### dispersive transport

(Update 2013 The images got lost by Google blogger, but the original Google docs is here

Google Doc version. Also see the online book The Oil Conundrum, where I removed many of the grandiose statements at the beginning of this blog post)

# Why is dispersion anomalous?

This post touches on the nature of theoretical and

experimental research and illustrates how a fundamental idea can take quite a circuitous route before it lodges in a remotely related application area. The acceptance of the original idea tends to create a momentum that makes it difficult to dislodge from the conventional wisdom and impenetrable to anyone but the

*cognoscenti*.

First off, prep yourself for some solid-state physics. But don't worry about the math as the commentary makes up for the potential MEGO. The extended narrative traverses the scale of applicability from statistical mechanics to environmental geology picking up arcs of connectivity along the way. I also buried a valuable nugget in here, presenting a surprisingly powerful analytical result that has laid dormant for over 30 years, perhaps even 20 years prior to that, and has huge implications for the analysis of solar cells, MOS technology, and quantum electronics. To put it another way, if I had to do another graduate thesis, I could easily defend this argument, and on top of that, I would have fun doing it. It all fits together tighter than a Peyton Manning spiral. The fact that it also connects across disparate domains of science makes it frankly

*mind-blowing*.

I call it mind-blowing from the fact that the actual argument derives from such a simple premise, and I have to seriously wonder why no one has picked up on this before. I actually question some of the belief systems inherent in these fields of study and assert the likelihood of a sunk cost effect getting in the way of a fundamental understanding. If this sounds familiar to those following our oil predicament, it should, as I have definitely seen such oversight play out before.

Often in physics, experimental observations are termed "anomalous" before they are understood. Once theory succeeds in explaining and illuminating the observations, they are no longer "anomalous" and instead come to be regarded as "obvious". A crucial paper can trigger such an "anomalous => obvious" transition, and in the present case that key role was played by a 1975 paper by Scher and Montroll. That landmark paper has become basic to our understanding of a striking characteristic of carrier motion (now called dispersive transport) which is a common occurence in amorphous semiconductors, though foreign to our experience with crystals.

--Richard Zallen, "The physics of amorphous solids", Wiley-VCH, 1998

###
Hmm ... Reserve growth also considered anomalous...^{1}

The term anomalous in scientific code-speak essentially means "dunno". We have to admit that we don't understand lots of things, largely due to issues of complexity, observability, or just too much noise. Yet that doesn't prevent us from trying to extract a fundamental meaning of some strange behavior that we observe.

In a crystalline semiconductor with a contact electrode at each end, a pulse of light incident at one electrode will, upon the effect of an electric field or potential drop between the electrodes, generate an almost immediate flow of current across the load lasting as long as the transit time of the photo-induced carriers.

^{2}The carriers, either electrons or holes depending on the polarity of the electric field bias relative to the absorbing electrode, will drift from the site of the photon-induced excitation, to the opposite contact. Experimentalists consider the behavior well-characterized; the mobility of the carriers at temperature, the strength of the electric field, and the contact separation,

**, determine the transit time,**

*d***, of the output current pulse. Because the carriers scatter against lattice imperfections, the speed does not continuously accelerate but instead achieves a bounded drift velocity,**

*t*_{T}**. This drift mobility has an intuitive real-world analog -- think in terms of a drag coefficient for the analogous situation of a falling body under gravitational forces, which eventually achieve what we call terminal velocity. For all practical purposes, the equivalent terminal velocity occurs almost immediately in a semiconductor (a short relaxation or quenching time) and sets the bound for the transit time duration.**

*v*_{0}Ideally, the pulse looks like a perfect square wave with temporal duration

**. In terms of a mathematical expression, the current behaves as I(**

*t*_{T}*t*) =

*K**[u(

*t*)-u(

*t-t*

_{T})] where u(

*t*) is the unit step operator with magnitude

**. See the figure to the right.**

*K*Because of carrier diffusion, the actual drop-off in current has rounded leading and trailing edges as the charged carrier pulse spreads out a bit into a Gaussian packet as it propagates. This relatively innocuous but well-understood form of dispersion, known as diffusion, occurs from random walk excursions as the carrier makes its way across the transit width. In the ideal case, the diffusion constant varies linearly with the mobility (or drag for particle systems) according to the Einstein relation. For high mobilities and small contact separations, the amount of diffusion that occurs does not appreciably round the pulse edges. The prized high mobility solid-state material allows device manufacturers to fabricate ultra-high speed photo-detectors as the sharp transition and short transit time generates an excellent and well-characterized frequency response.

