Fluid Transport In Porous Media Research Articles

Experimental Study on Forced Imbibition and Wettability Alteration of Active Carbonated Water in Low-Permeability Sandstone Reservoir

Summary Imbibition is one of the main mechanisms for fluid transport in porous media. A combination of carbonated water and active water [active-carbonated water (ACW)] has great prospects in enhanced oil recovery (EOR) and carbon reduction processes. To date, the law of hydrocarbon recovery induced by ACW imbibition is not clear. In this paper, the optimal surfactant concentration was first selected through a spontaneous imbibition experiment, and on this basis, CO2 was dissolved to form ACW. The imbibition effects of formation water (FW), surfactant solution DX-1, and ACW under different pressures were compared. The changes in rock wettability in the three imbibition solutions during imbibition were studied by measuring the contact angle. The effect of fracture on ACW imbibition was studied. Finally, the improved NB−1 was calculated to elucidate the mechanism of forced imbibition for EOR. The results show that 0.1% DX-1 produces the optimal imbibition effect. Pressure is positively correlated with imbibition recovery. ACW can significantly improve the imbibition effect due to its wettability reversal ability being better than those of FW and DX-1. CO2 in ACW can be trapped in the formation through diffusion into small rock pores. The contact angles of the three imbibition solutions decrease with increasing pressure. The contact angle between the rock and oil droplet in the ACW is as low as 38.13°. In addition, the fracture increases the contact area between the matrix and the fluid, thereby improving the imbibition effect. The alteration of NB−1 indicates that FW imbibition is gravity-driven cocurrent imbibition. DX-1 and ACW imbibitions are countercurrent imbibitions driven by capillary force and gravity. The above results demonstrate the feasibility of ACW in low-permeability reservoir development and carbon reduction.

Experimental and model analysis of the effect of pore and mineral characteristics on fluid transport in porous soil media

The fluid transport in porous media is a critical property for oil and gas exploitation, construction engineering, and environmental protection. It is profoundly influenced by pore geometry and mineral properties. Currently, the Kozeny–Carman equation serves as the permeability prediction equation for porous media, established on the circular pores model. However, it fails to fully account for the impact of pore shape and mineral properties of the soil, leading to significant deviations between predicted and measured soil permeability results. In this paper, based on scanning electron microscope image and mercury intrusion porosimetry, the pores were divided into circular pores and narrow slit pores according to the ratios of pore area and circumference. Then, the quantitative expression of the two types of pores and their connectivity and tortuosity were given, and the circular and narrow slit composite pore model was used to describe the soil pore. Subsequently, the electrostatic potential of pore water was calculated by the Poisson–Boltzmann equation to consider the adsorption effect of minerals on pore water. Combined with the Navier–Stokes equation, the permeability prediction equation considering pore geometry, pore connectivity, and tortuosity and mineral properties was established. Finally, the experimental results illustrated that the theoretical prediction results were in good agreement with the experimental results. The proposed permeability prediction equation proves valuable for assessing and predicting the fluid transport in porous media.

Fluid transport in porous media based on differences in filter media morphology and biofilm growth in bioreactors

To elucidate the effect of pore structure on bioclogging and seepage flow in bioreactors, we used X-ray computed tomography (X-CT) to investigate the changes in seepage flow of porous media in zeolite, gravel and ceramsite bioreactors with biofilm growth by injecting a non-ionic contrast medium iohexol. Based on the X-CT images using a ball-and-stick model, the highest average pore radius (R‾) and the average pore throat radius (r‾) in the ceramsite column were found under the initial conditions, which facilitated its permeability. The pore and throat of the gravel column were small and homogeneous relatively. Biofilm growth decreased the pore and pore throat in the columns. The total throat area of zeolite, gravel and ceramsite columns declined by 74%, 73% and 79% respectively. The zeolite column had the highest average pore throat, which contributed to its maximum conductivity subsequently after biofilm growth. Further, the fractal dimensions of the pore structure increased with biofilm growth, especially in the zeolite and ceramsite columns. The heterogeneity of the porous media was reinforced by the biofilm growth in the zeolite and ceramsite columns due to their higher heterogeneity initially. We also observed that an increase in heterogeneity of porous medium amplified the preferential flow and flow heterogeneities, especially in the zeolite and ceramsite columns.

