VOL. 15, NO. 3 WATER RESOURCES RESEARCH JUNE 1979 Diffusion of Dissolved Gas in Consolidating Porous Media M. Y. CORAPCIOGLU Department of Geological Engineering, Middle East Technical University, Ankara, Turkey The diffusion of dissolved gas in liquid resulting from soil consolidation is analyzed as a functionof time and space. The medium is assumed to be isotropic, finite, and filled with a homogeneous and incompressible fluid. Two cases of consolidation have been considered to determine the fluid velocity expression. The model is useful to predict the concentration of dissolved gas in liquid for laboratory problems. INTRODUCTION Entrappedgasin porousmedia has significance with desir- able or undesirableresultsbecause of the changes in the hy- draulic behavior of the liquid in the porousmedia. Problems arise in soil samples which often are assumed to be saturated but are likely to have small amounts of air or other gases within their pores. Entrappedgas, even in very small quan- tities,hasa markedeffect on the coefficient of permeability; for example, the permeability is reduced by about 30% of its maximum value for a 10% reduction in saturation [Roberts, 1967]. Also, in groundwater wells the discharge is greatly reduced by trappedair, thus retardingthe pumpingcapacity. Sometimes air is utilized as a barrier to the migration of saline and polluted water. Repressurizing petroleum reservoirs to recover the lockedoil is alsoa common practice of utilization of air [Roberts, 1967]. I n addition to entrapped gas,certainamounts of gasalways exist in solution with the liquid in pores. It is known that permeability will not be accurately determined unless the amount of dissolved gasis kept nearlyconstant duringtesting [Taylor, 1948]. Owing to the practicalimportance of dissolved gasthe sub- ject has received considerable attention, especially from the scientists at Colorado State University.Works by Bloomsburg and Corey [1964] and Adam et al. [1969] have developed ap- proximate methods to estimate the concentration of dissolved gas in liquid by molecular diffusion. The experimental results were consistent. Later, McWhorter et al. [1973] developed a theoretical model which eliminated the assumption of the lin- ear concentrationdistribution of Adam et al. [1969]. In fact, both solutions yield identicalresults when a constant parame- ter is approximately smallerthan 0.15. In all these studies the basic problemwasthe diffusion of the gasin a staticliquid by molecular diffusion but not in a flowing liquid where the phenomenon is alsodominated by the fluid velocity. The fluid velocityis either seepage velocityin aquifers or velocitydue to the compression of soil samples. It is the purpose of this paper to analyze the diffusion of dissolved gas in consolidating porous media. The governing diffusion equation will be solved for the classical consolidation problem. Also, the one-dimensional diffusion problem de- scribed by Adam et al. [1969] will be analyzed under one- dimensional deformation. pin's [1960] continuum theory of mixtures. We will regardthe unsaturated soil as a mixture of a solid phase, an aqueous phase,and a gaseous phase. The derivation of the balance equations hasbecome basic in variousfields. Verruijt [1969] and Brutsaert and Corapcioglu [1976] have studied the subjectfor the groundwater flow in compressible aquifers.Muskat and Meres [1936] and Florin [1961] have beeninterested in the subject from the pointsof view of petroleum engineering and soil mechanics, respec- tively. The basic principlewhich the specific mass discharges have to satisfy is the conservation of mass, and this principle shouldhold for every phaseof the mixture. For the solid phase the balance equationof mass is 8--7 = V'qs (1) wheren is the porosity and q8is the specific volumedischarge of the solid phase. The variableq8can be formulatedas q, = (l - n)g, (2) where g• is the velocity of the solid phase, i.e., rate of dis- placementof the solid phase. Note that (1) is based on the assumption that the density of the solid phase is constant [Brutsaert and Corapcioglu, 1976]. The balance of mass equation for the aqueous phase can be written as _.O(ornS) = V.jr (3) 8t where Orand Jr are the density and the specific mass discharge of the aqueousphase, respectively, and S is the degree of saturation.It is known that Jr is equalto the product of or and qr, which is the specific volume discharge of the fluid. Then under the assumption of a homogeneous and incompressible fluid density(i.e., Vpr = O, 8pt/St = 0), (3) reduces to 8(nS) t•t - V.qr (4) Conservation of mass for the gaseous phase is given by 8Lo,•n(l- S)] 8(riSC) = + (5) ot ot GENERAL DIFFUSION EQUATION In this section we develop the equations of balanceof mass for a partly saturated soil by making use of the concepts introducedby Raats and Klute [1968] after Truesdell and Tou- Copyright ¸ 1979by the American Geophysical Union. Paper number 9W0071. 0043-1397/79/009W-0071 $01.00 where 0a and Ja are the density and the specific mass discharge of the gaseous phase, respectively, and C is the concentration of dissolved gas mass per unit volume of the aqueous phase. J, can be formulated as J,• = p,•q,• + Cqr + J (6) 563