Analytical Modeling in Structural Geology

R. C. FletcherPGP, University of Oslo

GEOMOD2010, Lisbon, Portugal 9/27/2010

with acknowledgements to:

Ezra Pound, poetGeorge Polya, mathematician

Bill Chapple, structural geologistOphelia, cat

Laccoliths of the Henry Mountains, Utah & Gilbert’s model (1875-1880)

2 2 0

2

M

M

p gZ R RZ

ZR

p gZ

Analytical modeling in structural geology (AMSG)

entire, iterative process connecting two elements:

(1) sets of observations of a naturally deformed rock mass

(2) description of the physics of deformation provided by continuum mechanics.

Connected through the formulation, implementation and application of a model.

Following Pappus of Alexandria (290-350), as discussed in the mathematician G. Polya’s book “How to Solve It,” a distinction may be made between analysis, or solution backward, or regressive reasoning, and synthesis, or constructive solution, or progressive reasoning. Going from (1) to (2) is chiefly analysis; going from (2) to (1) is chiefly synthesis.

1 FAULTS 25%2 “FOLDS” 13%3 FABRIC 25%4 EXP’TS – RHEOLOGY & ANALOG MODELS 10%5 KINEMATIC MODELS 11%6 AMSG (?) 16%

N = 111 papers in JSG

Theory“Theory” is the underlying physics. There is

no “theory of folding”; no “theory for metamorphic differentiation”

A theory is closed.An application of continuum mechanics is to

model folding or metamorphic differentiation. Continuum mechanics is the theory.

Exhibit: AMSG in the study of rock folding

Ramberg (1960): Ld/H = 2p/(2/3R)1/3; a figure of two folds gives L/H ~ 6.2. Using Ld/H ~ L/H, R = 1.4. Equation not used; folds used to illustrate “contact strain” within a zone of thickness ~ Ld.

Biot (1961) : Ld/H = 2p/(6/R)1/3 : R = 6. Thick-plate solution: R = 1.

Biot, Ode and Roever (1961) demonstrated Biot results in experiments, so the geologists would “believe” them. Folds in a pitch- corn syrup assembly for which R ~ 2200, yielded L/H = Ld/H = 45; a value yet to be seen in nature.

Currie, Patnode & Trump (1962) report L/H ~ 27, but where this value comes from remains a mystery.

Sherwin & Chapple (1968) test Ld/H ~ L/H. Chapple surprised with small L/H measured by Sherwin; he had seen many single-layer folds. Their AMSG application requiredincorporation of layer thickening during wavelength selection. R ~ 14 – 35.

Chapple (1968), Dieterich & Onat (1968), Dietrich & Carter (1969) & Parrish (1973:power-law) : FEM; relationships between folding and cleavage; distributions of stress and strain. [Re: “folding mechanisms”] Fletcher (1974) & Smith (1977) single-layer folding in power-law materials, extended interpretation of small values L/H ~ 4 – 6. Biot (1961) treated nonlinearity in thin-plate approximation.Fletcher got 3 < n < 12 for limestone layer in shale. Fletcher & Sherwin (1978): distribution of L/H provides another fitting parameter.

Hudleston & Holst (1984) estimate rheological parameters for single-layer folds of limestone in shale; n was poorly-constrained. Petrographic study of deformation mechanisms: twinning, stylolites, veins & of strain distribution.

Exhibit: Castile microfolds

Field observations [1]Cylindrical microfolds

Castile Formation, Permian Basin

west Texas & New Mexico

Stacked SLUs from “infinite” multilayer

1 /2

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21

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j

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i j

n

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kl kl

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v vD

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D BJ s

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Unknowns Equations

6 3

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THEORY [2]

SLU VI

SLU VII

SLU VIILd/H = 4.2Hi: 1, 1, 1Ai: 0.4, 1, 0.6

Link (1) and (2): The model

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work.

