Flow Problems In Porous Media Research Articles

Adaptive space-time model order reduction with dual-weighted residual (MORe DWR) error control for poroelasticity

In this work, the space-time MORe DWR (Model Order Reduction with Dual-Weighted Residual error estimates) framework is extended and further developed for single-phase flow problems in porous media. Specifically, our problem statement is the Biot system which consists of vector-valued displacements (geomechanics) coupled to a Darcy flow pressure equation. The MORe DWR method introduces a goal-oriented adaptive incremental proper orthogonal decomposition (POD) based-reduced-order model (ROM). The error in the reduced goal functional is estimated during the simulation, and the POD basis is enriched on-the-fly if the estimate exceeds a given threshold. This results in a reduction of the total number of full-order-model solves for the simulation of the porous medium, a robust estimation of the quantity of interest and well-suited reduced bases for the problem at hand. We apply a space-time Galerkin discretization with Taylor-Hood elements in space and a discontinuous Galerkin method with piecewise constant functions in time. The latter is well-known to be similar to the backward Euler scheme. We demonstrate the efficiency of our method on the well-known two-dimensional Mandel benchmark and a three-dimensional footing problem.

Importance Weighting in Hybrid Iterative Ensemble Smoothers for Data Assimilation

AbstractBecause it is generally impossible to completely characterize the uncertainty in complex model variables after assimilation of data, it is common to approximate the uncertainty by sampling from approximations of the posterior distribution for model variables. When minimization methods are used for the sampling, the weights on each of the samples depend on the magnitude of the data mismatch at the critical points and on the Jacobian of the transformation from the prior density to the sample proposal density. For standard iterative ensemble smoothers, the Jacobian is identical for all samples, and the weights depend only on the data mismatch. In this paper, a hybrid data assimilation method is proposed which makes it possible for each ensemble member to have a distinct Jacobian and for the approximation to the posterior density to be multimodal. For the proposed hybrid iterative ensemble smoother, it is necessary that a part of the mapping from the prior Gaussian random variable to the data be analytic. Examples might include analytic transformation from a latent Gaussian random variable to permeability followed by a black-box transformation from permeability to state variables in porous media flow, or a Gaussian hierarchical model for variables followed by a similar black-box transformation from permeability to state variables. In this paper, the application of weighting to both hybrid and standard iterative ensemble smoothers is investigated using a two-dimensional, two-phase flow problem in porous media with various degrees of nonlinearity. As expected, the weights in a standard iterative ensemble smoother become degenerate for problems with large amounts of data. In the examples, however, the weights for the hybrid iterative ensemble smoother were useful for improving forecast reliability.

A Dimension-Reduced Line-Element Method to Model Unsaturated Seepage Flow in Porous Media

Contrary to the continuum hypothesis, which averages water flow across the entire domain, including both grains and pores, the line-element model concentrates unsaturated flow in the pore space in the intermediate region of horizontal and vertical channels. The flux equivalent principle is used to deduce the equivalent unsaturated hydraulic conductivity, the flow velocity and the continuity equations. It is found that the relative hydraulic conductivities derived from the line-element model and the continuum model are identical. The continuity equations in the two models are also similar, except that the coefficient in the water content term is half that in the line-element model. Thus, the unsaturated flow problem in porous media is transformed into a one-dimensional problem. A dimension-reduced finite line-element method is proposed that includes a complementary algorithm for Signorini’s-type boundary conditions involving the seepage-face boundary and the infiltration boundary. The validity of the proposed model is then proved by good agreement with analytical, experimental and simulated results for one-dimensional infiltration in a vertical soil column, unsaturated flow in a sand flume with drainage tunnels, and transient unsaturated flow water-table recharge in a soil slab, respectively. In general, the proposed method has good computational efficiency, especially for smaller mesh sizes and short time intervals.