Use of Boundary-Driven Nonequilibrium Molecular Dynamics for Determining Transport Diffusivities of Multicomponent Mixtures in Nanoporous Materials.

The boundary-driven molecular modeling strategy to evaluate mass transport coefficients of fluids in nanoconfined media is revisited and expanded to multicomponent mixtures. The method requires setting up a simulation with bulk fluid reservoirs upstream and downstream of a porous media. A fluid flow is induced by applying an external force at the periodic boundary between the upstream and downstream reservoirs. The relationship between the resulting flow and the density gradient of the adsorbed fluid at the entrance/exit of the porous media provides for a direct path for the calculation of the transport diffusivities. It is shown how the transport diffusivities found this way relate to the collective, Onsager, and self-diffusion coefficients, typically used in other contexts to describe fluid transport in porous media. Examples are provided by calculating the diffusion coefficients of a Lennard-Jones (LJ) fluid and mixtures of differently sized LJ particles in slit pores, a realistic model of methane in carbon-based slit pores, and binary mixtures of methane with hypothetical counterparts having different attractions to the solid. The method is seen to be robust and particularly suited for the study of study of transport of dense fluids and liquids in nanoconfined media.

Upscaling non-Newtonian rheological fluid properties from pore-scale to Darcy’s scale

Continuity and momentum equations govern the pore-scale flow properties of the fluid in the porous media, whereas Darcy’s law governs Darcy scale properties of the fluid transport in porous media. The empirical shift factor relates the steady shear-dependent viscosity of non-Newtonian fluids to the Darcy viscosity in porous media. The reported values of the empirical shift factor cover three orders of magnitude depending on the considered fluid-medium configurations. This creates a challenge to upscale non-Newtonian rheology from pore-scale to Darcy’s scale. We upscale Darcy viscosity based on pore-scale shear viscosity of Meter model fluid and viscoelastic linear Phan-Thien-Tanner fluid. We propose a Bundle-of-Capillaries model modified with pore-correction coefficient and fluid-correction coefficient. We numerically simulate the flow of polyacrylamide fluid, modelled using Meter model and linear Phan-Thien-Tanner model, through 3D symmetric micro-channel, 2D porous medium and 3D Mt Simon sandstone. Pore-scale direct numerical simulation using linear Phan-Thien-Tanner model showed viscoelastic instability in heterogeneous Mt Simon sandstone at low Reynolds number flow. Direct numerical simulations overestimate Darcy viscosity due to the presence of stagnant zones without active flow.

Comprehensive study and comparison of equilibrium and kinetic models in simulation of hydrate reaction in porous media

Coupling hydrate reaction in fluid transport in porous media is essential for simulation of gas hydrate production, as well as carbon sequestration in deep-sea sediments. It can be conceptualized as multiphase, multicomponent, and non-isothermal reactive transport. Two types of models have been developed for the description of hydrate reaction in numerical models. The equilibrium model (EM) assumes instantaneous chemical equilibrium among species, and thus ignores reaction kinetics. The kinetic model (KM) incorporates reaction kinetics by introducing a term of reaction rate dependent on fugacity difference. Although KM achieves a more accurate description of the reaction, it is potentially more computationally intensive due to the increased degree of freedom in the nonlinear equation system. Although several previous studies have investigated the similarities and differences between these two models, inconsistencies exist in their studies, and the mechanisms to distinguish these two models are still not well understood. In our study, in order to identify the reasons for these inconsistencies and elucidate the mechanisms that control the differences between the two reaction models, we provide a comprehensive investigation and comparison of these two models via theoretical analysis and numerical experiments. Through comparing the basic assumptions, mathematical model and the equation systems, we pointed out the basic difference of these two models, which results from the different calculations of physical parameters. Dimensionless analysis of the governing equations of KM yields several characteristic numbers, representing the relative strength of different physical processes. We performed numerical experiments to investigate the effect of characteristic numbers on the difference between the two models, and the results indicate that there exist critical values for these characteristic numbers, lower than which lead to an obvious gap between the results of EM and KM. It turns out that the relative strength of the hydrate reaction and other physical processes controls the magnitude of difference between these two models. EM is essentially a special case of KM when the time scale of the hydrate reaction is sufficiently smaller than other physical processes, such as convective and diffusive transport of mass and heat. Comparison between the time and iteration steps in the two reaction models provides insights into the computational efficiency of the two models.

