidth=3, label='Ferguson-Church')
# data digitized from Southard textbook, figure 2-2:
Re_exp = [0.04857,0.10055,0.12383,0.15332,0.25681,0.3343,0.62599,0.77049,0.94788,1.05956,
Cd_exp = [479.30811,247.18175,199.24072,170.60068,112.62481,80.21341,45.37168,39.89885,34.56996,
loglog(Re_exp, Cd_exp, 'o', markerfacecolor = [0.6, 0.6, 0.6], markersize=8)

# Reynolds number for golf ball:
rof_air = 1.2041 # density of air at 20 degrees C
u = 50 # velocity of golf ball (m/s)
d = 0.043 # diameter of golf ball (m)
visc_air = 1.983e-5 # dynamic viscosity of air at 20 degrees C
Re = rof_air*u*d/visc_air
loglog([Re, Re], [0.4, 2], 'k--')
text(3e4,2.5,'$Re$ for golf ball',fontsize=13)
xlabel('particle Reynolds number ($Re$)', fontsize=15)
ylabel('drag coefficient ($C_d$)', fontsize=15);


The grey dots are experimental data points digitized from the excellent textbook by John Southard, available through MIT Open Courseware. As turbulence becomes dominant at larger Reynolds numbers, the drag coefficient converges to a constant value (which is equal to C2 in the equations above). Note however the departure of the experimental data from this ideal horizontal line: at high Reynolds numbers there is a sudden drop in drag coefficient as the laminar boundary layer becomes turbulent and the flow separation around the particle is delayed, that is, pushed toward the back; the separation wake becomes smaller and the turbulent drag decreases. Golf balls are not big enough to reach this point without some additional ‘help’; this help comes from the dimples on the surface of the ball that make the boundary layer turbulent and reduce the wake.

You can view and download the IPython notebook version of this post from the IPython notebook viewer site.

Ferguson, R. and Church, M. (2004) A simple universal equation for grain settling velocity. Journal of Sedimentary Research 74, 933–937.

Salt and sediment: A brief history of ideas

Salty weirdness
Salt is a weird kind of rock. At first sight, it behaves like most other rocks: if you pick up a piece, it is hard, it is heavy, and it breaks if hit with a hammer. But put it under stress for thousands of years, and salt will behave like a fluid: relatively small forces can cause it to flow toward less stressful surroundings. This often means it will try to find its way to the surface.

When deposited, sand and mud have lots of pore space filled with water and have relatively low density. However, as they get buried by more sediment, much pore space is lost, both through compaction and cementation. Sediments turn into sedimentary rocks, become harder, and their density increases. In contrast, salt doesn’t have much pore space to begin with; its density will stay the same, regardless of depth of burial. As both salt and sediment are buried to greater depths, an unstable condition develops: lighter salt lying under denser material. In addition, the location of the salt layer in the sediment column is not entirely random: it is in the nature of sedimentary basins to initially place salt at the bottom of the sediment pile. Extensive salt layers usually form early in a basin’s lifetime, when seawaters invade for the first time shallow depressions on a continent that is about to split into two along a rift zone. The Dead Sea is an obvious example that comes to mind.

Layering salt and sediment in this unstable order is a recipe for a spectacular geological show. As salt is trying to find its way to the surface, it forms drop-shaped blobs called diapirs; but also ridges, walls, and salt sheets. Several sheets can connect laterally into a huge salt canopy, a new salt layer that is entirely out-of-place or allochtonous. Salt can also act as a lubricating layer at the base of a thick sequence of sedimentary rocks. But I am rushing ahead a little bit; salt tectonics is such a new – but rapidly growing – science that salt canopies, despite their widespread presence in the subsurface Gulf of Mexico, were not recognized and described until the 1980s.

Tectonics vs. buoyancy, Europe vs. America
Before the beginning of the twentieth century, even with the role that salt played in human history, little was known about how salt domes formed. This was an age of rampant speculation; surface data was scarce because salt does not last very long after exposed as it quickly gets dissolved and washed away by precipitation. Many geologists thought that formation of salt domes didn’t require any significant salt deformation or displacement. But things have changed dramatically in 1901, with the discovery of the Spindletop oil field on top of a salt dome in southeastern Texas. The recognition that oil is often found on top of and around salt domes created a much stronger interest in understanding how exactly salt formations are put in place.

