Numerical manifold method for dynamic consolidation of saturated porous media with three‐field formulation

Abstract

In terms of the three-field formulation of Biot's dynamic consolidation theory, the numerical manifold method (NMM) is developed, where the same approximation to skeleton displacement (u) and fluid velocity (w) is employed and able to reflect incompressible as well as compressible deformation, but the approximation to pore pressure (p) takes two different types, respectively. The first type of approximation to p is continuous piecewise linear interpolation and the second type assumes that p is a constant within each element. It is verified that using the second type of approximation to p naturally satisfies the inf-sup condition even in the limits of rigid skeleton and very low permeability, avoiding the locking problem accordingly. Energy components done by various forces are calculated to verify the accuracy and stability of the time integration scheme.A mass lumping technique in the NMM framework is employed to effectively reduce the unphysical oscillations and increase computational efficiency, which is another unique advantage of NMM over other numerical methods. A number of numerical tests are conducted to demonstrate the robustness and versatility of the proposed mixed NMM models. KEYWORDSdynamic consolidation, inf-sup condition, mass lumping, numerical manifold method 768 769 u − p models are established. When the liquid in the porous media is deemed compressible, pore pressure p can be eliminated instead of fluid velocity w, leading to the two-field u − w models that are advantageous in dealing with issues of porous mixture induced by earthquake. 1,2 On the other hand, to solve the coupling systems of equations for poroelasticity problems, three approaches are frequently employed, ie, fully implicit, loose or explicit coupling, and iterative coupling schemes. Fully implicit approach generally presents more accurate results but requires more expensive equation solvers in comparison with the loose or explicit coupling and iterative coupling approaches. A fully implicit solving approach is employed in this study and other two efficient implicit solving schemes can be found in the works of Saett and Vitaliani 14 and Zienkiewicz et al. 15 Several effective and commonly used iterative coupling procedures are presented in the works of Mikelić and Wheeler, 16 Both et al, 17 and Kim et al, 18 where their stability and convergence are also demonstrated, which are quite helpful for increasing computational efficiency. For nonlinear analysis of porous media, de Boer and Kowalski 19 presented a plasticity theory of saturated porous mixture and a more general plastic formulation can be found in the work of Zienkiewicz and Shiomi. 1 Besides fully implicit approach, the iterative coupling approach, especially the splitting scheme, 20,21 is also applied to solve the nonlinear Biot model. For more details of the applications of iterative coupling approach in nonlinear Biot model, please see the work of Borregales et al 22 where nonlinear mechanics and nonlinear fluid flow are both considered. Moreover, the splitting schem...

An energy stable, conservative and bounds‐preserving numerical method for thermodynamically consistent modeling of incompressible two‐phase flow in porous media with rock compressibility

“…Finally, we note that the skeleton deformation analysis plays an important role in many geological engineering problems. [13][14][15][16][17][18][19][20][21] In our ongoing work, we will combine the proposed model with Biot's model to calculate the solid/rock deformation. In the context of two-phase flow, the proposed formulation for the effective pore fluid pressure can be applicable to Biot's model.…”

“…1,2 To take into consideration the changes of rock properties, the fluid and poroelasticity coupling models have been extensively investigated in the literature, for instance. [13][14][15][16][17][18][19][20][21] In such models, Biot-type equations are usually used to obtain the solid displacements and subsequently the porosity changes are derived from the displacements. A distinct advantage of Biot-type model is that it can describe both the compressibility and deformation of rocks.…”

“…Zheng et al 52 proposed a variational inequality formulation for seepage problems with free surfaces, which can reduce the mesh dependency effectively and improve the numerical stability significantly. Wu et al 16,17,[19][20][21] and Zheng et al 18 developed a series of novel and effective numerical manifold methods to simulate multiphase flow problems in heterogeneous, fractured, deformable or elasto-plastic porous media. Nevertheless, due to the lack of thermodynamically consistent models of two-phase flows in incompressible or compressible porous media, energy stable methods have been scarcely studied in the past.…”

“…In this context, the porosity change should not be negligible although fluids can be still deemed to be incompressible 1,2 . To take into consideration the changes of rock properties, the fluid and poroelasticity coupling models have been extensively investigated in the literature, for instance 13‐21 . In such models, Biot‐type equations are usually used to obtain the solid displacements and subsequently the porosity changes are derived from the displacements.…”

Discontinuity Capture in One-Dimensional Space Using the Numerical Manifold Method with High-Order Legendre Polynomials

“…At present, the numerical manifold element method is already a very effective method in the solid field, and it is increasingly used in the field of engineering simulation. For example, hydraulic fracturing [20,21], pore medium problem [22][23][24], dynamic consolidation [25], fracture problem [26], slope stability analysis [27].…”