A new mathematical model to calculate sand-packed fracture conductivity

- 2 phases are in sand-packed fracture change: proppant rearrangement, and embedment and deformation in the rhombic proppants.
- Different ways were proposed to calculate the conductivity including the changes in width, porosity, seepage channel.
- The reasonable mathematical models considered more factors when compared with other existing models.
- It is convenient to apply the models because all the parameters for computation could be obtained from proppant tests.

The conductivity of sand-packed fractures is influenced by the fracture width and permeability. Proppants become rearranged, embedded in formations and even deformed under certain pressures. Thus, the fracture width, the porosity and the radius of seepage channels decrease with increasing pressure. Currently, sand-packed fracture conductivity is measured in the laboratory but rarely calculated by mathematical methods or based on ideal conditions, which may not always be true in practice. This paper demonstrates proppant rearrangement under low and high pressure and presents a rhombohedron model to calculate sand-packed fracture conductivity that considers the looseness coefficient, deformations of rock and proppants, particle size, crushing rate, number of proppant layers, closure pressure and so on. The mathematical model is based on the principle of elastic-plastic sphere contacts, the capillary bundle model and the Carman-Kozeny equation. In this case, the changing width and permeability of sand-packed fractures during compression can be calculated. Verification experiments and comparisons with existing models indicate that the mathematical models can reasonably predict the sand-packed fracture width and conductivity under real conditions.