European geologists thought that the main driving force was compression, the force that causes folding and thrusting and builds mountains. In Romania, where the Eastern Carpathians take a sharp turn toward the southwest, salt was found in the cores of oil-bearing anticlines. The contacts with the surrounding rocks were clearly discordant. These are the structures that prompted Ludovic Mrazec, professor of geology at University of Bucharest, to coin the term “diapir” in 1907.

Mrazec’s explanation of how salt diapirs form. From Barton (1925).

Salt in Germany and Poland also seemed to occur invariably in a compressional setting, in the cores of folds, next to folds that had no salt associated. It seemed obvious that salt was ‘pushed up’ by tectonic forces, and it appeared unlikely that the rise of salt itself was causing the folding.

But the discovery of a multitude of salt diapirs in the Gulf of Mexico made it clear that they can occur far away from any mountains and compressive tectonic forces. The much simpler setting and relative lack of deformation in the Gulf proved informative. “The Roumanian salt-dome geologist possibly may have more to learn from the American salt domes than the American salt-dome geologist has to learn from the Roumanian domes. The occurrence of the American domes in a region of tectonic quiescence suggests that tectonic thrust cannot have the importance postulated by Mrazec” – wrote Donald Barton in 1925.

This was also the time when the density difference between salt and sediment came into discussion. Gravity measurements in the Gulf of Mexico showed anomalies above salt domes that were due to the lower density of salt. It was increasingly recognized that density inversion must play an important role in diapirism, especially where compressive tectonic forces were absent. In addition, by the 1930s geologists have reached a consensus that salt diapirs must somehow punch through the overlying sediment. They seemed to ignore the fact that, as Wade (1931) put it, you cannot drive a putty nail through a wooden board. As mentioned before, salt does behave like a fluid over geological time scales. But how can it penetrate thick layers of hardened sedimentary rock?

A brilliant idea: downbuilding
The solution to this problem came in 1933, from the same Donald Barton who was discussing the differences between European and American salt domes in 1925. He suggested that diapirs can form without much piercement of the sediment above. Instead, once a small dome is initiated, it simply can stay in place, always at or close to the surface, while sediment is deposited around it and the source salt layer subsides: “it is the sediments which move, and not the salt core. The energy requirement (…) is very much less than if there were actual upward movement of the salt.”

The evolution of salt diapirs through ‘downbuilding’. Salt domes are always close to the surface and diapirism goes hand-in-hand with sedimentation. From Barton (1933).

This was a key insight: it got rid of the “room problem”, the need for moving huge volumes of hard rock out of the way of the rising salt. It also highlighted that salt movement can happen at the same time with sedimentation, a fact that became abundantly obvious later as high-quality seismic data became available. But the concept of ‘downbuilding’ was ignored for the next fifty years.

Animation showing how downbuilding works. Blue represents salt, yellow is sediment. To mimic mass balance for salt (-- what is lost from the source layer must go into the salt dome), the blue area is kept constant through the animation.

The beauty of instabilities

The main reason for conveniently forgetting Barton’s idea was that density inversion between two fluids could be nicely studied in the lab and described with elegant equations. In one of the papers that kicked off this fascination with Rayleigh-Taylor instabilities, Nettleton (1934) used corn syrup and less dense crude oil to visualize diapir-like blobs of fluid in a transparent cylinder and to show that gravity alone, without any help from contractional forces, was enough to generate structures similar to salt domes.

Less dense crude oil (black) forming diapir-like blobs as rising through higher-density corn syrup (yellow). Redrawn from Nettleton (1934).

