A robust mesh optimisation method for multiphase porous media flows

Flows of multiple fluid phases are common in many subsurface reservoirs. Numerical simulation of these flows can be challenging and computationally expensive. Dynamic adaptive mesh optimisation and related approaches, such as adaptive grid refinement can increase solution accuracy at reduced computational cost. However, in models or parts of the model domain, where the local Courant number is large, the solution may propagate beyond the region in which the mesh is refined, resulting in reduced solution accuracy, which can never be recovered. A methodology is presented here to modify the mesh within the non-linear solver. The method allows efficient application of dynamic mesh adaptivity techniques even with high Courant numbers. These high Courant numbers may not be desired but a consequence of the heterogeneity of the domain. Therefore, the method presented can be considered as a more robust and accurate version of the standard dynamic mesh adaptivity techniques.     

A developed technique for measuring water content in oil-contaminated porous media

Water content is an important physical parameter for soil, vadose zone, and porous aquifer. Accurate measurement of water content in oil-contaminated porous media is critical for the research on oil pollution process and remediation in soil and groundwater systems. In this study, an improved water content calculation formula for oil-contaminated porous media was proposed based on the theory of oven-drying method, and laboratory experiments were conducted to test the applicability and accuracy of the formula for several types of manually prepared oil-contaminated porous media with different water contents. Furthermore, the measuring method and calculation formula, which can be used to determine the water content of porous media sampled from the oil-contaminated sites, were proposed for the first time in this study based on the improved formula. The experimental results showed that the improved formula was very accurate when used to calculate the water contents of diesel-contaminated sand, gasoline-contaminated mild clay, and engine oil-contaminated sand, indicating that it was widely applicable to oils with different volatile ability as well as porous media with different texture. This study meets the urgent need for accurate determination of water content in oil-contaminated porous media, and it solves the technical problem that the existing water content measuring methods cannot be applied directly in the field study.     

Multiscale simulations of porous media flows in flow-based coordinate system

In this paper, we propose a multiscale technique for the simulation of porous media flows in a flow-based coordinate system. A flow-based coordinate system allows us to simplify the scale interaction and derive the upscaled equations for purely hyperbolic transport equations. We discuss the applications of the method to two-phase flows in heterogeneous porous media. For two-phase flow simulations, the use of a flow-based coordinate system requires limited global information, such as the solution of single-phase flow. Numerical results show that one can achieve accurate upscaling results using a flow-based coordinate system.     

Smooth particle hydrodynamics simulations of low Reynolds number flows through porous media

In this paper, a three‐dimensional smooth particle hydrodynamics (SPH) simulator for modeling grain scale fluid flow in porous media is presented. The versatility of the SPH method has driven its use in increasingly complex areas of flow analysis, including the characterization of flow through permeable rock for both groundwater and petroleum reservoir research. SPH provides the means to model complex multi‐phase flows through such media; however, acceptance of the methodology has been hampered by the apparent lack of actual verification within the literature, particulary in the three‐dimensional case. In this paper, the accuracy of SPH is addressed via a comparison to the previously recognized benchmarks of authors such as Sangani and Acrivos (Int. J. Multiphase Flow 1982; 8 (4): 343–360), Zick and Homsy (J. Fluid Mech. 1982; 115 :13–26) and Larson and Higdon (Phys. Fluids A 1989; 1 (1):38–46) for the well‐defined classical problems of flow through idealized two‐ and three‐dimensional porous media. The accuracy of results for such low Reynolds number flows is highly dependent on the implementation of no‐slip boundary conditions. A new, robust and numerically efficient, method for implementing such boundaries in SPH is presented. Simulation results for friction coefficient and permeability are shown to agree well with the available benchmarks. Copyright © 2010 John Wiley & Sons, Ltd.     

Analysis and numerical approximation of Brinkman regularization of two-phase flows in porous media

We consider a system of nonlinear partial differential equations that arises in the modeling of two-phase flows in a porous medium. The phase velocities are modeled using a Brinkman regularization of the classical Darcy’s law. We propose a notion of weak solution for these equations and prove existence of these solutions. An efficient finite difference scheme is proposed and is shown to converge to the weak solutions of this system. The Darcy limit of the Brinkman regularization is studied numerically using the convergent finite difference scheme in two space dimensions as well as using both analytical and numerical tools in one space dimension. The results suggest that the Brinkman regularization may not approximate the accepted entropy solutions of the Darcy model and raise fundamental questions about the use of Brinkman type models in two-phase flows.     

