One-dimensional Consolidation Problem Research Articles

A Poroviscoelastic Model at Finite Strains for a Viscous Compressible Solid Skeleton

In this paper, we propose a constitutive model for describing the poroviscoelastic behavior of fluid-saturated porous solids at finite strains. Rather than a viscous incompressible solid skeleton as assumed in previous works, a viscous compressible solid skeleton is considered in this research. To show the poroviscoelastic effect, we apply the model to the study of the classical Terzaghi’s one-dimensional consolidation problem and compare the numerical results with those from poroelasticity. From the numerical results, the secondary consolidation behavior induced by the creep of the porous solid can be observed. Besides, it is shown that the creep of the solid skeleton causes the delay of pore pressure diffusion.

Affine transformations accelerate the training of physics-informed neural networks of a one-dimensional consolidation problem

Physics-informed neural networks (PINNs) leverage data and knowledge about a problem. They provide a nonnumerical pathway to solving partial differential equations by expressing the field solution as an artificial neural network. This approach has been applied successfully to various types of differential equations. A major area of research on PINNs is the application to coupled partial differential equations in particular, and a general breakthrough is still lacking. In coupled equations, the optimization operates in a critical conflict between boundary conditions and the underlying equations, which often requires either many iterations or complex schemes to avoid trivial solutions and to achieve convergence. We provide empirical evidence for the mitigation of bad initial conditioning in PINNs for solving one-dimensional consolidation problems of porous media through the introduction of affine transformations after the classical output layer of artificial neural network architectures, effectively accelerating the training process. These affine physics-informed neural networks (AfPINNs) then produce nontrivial and accurate field solutions even in parameter spaces with diverging orders of magnitude. On average, AfPINNs show the ability to improve the {mathscr {L}}_2 relative error by 64.84% after 25,000 epochs for a one-dimensional consolidation problem based on Biot’s theory, and an average improvement by 58.80% with a transfer approach to the theory of porous media.

An improved semi-implicit material point method for simulating large deformation problems in saturated geomaterials

This study presents an improved semi-implicit MPM formulation for coupled large deformation problems in geotechnics based on the fractional step method. The main objective of this study is to make the choice of the time step size independent of the permeability and to reduce the pressure oscillations by applying the special splitting procedure and the stabilized technique. The proposed coupled MPM is validated by comparing numerical results with analytical solutions of the one-dimensional consolidation problem in both small and large deformation regimes. Three numerical tests are performed to further evaluate the performance of the proposed method, including a self-weight slumping block, 2D consolidation test, and the stability of a saturated elasto-plastic slope. The numerical results indicate that the proposed method is suitable for simulating fluid-saturated porous media involving large deformation.

Porous Media Model of Reservoir considering Seepage Stress Coupling and Seepage Field Analysis

In the process of reservoir exploitation, the range of rock stress field changes is much larger than the reservoir area where the seepage occurs; so the amount of calculation of the existing seepage stress full coupling method is often very huge. Based on the theory of seepage mechanics and rock mechanics, a coupling analysis model of reservoir multiphase seepage and stress is established, and a numerical solution is established by using finite element and finite difference methods. The evolution law of the seepage field and stress field and the change of rock mechanics parameters can be studied, with emphasis on the readjustment of rock stress distribution and its deformation characteristics under the influence of the seepage field. At the same time, to improve the calculation efficiency of numerical simulation, considering the difference between the seepage calculation area and stress calculation area, the finite element method is improved. The percolation area of the reservoir is calculated with a fine grid to obtain a more accurate distribution of underground fluid. The coarse grid is used to calculate the stress calculation area and to reduce the calculation time. The mechanical equilibrium equation in the fully coupled theory is discretized on the coarse grid by the finite element method. The mass conservation equation is discretized on the fine grid by the finite volume method. The numerical simulation of Terzaghi’s one-dimensional consolidation problem and Mandel’s two-dimensional consolidation problem shows that the calculation results of this method are in good agreement with the analytical solution. Through the numerical calculation of a two-dimensional single-phase flow single well production problem and a three-dimensional two-phase flow five-point well pattern production problem, the influence of the seepage stress coupling effect in reservoir numerical simulation is analyzed.