One problem with this approach was that oil and syrup can be photographed during deformation, but the transient structures could not be carefully dissected and analyzed later. Materials of higher viscosity were needed for that; however, increasing the viscosity resulted in a density difference too small to get the fluids moving in the first place. The trick was to place the whole experiment in a centrifuge and use the centrifugal force to imitate a larger-than-normal gravitational force. This approach formed the basis of a productive line of research on gravity tectonics in the laboratory of the Norwegian-Swedish geologist Hans Ramberg. The results are probably more relevant to what is happening deeper in the Earth, at higher temperatures and pressures, where most rocks become more similar in behavior to salt.

Modern salt tectonics
By the late 1980s it has become quite obvious that kilometer-thick piles of sedimentary rock cannot be treated as fluids and salt-sediment interaction is more similar to placing and deforming slabs of brittle material on top of a viscous fluid. Seismic from salt-bearing sedimentary basins suggested that the history of salt movement and sedimentation were highly interconnected and Barton’s downbuilding concept was strongly relevant.

Three-dimensional seismic data also showed the variety and complexity of allochtonous salt bodies in salt-rich sedimentary basins. Sandbox experiments with more realistic material properties and ongoing sedimentation during deformation were performed and the results beautifully visualized. The behavior of turbidity currents flowing over complex salt-related submarine topography was investigated. Hundreds of scientific papers were written on salt tectonics, both by industry geoscientists and researchers in the academia.

N-S cross section in the Gulf of Mexico. Large volumes of the Jurassic Louann salt have been displaced and squeezed into a salt canopy surrounded by much younger sediments. From Pilcher et al., 2011

And there is quite a bit left to explore and understand.

References and further reading
Barton, D. C. (1926) The American Salt-Dome Problems in the Light of the Roumanian and German Salt Domes, AAPG Bulletin, v. 9, p. 1227–1268.

Barton, D. C. (1933) Mechanics of Formation of Salt Domes with Special Reference to Gulf Coast Salt Domes of Texas and Louisiana, AAPG Bulletin, v. 17, 1025–1083.

Hudec, M., & Jackson, M. (2007) Terra infirma: Understanding salt tectonics. Earth Science Reviews, 82(1-2), 1–28.

Jackson, M. (1996) Retrospective salt tectonics, in M.P.A. Jackson, D.G. Roberts, and S. Snelson, eds., Salt tectonics: a global perspective: AAPG Memoir 65, p. 1–28. [great summary of the history of salt tectonics]

Mrazec, L. (1907) Despre cute cu sȋmbure de străpungere [On folds with piercing cores]: Bul. Soc. Stiint., Romania, v. 16, p. 6–8.

Nettleton, L. L. (1934) Fluid Mechanics of Salt Domes, AAPG Bulletin, v. 18, p. 1–30.

Pilcher, R. S., Kilsdonk, B., & Trude, J. (2011) Primary basins and their boundaries in the deep-water northern Gulf of Mexico: Origin, trap types, and petroleum system implications. AAPG Bulletin, v. 95(2), p. 219–240.

Wade, A. (1931) Intrusive salt bodies in coastal Asir, south western Arabia: Institute of Petroleum Technologists Journal, v. 17, p. 321–330, 357–361.

The complexity of sinuous channel deposits in three dimensions

ResearchBlogging.org The beauty of the shapes and patterns created by meandering rivers has long attracted the attention of many geomorphologists, civil engineers, and sedimentologists. Unless they are fairly steep or have highly stable and unerodible banks, rivers do not like to follow a straight course and tend to develop a sinuous plan-view pattern. The description and mathematical modeling of these curves is a fascinating subject, but that is not what I want to talk about here and now. It is hard enough to understand the plan-view evolution of rivers, especially if one is interested in the long-term results – when cutoffs become important -, but things get really complicated when it comes to the three-dimensional structure of the deposits that meandering rivers leave behind. The same can be said about sinuous channels on the seafloor, created and maintained by dirty mixtures of water and sediment (called turbidity currents). An ever-increasing number of seafloor and seismic images show that highly sinuous submarine channels are almost as common as their subaerial counterparts, but much remains to be learned about the geometries of their deposits that accumulate through geological time.