Modified Picard with multigrid method for two‐phase flow problems in rigid porous media

AbstractTwo‐phase flow problems in porous media can be found in several areas, such as Geomechanics, Hydrogeology, Engineering and Biomedicine, for example. Typically, these processes are mathematically modeled by a highly nonlinear system of coupled partial differential equations. The nonlinearity of the system makes the design and implementation of robust numerical solvers a challenging task. In this work we consider the flow of two immiscible and incompressible fluids within a non‐deformable porous medium. A mixed pressure‐saturation formulation is adopted, allowing the transition from the unsaturated to saturated zones and maintaining numerical mass conservation. A cell‐centered finite volume method and an implicit Euler scheme are considered for the spatial and time discretization of the problem. In this work, we propose a solution method for two‐phase flow problems which is based on the combination of the modified Picard linearization method and a very simple cell‐centered multigrid algorithm that performs efficiently even for heterogeneous random media. This is shown in the numerical experiments, where two test problems are presented to demonstrate the robustness of the proposed solver.

Nonlinear parallel-in-time simulations of multiphase flow in porous media

As the number of processors on supercomputers has increased dramatically, there is a growing interest in developing scalable algorithms with a high degree of parallelism for extreme-scale reservoir simulation. However, traditional simulators and algorithms for such nonlinear problems are usually based on the family of time-marching methods, where parallelization is restricted to the spatial dimension only. In this paper, we propose a family of parallel-in-time (PinT) reservoir algorithms for solving multiphase flow problems in porous media to fully exploit the parallelism of supercomputers, in which the design of efficient nonlinear and linear space-time algorithms plays an essential role. More precisely, we first introduce a space-time mixed finite element method for the fully-implicit discretization. The nonlinear system arising at each parallel-in-time step is solved by a variant of bound-preserving Newton methods with a nonlinear elimination preconditioner, where the corresponding linear system is solved by the space-time restricted additive Schwarz (stRAS) method. Since the straightforward extension of one-level stRAS to multilevel does not work because of the pollution effects from the temporal direction, we accordingly present a nonstandard V-cycle multilevel stRAS method with a pollution-removing strategy to accelerate the convergence and improve the robustness. Numerical experiments are presented to demonstrate that the aforementioned parallel-in-time algorithm can not only achieve a high degree of parallelism in accurately resolving reservoir transport features in a heterogeneous medium, but also successfully circumvent the convergence and scalability issues associated with the constraint of large time steps and the boundedness requirement of the solution.

A Dimension-Reduced Line Element Method for 3D Transient Free Surface Flow in Porous Media

In order to reduce the numerical difficulty of the 3D transient free surface flow problems in porous media, a line element method is proposed by dimension reduction. Different from the classical continuum-based methods, homogeneous permeable pores in the control volume are conceptualized by a 3D orthogonal network of tubes. To obtain the same hydraulic solution with the continuum model, the equivalent formulas of flow velocity, continuity equation and transient free surface boundary are derivable from the principle of flow balance. In the solution space of transient free surface flow, the 3D problem is transformed into 1D condition, and then a finite element algorithm is simply deduced. The greatest advantage of the line element method is line integration instead of volume/surface integration, which has dramatically decreased the integration difficulty across the jump free surface. Through the analysis of transient free surface flow in the unconfined aquifer, trapezoidal dam, sand flume and wells, the transient free surface locations predicted from the proposed line element method generally agree well with the analytical, experimental and other numerical data in the available literatures, the numerical efficiency can also be well guaranteed. Furthermore, the hydraulic anisotropy has significant effect on the evolution of free surface locations and the shape of depression cones in spatial. The line element method can be expanded to model the 3D unsaturated seepage flow, two-phase flow and thermos problems in porous media because of the similarity between the similarity of Darcy’s law, Buckingham Law and Fourier’s law.