This class of semiconductor has an ordered structure due to the crystalline lattice structure and it has properties such as carrier mobility which remain uniform through the sample. Such behavior shows little dispersion, either through diffusion or disorder. The narrower the distribution of velocities, the sharper the transition. Scientists have generally understood this for years and no one raises the spectre of anomalous behavior.

The truly anomalous behavior observed occurs in amorphous versions of certain semiconductors. The narrow pulse of carriers seen in an ordered sample now shows a huge spread in its concentration profile as it makes its way between the contacts. Obviously dispersion plays some role in this behavior, as it goes by the name "dispersive transport". Scientists had known about this "anomalous dispersion" since 1957 but it took nearly two decades before Scher and Montroll presented the solution to the problem mathematically.

^{3}

: (a) Normal transport shows a drifting packet of carriers. AsFigure 1long as the packet travels, we can detect a current proportional to the amount of carriers active. As they reach the opposing contact, the current rapidly declines to zero. The Gaussian-shaped packet widens slightly as it travels due to diffusion about the mean. A few extra fast carriers reach the far contact sooner than the bulk of the carriers, while a few stragglers take up the rear. (b) In dispersive transport, the velocity of the carriers varies over a wide range so that the original narrow impulse of carriers quickly spreads out as it drifts and diffuses across the width. This gives a long tail to the photo-response profile. ^{4}The stragglers keep arriving in this "fat tail" world, with progressively fewer in number as though they had joined and tried to complete a long marathon race (see Marathon Dispersion). | |

: The typical measurement in the photo-response current starts with aFigure 2spike followed by a soft plateau, then a shoulder or transition region, followed by the ubiquitous long tail ^{5}.Shapes of the current profile taken over a range of experimental conditions show invariance in the general shape with respect to the electric field and specimen thickness. Importantly, this profile does not follow from the expected spread of the Gaussian packet. Scale invariance or universality manifests itself in statistics if one can first transform the ordinates into dimensionless quantities while conserving the moments. ^{6} Transport of holes and electronsnear the absorption region electrode contributes to the initial spike, which has no impact on the longer tail due to the complementary carrier type (this transient spike is also known as prompt transport ^{7}) |

Figures 1 and 2 give a qualitative view of the dispersive transport that occurs in a disordered semiconductor. The first figure describes what we think happens internally and the second figure provides a view of the observable result. Figures 3 and 4 illustrate a couple of experimental results of widely studied amorphous (a-As

_{2}Se

_{3}) and organic materials (TNF-PVK).

: Typical experimental Time-Of-Flight curve shows a set of superimposed measurements from an earlyFigure 3Scher-Montroll behavior which exhibited the "universality" property of the scaling across different measurement conditions (the applied voltage in this case). | : The TOF curve of an organic semiconductorFigure 4illustrates the characteristic knee that Scher and Montroll had predicted; the two slopes differ but must sum to -2 according to their theory. |

The anomaly in the title of the Scher-Montroll paper

^{8}referred to

the fact that no one previously could formally explain the long tails in

the response (so called "time of flight") measurements. Other researchers

clearly had an inkling that it had something to do with the high

amounts of disorder leading to greater amounts of diffusion and dispersive spread than in an

ordered material. Amorphous materials naturally have many inhomogeneities, defects, and

carrier traps that can lead to varying delays in transit time. Scher and Montroll derived a statistical formulation of random walk

called the Continuous Time Random Walk (CTRW) that they then applied to

the experimental results.

Most experimentalists around that

time got good results using the CTRW formulation so that it has become

fairly well accepted in semiconductor circles for the last 30 years.

The math gets fairly hairy in spots, and soon experimentalists

began to simply use the empirical sloped lines to get at the Scher-Montroll disorder parameter (ɑ = alpha). High values of alpha indicate more order and low values more disorder (0 < ɑ < 1).

Research continues in understanding dispersive transport as new electronic materials come on line. Physicists have had a long-standing interest in disorder, as the finding of an order/disorder transition easily classifies as a type of "holy grail" discovery, certain to elicit

*oohs*and

*aahs*from their colleagues. As the figure to the right shows, one can add a controlled amount of disorder to a sample and observe the results of diffusive transport. The upper TOF trace shows linear transport while the lower trace shows the effects of dispersive transport. In the latter case the disorder comes in the way of the intentional introduction of impurities, apparently forming electronic traps which slow down the carrier motion as it traverses the width of the sample.

^{9}

Generally I have noticed that the basis for much of the current research has to do with finding some novel aspect of dispersion relating somehow to material properties. In reality, the rather mundane effect of randomness due to heterogeneity likely plays a far more important role. For many of these materials, we simply can't control the distribution of defects and traps and the disorder evolves into a garden variety randomness, with which we have a single mean rate, say average drift velocity, to characterize the behavior.

