Poroelasticity

In Codex Leicester (da Vinci L, Codex leicester, 1506–1510), da Vinci (see Sect. F.1 for a biography) drew the analogy between the Earth and the human body, to argue that Earth too, is a living body. He compared soils, rocks, and tufa (volcanic rock composed of fused detritus) to flesh, bones, cartilages, heart, arteries, and veins. We may ask, what do these materials have in common? They are all porous materials, that is, solid materials containing void space (pores) in them. Particularly, the pores can be occupied by a fluid, such be water, air, oil, methane gas, blood, body fluid, or a mixture of these.This book is dedicated to the study of mechanics of porous materials, especially those infiltrated by a fluid. Porous materials can be found in nature as inanimate objects such as sand, soil, and rock, as living bodies such as plant tissue and animal and human flesh and bones, or as man-made materials for various industry or biomedical applications. These materials can look much different in their appearances due to their origin, but the underlying physical principles governing their mechanical behaviors can be the same. We are interested in the static and dynamic responses of these materials subject to mechanical as well as other type of forces, such as those of thermal and chemical origin. These studies are generally known as poromechanics (The term “poromechanics” was first created for the Biot Conference on Poromechanics (Thimus et al (eds), Poromechanics–a tribute to Maurice A. Biot. Balkema, Rotterdam/Brookfield, 648pp, 1998)).The modeling of a full range of porous material responses, ranging from quasi-static to dynamic, from linear to nonlinear, and from partial uncoupling to full coupling, can be complex and unwieldy. The goal of this book is limited. It focuses largely on the linear theories, known as poroelasticity (The term “poroelasticity” was first used by Deresiewicz and Skalak (Deresiewicz and Skalak, Bull Seismol Soc Am 53(4):783–788, 1963)), as in the linear theory of elasticity and Darcy flow; hence the book is of an introductory nature.

A constitutive equation is a mathematical relation between two or more physical quantities. To define such mathematical relations, coefficients that are specific to a material, or to a composite material, known as material constants, are needed.As quoted above, Robert Hooke (see Sect. F.2 for a biography) was the first to establish a constitutive relation for elastic bodies by observing that the elongation of a coil spring, a spiral spring, a wire string, and also the bending of a straight piece of wood, are directly proportional to the weight attached to them (Hooke R, De Potentia Restitutivâ, or of spring, explaining the power of springy bodies. Martyn, London, 1678, 56pp). When this observation is used to establish linear relations between the stress and strain of an elastic body, it is called Hooke’s law, to honor the original contribution by Hooke.Since Hooke’s observation, it took almost 130 years, until 1807, that a material constant was explicitly defined by Thomas Young (Young T, A course of lectures on natural philosophy and the mechanical arts, vol 2. Joseph Johnson, London, 1807, 738pp) (see Sect. F.4 for a biography), and he stated:

The modulus of the elasticity of any substance is a column of the same substance, capable of producing a pressure on its base which is to the weight causing a certain degree of compression, as the length of the substance is to the diminution of its length.

In the above, Young first defined a weight of modulus of elasticity, which was dependent on the cross-sectional area of the column. When the weight of modulus is divided by the cross-sectional area to yield a height of modulus, then it becomes a material property only, which in present-day terminology is called the Young’s modulus. In fact, it was quite amazing that Young was able to determine the Young’s modulus of a steel to be 2. 9 × 10⁷ psi, which is the same as the present-day determined value, by finding the frequency of vibration of a tuning fork (Timoshenko SP, History of strength of materials. Dover, New York, 1983, 452pp; Young T, A course of lectures on natural philosophy and the mechanical arts, vol 2. Joseph Johnson, London, 1807, 738pp)!Other examples of constitutive equations include Newton’s law of viscosity, relating the rate of fluid shear strain to applied shear stress, through a material constant called viscosity, and Darcy’s law of porous medium, relating the fluid specific flux to the applied head gradient, through a constant called hydraulic conductivity, which is a combined property of the porous medium void space geometry and the fluid residing in it.These constitutive equations (laws) are often combined with other physical laws of more general nature, such as mass, momentum, and energy conservation, which are not specific to a material, to form a set of governing equations to predict the physical response of matters subject to disturbances (see Chap. 6).In this chapter, we shall construct the constitutive equations for poroelasticity. Particularly, we are interested in the deformation of the porous solid and the fluid, subjected to applied forces and pressure.