Non-Overlapping Schwarz Waveform-Relaxation for Nonlinear Advection-Diffusion Equations

Nonlinear advection-diffusion equations often arise in the modeling of transport processes. We propose for these equations a non-overlapping domain decomposition algorithm of Schwarz waveform-relaxation (SWR) type. It relies on nonlinear zeroth-order (or Robin) transmission conditions between the sub-domains that ensure the continuity of the converged solution and of its normal flux across the interface. We prove existence of unique iterative solutions and the convergence of the algorithm. We then present a numerical discretization for solving the SWR problems using a forward Euler discretization in time and a finite volume method in space, including a local Newton iteration for solving the nonlinear transmission conditions. Our discrete algorithm is asymptotic preserving, i.e., robust in the vanishing viscosity limit. Finally, we present numerical results that confirm the theoretical findings, in particular the convergence of the algorithm. Moreover, we show that the SWR algorithm can be successfully applied to two-phase flow problems in porous media as paradigms for evolution equations with strongly nonlinear advective and diffusive fluxes.

Wasserstein model reduction approach for parametrized flow problems in porous media

The aim of this work is to build a reduced order model for parametrized porous media equations. The main challenge of this type of problems is that the Kolmogorov width of the solution manifold typically decays quite slowly and thus makes usual linear model order reduction methods inappropriate. In this work, we investigate an adaptation of the methodology proposed in [Ehrlacher et al., Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces, ESAIM: Mathematical Modelling and Numerical Analysis (2020)], based on the use of Wasserstein barycenters [Agueh & Carlier, Barycenters in the Wasserstein Space, SIAM Journal on Mathematical Analysis (2011)], to the case of non-conservative problems. Numerical examples in one-dimensional test cases illustrate the advantages and limitations of this approach and suggest further research directions that we intend to explore in the future.

A Novel Space-Time Marching Method for Solving Linear and Nonlinear Transient Problems

In this study, a novel space-time (ST) marching method is presented to solve linear and nonlinear transient flow problems in porous media. The method divides the ST domain into subdomains along the time axis. The solutions are approximated using ST polyharmonic radial polynomial basis functions (RPBFs) in the ST computational domain. In order to proceed along the time axis, we use the numerical solution at the current timespan of the two ST subdomains in the computational domain as the initial conditions of the next stage. The fictitious time integration method (FTIM) is subsequently employed to solve the nonlinear equations. The novelty of the proposed method is attributed to the division of the ST domain along the time axis into subdomains such that the dense and ill-conditioned matrices caused by the excessive number of boundary and interior points and the large ST radial distances can be avoided. The results demonstrate that the proposed method achieves a high accuracy in solving linear and nonlinear transient problems. Compared to the conventional time marching and ST methods, the proposed meshless approach provides more accurate solutions and reduces error accumulation.

Iterative Methods with Nonconforming Time Grids for Nonlinear Flow Problems in Porous Media

Iterative Methods with Nonconforming Time Grids for Nonlinear Flow Problems in Porous Media

Generalized finite difference method based meshless analysis for coupled two-phase porous flow and geomechanics

This paper applies GFDM to analyze coupled two-phase flow and geomechanics for the first time, to our knowledge, is also the first meshless solver for two-phase fluid-solid coupling problems in porous media. Firstly, the implicit GFDM-based discrete schemes of the hyperbolic two-phase flow equation and elliptic stress equilibrium equation coupled by Biot's poroelastic theory are derived. Then the nonlinear solver based on Newton's method is used to solve the fluid pressure, water saturation, and displacement at each node simultaneously. Two technical details are found critical to the implementation of this method. The one is using different radii of the node influence domain to discretize the flow equation and the stress equilibrium equation respectively, and the other one is the selection criterion of the two radii of the node influence domain, which improves the implementation flexibility of the multi-physics coupling calculation, reduces the dissipation error of the calculation results of the convection-dominated water saturation distribution, and ensure a low computational cost. Several numerical test cases are implemented to analyze the computational efficiency and accuracy, and illustrate that GFDM can achieve good computational performance in case of regular domains, irregular domains, and different point clouds.Overall, this work may provide an important reference for building a GFDM-based general-purpose numerical simulator for multi-physics flow problems in porous media.