A coupled hydro-mechanical analysis for prediction of hydraulic fracture propagation in saturated porous media using EFG mesh-less method

The details of the Element Free Galerkin (EFG) method are presented with the method being applied to a study on hydraulic fracturing initiation and propagation process in a saturated porous medium using coupled hydro-mechanical numerical modelling. In this EFG method, interpolation (approximation) is based on nodes without using elements and hence an arbitrary discrete fracture path can be modelled.The numerical approach is based upon solving two governing partial differential equations of equilibrium and continuity of pore water simultaneously. Displacement increment and pore water pressure increment are discretized using the same EFG shape functions. An incremental constrained Galerkin weak form is used to create the discrete system of equations and a fully implicit scheme is used for discretization in the time domain. Implementation of essential boundary conditions is based on the penalty method. In order to model discrete fractures, the so-called diffraction method is used.Examples are presented and the results are compared to some closed-form solutions and FEM approximations in order to demonstrate the validity of the developed model and its capabilities. The model is able to take the anisotropy and inhomogeneity of the material into account. The applicability of the model is examined by simulating hydraulic fracture initiation and propagation process from a borehole by injection of fluid. The maximum tensile strength criterion and Mohr–Coulomb shear criterion are used for modelling tensile and shear fracture, respectively. The model successfully simulates the leak-off of fluid from the fracture into the surrounding material. The results indicate the importance of pore fluid pressure in the initiation and propagation pattern of fracture in saturated soils.     

A pore-scale numerical model for flow through porous media

A pore-scale numerical model based on Smoothed Particle Hydrodynamics (SPH) is described for modelling fluid flow phenomena in porous media. Originally developed for astrophysics applications, SPH is extended to model incompressible flows of low Reynolds number as encountered in groundwater flow systems. In this paper, an overview of SPH is provided and the required modifications for modelling flow through porous media are described, including treatment of viscosity, equation of state, and no-slip boundary conditions. The performance of the model is demonstrated for two-dimensional flow through idealized porous media composed of spatially periodic square and hexagonal arrays of cylinders. The results are in close agreement with solutions obtained using the finite element method and published solutions in the literature. Copyright © 1999 John Wiley & Sons, Ltd.     

Discretization of flow contain nitrate in porous media by finite volume technique

One of the most important pollutants of groudwaters is nitrate. Different human activities including the application of chemical fertilizers in agriculture, causes the emission of nitrate into groudwaters. In this paper, the dynamic effect of soil moisture on carbon and nitrogen cycles has been analyzed by presenting a connection between soil moisture sample and nonlinear differential equations. At present, wide researches are carried out on modeling soil moisture control in solution flows contain nitrate. In order to do so, separation of energy conservation law equations is carried out by a particular method. The mathematical model governing the nitrate containing current in non-isotropic environment has been presented in the form of combined equations. Equation for distribution in multiple environments and Darcy rule has been considered in this model. Then, using finite volume method, separation of flows contain nitrate in porous media is carried out. The current flux is obtained from central difference approximations or upwind approximation. Mashad plain has been considered for case study at this research. Carrying out calibration operation, the measured results have been contrasted with numerical results of finite volume method. After testing the model, it is possible to foresee the way of nitrate changes in other nodes of calculation network. Using these forecasts, the quality of drinking water for several next years is determined. Carrying out numerical modeling by finite volume method, it is found out that the quality of drinking water of Mashad plain would be suitable for the next ten years.     

A general reduction scheme for reactive transport in porous media

We present a method to transform the governing equations of multispecies reactive transport in porous media. The reformulation leads to a smaller problem size by decoupling of equations and by elimination of unknowns, which increases the efficiency of numerical simulations. The reformulation presented here is a generalization of earlier works. In fact, a whole class of transformations is now presented. This class is parametrized by the choice of certain transformation matrices. For specific choices, some known formulations of reactive transport can be retrieved. Hence, the software based on the presented transformation can be used to obtain efficiency comparisons of different solution approaches. For our efficiency tests, we use the MoMaS benchmark problem on reactive transport.     

A semi-implicit method for incompressible three-phase flow in porous media

In this paper, we present a semi-implicit method for the incompressible three-phase flow equations in two dimensions. In particular, a high-order discontinuous Galerkin spatial discretization is coupled with a backward Euler discretization in time. We consider a pressure-saturation formulation, decouple the pressure and saturation equations, and solve them sequentially while still keeping each equation implicit in its respective unknown. We present several numerical examples on both homogeneous and heterogeneous media, with varying permeability and porosity. Our results demonstrate the robustness of the scheme. In particular, no slope limiters are required and a relatively large time step may be taken.     