The bulk continuum model presented in the preceding chapter relates the externally observable quantities of a deforming porous specimen, such as the total stress, pore pressure, frame deformation, and fluid expelled from the frame, to each other, in order to construct constitutive relations that can be used to predict material behaviors. There are times, however, it is desirable, or even necessary, to learn what is happening inside a porous medium in terms of the solid and fluid phase, and the porous structure, such as change in porosity. For example, when we observe an external volume change of a porous frame, we may want to know how much of it is derived from the solid deformation, and how much is due to the pore space being taken out? When we measure a volume of fluid being expelled from a porous frame, how much of it is due to the reduction of the internal pore space, and how much is due to the expansion of fluid itself? For material constants, such as the undrained bulk modulus, how much of its apparent compliance is attributed to the compressibility of the solid constituent (which is typically small), and how much is to the pore space (which can be much larger)? Or, given an undrained bulk modulus, in what proportion does it draw its strength from the porous frame and the fluid?Gassmann in 1951 (Veirteljahrsschrift der Naturforschenden Gesellschaft in Zürich 96:1–23, 1951) presented a model intended to partially answer these questions. In the model, Gassmann partitioned the total volume of the frame into a part occupied by the solid, and a part by the pores. In an effort to construct constitutive equations that relate the volumetric deformations to the applied stresses, he identified three micromechanical material constants, a solid, a fluid, and a pore compressibility. The Gassmann model, however, assumed that at the grain level (microscopic scale), the solid phase is homogeneous and isotropic, though at the macroscopic scale, the material can be heterogeneous and anisotropic. This model has been called the ideal porous medium model. Many porous material, particularly geomaterials, however, are not homogeneous and/or isotropic at the grain level. For example, rocks at the microscopic level are made of grains of different minerals, such as quartz, calcite, mica, and even clay minerals; hence are heterogeneous at that level. This suggests that Gassmann model is a special model.For a general model, the microhomogeneity and microisotropy assumptions were removed by Biot and Willis in 1957 (J Appl Mech ASME 24:594–601, 1957). The resultant micromechanics model contains four independent material constants associated with volumetric deformation, one more than the ideal porous medium model. This micromechanical analysis has been widely accepted, and reformulated by many others (Brown, Korringa, Geophysics 40(4):608–616, 1975; Carrell, Mechanical response of fluid-saturated porous materials. In: Rimrott FPJ, Tabarrok B (eds) Theoretical and applied mechanics, 15th international congress on theoretical and applied mechanics, Toronto, pp 251–262, 1980; Detournay, Cheng, Fundamentals of poroelasticity. In: Fairhurst C (ed) Comprehensive rock engineering: principles, practice and projects, vol. II, Analysis and design method. Pergamon Press, Oxford/New York, pp 113–171, 1993; Nur, Byerlee, J Geophys Res 76(26):6414–6419, 1971; Rice, Cleary, Rev Geophys 14(2):227–241, 1976; Wang, Theory of linear poroelasticity: with applications to geomechanics and hydrogeology. Princeton University Press, Princeton, 287pp, 2000), in ways that are consistent with the original model. In this chapter, the Biot-Willis micromechanics model is presented.