Simulation of non-Newtonian fluid seepage into porous media stacked by carbon fibers using micro-scale reconstruction model

Ceramic matrix composite of carbon fiber framework reinforced by silicon carbide (Cf/SiC) is a new material with excellent properties. In order to improve its preparation process, the flow of polycarbosilane solution in carbon fiber is studied based on the computational fluid dynamics method. CT images of carbon fiber porous media are obtained by Micro-CT imaging technology and optimized by means of ImageJ software. The 3D solid matrix and pore space models of carbon fiber porous media are established by using Mimics software, and the pore space model is meshed. Using the computational fluid dynamics software Fluent, the VOF multiphase flow model is adopted, the surface tension model and contact angle model are introduced considering the micro-pore effect of porous media, and a suitable mathematical model for the transport of gas–liquid two-phase flow in carbon fiber porous media is established. The validity of CFD mathematical model is verified by comparing experimental data with simulation results. Based on the CFD simulation results, the effects of feed pressure, surface tension coefficient and contact angle on fluid flow are studied. It is found that the larger the feed pressure and contact angle, the more uniform the distribution of liquid phase, pressure and velocity in the medium, which is conducive to the flow of liquid phase in porous media. The larger the surface tension coefficient, the longer the breakthrough time of liquid, and the more uneven the liquid distribution. This study is helpful to understand the law of fluid transport in porous media and provides theoretical basis for the preparation of Cf/SiC composites.

Unsteady-State Coreflooding Monitored by Positron Emission Tomography and X-ray Computed Tomography

Summary The forced-imbibition displacement process—a complex function of multiple physical properties of rock, fluids, and flow conditions—is of vital importance. For this paper we used imaging techniques to detect the process in three dimensions with high temporal resolution. Positron emission tomography (PET) is a powerful tool for imaging fluid transport in porous media because it has high temporal resolution and is not affected by complex lithology and pore structures. In this study, we conducted a detailed experimental and numerical analysis to prove how PET can be used with microcomputed tomography (micro-CT) to complement the imaging of water displacement in the porous space of sandstone. To characterize the rock properties, a dry sample was imaged with a high-resolution micro-CT scanner. The image was reconstructed and segmented into two phases (solid and void) to calculate the porosity/permeability relationship on cubic subsets. The Navier-Stokes equation was solved for permeability calculation. The porosity/permeability relationship was used as the input to simulate the forced imbibition process on an upscaled micro-CT image using the Matlab Reservoir Simulation Toolbox (MRST) (Lie et al. 2011; Krogstad et al. 2015; Lie 2016; Bao et al. 2017). The result was the evolution of invading-fluid saturation in every upscaled voxel. In the following step, water tracer labeled with fluorodeoxyglucose (18F-FDG) was injected into the dry sample and a series of PET images was acquired to detect fluid pathways. Fluid-front topology and a propagation procedure were captured. We integrated saturation data across the core cross section for both data sets (direct PET-based and MRST-simulated on upscaled tomograms) to reduce the complexity of the analysis. Comparison of the resulting 1D saturation profiles demonstrated the capability of PET imaging to monitor and understand dynamic flow processes in reservoir rocks.