Using simple modeling of how channel surfaces migrate through time, two recent papers attempt to illustrate the three-dimensional structure of sinuous fluvial and submarine channel deposits. In the Journal of Sedimentary Research, Willis and Tang (2010) show how slightly different patterns of fluvial meander migration result in different deposit geometries and different distribution of grain size, porosity and permeability. [These properties are important because they determine how fluids flow - or don't flow - through the pores of the sediment.] River meanders can either grow in a direction perpendicular to the overall downslope orientation, or they can keep the same width and migrate downstream through translation. In the latter case – which is often characteristic of rivers incising into older sediments -, deposits forming on the downstream, concave bank of point bars will be preferentially preserved. These deposits tend to be finer grained than the typical convex-bank point bar sediments. In addition to building a range of models and analyzing their geometries, Willis and Tang also ran simulations of how would oil be displaced by water in them. One of their findings is that sinuous rivers that keep adding sediment in the same area over time (in other words, rivers that aggrade) tend to form better connected sand bodies than rivers which keep snaking around roughly in the same horizontal plane, without aggradation.

Map of deposits forming as river meanders grow (from Willis and Tang,  2010).
Cross sections through the deposits of two meander bends (locations shown in figure above). Colors represent permeability, red being highly permeable and blue impermeable sediment. From Willis and Tang, 2010.

Check out the paper itself for more images like these, plus discussions of concave-bank deposition, cutoff formation, and filling of abandoned channels.

The second paper (Sylvester, Pirmez, and Cantelli, 2010; and yes, one of the authors is also the author of this blog post, so don’t expect any constructive criticism here) focuses on submarine channels and their overbank deposits, but the starting point and the modeling techniques are similar: take a bunch of sinuous channel centerlines and generate surfaces around them that reflect the topography of the system at every time step. However, we know much less about submarine channels than fluvial ones, because it is much more difficult to collect data at and from the bottom of the ocean than it is from the river in your backyard. The result is that some of the simplifications in our model are controversial; to many sedimentary geologists, submarine channels and their deposits are fundamentally different from rivers and point bars, and there is not much use in even comparing the two. Part of the problem is that not all submarine channels are made equal, and, when looking at an outcrop, it is not easy – or outright impossible – to tell what kind of geomorphology produced the  stratigraphy. In fact, the number of exposures that represent highly sinuous submarine channels, as observed on the seafloor and numerous seismic images, is probably fairly limited. One thing is quite clear, however: many submarine channels show plan-view migration patterns that are very similar to those of rivers, and this large-scale structure imposes some significant constraints on the geometry of the deposits as well.

That being said, nobody denies that there are plenty of significant differences between real and submarine ‘rivers’ [note quotation marks]. A very important one is the amount of overbank – or levee – deposition: turbidity currents often overflow their channel banks as thick muddy clouds and form much thicker deposits than the overbank sediment layers typical of rivers. When these high rates of levee deposition combine with the strong three-dimensionality of channel migration, complex geometries result that are quite tricky to understand just by looking at a single cross section.

Cross section and chronostratigraphic diagram through a submarine channel system with inner and outer levees (from Sylvester et al., 2010).

One of the consequences of the channel migration is the formation of erosional surfaces that develop through a relatively long time and do not correspond to a geomorphologic surface at all (see the red erosional zones in the Wheeler diagram above). This difference between stratigraphic and geomorphologic surfaces is essential, yet often downplayed or even ignored in stratigraphy. In terms of geomorphology, the combination of channel movement in both horizontal and vertical directions and the extensive levee deposition results in a wide valley with scalloped margins and numerous terraces inside:

Three-dimensional view of an incising channel-levee system (from Sylvester et al., 2010).

This second paper is part of a nice collection focusing on submarine sedimentary systems that is going to be published as a special issue of Marine and Petroleum Geology, a collection that originated from a great conference held in 2009 in Torres del Paine National Park, Southern Chile.

PS. As I am typing this, I see that Brian over at Clastic Detritus is also thinking about submarine channels and subaerial rivers… Those channels formed by saline density currents on the slope of the Black Sea are fascinating.