One-Dimensional Nonlinear Consolidation for Soft Clays with Continuous Drainage Boundary Considering Non-Darcy Flow

Adopting the non-Darcy flow presented by Hansbo and considering the nonlinear compression and permeability characteristics of soils, the one-dimensional nonlinear consolidation problem of soft clays is investigated by means of a continuous drainage boundary. The numerical solutions of average consolidation degrees defined by settlement and excess pore water pressure are derived by using the finite difference method, and the correctness of these solutions is verified by comparing them with existing analytical and numerical solutions. Based on the proposed solutions, a parametric study is conducted to study the influence of interface parameter, non-Darcy flow parameter and soil nonlinearity on the consolidation behavior of soft clays. The results show that the solutions based on the continuous drainage boundary can be degenerated into the solutions based on the Terzaghi drainage boundary if the interface parameter is taken as a reasonable value. The soil consolidation behavior considering both non-Darcy seepage and nonlinear characteristics of soil is very complex.

Coupled heat-fluid-solid numerical study on heat extraction potential of hot dry rocks based on discrete fracture network model

Fracture networks within hot dry rock (HDR) geothermal reservoirs are complex, and heat extraction via water injection is thus a coupled process of heat-fluid-solid multifield. In this paper, utilizing the theory of normally distributed random functions, we develop a corresponding pre-processing subprogram to establish a discrete network model of complex fracture distribution in HDR reservoirs; then construct a heat-fluid-solid finite element model for heat extraction via water injection and compare the numerical solution with the analytical solution of the one-dimensional non-isothermal consolidation problem for verification. The numerical simulation results show that the main factors affecting the heat extraction efficiency of HDR reservoirs include fracture width, fracture density, fracture permeability, and matrix permeability. When a HDR reservoir is injected with water for heat extraction, there is a certain threshold value of these influential parameters, beyond which the outlet temperature drops significantly, resulting in an obvious thermal breakthrough. When injecting water for heat extraction, the values of these parameters should be controlled and kept at a reasonable level, otherwise, the HDR reservoir may enter a thermal breakthrough stage in advance, which is not conducive for long-period heat extraction. Influenced by the random distribution of complex fractures, the leading edge of the cold front may present an irregular distribution. During the process of heat extraction, the stress gradually changes from a compressional state to a tensile state, which induces further damage to the HDR reservoir.

An extended numerical manifold method for two-phase seepage–stress coupling process modelling in fractured porous medium

To investigate the two-phase seepage–stress coupling process in fractured porous medium, this study extends the cohesive element-based numerical manifold method (Co-NMM) by incorporating a two-phase seepage–stress coupling model considering the effect of matrix-fracture interface on the two-phase flow and fracture propagation induced by the two-phase seepage pressure. The proposed two-phase flow solving framework implicitly calculates the fluid pressure and saturation of the two-phase flow based on a two-phase unified pipe network method. Furthermore, to more realistically model the hydraulic behaviour of two-phase flow in fractured porous medium, a matrix-fracture interface condition called the extended capillary pressure condition is incorporated into the two-phase flow solving framework to capture the interactions among fluid flow in the fractures and matrix. Due to the inheritance of the Co-NMM, one key advantage of the extended method is the simulation of complex multi-fracture propagation caused by the two-phase seepage–stress coupling. The two-phase flow solving framework is first validated by reproducing the water displacing oil in a single fracture and the gas displacing water in a single-fractured porous medium against analytical and numerical solutions, respectively. The two-phase seepage–stress coupling procedure is then verified by performing a one-dimensional consolidation problem of soil column, in which comparisons between the numerical and analytical results regarding the pore pressure and compression displacement are presented. Finally, with the extended method, CO2-enhanced oil recovery in fractured reservoir is preliminarily studied by considering the effect of gas injection rate and capillary pressure on the evolution of two-phase flow and fracture propagation. The results elucidate that high CO2 injection rate can lead to fracture propagation in the reservoir, and both capillary pressure and fractures have a significant effect on the CO2 distribution.

