Laboratory samples of consolidated rock often have broken grain contacts and/or microcracks in the grains. Much of this damage occurs as the rock is brought from depth to the surface. Since diagenetic processes in a sedimentary basin tend to cement microcracks and grain contacts, it is uncertain whether in situ rocks have significant numbers of open microcracks. Nonetheless, when such grain-scale damage is present, as it always is in laboratory rock samples at ambient pressures, the fluid-pressure response in the microcracks will be greater than in the principal porespace when the rock is compressed by a P-wave. The resulting flow from crack to pore is called ``squirt flow'' and Dvorkin et al. (1995) have obtained a quantitative model for fully-saturated rocks.
In the squirt model of Dvorkin et al. (1995), the grains of a porous material are themselves allowed to have porosity in the form of microcracks. The effect of each broken grain contact is taken as equivalent to a microcrack in a grain. The number of such microcracks per grain is thus limited by the coordination number of the packing and so the total porosity contribution coming from the grains is always negligible compared to the porosity of the main porespace.
Our modeling of squirt is also based on this idea but we use the double-porosity framework of the previous sections. Phase 1 is now defined to be the pure fluid within the main porespace of a sample and is characterized elastically by the single modulus K_{f} (fluid bulk modulus). Phase 2 is taken to be the porous (i.e., cracked) grains and characterized by the poroelastic constants K^{d}_{2} (the drained modulus of an isolated porous grain), (the Biot-Willis constant of an isolated grain), and B_{2} (Skempton's coefficient of an isolated grain) as well as by a permeability k_{2}. The overall composite of porous grains (phase 2) packed together within the fluid (phase 1) has two distinct properties of its own that must be specified; an overall drained modulus K, and an overall permeability k associated with flow through the main porespace. The volume fractions occupied by each phase are again denoted v_{i} where is the porosity associated with the main porespace.
The theoretical approach is to again obtain the average fluid response in each of these two phases and then to make an effective Biot theory by saying that the fluid within the grains cannot communicate directly with the outside world; i.e., the fluid in the grains can only communicate with the main pores. Equations (11)-(12) again define the effective poroelastic moduli in the squirt model and we need only determine the a_{ij} constants and internal transport coefficient that are appropriate to squirt.