基于精细时程积分的饱和两相介质波动问题时域解法

针对u-p形式的饱和两相介质的波动方程,采用精细时程积分方法计算固相位移u,采用向后差分算法求解流体压力p,建立了基于精细时程积分技术的饱和两相介质波动问题的时域求解方法。针对标准算例,将该方法的计算结果与可视为标准结果的Zienkiewicz隐式算法的计算结果进行比较分析,二者符合较好,表明了该方法具有良好的计算精度。同时,该方法的计算过程为交替迭代求解,避免了在每个时间分析步上求解耦联方程组,因而具有较高的计算效率。该方法具有时域显式计算方法的基本特点,是进行饱和两相介质动力问题计算与分析的一种有效方法。     

A prediction model for debris flows triggered by a runoff-induced mechanism

Many debris flows were triggered within and also outside the Dayi area of the Guizhou Province, China, during a rainstorm in 2011. High-intensity short-duration rainfall was the main triggering factor for these gully-type debris flows which are probably triggered by a runoff-induced mechanism. A revised prediction model was introduced for this kind of gully-type debris flows with factors related to topography, geology, and hydrology (rainfall) and applied to the Wangmo River catchment. Regarding the geological factor, the “soft lithology” and “loose sediments” in the channel were added to the list of the average firmness coefficient for the lithology. Also, the chemical weathering was taken into account for the revised geological factor. Concerning the hydrological factor, a coefficient of variation of rainfall was introduced for the normalization of the rainfall factor. The prediction model for debris flows proposed in this paper delivered three classes of the probability of debris flow occurrence. The model was successfully validated in debris flow gullies with the same initiation mechanism in other areas of southwest China. The generic character of the model is explained by the fact that its factors are partly based on the initiation mechanisms and not only on the statistical analyses of a unique variety of local factors. The research provides a new way to predict the occurrence of debris flows initiated by a runoff-induced mechanism.     

孔隙介质毛细滞回简化模型研究

土-水特征曲线的研究是非饱和孔隙介质研究的重要部分。完整的土-水特征曲线包括初始脱湿曲线（IDC）、主吸湿线(MWC)和主脱湿线(MDC)3部分。传统的土-水特征曲线实验需要花费较长的时间和精力。以文献中的试验数据为依据,对非饱和土土-水特征关系和岩石的汞注入抽出试验的滞回现象进行了深入研究,提出一个经验模型来模拟MDC曲线。如果已根据试验得到了孔隙介质的IDC和MWC,此模型只需要一个参数,即可得到增量形式的MDC。通过与试验结果相比较,验证了模型对MDC模拟的有效性。     

Thermo-hydro-mechanical modeling of fracturing porous media with two-phase fluid flow using X-FEM technique

In this paper, a fully coupled thermo-hydro-mechanical model is presented for two-phase fluid flow and heat transfer in fractured/fracturing porous media using the extended finite element method. In the fractured porous medium, the traction, heat, and mass transfer between the fracture space and the surrounding media are coupled. The wetting and nonwetting fluid phases are water and gas, which are assumed to be immiscible, and no phase-change is considered. The system of coupled equations consists of the linear momentum balance of solid phase, wetting and nonwetting fluid continuities, and thermal energy conservation. The main variables used to solve the system of equations are solid phase displacement, wetting fluid pressure, capillary pressure, and temperature. The fracture is assumed to impose the strong discontinuity in the displacement field and weak discontinuities in the fluid pressure, capillary pressure, and temperature fields. The mode I fracture propagation is employed using a cohesive fracture model. Finally, several numerical examples are solved to illustrate the capability of the proposed computational algorithm. It is shown that the effect of thermal expansion on the effective stress can influence the rate of fracture propagation and the injection pressure in hydraulic fracturing process. Moreover, the effect of thermal loading is investigated properly on fracture opening and fluids flow in unsaturated porous media, and the convective heat transfer within the fracture is captured successfully. It is shown how the proposed computational model is capable of modeling the fully coupled thermal fracture propagation in unsaturated porous media.     

A boundary integral method for steady flow in unsaturated porous media

The governing equation for steady flow in a homogeneous, partially saturated, porous medium can be written in a linear form if one adopts a hydraulic conductivity function which varies exponentially with capillary-pressure head. The resulting linear field equation is a steady Fokker–Planck equation and is well-suited to numerical solution by the boundary integral equation method (BIEM). The exponential conductivity function is often used in soil physics and is known to be a reasonable approximation over limited ranges of pressure head. A computer code based on the BIEM for obtaining numerical solutions is described and tested. The BIEM is found to exhibit quadratic convergence with element size reduction on smooth solutions and on singular problems, if mesh grading is used. Agreement between results from the BIEM code and a finite element code that solves the fully non-linear problem is excellent, and is achieved at a substantial advantage in computer processing time. As an illustrative example, the code is applied to determine the distribution of moisture in the vicinity of a tunnel.     

