Subsurface flow processes may take place at many different scales. The different scales refer to rock pore structure, micro-fractures, distinct fracture networks ranging from small to large fracture spacing, and even faults. Presently, there is no satisfactory methodology for describing quantitatively flow and reactive transport in multi-scale media. Approaches commonly applied to model fractured systems include single continuum models (SCM), equivalent continuum models (ECM), discrete fracture models (DFM), and various forms of dual continuum models (DCM). The SCM describes flow in the fracture network only and is valid in the absence of fracture-matrix interaction. The ECM, on the other hand, requires pervasive interaction between fracture and matrix and is based on averaging their properties. The ECM is characterized by equal fracture and matrix solute concentrations, but generally different mineral concentrations. The DFM is perhaps the most rigorous, but would require inordinate computational resources for a highly fractured rock mass. The DCM represents a fractured porous medium as two interacting continua with one continuum corresponding to the fracture network and the other the matrix. A coupling term provides mass transfer between the two continua. Vidues for mineral and solute concentrations and other properties such as liquid saturation state maybe assigned individually to fracture and matrix. Two forms of the DCM are considered characterized by connected and disconnected matrix blocks. The former is referred to as the DCCM (dual continuum connected matrix) model and the latter as the DCDM (dual continuum disconnected matrix) model. In contrast to the DCCM model in which concentration gradients in the matrix are allowed only parallel to the fracture, the DFM provides for matrix concentration gradients perpendicular to the fracture. The DFM and DCCM models can agree with each other only in the case where both reduce to the ECM. The DCCM model exhibits the incorrect behavior as the matrix block size increases, resulting in reduced coupling between fracture and matrix continua. The DCDM model allows for matrix gradients within individual matrix blocks in which the symmetry of the surrounding fracture geometry is preserved. However, the DCDM model breaks down for simultaneous heat and mass transport and cannot account for significant changes in porosity and permeability caused by chemical reactions.
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