If you look at the curve to the right, the red line shows my simple assumption for maximum disorder. I had noticed this same shaped curve in my studies of dispersive discovery that I posted to TOD [], and had a hunch that I could use the same formulation in the semiconductor case. After all, as I assert that dispersion is just dispersion, and I have enough experience dealing with semiconductor physics that I didn't expect any gotchas. As for the ideas of Scher and Montroll, I turn their formulation upside down and don't even consider a random walk premise, as this leads to overly complex math.

## Breakthrough

I started to look at this problem because I had a nice intuitive way of modeling dispersive behavior in oil production [google links]

and figured that I could try applying my general dispersive model to dispersive

transport.

^{10}And that I could do it much more simply than the approach by Scher and Montroll. At certain places in their seminal papers and review articles, I find passages that amount to "

*... and then a miracle occurs*" and knew that this meant some messy first-principles work had gone missing. The way I turned their model on its head basically amounted to working in the

*rate domain*, corresponding to velocities, instead of the

*time domain*that corresponded to random-walk hopping (see figure).

^{11}The latter derives from the classical work used to describe everything from Brownian motion to large scale diffusion. The former relates to a more or less pragmatic view of the world which relies on entropy considerations instead of the statistics of hopping over energy barriers with small probabilities.

As a basic premise, I use the Maximum Entropy Model (MEM) to select a stochastic rate Probability Density Function (PDF or more precisely PMF for probability mass function) in which I can then derive dispersive transport.

One way to choose the “right” distributionpis by

using the principle of maximum entropy. This principle states that the

least biased probability assignment is that which maximizes the system

entropy subject to the constraints supplied by the available

information.^{12}

For the constrained system of interest, all we really know is the mean carrier transport velocity. If we don't know the higher order moments, the MEM says to use a damped exponential as the PDF to maximize entropy. In a general sense, this maximizes the amount of disorder that exists in this quasi-equilibrium system. It says that many slow carriers exist, with an exponentially diminishing supply of fast carriers. For the fixed geometry shown in the schematics at the top, the normalized expression for the time dependence of dispersed current reaching the far contact derives as follows:

I(

This essentially describes the integral over a PDF of normalized velocities

for carriers that have not yet swept through the transport layer. We assign

I pulled out the fundamental transport coefficients such as mobility and the diffusion constant for the time-being as this assumes that a uniform drift plays the prominent role. In the normalized case, I show the response profile below superimposed on the figure from Kao. At a subjective level, it follows the qualitative plateau/decline behavior quite well.

*t*) = I_{0}[1 - e^{-1/t}(1 + 1/*t*)] (EQ 1)This essentially describes the integral over a PDF of normalized velocities

p(

*v*) = (1/*v*)exp(-_{0}*v/v*) (EQ 2)_{0}for carriers that have not yet swept through the transport layer. We assign

*t*= 1 to show the scale invariance of the result of Equation 1. Crucially the formulation maintains the moments of the distribution. If the velocity distribution becomes dispersed as a damped exponential then the cumulative position distribution of a particle/carrier also advances by a damped exponential. Nothing more to it than that!

_{T}= d/v_{0 }I pulled out the fundamental transport coefficients such as mobility and the diffusion constant for the time-being as this assumes that a uniform drift plays the prominent role. In the normalized case, I show the response profile below superimposed on the figure from Kao. At a subjective level, it follows the qualitative plateau/decline behavior quite well.

: The maximum entropy dispersion for time of flight according to the normalized (EQ 1)Figure 5 |

Since we know that dispersion plays a role in the transport, we just have to figure out how to use the much simpler dispersion formulation instead of the hideous Scher-Montroll derivation. The key to understanding physics is to

*keep it simple, but not too simple*(quoting Einstein I believe). In fact the maximum entropy formulation that I had used previously in the dispersion analysis for oil field sizes, discovery, and reserve growth, I retain in this analysis

^{13}. Also known as the "method of least information", it essentially relies on using common sense in not trying to under- or over-estimate the variance of the dispersive spread. In one sense, the interpretation I make looks similar to the schematic at the right. I assume the equivalence of multiple mobility pathways through the device.

For any one pathway, the advance in the particles motion has a diffusive component as well as a drift component. This leads to an expression involving time as shown below

^{14}

<

*x*> = sqrt(*Dt*+ (*v*)_{0}t^{2}) (EQ 3)This essentially incorporates the concurrent diffusion component along with the drift component of the velocity and we can make an implicit transform into the actual timeline. The drift velocity

*v*relates to the electric field by

_{0}*v*, where

_{0 }= uE*u*is the carrier mobility (pronounced "

*mu*") and

*E*is the electric field strength. I would consider this a routine parametrization into a Hilbertian space where we can maintain moments of the distributions across dimensions, <

*t>*/

*t*/

_{T}= <x>*w*.