Willis, B., & Tang, H. (2010). Three-Dimensional Connectivity of Point-Bar Deposits Journal of Sedimentary Research, 80 (5), 440-454 DOI: 10.2110/jsr.2010.046

Sylvester, Z., Pirmez, C., & Cantelli, A. (2010). A model of submarine channel-levee evolution based on channel trajectories: Implications for stratigraphic architecture Marine and Petroleum Geology DOI: 10.1016/j.marpetgeo.2010.05.012

Hillslope diffusion

Modeling erosion and deposition of sediment using the diffusion equation is among the important subjects that are usually omitted from sedimentary geology textbooks. Part of the reason for this is that ‘conventional’ sedimentary geology tended to only pay lip service to earth surface processes and was more interested in describing the stratigraphic record than figuring how it relates to geomorphology. Nowadays, a good discussion of stratigraphy and sedimentology cannot ignore anymore what geomorphologists have learned about landscape evolution. (One textbook that clearly recognizes this is this one.)

But let’s get back to the subject of this post. Hillslope evolution can be modeled with the diffusion equation, one of the most common differential equations in science, applied for example to describe how differences in temperature are eliminated through heat conduction. In the case of heat, the heat flux is proportional to the rate of spatial temperature change; on hillslopes, the sediment flux is proportional to the spatial rate of change in elevation. This last quantity of course is the slope itself. In other words,

q = -k*dh/dx,


q = -k*slope,

where q is the volumetric sediment flux per unit length, k is a constant called diffusivity, h is the elevation, and x is the horizontal coordinate.

We also know that sediment does not disappear into thin air: considering a small area of the hillslope, the amount of sediment entering and leaving this area will determine how large the change in elevation will be:

dh/dt = -dq/dx,

in other words, deposition or erosion at any location is determined by the change in sediment flux.

Combining this equation with the previous one, we arrive to the diffusion equation:

dh/dt = k*d2h/dx2.

Note that the quantity on the right side is the second derivative (or curvature) of the slope profile. Large negative curvatures result in rapid erosion; places with large positive curvature have high rates of deposition. Through time, the bumps and troughs of the hillslope are smoothed out through erosion and deposition.

The simplest possible case is the diffusion of a fault scarp. The animation below illustrates how a 1 m high fault scarp gets smoothed out through time; the evolution of slope and curvature are also shown. The dashed line indicates the original topography, at time 0. [The plots were generated using Ramon Arrowsmith's Matlab code].

More complicated slope profiles can be modeled as well; here is an example with two fault scarps:

Note how both erosion and deposition get much slower as the gradients become more uniform.

The simplicity of the diffusion equation makes it an attractive tool in modeling landscape evolution. In addition to hillslopes and fault scarps, it has been successfully applied in modeling – for example – river terraces, deltaic clinoforms, cinder cones, fluvial systems, and foreland basin stratigraphy. However, it is important to know when and where the assumptions behind it become invalid. For example, steep slopes often have a non-linear relationship between sediment flux and slope as mass movements dramatically increase sediment flux above a critical slope value. Also, the models shown here would fail to reproduce the topography of a system where not all sediment is deposited at the toe of the steeper slope, but a significant part is carried away by a river. And that brings us closer to advection; a subject that I might take notes about at another time.

Further reading: 1) The book “Quantitative Modeling of Earth Surface Processes” by Jon Pelletier has a chapter with lots of details about the diffusion equation. 2) Analog and numerical modeling of hillslope diffusion- a nice lab exercise.

Climbing Ripples I.

Ripples, dunes, cross bedding and cross lamination have always been some of the sexiest subjects in sedimentary geology. They are certainly responsible (in part) for my choice of a certain walk of life that consists of studying dirt. You might say that everything has been already said about ripples and dunes, and you clearly get that feeling if you read some of J.R.L. Allen’s work on the subject (and that can be a lot of reading, by the way) or look at the fantastic multimedia material that David Rubin at the USGS put together. [Of course, there are numerous other authors who have written great papers on the subject, but it is not my purpose here to write a history of bedform sedimentology. Although that would be an interesting subject, if somebody had the time for it.]

