Flow and Transport in Porous Formations

For easiness of reference and for readers convenience, we present here a brief recapitulation of elements of probability theory and of theory of random functions needed in this book. This subject is covered by many text books. A handy reference for introduction to probability theory is, for instance, Mood & Graybill (1963). For the theory of random functions one may consult the books by Papoulis (1965) and Yaglom (1962), or the monograph of Matheron (1965), which is close in spirit with the material presented herein. Recent texts oriented toward hydrologic applications are Bras and Rodriguez-Iturbe (1985) and Vanmarcke (1983).

A porous medium is a two-phase material in which the solid matrix constitutes one phase and the interconnected void (pores) constitutes the other. The solid matrix is either rigid or it undergoes small deformations. With a proper definition of the pore scale d, the theory of flow through porous media is concerned with the behaviour of samples of typical length scales L much larger than d. In the present part we consider the scale pertaining to laboratory samples, which are obtained for instance by extracting cores from natural formations or by packing laboratory columns. These serve as experimental support to validate the basic laws governing the flow, to be applied subsequently to large scale formations.

Beginning with the present part, and in the rest of this book, we shall study flow and transport in natural porous formations. These formations differ considerably from the laboratory samples (considered in Part 2) in a few respects, but mainly in their dimensions. Thus, natural formations, or aquifers, appear generally as predominantly horizontal layers whose thickness may vary between a few to hundreds of meters. Their planar extent is generally much larger than the thickness, the horizontal size being as large as tens of kilometers. At these vast scales, as compared to the porescale, the properties and flow variables of interest are the macroscopic ones discussed in Part 2. In other words, measurements or computations are carried out for variables averaged over volumes or surfaces which are large enough to warrant the use of macroscopic quantities regarded as deterministic and attached to each point in space, in the spirit of the discussion of Chap. 2.1. The heterogeneity at pore-scale is, therefore, ignored and its presence manifests indirectly in the equations satisfied by macroscopic variables and in the coefficients (permeability, effective storativity, etc) which characterize the macroscopic properties of the porous medium. Along these lines, the porous medium and the fluid are regarded as continua and the macroscopic properties and variables are represented mathematically as continuous spatial scalar, vector or tensor fields. To illustrate the concept, let us consider the meaning of permeability k at a point x: it is the one of a core surrounding the point, which is hypothetically or actually extracted from the formation and brought to the laboratory for measurement.

In this part we shall study the transport of solutes by water in saturated flow through porous formations. Unlike the case of transport under laboratory conditions, discussed in Chap. 2.10, we are concerned here with motion and spread of solutes at the large scale and under the natural conditions prevailing in aquifers. This subject has important applications to pollution processes, salt water intrusion, recharge of miscible fluids, waste disposal, etc. Its experimental investigation under field conditions is hampered by a few major difficulties, which are not encountered in the laboratory. First, the monitoring of solute distribution in space and time requires drilling a large number of observation wells and frequent measurements of the solute concentration. These are costly operations which cannot be justified in most standard cases. Furthermore, the transport under natural flow is a slow process and the monitoring of the motion of a solute body or a plume, to obtain a comprehensive picture of its development, may require many years of continuous measurements over a large area. Besides the large expenditure, this extended period may not be available to the practitioners. In view of these difficulties one should not wonder that comprehensive field studies (see next section) have been carried out only recently.

As we have mentioned already in the preceding parts, natural formations are generally of a much larger horizontal extent than their vertical dimensions. Whereas the thickness of the aquifers is usually in the range 10⁰–10² meters, the planar dimension is of the order 10³–10⁵ meters. In many applications we are interested in investigating the behaviour of the formation as a whole unit, e.g. to compute the change of water head caused by natural or artificial recharge applied to the entire aquifer, to predict the yield of wells to be operated in the aquifer or to predict the fate of a solute plume which traverses the aquifer over an extended period of time. The problems we are addressing now are at a much larger scale than those of the local one, of the order of the depth, considered in Parts 3 and 4. This new scale, of the order of hundreds to thousands formation thicknesses, is termed here the regional scale, by following the terminology adopted in Dagan (1986).