Homogenized lattice Boltzmann model for simulating multi-phase flows in heterogeneous porous media

A homogenization approach for the simulation of multi-phase flows in heterogeneous porous media is presented. It is based on the lattice Boltzmann method and combines the grayscale with the multi-component Shan–Chen method. Thus, it mimics fluid–fluid and solid–fluid interactions also within pores that are smaller than the numerical discretization. The model is successfully tested for a broad variety of single- and two-phase flow problems. Additionally, its application to multi-scale and multi-phase flow problems in porous media is demonstrated using the electrolyte filling process of realistic 3D lithium-ion battery electrode microstructures as an example. The approach presented here shows advantages over comparable methods from literature. The interfacial tension and wetting conditions are independent and not affected by the homogenization. Moreover, all physical properties studied here are continuous even across interfaces of porous media. The method is consistent with the original multi-component Shan–Chen method (MCSC). It is as stable as the MCSC, easy to implement, and can be applied to many research fields, especially where multi-phase fluid flow occurs in heterogeneous and multi-scale porous media.

Existence and convergence of a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media

Abstract This paper presents and analyzes a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media. We use a first-order time extrapolation, which allows us to solve the equations implicitly and sequentially. We show that the discrete problem is well posed, and obtain a priori error estimates. Our numerical results validate the theoretical results, i.e., the algorithm converges with first order.

Generalized finite difference method (GFDM) based analysis for subsurface flow problems in anisotropic formation

This paper applies the meshless generalized finite difference method (GFDM) to analyze the subsurface flow problem in anisotropic formations for the first time, and develops the treatment methods of anisotropic permeability tensor in continuous function and discrete distribution expressions respectively. In particular, in case of the common discrete distribution expression of anisotropic permeability tensor in reservoir simulation, this paper suggests combining general difference operators and harmonic average scheme to simply and efficiently obtain the meshless discrete scheme of the anisotropic flow equation. Three numerical examples of steady and transient flow in the rectangular computational domain, transient flow with complex boundary, and steady flow with strongly discontinuous anisotropic permeability are used to verify the high accuracy and good convergence of GFDM to handle anisotropic flow problems. Compared with the traditional mesh-based reservoir numerical simulation method that replies on laborious gridding to discretize the reservoir domain and complicated multi-point flux approximation (MPFA) to handle discrete distribution anisotropic permeability tensor, the meshless GFDM can be more practically applied to the anisotropic flow problems in the reservoir domain with complex geometry and complex boundary conditions. Besides, it is found that the radius of the influence domain has little impact on the accuracy of steady flow. For the transient flow, it generally holds that the larger the radius of the influence domain, the lower the calculation accuracy in case of Cartesian collocation. In sum, this work provides an efficient and simple meshless solver to handle anisotropic flow problems in porous media under GFDM framework, which reveals the great application potential of GFDM in reservoir numerical simulation.

Determination of the spatial distribution of wetting in the pore networks of rocks

HypothesisThe macroscopic movement of subsurface fluids involved in CO2 storage, groundwater, and petroleum engineering applications is controlled by interfacial forces in the pores of rocks. Recent advances in modelling these systems has arisen from approaches simulating flow through a digital representation of the complex pore structure. However, further progress is limited by difficulties in characterising the spatial distribution of the wetting state within the pore structure. In this work, we show how observations of the fluid coverage of mineral surfaces within the pores of rocks can be used as the basis for a quantitative 3D characterisation of heterogeneous wetting states throughout rock pore structures. ExperimentsWe demonstrate the approach with water–oil fluid pairs on rocks with distinct lithologies (sandstone and carbonate) and wetting states (hydrophilic, intermediate wetting, and heterogeneously wetting). FindingsFluid surface coverage the within rock pores is a robust signal of the wetting state across varying rock types and wetting states. The wetting state can be quantified and the resulting 3D maps can be used as a deterministic input to pore scale models. These may be applied to multiphase flow problems in porous media ranging from soil science to fuel cells.