John von Neumann

What is the question? Rheological parameters for a fit & are they plausible? Two alternate models for SLU VII

3 layers, equal thickness,bounded byrigid, frictionless platensslip parameter Q*=h*H/hd*stress exponent n

3 layers, equal thickness,bounded byviscous ½-spaceR = h/h2

Note: actual thicknesses & amplitudes shown;models are symmetric equivalents

SLU VII 3 equal layers L/H = 4.2 Bounded by rigid, frictionless platens

Interfacial viscous slip Q = (h*H)/(hd*)

Stress exponent n

Contours of Ld/H (black) & qd (red)

Fit: n > 50 Q ~ 0.1

n > 12

Consistency:R ~ (1/Q)1/2

Lake Char-gogg-agogg-manchaugg-agogg-

chau-bun-a-gun-ga-maugg

Webster, MassachusettsNipmuc Native American:

“You fish on your side. I fish on my side. Nobody fishes in the middle.”

The descendents of the fishing Nipmucs now run the biggest gambling casino in Connecticut

The Four Tribes – warlike & independentfought for “research wampum”

Buffalo Hunters: Field-based structural geologistsLived in teepeesMeat eatersOutcrop scalpers

Mound Builders: Model buildersUrban dwellersAte pumpkins & corn mealBuild large mounds of ??? For burial of the dead? To escape floods? On which to erect fancy homes?

Coastal Indians: Rock rheologistsBuilt totem polesAte oysters & lobsters, the local cheesesDrank expensive alcoholic beverages

Paleo-Indians: PetrofabricatorsWhere they lived now covered with tiny rock chipsLeft untranslatable manuscripts on moose hide

Analytical solutions

1. Infinite space internal folding

2. Layers 3. Wedges4. Inclusions – in infinite space

SCA in polycrystals & other composites

5. Cracks, anticracks Joints, faults, stylolites

6. Pipesmagma flow/wall erosion

Should kinematic modeling go the way of the Ptolemy’s Epicycles? [YES]

Is it based on theory?NO – No closure

Does it address field observations? Over-constrains form

Does it provide insight into processes operative in development of structure & fabric?

NO – Excludes quantities containing dimension of mass

TRISHEAR: deformation associated with propagating basement fault

Governing equation:

KINEMATIC VISCOUS

10

0

r r

r r r

vv v

r r rv

v v v

What does pursuit of a model that can be treated by an analytical solution, as a part of AMSG, do for you?

All modeling, including AMSG, centers on posing interesting questions

Write out the equations = physicsSketch the field exampleAssemble a set of measurementsSimplifyPose a tractable boundary-value problem

Intuitive ability closely resembles artistic talent (M. A. Biot, 1963)

“Though I haven't applied your work in exactly the way you did, what you describe for the "effective aspect ratio" is a perfectly sound approach to me as well, but is in a way an a posteriori method.  The core challenge for an analytical solution in my mind remains an a priori approach, that is to say to look at a distribution of hard and soft phases and predict/know/easily measure what the effective aspect ratio is.  For a conglomerate or similar, it's straightforward, but I am not sure how to know for situations such as anastomosing shear zones (with various degrees of connectedness).”

Postmodernism

a tendency in contemporary culture characterized by rejection of objective truth. Emphasizes roles of language, power relations, and motivations; attacks use of sharp classifications such as white versus black.

Postmodernist thought is an intentional departure from modernist approaches previously dominant. Term "postmodernism" comes from its critique of the "modernist" scientific mentality of objectivity and progress associated with the Enlightenment.

Modernism and postmodernism are understood as cultural projects."Postmodernism" in critical theory refers to a point of departure for works of literature, …, and design, as well as in marketing and business and in the interpretation of law, culture, and religion. Indeed, postmodernism … can be understood as a reaction to modernism….Whereas modernism was primarily concerned with principles such as identity, unity, authority, and certainty, postmodernism is often associated with difference, plurality, textuality, and skepticism.

Four worldviews are the Postmodern-ironist, which sees truth as socially constructed; the scientific-rational, in which truth is found through methodical, disciplined inquiry; the social-traditional, in which truth is found in the heritage of American and Western civilization; and the neo-romantic, in which truth is found through attaining harmony with nature and/or spiritual exploration of the inner self.