In Chap. 2, the constitutive equations for poroelasticity were constructed using the phenomenological approach. In such approach, we attempt to model a new phenomenon, such as the deformation of a saturated porous body, by drawing an analogy with a familiar phenomenon, such as the deformation of an elastic body. In the analogy, we define stresses and strains for a porous body following the elasticity concept, even though their interpretation may not be clear. (For example, see the illustration in Fig. 2.​1 for the lack of a clear definition for a continuous stress field in a porous body.) We then bring in the additional force component, namely the pore pressure, and its conjugate deformation, the fluid strain (or actually the relative fluid to solid strain), to build a linear relation that is similar to that of the elasticity theory. This type of ad hoc construction of a working theory is typically motivated first by the observation, and then supported by physical insight, without formally resorting to the laws of physics. To gain additional physical insight, in Chap. 3 we utilized the micromechanics approach to explicitly model the material phases in a porous medium, not only the solid and the fluid, but also the pore space as an additional “phase”, together with their interactions. These micromechanical constitutive laws were assembled and matched up with the bulk continuum theory to provide physical insight. However, these constitutive laws were largely constructed using the “effective stress” concept; hence their theoretical basis is still phenomenological. The material constants associated with the theory, such as K, K′ _s , and K″ _s , are still empirical constants, as their physical mechanisms are based on composite responses, and are not fully isolated to tie to the equation of the state of the phases. There are many attempts to build porous medium constitutive models that go beyond the phenomenological model. Among the more widely pursued approaches are the theory of mixtures (Atkin and Craine, Q J Mech Appl Math 29:209–244, 1976; Bowen, Int J Eng Sci 18(9):1129–1148, 1980; Bowen, Int J Eng Sci 20(6):697–735, 1982; Crochet and Naghdi, Int J Eng Sci 4:383–401, 1966; de Boer R, Theory of porous media, highlights in the historical development and current state. Springer, Brelin/New York, 2000, 618pp; Morland, J Geophys Res 77(5):890–900, 1972), and the homogenization theory (Auriault and Sanchez-Palencia, J de Méc 16(4):575–603, 1977; Chateau and Dormieux, Int J Numer Anal Methods Geomech 26(8):831–844, 2002; Mei and Auriault, Proc R Soc Lond Ser A Math Phys Eng Sci 426(1871):391–423, 1989; Terada et al., Comput Methods Appl Mech Eng 153(3–4):223–257, 1998). The theory of mixtures uses the mathematical assumption that the solid and fluid phases occupies the same space, whose presence and influence are weighted by a volume fraction, in order to fulfill the requirement of a continuous mathematical function. This approach is largely mathematical, and the “material coefficients” generally do not possess physical meaning until a comparison is made with a theory that is in use, such as a phenomenological theory. The homogenization approach, on the other hand, explicitly recognizes solid and fluid phases occupying different space at the microscopic level, with an assumed periodic pore geometry. The full partial differential equations, such as Navier-Stokes equation for the fluid and elasticity equation for the solid, are prescribed, together with boundary conditions. Effort is then made to simplify these equations based on the perturbation of small parameters, in order to extract mathematical terms, and physical phenomena, of the first order, second order, etc. The homogenization approach typically involves heavy mathematics. Different judgement on the mathematical treatment sometimes leads to different, and even inconsistent results. Its ultimate justification still requires the validation by physical experiments. In this chapter we take a different approach from those above. We shall build the poroelasticity theory based on the thermodynamics principles of work and energy, first based on the reversible processes for the elastic constitutive laws, and then in the later chapters on the irreversible processes for the fluid, heat flux, and chemical diffusion laws. To cope with the highly heterogeneous nature of porous materials, the volume averaging method developed in the flow through porous medium theory (Bear J, Bachmat Y, Introduction to modeling phenomena of transport in porous media. Kluwer, Dordrecht/Boston, 1990, 553pp), and the theory of heterogeneous (composite) materials (Christensen RM, Mechanics of composite materials. Wiley Interscience, New York, 1979, 384pp; Nemat-Nasser S, Hori M, Micromechanics: overall properties of heterogeneous materials, 2nd edn. North-Holland, 1999, 810pp), are used to define the continuous and smooth functions needed in the construction of partial differential equations. The porous materials are considered as consisting of a heterogeneous solid phase and a homogeneous fluid phase (Lopatnikov and Cheng, Mech Mater 34(11):685–704, 2002). The macroscopic stresses and strains are defined as microscopically volume or surface averaged quantities depending on whether internal energy or external work is concerned. Constitutive equations are constructed using the variational energy principle stemming from the classical physics law of minimum potential energy (Landau LD, Lifshitz EM, Theory of elasticity, 3rd edn. Course of theoretical physics, vol 7. Butterworth-Heinemann, Oxford/Burlington, 1986, 195pp). The resultant model consists of a set of intrinsic material constants (Cheng and Abousleiman, Int J Numer Anal Methods Geomech 32(7):803–831, 2008), which are directly associated with the equations of state of the solid and fluid phase, and the fundamental deformation modes of the pore structure.

In the preceding chapters, the constitutive law developed for poroelastic materials assumes that the materials do not exhibit directional properties at the macroscopic level, and are isotropic. Geomaterials however are often anisotropic due to the existence of bedding surfaces in sedimentary rocks, foliations in metamorphic rocks, and microcracks aligned in the direction of stresses. Biomaterials such as cortical and trabecular bones are also anisotropic, due to their growth oriented in the direction of the physiological load (Yoon and Cowin, Biomech Model Mechanobiol 7(1):13–26, 2008). In this chapter we shall develop the constitutive laws for the general material anisotropy, which are also simplified to special cases.

In Chaps. 2 through 5 we have constructed the constitutive laws that relate the forces (stresses) applied to a porous body to its deformation (strains). In addition to these constitutive laws, there are other physical laws that are relevant to the deformation and motion of porous materials. These are presented in this chapter.

