Nonlinear Elasticity and Hysteresis

Fluid-Solid Coupling in Porous Media

Alicia H. Kim, Robert A. Guyer

• Publisher: Wiley-VCH
• Year: 2014
• Print ISBN:
  9783527333028
• Edition: 1
• eBook ISBN:
  9783527665082
• Topic: Technology & Engineering
• Subtopic: Materials Science
• Index: Technology & Engineering

The book provides the reader with the knowledge, tools, and methods to understand the phenomenon of hysteresis in porous materials.

As many challenges have been met only recently, the book summarizes the research results usually found only scattered in the literature, connecting knowledge from traditionally separated research fields to provide a better understanding of the physical phenomena of coupled elastic-fluid systems.

The result is an invaluable self-contained reference book for materials scientists, civil, mechanical and construction engineers concerned with development and maintenance of structures made of porous materials.

1
Dynamic Pressure and Temperature Responses of Porous Sedimentary Rocks by Simultaneous Resonant Ultrasound Spectroscopy and Neutron Time-of-Flight Measurements

James A. TenCate, Timothy W. Darling, and Sven C. Vogel

1.1 Introduction and Background

Rocks are everywhere, yet there are still surprising puzzles about their peculiar dynamic elastic properties, especially their hysteresis, non-Hookean response, and rate-dependent behavior. Since before recorded history, mankind has been making dwellings, hammering out monuments, and even constructing huge buildings out of rock, for example, the famous Strasbourg Cathedral built in the Middle Ages is made almost entirely from Vosges sandstone. Nowadays, one extracts oil and gas from rocks, explores ways to store excess CO₂ in them, and tries to mimic their resilience and durability with concrete. The imperfect way in which mineral grains end up cemented into rocks dictates how fluids move in oil or gas reservoirs or in aquifers. Indeed, these very fluids are often a key mechanism for that cementation. The diagenesis of rocks, their formation, and cementation history are of great geological interest as well. Hence, the dynamic elastic properties of rocks have been a topic of continuing scientific study for well over a century.

To narrow the focus of this chapter, the subject is primarily the behavior of rocks that have commercial interest. These rocks may contain oil and gas, or might be considered as a reservoir for CO₂ storage. These rocks are primarily sedimentary, and the focus of this chapter will sharpen even more, dealing exclusively with sandstones. A sandstone is an imperfectly cemented collection of quartz grains, which is porous and permeable to fluids (which often play a key role in the cementation) and may contain significant amounts of clays and other materials. In the experiments described here, the rocks studied will be extremely pure and clean sandstones, 99+% pure SiO₂ formed from quartz (prehistoric, 77 MYBP) Aeolian beach sand, known as Fontainebleau sandstone. Such rocks are simply composed of the grains and cementation.

How does one describe and examine such a sandstone and how is it different from man-made materials? A thin section examined under a polarizing microscope can show the crystallographic orientation and the nature of the grains, bonds, and cementation. A thin section of Fontainebleau sandstone is shown in Figure 1.1. All the grains are roughly of the same size (about 150–200 µm), and none of the material in the section shown here has any significant polycrystalline components and very little of it is amorphous or glassy (which shows up as black under crossed polarizers). However, the reality is that a thin slice of a rock really does not give a very good representation of the porosity and permeability or even of the cementation. Care must be taken in extracting distributions of pore and grain sizes from sections, and often the cementation at grain contacts is difficult to identify. Amorphous cement, for example, can easily be missed and more advanced petrographic techniques such as cathodoluminescence or electron backscatter diffraction in an SEM must be used [1]. Occasionally, pore casts – where an epoxy is spun into the pore space and the sand dissolved away with an acid – are made to study the three-dimensional network of pores [2]. X-ray micro-CT images on very small samples are made as well, originally to provide an input for modeling, but similar to thin sections, they contain no information on the mechanical properties of the system. The contact network of grains, the pore space they can rotate into, and the fluids that can move around in that pore space (e.g., water, oil, and gas) all couple to the dynamics of a rock; direct measurements on the scale of these features is extremely difficult.

Figure 1.1 An image of a thin section of Fontainebleau sandstone in a polarizing microscope. The largely single crystal grains are rounded (convex) while porosity, filled with epoxy, has more concave boundaries. The largest visible grains are taken as characteristic of the grain size, about 200 µm across. With very porous rocks, the possibility of grain movement during polishing must be considered, possibly distorting apparent boundaries.