Experimental Investigation into the Screening of New Surfactant Flowback Aids and Statistical Interpretation of Their Imbibing Capacity for Selected Core Substrates

AbstractSpontaneous imbibition is a pervasive part of many natural and industrial processes. As an inherent feature of fluid transport in porous media, it is a driver for oil recovery. Understanding spontaneous imbibition and leveraging surface science is fundamental for fluid recovery; specifically, the role of the surfactant in the imbibition processes and the potential to alter capillarity and wettability of reservoir rock. Surfactant success relies on the understanding of the factors governing the interfacial phenomena among crude oil, and formation properties under reservoir conditions. Developing a methodology coupling chemical performance with analytical techniques, and statistical interpretation of core/surfactant/oil interactions, can help establish workflows to advance new chemistries and enhance oil recovery. This article discusses a study of flowback aids formulated as microemulsions corresponding to the thermodynamically stable Winsor Type IV solutions. Neat formulated microemulsions, when dosed at field treatment concentrations, provide either oil‐in‐water droplet microemulsions or nanoemulsions. The solvency potential was measured, and the Kauri‐butanol (Kb) value was determined. Parameters such as critical micelle concentration (CMC) and interfacial tension (IFT) were determined to characterize microemulsion solutions. These systems were tested using either in the column flow test with formation material sieved to match mineral grain size, or sandstone cores of various permeabilities. The results indicate that surfactant‐based flow‐enhancing aids are desirable for improved oil recovery when compared to the control fluid. The statistical analysis of core‐fluid interaction includes an ANOVA followed by assumption evaluations and model interpretation, which demonstrates that the core permeability term, followed by the surfactant term, has the highest contribution whereas oil has no statistical significance to the model.

Aspects of finite element formulations for the coupled problem of poroelasticity based on a canonical minimization principle

This work presents a new finite element treatment of the coupled problem of Darcy–Biot-type fluid transport in porous media undergoing large deformations, that is free from any stabilization techniques. The formulation bases on an incremental two-field minimization principle that is constrained by the equation of continuity for the fluid mass content and determines at a given state the deformation and the fluid mass flux vector. The big advantage of the minimization formulation over classical saddle point principles of poroelasticity is the omission of the inf-sup condition—a condition that makes the construction of stable and computationally efficient finite element formulations difficult. Due to the $$H(\hbox {Div}, {\mathcal B}_0)$$ variational structure of the minimization principle on the fluid side, lowest order Raviart–Thomas elements are used for the conforming approximation of the fluid mass flux. Furthermore, a standard nodal-based element using bilinear interpolation for both fields combined with a reduced numerical integration of the (volumetric) coupling term is analyzed and used for the solution of the minimization principle. Representative numerical examples demonstrate the performance of the proposed finite element designs of the minimization principle and clearly underline advantages over finite element formulations of the classical two-field saddle point principle formulated in deformation and fluid potential.

A further study on Welge equation

Buckley–Leverett theory and Welge equation is one of the most widely used methods for predicting fluid transport in porous media. However, the average water saturation in Welge equation is a function of both water saturation and water cut, which is inconvenient to be used. While the linear relationship (the slope is 2/3) of the average water saturation and outlet water saturation proposed by Зфрос provides a simple method the oil–water viscosity ratio greatly limits the application. In this paper, based on the analytical relative permeability model and Welge equation, a new average water saturation equation with a variable (the outlet water saturation) was derived and applied in some cases under different relative permeability parameters and oil–water viscosity. Moreover, a simplified equation was proposed through certain data regression. Thus, the coefficient calculation method of the linear function was introduced based on the actual production data. Results showed a nonlinear relationship between the average water saturation and outlet water saturation, which is mainly related to water and oil relative permeability, and oil–water viscosity ratio. For the outlet water saturation higher than frontal saturation, the equation can be simplified to a linear function, which is the derivation of Зфрос equation. However, it is a function of both relative permeability and oil–water viscosity ratio instead of a constant slope of 2/3. In view of the fact that oil–water viscosity is mostly bigger than 10 in most reservoirs, the new model is an important supplement to Buckley–Leverett theory and Welge equation.