A mixed solution for two-dimensional unsteady flow in fractured porous media

A mixed finite element–boundary element solution for the analysis of two-dimensional flow in porous media composed of rock blocks and discrete fractures is described. The rock blocks are modelled implicitly by using boundary elements whereas finite elements are adopted to model the discrete fractures. The computational procedure has been implemented in a hybrid code which has been validated first by comparing the numerical results with the closed-form solution for flow in a porous aquifer intercepted by a vertical fracture only. Then, a more complex problem has been solved where a pervious, homogeneous and isotropic matrix containing a net of fractures is considered. The results obtained are shown to describe satisfactorily the main features of the flow problem under study. © 1997 by John Wiley & Sons, Ltd.     

Weak discontinuity in porous media: an enriched EFG method for fully coupled layered porous media

In this paper, a series of multimaterial benchmark problems in saturated and partially saturated two‐phase and three‐phase deforming porous media are addressed. To solve the process of fluid flow in partially saturated porous media, a fully coupled three‐phase formulation is developed on the basis of available experimental relations for updating saturation and permeabilities during the analysis. The well‐known element free Galerkin mesh‐free method is adopted. The partition of unity property of MLS shape functions allows for the field variables to be extrinsically enriched by appropriate functions that introduce existing discontinuities in the solution field. Enrichment of the main unknowns including solid displacement, water phase pressure, and gas phase pressure are accounted for, and a suitable enrichment strategy for different discontinuity types are discussed. In the case of weak discontinuity, the enrichment technique previously used by Krongauz and Belytschko [Int. J. Numer. Meth. Engng., 1998; 41:1215–1233] is selected. As these functions possess discontinuity in their first derivatives, they can be used for modeling material interfaces, generating only minor oscillations in derivative fields (strain and pressure gradients for multiphase porous media), as opposed to unenriched and constrained mesh‐free methods. Different problems of multimaterial poro‐elasticity including fully saturated, partially saturated one, and two‐phase flows under the assumption of fully coupled extended formulation of Biot are examined. As a further development, problems involved with both material interface and impermeable discontinuities, where no fluid exchange is permitted across the discontinuity, are considered and numerically discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Upscaling of the permeability by multiscale wavelet transformations and simulation of multiphase flows in heterogeneous porous media

We describe a new approach for simulation of multiphase flows through heterogeneous porous media, such as oil reservoirs. The method, which is based on the wavelet transformation of the spatial distribution of the single-phase permeabilities, incorporates in the upscaled computational grid all the relevant data on the permeability, porosity, and other important properties of a porous medium at all the length scales. The upscaling method generates a nonuniform computational grid which preserves the resolved structure of the geological model in the near-well zones as well as in the high-permeability sectors and upscales the rest of the geological model. As such, the method is a multiscale one that preserves all the important information across all the relevant length scales. Using a robust front-detection method which eliminates the numerical dispersion by a high-order total variation diminishing method (suitable for the type of nonuniform upscaled grid that we generate), we obtain highly accurate results with a greatly reduced computational cost. The speed-up in the computations is up to over three orders of magnitude, depending on the degree of heterogeneity of the model. To demonstrate the accuracy and efficiency of our methods, five distinct models (including one with fractures) of heterogeneous porous media are considered, and two-phase flows in the models are studied, with and without the capillary pressure.     

Numerical solutions for flow in porous media

A numerical approach is proposed to model the flow in porous media using homogenization theory. The proposed concept involves the analyses of micro‐true flow at pore‐level and macro‐seepage flow at macro‐level. Macro‐seepage and microscopic characteristic flow equations are first derived from the Navier–Stokes equation at low Reynolds number through a two‐scale homogenization method. This homogenization method adopts an asymptotic expansion of velocity and pressure through the micro‐structures of porous media. A slightly compressible condition is introduced to express the characteristic flow through only characteristic velocity. This characteristic flow is then numerically solved using a penalty FEM scheme. Reduced integration technique is introduced for the volumetric term to avoid mesh locking. Finally, the numerical model is examined using two sets of permeability test data on clay and one set of permeability test data on sand. The numerical predictions agree well with the experimental data if constraint water film is considered for clay and two‐dimensional cross‐connection effect is included for sand. Copyright © 2003 John Wiley & Sons, Ltd.