A macro–microscopic coupled consolidation model for saturated porous media with compressible constituents

Porosity is an important variable in consolidation theory. However, porosity information is missing in Biot’s model when the solid constituent of the porous media is assumed to be compressible. This paper develops a macro–microscopic coupled consolidation model in order to track the evolution of the porosity for saturated porous media with compressible constituents. The proposed model exploits a u−εvsr−p form with the aim to incorporate the coupled behavior between the macroscopic and microscopic volumetric compressibility. Subsequently, a specific study on the analytical solutions for the case of a linear-loaded one-dimensional consolidation problem based on the proposed model is presented. Two examples are shown in order to demonstrate how to apply the proposed model. The proposed model paves the way to study the consolidation problem with porosity-dependent coefficients for saturated porous media with compressible constituents.

DISCONTINUOUS GALERKIN FEM METHOD FOR THE COUPLING OF COMPRESSIBLE TWO-PHASE FLOW AND POROMECHANICS

Understanding the coupled multiphase flow and solid deformation processes in porous media is a significant issue in the area of developing and utilizing underground resources. This study first established the coupled modeling of compressible two-phase flow and deformation of porous media, which considers capillarity and gravity. Meanwhile, the strong form and the corresponding weak form of coupled multiphase flow and solid deformation model were presented. Then, the capacity of the proposed DG method for the coupled hydromechanical model was verified by comparison with analytical and numerical results of the one-dimensional Terzaghi consolidation problem. Subsequently, the two- and three-dimensional cases were performed to study the flow behaviors and deformation characteristics, respectively. In addition, the effects of the penalty factors \begin{document}$ {\delta }_{\rm{s}} $\end{document} and \begin{document}$ {\delta }_{\rm{f}} $\end{document} on the stability of the numerical results were analyzed. The simulation results show that gas saturation and pore pressure continually increase with the injection of gas. The increment of pore pressure reduces the effective stress, which results in deformation and expansion of the porous medium. The gas floats up and gathers at the top boundary due to gravity. The decrease of the penalty factors \begin{document}$ {\delta }_{\rm{s}} $\end{document} and \begin{document}$ {\delta }_{\rm{f}} $\end{document} trends to cause the fluctuation of saturation, pressure, effective stress, and displacement. The increases in penalty factors are beneficial to suppress the discontinuity of the finite element function crossing the elements.

Possibilistic Uncertainty Quantification in One-Dimensional Consolidation Problems

Abstract In this study, we use possibility distribution as a basis for parameter uncertainty quantification in one-dimensional (1D) consolidation problems. A possibility distribution is the one-point coverage function of a random set and is viewed as containing both partial ignorance and uncertainty. Vagueness and scarcity of information needed for characterizing the coefficient of consolidation in clay can be handled using possibility distributions. Possibility distributions can be constructed from existing data, or based on the transformation of probability distributions. An attempt is made to set a systematic approach for estimating uncertainty propagation during the consolidation process. The measure of uncertainty is based on Klir's definition (1995). We make comparisons with results obtained from other approaches (probabilistic…) and discuss the importance of using possibility distributions in this type of problem.

One-Dimensional Consolidation Analysis Considering Non-Darcian Flow and Self-Weight Based on Continuous Drainage Boundary

The one-dimensional consolidation problem of soft soil with self-weight and non-Darcian flow is studied by introducing continuous drainage boundary. A numerical solution is derived by using finite difference method and its correctness is proved by comparing with existing analytical and numerical solutions. Based on the present solution, the consolidation behavior of soil is detailedly analyzed by adjusting the different parameters. The results show that, the soil consolidation rate increases with the increase of interface parameters or external loading rate. The difference between the average consolidation degree obtained by the solution with continuous drainage boundary and that with traditional boundary is mainly in the early stage of consolidation, but in the late stage of consolidation, the difference between these two solutions becomes smaller. Compared with the soil consolidation solution considers non-Darcian flow, the soil consolidation rate with Darcy flow is faster.