I(

*t*) = I_{0}[1 - exp(-*w*/sqrt(*Dt*+ (*vt*)^{2})) (1 +*w*/sqrt(*Dt*+ (*vt*)^{2}))] (EQ 4)Qualitatively the constant (drift) velocity drops as

*t*

^{-2}while diffusional velocity drops by

*t*

^{-1}. I am not certain whether the formulation by Scher and Montroll take this into account. They simply say that long-range correlations go as t

^{-a-1}when they set up their CTRW model. I believe this step links my exponentially damped rate dispersion to their long range time correlations.

Many experimental results show the knee in the curve of Scher and Montroll, but with usually not much dynamic range. I looked at a few material studies done fairly recently to see how well the simple theory works.

###
Transport in SiO_{2}

For verifying any theoretical formulation, you usually want to match

the behavior to as wide a dynamic range as experimentally feasible. The

larger the dynamic range in the measured quantities, the more

confidence that you have in its worth or value.

: Dispersive transport via the MEM model compared to SiO2 measurements. This shows mainly diffusion with the drift catching up at longer times.Figure 6 | : The effect of changing the width of the transport layer. The Montroll-Scher knee does not show up prominently.Figure 7 |

The case of carrier transport across SiO

_{2}insulating layers for MOS devices provides some cases of amazing dynamic range, up to 8 orders of magnitude in current. I took data from the text "

*Ionizing radiation effects in MOS devices and circuits*" by Ma and Dressendorfer. The fundamental idea remains the same in this situation as the photo-response experiment, although a different form of ionizing radiation supplies the pulse of carriers -- in this case holes become the charge carrier instead of electrons. Otherwise, the same diffusive transport occurs, with the authors trying to explain the results by applying the same unwieldy Scher-Montroll formulation. As a side note, these kinds of measurements need a delicate touch as the dose of the radiation can actually effect the field due to space charge formation. I did some pioneering work on a similar experiment years ago where I tried to force dopant concentrations via ion bombardment into a growing junction and the bias of the junction alone pulled the mobile dopants from one side of the junction to the other. The key is that even though you see

*weird*stuff happen, you can always explain it via some rather elementary considerations.

In any case the fits to the data using the simple diffusive transport model works over a large dynamic range in ordinates. The sharp bend near the top indicates the potential start to the plateauing, and one can observe that some of the pairs of data indeed do flatten out. The other gradual bend indicates the transition between diffusion transport and drift transport. The universality of this bend does not scale perfectly as drift does depend on the electric field whereas the diffusion doesn't. And as we will see in the next example, the temperature may not play a big role in deviations from universality.

### Transport in a-Si:H

Amorphous

semiconductors have a huge influence on the solar cell and photovoltaic

industry. In general, it costs much less to manufacture amorphous

materials as the fabrication facilities do not have to follow as strict a material process. Unfortunately the performance characteristics of the amorphous silicon in comparison to its crystalline brethren leaves lots of room for improvement. Although not as important for solar cells, the photo-response time for an incident light stimulus shows the long tails characteristic of diffusive transport.

I culled the data from a 2005 paper by Emelianova, et al studying the photo-response of amorphous hydrogenated Silicon (a-Si:H). This material was undoped and the investigation looked at hole carriers. I found this study very comprehensive and it leans toward questioning the applicability of Scher and Montroll's original formulation in terms of an alternate model that they formulate.

*I have never seen a spanking new model that popped out with such obvious agreement in my life.*This basically should set the hairs on end; I really believe that after 50 years of first discovering the anomalous dispersive transport that a simple equation, suitable for spreadsheet entry, would agree so well.

Mobility (u) | Width (w) | Diffusion constant (D) | Temperature (T) | Current Scaling (C) | Electric Field (E) |

0.00193 cm^{2}/V/s | 2.4 microns | 0.69 * u | 264 | 1.2e-13 | varies as in figure (V/cm) |

*t/t*= normalized time = sqrt (2

_{0}*Dt*+

*(uEt)*

^{2})/

*w*(EQ 5)

I(

*t*) = I

_{0}[1 - exp (-

*t*)(1 +

_{0}/t*t*

_{0}

*/t*)] (EQ 6)

where I

_{0}=

*C*E*E*

The idealness of the fit should preclude me from over-analyzing the results but a few interesting issues remain.