So the cementation, porosity, and permeability are important for many reasons: what can be done to study and understand how a rock is put together in a laboratory setting? The following lists several possible applied external fields [3] that will guide the experimental discussion that follows:

1. Pressure/stress
2. Temperature
3. “Humidity” resulting in the movement of fluid in a rock
4. Electric or magnetic fields
5. Vibration or acoustic energy.

The first two parameters, pressure — and the associated stress-strain measurements – and temperature – and the associated sound speed versus temperature measurements, seem to have been motivated by oil and gas exploration at the turn of the last century. “Humidity” measurements came out of interest in ultra dry lunar/moon rocks in the 1970s. Interest in electric and magnetic fields came a bit later and the last, vibration and acoustic energy, came later still. Vibration as an applied field is a bit unusual and is distinct from the use of a low amplitude stress wave to measure sound speed. “Vibration” has to do with changing the internal arrangement of strain fields with an acoustic AC drive – dubbed slow dynamics by TenCate and Shankland [4] – in contrast with the other DC applied fields. Other experiments and external fields may be possible. However, in this chapter just pressure and temperature measurements will be considered. These measurements are rate dependent and show a nonreversible response normally thought of as “hysteresis.” These are all “macroscopic” measurements, done at sample scales of a few to tens of centimeters. Some discussion of each kind of measurement is appropriate at this point before we delve into the combined macroscopic and atomic (neutron) measurements reported in the bulk of the chapter.

1.2 Macroscopic Measurements

1.2.1 Stress-Strain Measurements

The hysteretic macroscopic strain response of rocks to uniaxial compressive stress (force/unit area) has been noted since the turn of the last century [5]: after being brought to a “state of ease” (conditioning) by application of high stresses, most rocks display a repeatable curved, stress–strain loop under cyclic loading, for example, [6, 7]. Fully recoverable hysteretic processes, driven by stress and dependent on the previous extreme stress values, produce multiple values of strain for stresses between the cycle and values. The details of these processes (which produce similar effects in many different kinds of rocks) are usually ascribed to grain contact and fluid effects.

The literature reports a number of models to describe this hysteretic behavior, such as the Hertz–Mindlin model [8] and the Preisach–Mayergoyz (P–M) space model [9–11]. In analogy with magnetic domains in a magnetic system, a hysteretic rock “domain” or unit in the P–M scheme opens at one stress and closes at another. While these models can reproduce qualitatively the observed responses, assignment of the model systems to real physical elements has some problems, such as requiring frictional surface slip at interfaces in sandstone, where we expect small-area, perhaps brittle, bonds of a solid silicate. The tensile and shear strength of rocks, however, suggest that macroscopic slip cannot occur at every contact.

Rate effects have been reported and are worth noting. Claytor et al. [12] discovered that the rate at which hysteresis loops are taken is important for certain rock samples, especially sandstones. Elastic aftereffect, in analogy with magnetic aftereffect, is a process whereby hysteretic elements do not stay “switched” but snap back to some “relaxed” state. Thus, if a hysteresis loop is taken slowly enough, the area between the loops vanishes and the stress–strain curve is merely nonlinear. Figure 1.2 shows two stress–strain hysteresis loops taken from Claytor's data: one quickly, 0.38 s between stress increments; and the other very slowly, over a weekend with 60 s between stress increments. The area within the error bars (not shown) is essentially zero for the very long stress–strain experiment. What physics is responsible? Fluids coupling to the rock grain skeleton could be one answer, and there may be many other possible explanations. There is simply no enough data on enough rocks yet to model how fluids ...

Table of contents

• Cover
• Related Titles
• Title Page
• Copyright
• Preface
• List of Contributors
• Chapter 1: Dynamic Pressure and Temperature Responses of Porous Sedimentary Rocks by Simultaneous Resonant Ultrasound Spectroscopy and Neutron Time-of-Flight Measurements
• Chapter 2: Adsorption, Cavitation, and Elasticity in Mesoporous Materials
• Chapter 3: Theoretical Modeling of Fluid–Solid Couplingin Porous Materials
• Chapter 4: Influence of Damage and Moisture on the Nonlinear Hysteretic Behavior of Quasi-Brittle Materials
• Chapter 5: Modeling the Poromechanical Behavior of Microporousand Mesoporous Solids: Application to Coal
• Chapter 6: A Theoretical Approach to the Coupled Fluid–Solid Physical Response of Porous and Cellular Materials: Dynamics
• Chapter 7: Swelling of Wood Tissue: Interactions at the Cellular Scale
• Chapter 8: Hydro-Actuated Plant Devices
• Index
• EULA