Experimental study for the impacts of flow rate and concentration of asphaltene precipitant on dynamic asphaltene deposition in microcapillary medium

Asphaltenes have been referred to ‘cholesterol of petroleum’ and the heaviest and the most complex components in crude oil. Asphaltene aggregation and deposition have always been a challenging issue on fluid transport in porous media for oil recovery and production processes in industries. Many experimental efforts have been done to study asphaltene aggregation and deposition from crude oil. Most of these works were carried out by coreflood tests or in bulk solution. Only several macroscopic parameters, such as pressure variation and permeability reduction caused by asphlatene deposition, were studied. Few experimental studies have been done to make the direct pore-scale visualization of dynamic asphaltene deposition behavior in capillary flow. In this study, we developed a new experimental approach to directly visualize and measure the dynamic asphaltene deposition process in capillary flow. Effects of concentration of asphaltene precipitant and flowrate were examined. The results indicate that the concentration of precipitant has a significant effect on the dynamic asphaltene deposition in capillary flow. The size of asphaltene deposit increases significantly with elapsed time and size distribution becomes wider at higher precipitant concentration. Higher concentration of asphaltene precipitant could lead to higher deposition rate. The results also indicate that higher flowrate causes faster growth of asphaltene deposits in capillary flow. Larger sizes of asphaltene deposits are formed and number of asphaltene deposits decreases with the elapsed time at higher flowrate. The results produced provide new sights into the dynamic asphaltene deposition in capillary flow at a pore scale. They could contribute to predict the growth rate of arterial asphaltene deposition in porous media and develop accurate numerical simulator for enhanced oil recovery purpose in industries.

Fractional-calculus-based formulation of the fractured wells in fractal radial composite reservoirs

This paper goes into the behavior of a vertically hydraulic fractured well in a radial composite system with complex structure. Using fractal geometry, a mathematical model of this type of reservoir is developed in terms of two important fractal parameters; i.e., fractal topological dimension (\(d_{\mathrm{f}}\)) and fractal dynamical index (\(\theta\)). Moreover, the fractional derivative approach is used to model the fluid transport in porous media and explain the dynamical properties of the system. An infinitesimally thin skin at the discontinuous boundary is considered to explain the actual behavior of radial composite reservoirs. Having applied the Laplace transformation approach and making use of a rigorous method, an analytical solution for hydraulic fractured wells is attained in a closed form. The importance of having a closed form solution and its application in analyzing the pressure-transient behavior of fractured wells in composite reservoirs are shown by investigating the effects of the reservoir parameters in seven synthetic examples.

Poiseuille‐Type Fluid Transport in Poro‐Elastic Solids at Fracture

AbstractHydraulic fracturing has led to controversial public debates about its risks due to environmental issues, such as increased seismic activity or drinking water contamination. The technique is primarily used to gain crude oil or natural gas from unconventional wells. To this end, a high pressurized fracking fluid is injected into a wellbore to fracture deep‐rock formations, leading to growing cracks, a subsequent increase of permeability in these formations and thus to an increased flow of natural gas and crude oil. To weight the risks and the perspectives of hydraulic fracturing a profound understanding of the involved processes is crucial. Numerical simulations are the most cost efficient way for the required studies, but demand a reliable model with a stable implementation.We propose a canonical minimization principle for the Biot‐type fluid transport in porous media based on only two constitutive functions, that is the free energy function $\hat{\psi}$, and a dissipation potential $\hat{\phi}$, [1]. This formulation is coupled to a phase‐field approach for fracture which characterizes an intuitive and descriptive regularization of a crack surface that converges for vanishing length‐scale parameter to a sharp crack. The crack phase‐field allows for a distinct incorporation of an extra fluid flow within cracked regimes of the solid [2]. This extra fluid flow is modeled according to Poiseuille law for laminar flow, yielding an implementation via a change of the permeability tensor, i. e., making it a function of the crack opening width, formulated itself in terms of the strain and the gradient of the crack phase field. © 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Phase‐Field Modeling of Hydraulic Fracture

