CONSOLIDATION OF ELASTIC POROUS MEDIA SATURATED BY Two IMMISCIBLE FLUIDS By Kagan Tuncayl and M. Yavuz Corapcioglu,z Fellow, ASCE ABSTRACT: A theory is presented to simulate the consolidation of elastic porous media saturated by two immiscible Newtonian fluids. The macroscopic equations, including mass and momentum balance equations and constitutive relations, are obtained by volume averaging the microscale equations. The theory is based on the small deformation assumption. In the microscale, the grains are assumed to be linearly elastic and the fluids are Newtonian. The bulk and shear moduli of the solid matrix are introduced to obtain the macroscopic constitutive equations. Momentum transfer terms are expressed in terms of intrinsic and relative permeabilities assuming the validity of Darcy's law. In one dimension, the governing equations reduce to two coupled diffusion equations in terms of the pore pressures of the fluid phases. An analytical solution is obtained for a column with a fixed impervious base and a free drainage surface. Results are presented for cases of practical interest, Le., columns saturated by oil-water and air-water phases. Results indicate that the presence of a second fluid phase affects pore water pressure and total settlement. INTRODUCTION Many geological formations contain more than one fluid phase in their pores such as air, water, and oil. The interfacial tension between these fluids is nonzero and a fluid-fluid inter- face separates the phases within the pores. Although most earth materials are saturated by more than one fluid, consoli- dation of porous media saturated by multiphase fluids received limited attention from researchers. Almost all studies deal with soils containing air and water in their pores, i.e., unsaturated soils. In early studies, Terzaghi's (1923) effective stress prin- ciple was extended to unsaturated soils to incorporate the ef- fect of air on pore water pressure. Among them, Bishop's (1959) principle is the most widely accepted one. Bishop in- troduced a material parameter to take the presence of air phase into consideration. Jennings and Burland (1962) showed that volume changes due to water wetting of the solid phase cannot be explained by Bishop's expression. For a detailed discussion of modified effective stress principle, the reader is referred to Bishop and Blight (1963), Kohgo et al. (1993), and Fredlund and Morgenstern (1977). However, we should note that these expressions were obtained based on practical considerations rather than theoretical ones. To explain the behavioral alterations in volume change such as collapse of soil structure, various researchers proposed two stress states that are treated independently (Jennings and Bur- land 1962; Coleman 1962; Fredlund and Morgenstern 1976; Fredlund and Hasan 1979). In this approach, stress states are chosen as algebraic combinations of the stresses in the solid, air, and water phases. Then, constitutive relations for volume changes are expressed in terms of these stress states, e.g., as a function of the difference between total stress and air pres- sure, and the difference between air pressure and water pres- sure (Fredlund and Hasan 1979; Lloret and Alonso 1980). Fredlund and Hasan (1979) proposed a one-dimensional con- solidation theory for unsaturated soils by considering volume changes and fluid flow due to variations in these two stress states. Their theory consists of two coupled diffusion equations 'Asst. Prof., Izmir Inst. of Technology, Facu. of Engrg., Gaziosman- pasa Bulvari, No. 16, Cankaya, IZmir, Turkey. 'Prof., Dept. of Civ. Engrg., Texas A&M Univ., College Station, TX 77843-3136. Note. Associate Editor: James T. Kirby. Discussion open until April I, 1997. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on November 21, 1994. This paper is part of the Journal of Engineering Mechanics, Vol. 122, No. II, November, 1996. ©ASCE, ISSN 0733-9399/96/0011- 1077-1085/$4.00 + $.50 per page. Paper No. 9619. in terms of air and water pore pressures. Fredlund and Hasan (1979) have expressed the flow of wetting (water) and non- wetting (air) phases by Darcy's and Fick's laws, respectively. Lloret and Alonso (1980) studied swelling and collapse be- havior of unsaturated soils based on the same principles with the exception of Darcy's law expressed in terms of respective relative permeabilities for each phase. Narasimhan and With- erspoon (1977, 1978a, 1978b) proposed a numerical model for water flow in deforming unsaturated porous media based on Richards' equation and Bishop's effective stress principle. Corapcioglu and Bear (1983) have noted the contribution of the unsaturated zone above the water table to subsidence of a deformable phreatic aquifer. Geraminegad and Saxena (1986) developed a model to predict the coupled transient flow of heat, water, and air. They solved the resulting equations by the finite element method. Brutsaert and EI-Kadi (1984) in- vestigated the relative importance of partial saturation in ground-water flow by extending Biot's theory to unsaturated porous media. Governing equations of compressible porous media can be obtained by using two different techniques. The first technique employs the mixture theory that treats the particulate volume fraction as a constitutive variable for multiphase media like compressible porous materials (Adkins 1963; Bowen 1982). The second technique volume averages the microscale gov- erning equations to obtain macroscale ones (Slattery 1981). In the present study, we present a theory to simulate the consol- idation of porous media saturated by two immiscible fluids. Mass and momentum balance equations as well as the consti- tutive relations are obtained by volume averaging the equa- tions and relations expressed at the microscopic scale. The volume averaging technique has been employed after the de- velopment of the theorem for volume average of a gradient (Slattery 1967; Anderson and Jackson 1967; MarIe 1967; Whi- taker 1967). Although the volume averaging technique has been used extensively to formulate the flow problems in rigid porous media (Slattery 1967, 1968; Whitaker 1967), it has been recently applied to deformable media [e.g., refer to Bear et al. (1984)]. De la Cruz and Spanos (1985) made the first attempt to formulate the constitutive relations and balance equations of wave propagation in saturated porous media. In a subsequent paper, de la Cruz and Spanos (1989) extended their theory to include the thermodynamic considerations. Pride et al. (1992) obtained Biot's (1941, 1956) equations for porous media by employing the volume averaging techmque. The resulting constitutive relations of Pride et al. (1992) contained the same parameters as those of Biot and Willis (1957). As seen in this brief review, there is no theory available to obtain constitutive relations as well as mass and JOURNAL OF ENGINEERING MECHANICS / NOVEMBER 1996/1077 J. Eng. 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