For the purpose of modeling, these laws are formulated in the form of mathematical equations. To reduce the size of the solution system, variables can be eliminated among the physical laws to produce governing equations that contain fewer variables. Given a complete set of governing equations, together with a set of well-posed boundary conditions, the mathematical system can be solved either analytically or numerically. There are times, however, that the mathematical system can be further reduced by the introduction of non-physical variables, known as potentials, to replace the physical ones. The physical variables are typically associated with the potentials as their spatial derivatives. We shall refer the mathematical system involving potentials as field equations. In this chapter we shall discuss these equations, as well as the initial and boundary conditions, leading to a complete mathematical solution system.

Once we set up a boundary value problem consisting of a set of partial differential equations, together with a set of well-posed boundary and initial conditions, our goal is to find the solution of the system. It is generally desirable to find analytical solutions, that is, solutions explicitly expressed in terms of known mathematical functions. However, for problems arising from engineering applications, this is generally not feasible; hence numerical solutions are sought. Nevertheless, there are a good number of analytical solutions that can be elegantly derived. In this chapter, we shall examine some of these solutions, and their applications.

In the mathematical terminology, a fundamental solution is a singular solution of a linear partial differential equation that is not required to satisfy boundary conditions. A Green’s function, on the other hand, is the singular solution tied to a certain domain geometry and boundary condition. For this reason, a fundamental solution is also called a free space Green’s function. Using the symbolism of a generalized function, known as the Dirac delta function, δ, a fundamental solution is a solution of linear partial differential equation with the Dirac delta as its right hand side. From physical considerations, Dirac delta can be used to approximate a forcing applied to a small region. For example, a fluid mass injected through an injection well, whose radius is small compared to the formation that it is flooding, can be considered as a point source. Similarly, a force acting on a small area can be considered as a point force, and a defect in crystal structure (dislocation) or a local slippage of a geological fault can be approximated as a displacement discontinuity, etc. Hence fundamental solution is not just an abstract mathematical construct; it has its root in physics and can be used to model physical phenomena. The use of fundamental solution to simulate a physical phenomenon and to solve mathematical problems can be traced to George Green (An essay on the application of mathematical analysis to the theories of electricity and magnetism, printed for the author, by T. Wheelhouse, Nottingham, 1828) (see Sect. F.7 for a biography). Green utilized the fundamental solution 1∕r of the Laplace equation to model the electrical and magnetic potential created by concentrated electrostatic and magnetic charges. The potential is a mathematical construct, whose derivative gives the force associated with the field. The use of fundamental solution to solve mathematical problems is closely tied to the integral equations known as Green’s identities. Particularly, the third identify, shown in its original form in the prologue of the chapter, provides the general solution to the boundary value problem known as the Dirichlet problem. In this chapter, we extend the classical work on fundamental solutions and integral equations for potential (satisfying Laplace equation) and elasticity problems to poroelasticity. We shall demonstrate that the several varieties of integral equations derived from different origins and the many different fundamental solutions are intricately related. The presentation begins with the integral equations for general anisotropic poroelastic materials. The fundamental solutions are then derived, but only for the case of isotropy. The presentation follows the work of Cheng and Detournay (Int J Solids Struct 35(34–35):4521–4555, 1998).