(1) For one, in this case the relation between mobility and diffusion constant does not obey the

Einstein relation but this rarely happens in non-ideal and disordered materials as the energy states get sufficiently smeared across the bandgap. The general Einstein relation relates the diffusion constant D to the energy distribution of the carrier states.

^{15}

*D*=

*u/q*[N(

*E*)/N'(

_{c}*E*)]

_{c}*E*around the Boltzmann distribution

_{c}^{16}, the relation

*D=u/q**

*kT*holds as an invariant, but as

*E*spreads out --

_{c}*and in the maximum entropy case of a large variance knowing only the mean*-- the diffusion constant tracks

*E*more than it does temperature,

_{c}*T*. I worked it out and

*D=u/q**(

*kT+E*) in that case. Since

_{c}

*E*_{c }typically exceeds the statistical value of thermal energy,

*, we will see a higher diffusivity than one would expect from an ordered solid (see Schiff).*

**kT**(2) Also, the initial transient spike has to do with the collection of complementary carriers at the near electrode, and has no influence on the results (undoped material generates equal number of oppositely charged carriers). As a probability exercise, the results also show that the integrated area under each curve is identical to within 0.2% for each voltage bias. In fact the cumulative charge collection based on Equation 1 becomes the following simple formula:

Q(

*t*) = I_{0}t*(1 - e^{-1/t}) (EQ 7)The build-up of charge starts linearly and then converges asymptotically to a value proportional to the total number of carriers generated during the pulse duration (excepting recombination and other losses).

The authors apply their own model to the results and suggest that the dispersion is wider than gaussian as the figure to the right shows, yet they also curiously indicate that is a gaussian non-dispersive transport. Much of the confusion arises from the original Scher-Montrose formulation which demarcates the curves into ordered or non-disordered instead of what I would like to see -- a dispersed diffusion-dominated regime versus a dispersed drift-dominated regime.

The upshot of the good agreement of my fundamental model with the results means that any smart electrical engineer can start using the simple formulation right now, and should that engineer want to calculate frequency response or impulse response of an amorphous material device, they just have to use Equation 6. They can do FFT or Laplace transforms or anything they want since they have an analytical result which they can plop into their notebook or spreadsheet or Matlab and work out. I guarantee no one would want to mess with the Montroll-Scher result as it gets way too unwieldy and I dare say that no one actually understands it. I consider this simplicity a huge benefit.

The only caveat: you need a disordered material to apply this to .... but, of course, that goes with the premise.

### Quantum Dots

Scientists have looked to unique materials including a variety of organic semiconductors in the hope of creating structures suitable for quantum dot devices. This paper Charge Carrier Transport in Poly(N-vinylcarbazole):CdS Quantum Dot Hybrid Nanocomposite provides a few time-of-flight curves in terms of a completely different material system. These TOF's appear to obey the same simple maximum entropy model for dispersive transport as you can see in Figure 8 and Figure 9.: TOF traces taken at different applied electric fields. The original diagram did not have dimensions on the axis so I guessed on the scaling based on the inset. I show the simple dispersive transport model as symbols with the electric field dependence as in Equation 6 Figure 8 | : All curves plotted on a universal scale. The tFigure 9^{-2} drift dependence extends beyond the range of the data. |

Of course good agreement means that the disorder in the systems has to agree with the maximum entropy model. Nothing precludes different diffusion mechanisms or even further disorder, implying even fatter tails than t

^{-1}. Some systems likely exist with a mix of order and disorder, such as crystalline semiconductors with many defects. In that case, one could conceivably separate out the effects.

## The Connection

*"When the weird gets going, the weird turn pro"*-- Hunter S. Thompson

I got sidetracked into the dispersive transport behavior of carriers in disordered solid-state materials as I searched for ideas that might substantiate the oil depletion models that I had worked on. I have long asserted that everything about the behavior of oil, from reservoir sizes, to oil discovery, and on to reserve growth has as a basis the effects of dispersion

^{17}. Just as the disorder in amorphous semiconductors causes a dispersion in carrier velocities, so too does the randomness and disorder in aspects of the fossil fuel process. Whether the randomness has to do with varying velocities in the drift of oil over eons or the variance of human search efforts (see figure at right), these all lead to the same fundamental formulation for dispersive analysis. Moreover, any chaotic or complex behavior gets smoothed out by the filter of dispersion. I essentially derived a new math shorthand to describe oil, and stumbled across the fact that this same derivation applies equally well to a field totally removed from the macroscopic. I essentially went from the macroscopic to the microscopic, and then back again to substantiate what I had earlier conjectured.