Numerical simulation of forerunning fracture in saturated porous solids with hybrid FEM/Peridynamic model

In this paper, a novel hybrid FEM and Peridynamic modeling approach proposed in Ni et al. (2020) is used to predict the dynamic solution of hydro-mechanical coupled problems. A modified staggered solution algorithm is adopted to solve the coupled system. A one-dimensional dynamic consolidation problem is solved first to validate the hybrid modeling approach, and both δ-convergence and mr-convergence studies are carried out to determine appropriate discretization parameters for the hybrid model. Thereafter, dynamic fracturing in a rectangular dry/fully saturated structure with a central initial crack is simulated both under mechanical loading and fluid-driven conditions. In the mechanical loading fracture case, fixed surface pressure is applied on the upper and lower surfaces of the initial crack near the central position to force its opening. In the fluid-driven fracture case, the fluid injection is operated at the centre of the initial crack with a fixed rate. Under the action of the applied external force and fluid injection, forerunning fracture behavior is observed both in the dry and saturated conditions.

A Novel Approach to the Analysis of the Soil Consolidation Problem by Using Non-Classical Rheological Schemes

The paper presents classical and non-classical rheological schemes used to formulate constitutive models of the one-dimensional consolidation problem. The authors paid special attention to the secondary consolidation effects in organic soils as well as the soil over-consolidation phenomenon. The systems of partial differential equations were formulated for every model and solved numerically to obtain settlement curves. Selected numerical results were compared with standard oedometer laboratory test data carried out by the authors on organic soil samples. Additionally, plasticity phenomenon and non-classical rheological elements were included in order to take into account soil over-consolidation behaviour in the one-dimensional settlement model. A new way of formulating constitutive equations for the soil skeleton and predicting the relationship between the effective stress and strain or void ratio was presented. Rheological structures provide a flexible tool for creating complex constitutive relationships of soil.

On the effective stress law and its application to finite deformation problems in a poroelastic solid

Certain materials, such as rubber and gel, can undergo relatively large deformations without failure; and elastic finite deformation theories have been developed. Porous materials, such as soil, clay, and sediment, can also undergo relatively large deformations, and the concept of effective stress is commonly introduced to formulate the constitutive laws. In this research, it is demonstrated that for a porous solid, the effective stress is work-conjugate only to the Green strain tensor and not to the other strain measures under the assumption of an incompressible solid constituent. Based on this finding, the previously reported hyperelastic model, in which the effective stress and the Hencky strain has an elastic linear relation, is questioned.Several alternative hyperelastic models developed for solid materials are tested for their suitability for porous solids. Certain constitutive models, such as the Saint Venant–Kirchhoff model, exhibit an apparent strain-softening in the elastic range, which is believed to be not consistent with the observed behavior of most porous materials. Other models, such as the Ogden model, show the potential for modeling finite deformations of fluid-saturated poroelastic materials. One-dimensional finite consolidation problems are numerically solved and compared with the analytical solution based on the infinitesimal deformation theory.

Physical and geometric non-linear analysis using the finite difference method for one-dimensional consolidation problem

Abstract This article presents a numerical model based on the finite difference method for the physical and geometric non-linear analysis of a one-dimensional consolidation problem regarding a saturated, homogeneous and isotropic soil layer with low permeability and high compressibility. The problem is formulated by adopting the void ratio as the primary variable, considering a Lagrangian movement description. The physical non linearity is introduced on the formulation by the constitutive law defined as effective stress and permeability void ratio functions. Based on this numerical model, a computational system named AC-3.0 was developed, which has been verified and validated in terms of the temporal variation of the void ratio distribution throughout the soil layer, by comparing the numerical results with analytical and numerical solutions found in literature for some specific scenarios. Knowing the void ration distribution,it is possible to obtain secondary variables such as: superficial settlement, effective stress and excess of pore water pressure.The importance of the non-linear formulation is highlighted for the analysis of problems related to material presenting high compression and a very high initial void ratio.