In the preceding chapters we have been dealing with poroelastic theories and problems under the assumptions similar to elastostatics; that is, at any instant of a loading, the poroelastic body is at a state of static equilibrium. In other words, for a body of any size, finite or infinitesimal, the summation of all forces, including surface and body forces, must equal to zero, \(\sum \vec{F} = 0\), such that there is no acceleration created by the imbalance of forces. This, however, does not mean that there is no motion. One of the characteristics of poroelastic body is that its deformation is time-dependent, giving the impression of a creeping-like motion, even if the applied load is constant in time. This transient behavior is the consequence of a fluid phase. Fluid has no shear strength to resist shear deformation, but has a viscosity that resists the rate of shear deformation. Hence the force equilibrium of a fluid can be accompanied by motion. So even without considering the acceleration caused by force imbalance, the poroelastic body is not exactly static, and the poroelastic theory presented in the preceding chapters can be called a quasi-static theory. When a force is rapidly applied, such as by an explosion in the air, by the impact of a solid body, or due to the slippage of a fault, the inertial effect, that is, the right hand side of Newton’s second law of motion \(\vec{F} = m\vec{a}\), cannot be neglected. A dynamic theory should be introduced. When the inertial effect is considered in a continuum body that is compressible, a wave phenomenon results. Particularly, the stress caused by the applied force is not instantly felt throughout the body—it has a finite speed of propagation. Sound propagation in the air as a wave phenomenon was recognized by philosophers and scientists as early as Aristotle (384–322 BC), and then by Galileo Galilei (1564–1642) (Imelda and Subramaniam, Phys Educ 42(2):173–179, 2007). In fact, Aristotle already recognized that sound is a longitudinal wave when he wrote “(the air) is set in motion …by contraction or expansion or compression” (Barnes J (ed), The complete works of Aristotle. Revised Oxford translation, vol 1. Princeton University Press, Princeton, 1984, 1256pp). This is indicative of the definition of a longitudinal wave, in which the particle motion is parallel to the wave propagation direction. In a solid, elastic medium, there exist two types of waves: in addition to the longitudinal, or compressional wave, there also exists a transverse wave, also called a shear wave, in which the particle motion is perpendicular to the wave propagation direction. These two waves propagate at different wave speed. The historical development of elastic wave theory, or elastodynamics, is well summarized in Love (A treatise on the mathematical theory of elasticity, vol 1. Cambridge University Press, Cambridge/New York, 1892, 354pp). In this chapter we are interested in the wave propagation in porous medium, or the theory of poroelastodynamics. In such medium, a third wave, called the second compressional wave, is observed, due to the existence of two phases, a solid and a fluid. The reasoning and theoretical demonstration of such waves were first presented by Yackov Frenkel (J Phys USSR 13(4):230–241, 1944. Republished, J Eng Mech ASCE 131(9):879–887, 2005) (see Sect. F.13 for a biography). Frenkel’s work was motivated by the field observation of Ivanov (Doklady Akademii Nauk SSSR 24(1):42–45, 1939; Izvestiya Akademii Nauk SSSR, Ser. Geogr. Geofiz 5:699–727, 1940), who discovered the so-called seismoelectric effect of the second kind (E-effect) generated by underground explosion—when a seismic wave is generated by an explosion, electric potential differences can be observed between electrodes situated at different distances from the source of the waves. Based on the continuum mechanics theory, Frenkel demonstrated that in a fluid infiltrated isotropic porous medium, in addition to a longitudinal and a shear wave, there existed a second longitudinal wave characterized by the out-of-phase movement between solid and fluid. He then showed that in the presence of electrolytes in liquids, electric current was generated due to the relative movement between the phases. The alternating directions of the electrical current in turn generate an electromagnetic wave. However, as quoted in the prologue of the chapter, after the proclamation of the discovery of a second wave, Frenkel did not further pursue its characteristics. Twelve years later, citing Frenkel’s original contribution, Biot (J Acoust Soc Am 28(2):168–178, 1956; J Acoust Soc Am 28(2):179–191, 1956) re-derived the theory of wave propagation in porous medium. Biot not only demonstrated the existence of the waves, but also presented the wave speeds. Particularly, it was shown that the second compressional wave is highly dissipative, and propagates at a much lower speed than the first compressional wave; hence they are respectively called the slow wave and the fast wave. In the higher frequency range, Biot also introduced a physical model of capillary flow in parallel plates or tubes, to account for the viscous-inertial attenuation. He also discovered a characteristic frequency at which the attenuation reaches its maximum. Biot’s model became immensely popular and the second wave has largely been referred to as the Biot second wave.

The poroelasticity theory presented so far assumes an isothermal condition, that is, temperature remains unchanged during the deformation and diffusion process. In practice, however, temperature of a porous medium can change. Not only it can change if it is in contact with a body of different temperature, and heat is transferred by conduction, but also the deformation itself can generate internal heat. In addition, heat can be transported in and out of the porous medium by a fluid flow.