AbstractHydraulically driven fracture has gained more and more research activity in the last few years, especially due to the growing interest of the petroleum industry. Key challenge for a powerful simulation of this scenario is an effective modeling and numerical implementation of the behavior of the solid skeleton and the fluid phase, the mechanical coupling between the two phases as well as the incorporation of the fracture process. Existing models for hydraulic fracturing can be found for example in [1], where the crack path is predetermined, or in [2] who use a phase field fracture model in an elastic framework, however without incorporating the fluid flow. In this work we propose a new compact model structure for the Biot‐type fluid transport in porous media at finite strains based on only two constitutive functions, that is the free energy function ψ and a dissipation potential ϕ that includes the incorporation of an additional Poiseuille‐type fluid flow in cracks. This formulation is coupled to a phase field approach for fracture and is fully variational in nature, as shown in [3]. In contrast to formulations with a sharp‐crack discontinuity, the proposed regularized approach has the main advantage of a straight‐forward modeling of complex crack patterns including branching. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Geothermal modelling of faulted metamorphic crystalline crust: a new model of the Continental Deep Drilling Site KTB (Germany)

The area of the 9.1-km-deep Continental Deep Drillhole (KTB) in Germany is used as a case study for a geothermal reservoir situated in folded and faulted metamorphic crystalline crust. The presented approach is based on the analysis of 3-D seismic reflection data combined with borehole data and hydrothermal numerical modelling. The KTB location exemplarily contains all elements that make seismic prospecting in crystalline environment often more difficult than in sedimentary units, basically complicated tectonics and fracturing and low-coherent strata. In a first step major rock units including two known nearly parallel fault zones are identified down to a depth of 12 km. These units form the basis of a gridded 3-D numerical model for investigating temperature and fluid flow. Conductive and advective heat transport takes place mainly in a metamorphic block composed of gneisses and metabasites that show considerable differences in thermal conductivity and heat production. Therefore, in a second step, the structure of this unit is investigated by seismic waveform modelling. The third step of interpretation consists of applying wavenumber filtering and log-Gabor-filtering for locating fractures. Since fracture networks are the major fluid pathways in the crystalline, we associate the fracture density distribution with distributions of relative porosity and permeability that can be calibrated by logging data and forward modelling of the temperature field. The resulting permeability distribution shows values between 10-16 and 10-19 m(-2) and does not correlate with particular rock units. Once thermohydraulic rock properties are attributed to the numerical model, the differential equations for heat and fluid transport in porous media are solved numerically based on a finite difference approach. The hydraulic potential caused by topography and a heat flux of 54 mW m(-2) were applied as boundary conditions at the top and bottom of the model. Fluid flow is generally slow and mainly occurring within the two fault zones. Thus, our model confirms the previous finding that diffusive heat transport is the dominant process at the KTB site. Fitting the observed temperature-depth profile requires a correction for palaeoclimate of about 4 K at 1 km depth. Modelled and observed temperature data fit well within 0.2 degrees C bounds. Whereas thermal conditions are suitable for geothermal energy production, hydraulic conditions are unfavourable without engineered stimulation.

