Porosity models

The oil industry has developed and used a double porosity model to describe flow in certain fracture systems (7). Little data on application to actual field problems are available in the literature suggesting that this model has had only limited success and has been used only after the fact, not in a predictive mode. 

A DUAL POROSITY MODEL FOR CONTAMINANT TRANSPORT IN EXPANSIVE CLAYS... 

A Dual Porosity Model for Contaminant Transport in Expansive Clays 175... 

Douglas, J. and Arbogast, T. (1990) Dual porosity models for flow in naturally fractured reservoirs, in Dynamics of Fluid in Hierarchical Porous Media, editor J.H. Cushman, Academic Press, New York, pp. 177-222... 

Disequilibrium single porosity models. In addition to complete equilibrium transport, several other variations to the basic model have been proposed. The first relaxes the assumption that moving melt remains in chemical equilibrium with the solid at all times (Spiegelman and Elliott, 1993), although instantaneous melts are assumed to be in chemical equilibrium with the mantle that produced them. For stable elements, this disequilibrium transport produces a residue that reflects perfect fractional melting and the melts have compositions identical to accumulated fractional melts. These models are similar to the dynamic melting models in the limit that but... 

It is also straightforward to extend the equations to allow for only partial equihbration during transport. Iwamori (1993a) presents a one-dimensional steady-state single-porosity model for stable elements that includes diffusive re-equili-bration between melt and solid. He does not extend it to radioactive nuclides in this paper but includes this effect in his two porosity model (Iwamori, 1994) (see Section 3.14.4.3.4). The expected effects of chemical disequilibrium should be similar to those in the Qin (1992) dynamic melting model, namely he effective bulk partition coefficients of all elements will be driven towards unity. 

The melt velocity estimated from transport models is a bit slower but still comparable to estimates from the dynamic melting models. For example, if we assume that Ra-excesses are produced at the bottom of a column 90 km deep and need to move to the surface in —3 half-lives, then wq —20myr. It should be stressed that this is a constraint on the average melt velocity across the entire melting column rather than a constraint on the maximum velocity near the surface. Moreover, the constraint from Equation (9) assumes that there is only a single porosity near the surface. Two-porosity models (next section) relax this constraint somewhat. 

Another problem with the transport models is that they assume complete equilibrium between melt and solid throughout the melting regime. This implies that the solid residues at the top of the column (e.g., near the Moho), should be in chemical equilibrium with MORE. However, another key observation of MORBs is that they are out of equilibrium with abyssal peridotities near the Moho for both major and trace elements. A quick fix for the transport models is to assume that melts remain in chemical equilibrium up to some depth and then melt fractionally for the remaining distance (e.g., see Kelemen et al., 1997). However, different attempts to explain both U-series and stable elements in melts and residues in the single-porosity transport models has motivated much of the development of the two-porosity models in next section. 

A fundamental observation of melt transport at mid-ocean ridges is that in both major and trace elements, MORBs are out of chemical equilibrium with the shallow upper mantle (e.g., see Kelemen et al, 1997, for review). One mechanism that has been suggested is that melt migration is localized into some form of channel network fed by porous flow at the grain scale. This basic idea has been incorporated into several different two-porosity models to try to reconcile U-series observations with those of stable trace elements. 

Figure 15 Porosity structure of a high-resolution single-channel calculation for an upwelling system undergoing melting by both adiabatic decompression and reactive flow (see Spiegelman and Kelemen, 2003). Colors show the porosity field at late times in the run where the porosity is quasi steady-state. The maximum porosity at the top of the column (red) is 0.8% while the minimum porosity at the bottom (dark blue) is 10 times smaller. Axis ticks are height and width relative to the overall height of the box. In the absence of channels this problem is identical to the equilibrium one-porosity transport model of Spiegelman and Elliott (1993). Introduction of channels, however, produces interesting new chemical effects similar to the two porosity models.

Two-porosity models can begin to explain both excesses and correlations by allowing multiple melt compositions to be produced for the channel and interchannel regions. The actual mechanism and structure of channel formation is somewhat ad hoc but the underlying behavior seems justified qualitatively. 

Full fluid-mechanically consistent melt transport models with reactive channeling extend the results of the two-porosity models and produce distributions of compositions for both stable and radiogenic tracers in melts and residues. These models suggest that much information on the structure and rates of magmatic process might be contained in the observed variability of mantle melts but they need to be explored more rigorously. 

We have stressed accounting for the striking local variations evident in MORE U-series systematics. There are some less robust variations with geophysical parameters. Equilibrium melting one-porosity models... 

The approach for unsaturated conductivity outlined in previous sections was extended to modeling the unsaturated hydraulic conductivity of rough fracture surfaces (Or Tuller, 2000). Flow on rough fracture surfaces is an essential component required for deriving constitutive relationships for flow in unsaturated fractured porous media (Or Tuller, 2001). The detailed derivations are obtained by consideration of a dual porosity model (matrix - fracture) and the proportional contributions to flow from these different pore spaces. 

Besides geochemistry, transport processes also exert significant controls on effluent concentrations. The dual porosity model (Decker, 1996 Dixon et al., 1993) for... 

Gerke, H.H. and van Genuchten, M.Th. (1993) A dual porosity model for simulating the preferential movement of water and solutes in structured porous media. Water Resources Research 29 305-319. 

Double porosity model. The double porosity model (Wang 1993 Skelton et al. 

Examples of double porosity models applied to stable isotope transport in regional and contact metamorphism are given by Bowman et al. (1994) (see discussion of Alta contact aureole below), though this is not explicitly stated. Curves calculated for onedimensional transport using a first order rate law were fitted to observed profiles with a... 

Murad, M.A. Cushman, J.H. 1997. A multiscale theory of swelling porous media II. dual porosity models for consolidation of clays incorporating physical chemical effects. Trans. Por. Media. 28(l) pp.69-108. 

The bulk modulus Ki of the rock matrix in terms of the grain density, Pj, the P- and S-wave velocities of the dry skeleton (v, vj, and the porosity allow us to constrain the double porosity model by use of the well log data and the equation. 

In a series of experiments, both the fracture- and the porosity-models were used the permeabilities of the models used were 0.26 Darcies and 5.2 Darcies. Two kinds of samples of liquids were used (a) samples of crude oil with different gas content, (b) samples of degassed oil recombined with gas with gas factors of 1.5, 4.6, 9.1, 14.4, and 20.8 m. The results of these experiments are given in Table 7 and Fig. 7. 

Patrick, L. 1995. Numerical simulator for 3-D unsteady-state dual-porosity modeling of gas flow within a coal structure. Lexington KY USA Soc for Mining, Metallurgy Exploration Inc. 

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