A Modeling Study of the Productivity of Horizontal Wells in Hydrocarbon-Bearing Reservoirs: Effects of Fracturing Interference

Commercial production from hydrocarbon-bearing reservoirs with low permeability usually requires the use of horizontal well and hydraulic fracturing for the improvement of the fluid diffusivity in the matrix. The hydraulic fracturing process involves the injection of viscous fluid for fracture initiation and propagation, which alters the poroelastic behaviors in the formation and causes fracturing interference. Previous modeling studies usually focused on the effect of fracturing interference on the multicluster fracture geometry, while the related productivity of horizontal wells is not well studied. This study presents a modeling workflow that utilizes abundant field data including petrophysical, geomechanical, and hydraulic fracturing data. It is used for the quantification of fracturing interference and its correlation with horizontal well productivity. It involves finite element and finite difference methods in the numeralization of the fracture propagation mechanism and porous media flow problems. Planar multistage fractures and their resultant horizontal productivity are quantified through the modeling workflow. Results show that the smaller numbers of clusters per stage, closer stage spacings, and lower fracturing fluid injection rates facilitate even growth of fractures in clusters and stages and reduce fracturing interference. Fracturing modeling results are generally correlated with productivity modeling results, while scenarios with stronger fracturing interference and greater stimulation volume/area can still yield better productivity. This study establishes the quantitative correlation between fracturing interference and horizontal well productivity. It provides insights into the prediction of horizontal well productivity based on fracturing design parameters.

BDDC for MHFEM discretization of unsteady two-phase flow in porous media

This work deals with the application of the Balancing Domain Decomposition by Constrains (BDDC) method to unsteady two-phase flow problems in porous media. We briefly describe the spatial discretization of the problem which is based on the mixed-hybrid finite element method (MHFEM) and semi-implicit time discretization. Then, we describe the BDDC method, in detail discuss the differences between the symmetric and nonsymmetric cases, and present the necessary modifications of the algorithm for the more complicated nonsymmetric case. We describe the parallel implementation of the method and highlight the critical steps of the algorithm that affect the performance and scalability. The parallel implementation is then tested on benchmark problems in 2D and 3D and its efficiency is investigated on various meshes. The numerical results indicate that the method preserves good computational efficiency for increasing number of processes and, therefore, allows solving problems on very fine meshes. In the case of unsteady problem, additional speed-up is achieved using the information from previous time steps for the solution in the current time step.

A finite element method for degenerate two-phase flow in porous media. Part II: Convergence

Abstract Convergence of a finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly the wetting phase pressure and saturation, which are the primary unknowns. Well-posedness is obtained in [J. Numer. Math., 29(2), 2021]. Theoretical convergence is proved via a compactness argument. The numerical phase saturation converges strongly to a weak solution in L 2 in space and in time whereas the numerical phase pressures converge strongly to weak solutions in L 2 in space almost everywhere in time. The proof is not straightforward because of the degeneracy of the phase mobilities and the unboundedness of the derivative of the capillary pressure.

A residual-driven adaptive Gaussian mixture approximation for Bayesian inverse problems

In this article, we develop a residual-driven adaptive Gaussian mixture approximation (RD-AGMA) for Bayesian inverse problems. The posterior distribution is often non-Gaussian in practical Bayesian inference. To obtain a good approximation of the posterior, we provide the adaptive Gaussian mixture approximation (GMA) based on a residual. For GMA, the clustering of ensemble samples provides the predictor of means, covariances and weights by smoothed expectation–maximization (SmEM). SmEM can overcome the singularity of covariance matrix for small ensemble size. Then the parameters of GMA are updated by an iterative ensemble smoother (IES). To enhance clustering efficiency, the ensemble samples for the clustering are updated by IES as well. Since the goal of inverse problems is to minimize the residual between the observation data and the model response, the adaptive GMA of the posterior is constructed through a residual threshold. The mixture components with large residuals will be discarded in the adaptive procedure. When the prior is incorporated into the likelihood model, small residuals can drive the AGMA close to the true posterior. In the proposed method, a large number of samples can be efficiently drawn from the posterior distribution of GMA. A few numerical examples are presented to demonstrate the efficacy of RD-AGMA with applications in multimodal inversion and channel identification for subsurface flow problems in porous media.

Adaptive modelling of variably saturated seepage problems

Summary In this article, we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy–Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.