Entropy generation of viscous dissipative flow in thermal non-equilibrium porous media with thermal asymmetries

The effect of thermal asymmetrical boundaries on entropy generation of viscous dissipative flow of forced convection in thermal non-equilibrium porous media is analytically studied. The two-dimensional temperature, Nusselt number and entropy generation contours are analysed comprehensively to provide insights into the underlying physical significance of the effect on entropy generation. By incorporating the effects of viscous dissipation and thermal non-equilibrium, the first-law and second-law characteristics of porous-medium flow are investigated via various pertinent parameters, i.e. heat flux ratio, effective thermal conductivity ratio, Darcy number, Biot number and averaged fluid velocity. For the case of symmetrical wall heat flux, an optimum condition with a high Nusselt number and a low entropy generation is identified at a Darcy number of 10−4, providing an ideal operating condition from the second-law aspect. This type of heat and fluid transport in porous media covers a wide range of engineering applications, involving porous insulation, packed-bed catalytic process in nuclear reactors, filtration transpiration cooling, and modelling of transport phenomena of microchannel heat sinks.

Minimization principles for the coupled problem of Darcy–Biot-type fluid transport in porous media linked to phase field modeling of fracture

This work develops new minimization and saddle point principles for the coupled problem of Darcy–Biot-type fluid transport in porous media at fracture. It shows that the quasi-static problem of elastically deforming, fluid-saturated porous media is related to a minimization principle for the evolution problem. This two-field principle determines the rate of deformation and the fluid mass flux vector. It provides a canonically compact model structure, where the stress equilibrium and the inverse Darcy's law appear as the Euler equations of a variational statement. A Legendre transformation of the dissipation potential relates the minimization principle to a characteristic three field saddle point principle, whose Euler equations determine the evolutions of deformation and fluid content as well as Darcy's law. A further geometric assumption results in modified variational principles for a simplified theory, where the fluid content is linked to the volumetric deformation. The existence of these variational principles underlines inherent symmetries of Darcy–Biot theories of porous media. This can be exploited in the numerical implementation by the construction of time- and space-discrete variational principles, which fully determine the update problems of typical time stepping schemes. Here, the proposed minimization principle for the coupled problem is advantageous with regard to a new unconstrained stable finite element design, while space discretizations of the saddle point principles are constrained by the LBB condition. The variational principles developed provide the most fundamental approach to the discretization of nonlinear fluid–structure interactions, showing symmetric systems in algebraic update procedures. They also provide an excellent starting point for extensions towards more complex problems. This is demonstrated by developing a minimization principle for a phase field description of fracture in fluid-saturated porous media. It is designed for an incorporation of alternative crack driving forces, such as a convenient criterion in terms of the effective stress. The proposed setting provides a modeling framework for the analysis of complex problems such as hydraulic fracture. This is demonstrated by a spectrum of model simulations.

Variational Phase Field Modeling of Fracture Caused by Fluid Transport in Poroelastic Solids

AbstractThe numerical modeling of failure mechanisms due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by recently developed diffusive crack modeling concepts, which are based on the introduction of a crack phase field. Such an approach is conceptually in line with gradient‐extended continuum damage models which include internal length scales. In this paper, we extend our recently outlined mechanical framework [1–3] towards the phase field modeling of fracture in the coupled problem of fluid transport in deforming porous media. Here, extremely complex crack patterns may occur due to drying or hydraulic induced fracture, the so called fracking. We develop new variational potentials for Biot‐type fluid transport in porous media at finite deformations coupled with phase field fracture. It is shown, that this complex coupled multi‐field problem is related to an intrinsic mixed variational principle for the evolution problem. This principle determines the rates of deformation, fracture phase field and fluid content along with the fluid potential. We develop a robust computational implementation of the coupled problem based on the potentials mentioned above and demonstrate its performance by the numerical simulation of complex fracture patterns. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Resistance Model for Newtonian and Power-Law Non-Newtonian Fluid Transport in Porous Media

A theoretical model based on the force balance between pressure, viscous force, and inertia force is proposed to predict the flow resistance of Newtonian and power-law non-Newtonian fluids through porous packed beds. The present model takes inertia effect into consideration, and the flow regime can be extended from Darcy flow to non-Darcy flow. It is demonstrated that the present model can predict most available experimental data well. The present results are also compared to the Ergun equation and other drag correlations.