Recall again the two scientists Scher and Montroll, who originally formulated the CTRW theory to explain dispersive carrier transport. They essentially worked as applied mathematicians and have gained quite a bit of recognition for their ideas. Montroll, arguably the more well-known of the two has since died

^{18}, but Harvey Scher has continued on applying the same formalism to other application areas.

Guess where he has applied it?

*Answer*: Transport of materials underground via porous structures ... as you may have guessed, pretty much the same life-cycle that petroleum operates under.

So Scher essentially transitioned from the

microscopic world of semiconductors to the macroscopic world of the earth. Currently Scher works as a consultant for a group of geologists and environmental scientists that use the CTRW theory to explain the way that contamination and other solutes spread over time via diffusive transport.

^{19}

I have problems with the CTRW theory in that at a certain step in the derivation, the authors invoke the legendary "and then a miracle occurs" argument into the proof. This turns into the essential observation that long-range correlations go as 1/t

^{alpha}.

^{20}Well, I can generate that just by invoking the Maximum Entropy assumption on the variance of velocities. In that case, the inverse time power-law behavior naturally takes over and an integral exponent depending on the mean velocity from the specific type of motion occurring -- either diffusion (t

^{-1}) or drift (t

^{-2}). If a combination of the behaviors occurs, just solve the classical equations of motion assuming Fick's Law and calculus, and the characteristic dispersive formula appears, just as for dispersive transport in amorphous semiconductors.

: Application of the dispersive transport to the motion of solute. This experiment showed a transition as the solute migrated over time.Figure 10Analysis of Tracer Test Breakthrough Curves in Heterogeneous | : Figure 11The illustration shows some of the causes of pore-scale dispersion. Solute traveling more tortuous pathways between sediment grains will move more slowly than that moving along more direct pathways. Diverging pathways will also cause the contaminant to spread perpendicular to the aquifer flow direction. |

### Double Breakthrough

*: Uranine dye moving downstream in*

**Figure 12**Fisher Creek after

injection to trace the

destination of the water as it disappears.

(Groundwater Tracing in the Woodville Karst Plain)

Most of the solute transport measurements use something called breakthrough curve analysis. "Breakthrough curves" enable a researcher to estimate the amount

of dispersion occurring in a flow of solute (or contamination or

whatever) in a media. For a non-dispersive flow, the breakthrough

curve looks like a unit step where the tracer material is detected

abruptly at a specific time at a certain point downstream. This has an analog to the Time-of-Flight measurements used in photo-response studies described earlier

^{21}. But due to

randomness and variability in the media due to pore structures (for

example), the dispersion smears the breakthrough curve over a broad

time window.

A very simple model for a breakthrough curve

involves solving the equation for a maximum entropy spread in

velocities (for a given mean velocity) at a specific distance L. Given

the average time taken is

*, and a random variate would take*

**T=L/v**time =

**, then the breakthrough curve looks like exp(**

*t**). If you plot*

**-L/vt**this curve it looks like what some people refer to as a "reciprocal

exponential". It isn't the classic exponential because the time

parameter goes in the denominator. That happens because we are dealing

with rates, and not time for the stochastic parameter.

Next, we must realize that an idealized breakthrough curve assumes a

*fixed*separation,

*, in a very controlled experimental environment.*

**L**In reality, the distance

*'s become spread out over space and for a*

**L**maximum entropy PDF of

*, an uncontrolled "breakthrough curve" will*

**L**have a temporal behavior that looks like 1/(1+

**), where**

*L/vt**becomes the*

**L***mean*separation. This looks

*exactly*like the formula for enigmatic reserve growth in oil discoveries that I derived before [ref].

I am satisfied with using a maximum entropy estimator for the dispersion because the effects could be due to many different possibilities. So, in a sense, variability overrules complexity and if we can concentrate on understanding the mean value, we have a very simple way to characterize the system. That is my premise and I have to be able to defend it from many angles.

Take a look at the figure to the right from a hydrogeology experiment.

^{22}Granted, I do not know anything about the particulars of the particular experiment, yet I assert that I can do a better job of fitting to the results solely because I do not place a bias on my estimator. I simply apply maximum entropy to randomize the effect, making the only assumption the mean transport rate.

I see some indication that Scher and his colleagues have at least considered this simple premise. From the following extract, note that they imply that some sort of ensemble average acts as a precondition to further analysis. In other words, they make the presupposition that geology is random and uncontrollable, just like amorphous semiconductors and human processes.