The poroelasticity theory presented in Chaps. 2 through 10 models the force and energy interaction of mechanical origin between two material phases, a solid and a fluid. The thermal energy and force field are introduced in Chap. 11 In the physical world, there exist other types of energy and forces, such as those of electrical, magnetic, and chemical origins, and their coupling, such as the electromechanical (piezoelectric), electromagnetic, electrochemical, and magnetohydrodynamic forces. The simultaneous presence of these multiple physical phenomena, particularly their interactions, is known as multiphysics.Depending on the engineering applications on hand and practical considerations, not all forces are present or significant, and need to be modeled. In geoscience, geophysical, and geomechanical applications, it is generally recognized that four processes, thermal (T), hydraulic (H), mechanical (M), and chemical (C), known as THMC processes (Taron J, Elsworth D, Thermal-hydrologic-mechanical-chemical processes in the evolution of engineered geothermal reservoirs. Int J Rock Mech Min Sci 46(5):855–864, 2009; Taron J, Elsworth D, Min K-B (2009) Numerical simulation of thermal-hydrologic-mechanical-chemical processes in deformable, fractured porous media. Int J Rock Mech Min Sci 46(5):842–854; Tsang C-F (2009) Introductory editorial to the special issue on the DECOVALEX-THMC project. Environ Geol 57(6):1217–1219), are more important. The poroelasticity theory considers the hydraulic and mechanical coupling (HM). The porothermoelasticity theory introduces the additional thermal coupling, and is a THM theory. In this chapter, we shall examine the THMC processes by including the chemical energy and force field and introduce the porothermochemoelasticity theory; though, for simplicity, we shall refer to it as the porochemoelasticity theory in this book.Many geotechnical, biological, and synthetic porous media are chemically active, and exhibit swelling or shrinking behaviors when brought in contact with aqueous solutions. This phenomenon, observed in clays (Bennethum et al., Transp Porous Media 39(2):187–225, 2000; Low, Langmuir 3(1):18–25, 1987; Sposito et al., Proc Natl Acad Sci 96(7):3358–3364, 1999; van Olphen, An introduction to clay colloid chemistry, 2nd edn. Wiley, New York, 318pp, 1977), shales (Ghassemi, Diek, J Pet Sci Eng 38(3–4):199–212, 2003; Nguyen, Abousleiman, Anais Da Academia Brasileira De Ciencias 82(1):195–222, 2010; Sherwood, Proc R Soc A–Math Phys Eng Sci 440(1909):365–377, 1993; Sherwood, Langmuir 10(7):2480–2486, 1994), cartilage (Gu et al., J Biomech Eng ASME 120(2):169–180, 1998; Lai et al. J Biomech Eng ASME 113(3):245–258, 1991; Wilson et al., J Biomech Eng 127(1):158–165, 2005) and gels (Hong et al., Int J Solids Struct 46(17):3282–3289, 2009; Marcombe et al., Soft Matter 6(4):784–793, 2010), is caused by electric charges fixed to the solid, counteracted by corresponding charges in the fluid. These charges result in a variety of features, including swelling, chemico-osmosis, electro-osmosis, streaming potentials, streaming currents, and electrophoresis (Mitchell, Soga, Fundamentals of soil behavior, 3rd edn. Wiley, New York, 592pp, 2005; Sachs, Grodzinsky, Physicochem Hydrodyn 11(4):585–614, 1989).In biological and medical applications, swelling behavior is observed in cartilage, glycogalyx, cell, and skin (van Meerveld et al., Transp Porous Media 50(1–2):111–126, 2003). Swelling of shales is of major problem for petroleum engineering. According to Steiger and Leung (SPE Drill Eng 7(3):181–185, 1992), shales make up more than 75 % of drilled formations and cause at least 90 % of wellbore-stability problems. If shale surrounding the wellbore swells, the wellbore diameter will be reduced, and the drill string and drill bit can become trapped, costing both time and money. Alternatively, if swollen shale disintegrates, motion of the drill string will be hindered by the soft, roughened walls of the well (Mody, Hale, J Pet Technol 45(11):1093–1101, 1993; Sherwood, Bailey, Proc R Soc Lond Ser A–Math Phys Eng Sci 444(1920):161–184, 1994). Although the use of oil based drilling mud can reduce the chemical effect, its use is much restricted due to the environmental concern of its disposal (van Oort et al., SPE Drill Complet 11(3):137–146, 1996). Clay is used as liner and buffer for containment of landfill leachate and underground burial of nuclear or hazardous wastes in environmental geotechnology. The integrity of the barrier can be much affected by the swelling or shrinkage of the material.Shales are fine-grained sedimentary rocks consisting of clay, silt, and mud. Clay minerals are basically crystalline and their properties are determined by the atomic structure of their crystals. When these minerals are exposed to a fluid with different physic-chemical properties, the microscopic changes take place, which manifest in macroscopic scale as swelling (or shrinkage) and an apparent pressure as osmotic pressure. Swelling mostly occurs in smectites and particularly montmorillonite, due to its expanding lattice and finely laminated structure, subject to cation exchange and water content.In the following, we shall examine the modeling of the chemical effects through the concept of a chemical potential. The constitutive relations and transport laws are constructed to form governing equations that allow the mathematical solution of the various applications.