However, little of this material gets into the standard sedimentology and stratigraphy textbooks. Maybe rightly so: after all, textbooks are not supposed to include all the details about any particular subject. And maybe there are higher-density issues out there, like whether we should call something a turbidite or a debrite. [Sorry, I could not refrain from typing that].

Take for example climbing ripples. They form when several trains of ripples are superimposed on each other and they seem to ‘climb’, by generating stratigraphic surfaces that are tilted in an upcurrent direction. [Note however that these surfaces are *not* topographic - or time - surfaces; more on that later]. Numerous textbooks and many papers mention climbing ripple cross lamination, but often the explanation is something like “they indicate high rates of deposition”, or “the steepness of the climb and stoss-side preservation are a function of the ratio between suspended-load and bedload”. The question is, what do we *exactly* mean by ‘high rates of deposition’? If we cannot put numbers on it, it is not that informative. Also, by ‘suspended load’, do we mean suspended load concentration? Or deposition from suspended load and bedload, respectively? Those statements are not necessarily wrong, but they do not do justice to the models that have been published many years ago, models that actually have some numbers and equations behind the “conclusion” section.

The key paper that I am talking about is “A quantitative model of climbing ripples and their cross-laminated deposit“, by J.R.L. Allen, published in 1970 in the journal Sedimentology.

The most important relationship that Allen has derived links the angle of climb ζ (see the sketch below) to the rate of deposition M (measured in units of mass over unit time and area), the rate of bedload sediment transport j, and the ripple height H:

tanζ = MH / 2j

This is simply based on decomposing the sediment flux to and through the bed into vertical and horizontal components (plus a relationship between the horizontal sediment transport rate in ripples and the horizontal migration rate of the bedforms). Note that the quantity j refers to the sediment mass that moves through a cross section perpendicular to the general current direction, and does this by being part of the ripples themselves. In other words, there is no direct equivalence between M and suspended load deposition, and j and bedload deposition. Although it is possible that in general suspended load contributes more to M than deposition from bedload, it says nowhere that grains transported within the bedload cannot be deposited on the stoss side of the ripples and thus contribute to the vertical growth of the bed.

Obviously, if the angle of climb is smaller than the dip of the stoss side, there will be no stoss side preservation and the resulting cross lamination will look like in the sketch below (which, by the way, was quite an effort to generate in Matlab; you can easily do this and much-much more with David Rubin’s Matlab code, but I wanted to understand things a little better by coding something simple myself):

This is often called ‘A-type’ (or subcritical) climbing ripple cross lamination, but everybody knows what you are talking about if you “simply” call it climbing ripple cross lamination with no stoss-side preservation.

In contrast, aggradation is much more prominent if the angle of climb is larger than the slope of the stoss side, and in this case deposition takes place on the stoss sides as well, resulting in ‘S-type’ (or supercritical) lamination:

Of course, it says nowhere that the rate of deposition M or the bedload transport rate j must stay constant through time. If the ratio of these quantities changes, the angle of climb will change as well. This sketch shows an example where the rate of deposition M increases through time:

One of the main points of the paper is that there is a fundamental difference between the rate of deposition M and the bedload sediment transport rate j. A rate of deposition larger than zero means that the sediment transport rate within the flow must decrease from an upcurrent position to a downcurrent position; a simple mass balance tells us that this change in the sediment transport rate has to equal the rate of deposition. In other words, the rate of deposition M is a derivative of the sediment transport rate, and as such, does not belong in the same drawer of physical quantities as the bedload transport rate.

Along the same line of thought, Allen emphasizes that climbing ripple lamination says something about flow uniformity and steadiness. A uniform and steady flow can only form a single train of ripples; either non-uniformity or unsteadiness is needed to have climbing-ripple deposition.

That’s it for now; to be continued. It’s time to do my taxes.

Further reading: Brian has a Friday Field Photo and a Geopuzzle on climbing ripples. Here are some pictures and a movie of climbing ripples generated by a turbidity current in a flume.