The point average of v and D can be very sensitive to small

changes in the local volume used to determine the average. Conversely,

if one fixes the volume to a practical pixel size (e.g., 10 m3) the use

of a local average v and D in each volume can be quite limited, i.e.,

the spreading effects of unresolved residual heterogeneities are

suppressed [e.g., Dagan, 1997]. We will return to this issue in a

broader context in section 4. It essentially involves the degrees of

uncertainty and its associated spatial scales..We start, at first, with an ensemble average of the entire medium and discuss the role of this approach in the broader context

-- "Physical Pictures of Transport in Heterogeneous Media:" Brian Berkowitz, Joseph Klafter, Ralf Metzler, and Harvey

Advection-Dispersion, Random Walk and Fractional Derivative

Formulations.

Scher

### Back to Oil

As a very generaltechnique we can apply the equivalent of breakthrough analysis across many domains. The usual problem remains that different application domains use different terminology. I never used breakthrough analysis terminology because no one does controlled experiments when they look for or extract oil. Oil exploration is a commercial enterprises after all and oil prospectors get what they can, while they can, and don't necessarily ponder any deeper meaning. Yet, I view the over-riding dispersion analysis as a very general concept and I simply apply the same technique in oil depletion by making the analogy to dispersion in human-aided discovery search rates. The fact that it also occurs for physical

processes such as contaminant flow in groundwater, carrier transport in amorphous semiconductors, or TCP dispersion should not surprise anyone.

^{23}

Over 50 years have lapsed since the day that Hubbert first sketched a Logistic curve to model oil depletion, and I think science has had a mental block on the dispersion problem all this time. We can easily and simply explain the dynamics of the oil production curve by using these same ideas from dispersion analysis.

^{24}

Getting meta for a moment, to

you I exist only as a blogger. I don't know if this analysis will go

anywhere. As you may realize, I have some good ideas on the way we can analyze oil

depletion. Yet, I have no credentials in that field. A pseudonymous

writer can only way sway an argument based on the logic of his

arguments. If you can follow the argument in this post, and believe it

applies, and that all experimental evidence backs up the theory, my

credibility builds. Someone will then say, "

*well, he got that part*

right, maybe this other part makes some sense". As far as I can tell,

right, maybe this other part makes some sense

no one has documented a similar

*simple*approach to what I have formulated via my blog postings. It takes a bit of intuition to determine the situations where disorder and diversity rules and where it does not. I say that where you can

appropriately apply these arguments you can start to understand the

dynamics. I can certainly understand the dynamics of the Hubbert curve via dispersion just as I can understand the transient of an amorphous semiconductor time-of-flight experiment by applying dispersion. The fact that no one else sees it this way turns my task into one of salesmanship, unfortunate, since no one likes a salesman.

Putting that aside, I assert that bottom-line we really can use fundamental concepts to understand the dynamics of these behaviors. Absolutely nothing about any of these empirical observations I would consider anomalous. No one resorts to calling the variability in rain "anomalous" and why we haven't universally figured out simple solutions as described here remains the real mystery. Or that is indeed the real anomaly.

</meta>

WHT

http://mobjectivist.blogspot.com

#### FOOTNOTES

- Or "enigmatic" as some have referred to the oil situation, see http://mobjectivist.blogspot.com/2008/07/solving-enigma-of-reserve-growth.html

Figure from "Physics of amorphous semiconductors", Kazuo Morigaki- "Transit-time measurements of charge carriers in disordered silicons: amorphous, nanocrystalline, and porous", E. A. Schiff
- "The physics of amorphous solids" Richard Zallen, Wiley-VCH, 1998
- Dielectric phenomena in solids, Kwan-Chi Kao, Elsevier,2004
- On a log-log

plot, as long as the orders of magnitude scale is maintained, one can

fit a curve simply by translation - "Ionizing radiation effects in MOS devices and circuits", T. P. Ma, Paul V. Dressendorfer
- Anomalous transit-time dispersion in amorphous solids, H. Scher, E. W. Montroll
__,__Phys. Rev. B 12, 2455 - 2477 (1975) - Computer simulation of photocurrent transients for charge transport in disordered organic materials containing traps

Proc. SPIE, Vol. 3799, 94 (1999)

Sergey V. Novikov, David H. Dunlap, Vasudev M. Kenkre, Anatoly V. Vannikov - One can use the Laplace transform to characterize the dimension of the disorder, see

http://mobjectivist.blogspot.com/2008/08/general-dispersive-discovery-laplace.html - "Excess electrons in dielectric media", C. Ferradine, J-P Jay-Gerin.
- A MAXIMUM ENTROPY ANALYSIS OF SINGLE SERVER QUEUING SYSTEM WITH SELF-SIMILAR INPUT TRAFFIC