Analytical and Numerical Modelling of One-Dimensional Consolidation of Stabilized Peat

The objective of the paper is to compare and evaluate analytical and numerical solutions of one-dimensional consolidation of stabilized peat. The type of analytical method used to solve the problem is exact method by separation of variables and utilization of Fourier series. Plaxis 2D 8.2 Professional version software was used to find numerical solution to the problem by employing the finite element method. One-dimensional consolidation problem of stabilized peat was solved numerically and validated with the one solved analytically based on laboratory experimental results. From the results, it was discovered that the consolidation characteristics of stabilized peat evaluated numerically were found to have close approximation to those evaluated analytically. There is a novel value in developing an accurate numerical prediction for the vertical consolidation of stabilized peat considering the complexity of the soil treatment method. It must be noted that peat is highly problematic because it is produced from plant decomposition with extremely high organic matter.

Analytical Solution for One-Dimensional Consolidation of Soil with Exponentially Time-Growing Drainage Boundary under a Ramp Load

By introducing the exponentially time-growing drainage boundary, this paper investigated the one-dimensional consolidation problem of soil under a ramp load. Firstly, the one-dimensional consolidation equations of soil are established when there is a ramp load acting on the soil surface. Then, the analytical solution of excess pore water pressure and consolidation degree is derived by means of the method of separation of variables and the integral transform technique. The rationality of this solution is also verified by comparing it with other existing analytical solutions. Finally, the consolidation behavior of soil is studied in detail for different interface parameters or loading scheme. The results show that the exponentially time-growing drainage boundary can reflect the phenomenon that the excess pore water pressure at the drainage boundaries dissipates smoothly rather than abruptly from its initial value to the value of zero. By adjusting the values of interface parametersbandc, the presented solution can be degraded to Schiffman’s solution, which can compensate for the shortcoming that Terzaghi’s drainage boundary can only consider the two extreme cases of fully pervious and impervious boundaries. The significant advantage of the exponentially time-growing drainage boundary is that it can be applied to describe the asymmetric drainage characteristics of the top and bottom drainage surfaces of the actual soil layer by choosing the appropriate interface parametersbandc.

Analytical solution for one-dimensional consolidation of double-layered soil with exponentially time-growing drainage boundary

In this article, the exponentially time-growing drainage boundary is introduced to study the one-dimensional consolidation problem of double-layered soil. First, the one-dimensional consolidation equations of soil underlying a time-dependent loading are established. Then, the analytical solution of excess pore water pressure and average consolidation degree is obtained by utilizing the method of separation of variables when the soil layer is separately undergone instantaneous load and single-stage load. The validity of the present solution is proven by the comparison with other existing analytical solution. Finally, the influence of soil properties and loading scheme on the consolidation behavior of soil is investigated in detail. The results indicate that, the present solution can be degraded to Xie’s solution utilizing Terzaghi’s drainage boundary by adjusting the interface parameter, that is to say, Xie’s solution can be regarded as a special case of the present solution. The interface parameter has a significant influence on the excess pore water pressure of soil, and the larger interface parameter means the better drainage capacity of the soil layer.

A general analytical solution to the one-dimensional consolidation problem for unsaturated soil under various loading conditions

A general analytical solution to the one-dimensional consolidation problem for unsaturated soil under various loading conditions

Analysis of consolidation of a soil layer with depth-dependent parameters under time-dependent loadings

In this paper, a Laplace transform method is proposed to solve the one-dimensional consolidation problems considering depth-dependent parameters of the soil layer, which is subjected to complicated time-dependent loadings at ground surface. The coefficient of volume compressibility and the coefficient of permeability for the soil layer are continuous functions of vertical coordinate. The solution is obtained and described in detail. The characteristics of consolidation under suddenly imposed constant loading are analysed. The results show that the excess pore pressure distribution of a soil layer with depth-dependent parameters is dissimilar to the solution Terzaghi has proposed. The characteristics of consolidation under time-dependent loadings are also analysed and the influences of depth-dependent parameters are considered. The results also suggest that the effective stress develops with time under time-dependent loading follows the trend under suddenly-imposed constant loading but exhibit certain oscillations. The oscillation amplitudes of the effective stresses don’t decrease with time. Therefore, it is inaccurate to simplify the cyclic loading to constant loading.