Model Geometry and solution method Yield Hardening/ softening Coupling Phases Sanding criteria Permeability alteration in the sanded zone Other features
Morita et al. [25 ], Morita et al. [26 ], Burton et al. [27 ] 3D; finite element (FE) (SAND3D software) Kinematic model with a cap yes (flow friction) Iteratively coupled (1) Fluid (2) Solid Maximum plastic strain limit No change (1) Only the onset of sanding (2) Burton applied it for gas reservoirs
(Vardoulakis et al. [7 ]) 1D; finite difference (FD) N/A N/A Fluid flow and erosion are coupled (1) Fluid (2) Fluidized solid Erosion
(Carman-Kozeny)(1) Only hydromechanical effects; equilibrium eqn. is not solved (2) Sand deposition is neglected in modeling
Skjaerstein et al. [28 ] 1D; FD N/A N/A Fluid flow and erosion are coupled. (1) Fluid (2) Fluidized solid Erosion
(found experimentally)Forcheimer’s law was used instead of Darcy’s law to account for turbulence
Papamichos and Malmanger [29 ] 2D Axial symmetry and 3D; FE; Newoton-Paphson (NR) iterations MC Yes Fully coupled (1) Fluid (2) Solid (3) Fluidized solid Erosion
Tension cut-off: function of both plastic strain and porosity; by the factor
Papamichos et al. [9 ] 2D Axial symmetry; FEM; NR iterations MC Yes Fully coupled (1) fluid (2) solid Erosion Carman-Kozeny Tension cut-off and Young mod changed by the factor
Yi [30 ] 2D axial symmetry; FD MC No (1) Fluid (2) Solid (3) Fluidized sand Erosion
Sand deposition in porous media is considered
Vaziri et al. [31 ] 2D axial symmetry; FE; fully-implicit Modified MC with tensile failure Yes Fully coupled (1) Fluid (2) Solid Tensile failure
Zero stiffness, compressibility and high k for the liquefied tensile zone
Wang and Xue [32 ] FE; Crank-Nicholson for time integration MC No Fully coupled (1) Oil (2) Water (3) Solid (4) Fluidized sand Erosion Kozeny-Poiseuille law and Carman-Kozeny
Chin and Ramos [33 ] 2D and 3D; FE; explicit; NR iteration Drucker-Prager No Coupled (1) Fluid (2) Solid Shear dilation Power law with porosity (exponent = 5.6) Porosity is changed as a function of plastic volumetric strain Nouri et al. [34 ] 2D plane strain; FD Bilinear MC Yes Fully coupled (1) Fluid (2) Solid Tensile failure or shear-failed element falls in tension 0 Capillary is considered as a residual cohesion
Wang et al. [35 ] 2D; FE Drucker-Prager No Fully coupled (1) Solid (2) Fluidized solid (3) Oil (4) Water (5) Gas
Cohesion and friction drop linearly with porosity
Servant et al. [36 ] 2D; FE MC No Iteratively coupled (1) Fluid (2) Solid (3) Fluidized solid Yielding Not mentioned Failed material is treated as a Poiseuille fluid. Constant viscosity for the slurry
Detournay et al. [11 ] 2D; FD MC Yes Iteratively coupled (1) Fluid (2) Solid Erosion
Bulk mod. Change by
Nouri et al. [12 ] 2D axial symmetry; FD Bilinear MC Yes Iteratively coupled (1) Fluid (2) Solid Tensile failure or a shear-failed element falls in tension
Nouri et al. [37 ] FE; NR iterations MC Yes Fully coupled (1) Fluid (2) Solid Tension 0 Adaptive mesh is used
Nouri et al. [38 ], Vaziri et al. [14 ] 2D plane strain and axial symmetry; FD Bilinear MC Yes Iteratively coupled (1) Fluid (2) Solid Complete degradation and tensile mean effective stress High permeability is assigned to infill materials (elements that satisfy sanding criteria) WH pulsing is included in the model, and stiffness changes with sanding (Vaziri et al. [14 ])
Detournay [15 ] 2D plane strain; FD Double-yield Yes Iteratively coupled (1) Fluid (2) Solid Same as (Detournay et al. [11 ]) Not mentioned Only cap yielding is considered
Nouri et al. [16 ]
All the features are the same as (Nouri et al. [38 ]) but fracture energy regularization is applied to remove mesh dependency and the calibration of hardening/softening behavior is discussed thoroughly
Kim et al. [17 ] 3D; FD MC Yes Not mentioned (1) Fluid (2) Solid Calculated from force balance on the element N/A (element removal) No calibration parameter to match the experiment is used and yet a good match is observed
Wang et al. [23 ] 2D; FE (ELFEN software) Soft rock model Yes Fully coupled (1) Fluid (2) Solid Displacement (the materials passing the wellface are removed) K changes as a function of dilation or volumetric strain:
(1) Adaptive mesh (2) Arbitrary Largrangian Eulerian formulation (3) Fracture Energy Regularization
Azadbakht et al. [39 ] 2D, FE for fluid and FD for solid Bilinear MC Yes Sequentially coupled (1) Fluid (2) Solid Erosion K changes linearly with porosityWater effect is considered by reducing the cohesion Bulk moduli change linearly with porosity After critical porosity, residual properties are assigned to the element as infill material