A. Asars, E. Petersons - I describe a comprehensive model for oil depletion and apply dispersion to three aspects of the model.

http://mobjectivist.blogspot.com/2008/11/comprehensive-oil-depletion-model-life.html

<img src="http://img224.imageshack.us/img224/448/comprehensivemodelsi5.gif"> - "Diffusion with drift on a finite line", M.Khantha, V. Balakrishnan, Journal Physics C: Solid State Physics, 16 (1983)
- Hydrogenated amorphous silicon, R. A. Street
- The Maxwell-Boltzmann is an approximation to the actual Fermi-Dirac distribution at higher temperatures.
- See theshock model for exceptions to this rule.
- Montroll held advanced research director positions at IBM Research, Institute for Defense Analysis, and Office of Naval Research.
- In the context of geological materials, CTRW theory has been developed and applied extensively. For an extensive review, see B. Berkowitz, A. Cortis, M. Dentz and H. Scher, Modeling non-Fickian transport in geological formations as a continuous time random walk, Reviews of Geophysics, 44, RG2003, doi:10.1029/2005RG000178, 2006.

"Dispersion in heterogeneous geological formations" Brian Berkowitz - They also refer to the alpha disorder parameter as beta.
- One significant difference is that solute does not induce a charge of current that we can measure. Instead the solute is measured directly as a concentration density.
- Colloid Mediated Transport of Contaminants in Shallow Aquifers, 2009 Institute of

Hydrochemistry,

Technische Universität München - Dispersion exists in the case of Network TCP latencies where collisions may occur. The

velocities disperse from the maximum according to the nominal transit

time and the mean latencies are known. Since transit times are related

to the inverse of speed for a fixed separation, the distribution of

times goes like T*exp(-T/t)/t^{2}. The breakthrough curve for TCP dispersion should look like exp(-T/t) beyond the fixed latency due to wave propagation. see http://mobjectivist.blogspot.com/2008/09/network-dispersion.html - Just apply an accelerating electric field to the the TOF experiment and I guarantee the output will start looking like a Logistic.

## 3 Comments:

Well, I've had a chance to read your post twice, the first time I skimmed, the second time I took a bit more care.

I'm impressed. Okay, let me just say it: 'Wow'. A beautiful and elegant argument. Feeling reading it was akin to the feeling I felt the first time I encountered generalised coordinates and the Lagrangian.

Million dollar question: can the theory be used to predict the eventual shape of the production tail for a particular field (from early production data and general known physical characteristics of field)?

Thanks.

Yes, exactly that can be done. I have a few posts on dispersive reserve growth. It turns out that the long reserve growth tail goes as 1/t(squared), or 1/t for the cumulative. The depletion analyst Laherrere calls this a "hyperbolic" and it describes single oil producing regions in many of the creaming curves published. The cumulative curve goes as 1/(1+tau/time), so the important thing is to determine the value of tau. Whether this can be done better by early extrapolation from reserve growth number or by some fundamental properties of the region is up in the air. In general tau is equivalent to volume of the region divided by the average velocity of the search volume explored. The dispersion in the velocities and the maximum entropy comes about because no one really knows exactly how fast to explore the region. So it occurs at different rates over time, leading to the characteristic curve. It looks very similar to the effect of diminishing returns.

The actual decline of a reservoir can also show a hyperbolic decline. I think this is a byproduct of the reserve growth and perhaps a dispersion of the exponential decline extraction rates. Declines are in general either exponential, hyperbolic, or harmonic (which is a degenerate mathematical condition). That area is wide open for further research because I don't think petroleum engineers understand the phenomena very well if at all. (Exponential is easy to explain, hyperbolic not so much apart via a slow diffusion and from gravity drainage of adjacent areas)

The problem right now is that a few things may go into the mix and it is difficult to separate the effects of incremental discovery and incremental extraction.

Regarding the Lagrangian comment, I think the Principle of Maximum Entropy and the Principle of Least Action are good bookends in explaining much of physics. Feynman's description of how to use the Lagrangian in terms of the Principle of Least Action is very elegant. So I appreciate that comment.

We have essentially two ideas working together -- the shortest distance between two points is a straight line (Principle of Least Action, evidenced by gravity or electric field) and the disturbances along the way (Maximum Entropy, diffusion+dispersion). I was seriously thinking about how to unify these two concepts for the sake of understanding, but haven't gotten past the philosophical stage. Fiddling around with the Einstein relation is about the furthest I got along with respect to making the connection. The Diffusion/Mobility Einstein relationship is really the equilibrium argument and dispersion essentially described departures from equilibrium, via macro-effects such as sample uniformity, etc.

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