Two-phase Porous Flow Research Articles

A novel meshless method based on the virtual construction of node control domains for porous flow problems

In this paper, a novel meshless method that can handle porous flow problems with singular source terms is developed by virtually constructing node control domains. By defining a connectable node cloud, this method uses the integral of the diffusion term and generalized finite-difference operators to derive overdetermined equations of the node control volumes. An empirical method of calculating reliable node control volumes and a triangulation-based method to determine the connectable point cloud are developed. The method focuses only on the volume of the node control domain rather than the specific shape, so the construction of node control domains is virtual and does not increase the computational cost. To our knowledge, this is the first study to construct node control volumes in the meshless framework, termed the node control domain-based meshless method (NCDMM), which can also be regarded as an extended finite-volume method (EFVM). Taking two-phase porous flow problems as an example, discrete NCDMM schemes satisfying local mass conservation are derived by integrating the generalized finite-difference schemes of governing equations on each node control domain. Finally, commonly used low-order finite-volume method (FVM)-based nonlinear solvers for various porous flow models can be directly employed in the proposed NCDMM, significantly facilitating general-purpose application. Theoretically, the proposed NCDMM has the advantages of previous meshless methods for discretizing computational domains with complex geometries, as well as the advantages of traditional low-order FVMs for stably handling a variety of porous flow problems with local mass conservation. Four numerical cases are implemented to test the computational accuracy, efficiency, convergence, and good adaptability to the calculation domain with complex geometry and various boundary conditions.

Investigations and Prevention Strategies on the Fracturing Water Pollution Triggering Permeability and Production Decrease in Underground Gas Reservoirs

The fracturing water injected into the underground gas reservoirs for development purposes has polluted the reservoirs, triggering a decrease in reservoir permeability and gas production. Here, we quantitatively investigate and provide preventions for the fracturing water pollution in the underground gas reservoir. We study the effects of fracturing water pollution on reservoir permeability with core experiments. According to the core experiments, we constructed an area-divided two-phase porous flow model to study the production of the underground gas reservoir considering fracturing water pollution. The simulation results are in good agreement with the field development data. It reveals that in the early, mid-term and late development, respectively, the fracturing water pollution accounts for 88%, 80% and 45% of the decline in permeability and production of the underground gas reservoir. In terms of the prevention of fracturing water pollution, reservoirs with an initial permeability over 0.20 mD are preferably produced by natural energy rather than fracturing. Once using the fracturing water, we suggest applying the propping agent with a large particle radius to reduce the pollution from the solid solute and adding the clay stabilizer and the surfactant to the fracturing water to reduce the pollution from the water solvent.

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.

An upwind generalized finite difference method for meshless solution of two-phase porous flow equations

This paper makes the first attempt to apply newly developed upwind GFDM for the meshless solution of two-phase porous flow equations. In the presented method, node cloud is used to flexibly discretize the computational domain, instead of complicated mesh generation. Combining with moving least square approximation and local Taylor expansion, spatial derivatives of oil-phase pressure at a node are approximated by generalized difference operators in the local influence domain of the node. By introducing the first-order upwind scheme of phase relative permeability, and combining the discrete boundary conditions, fully-implicit GFDM-based nonlinear discrete equations of the immiscible two-phase porous flow are obtained and solved by the nonlinear solver based on the Newton iteration method with the automatic differentiation, to avoid the additional computational cost and possible computational instability caused by sequentially coupled scheme. Two numerical examples are implemented to test the computational performances of the presented method. Detailed error analysis finds the two sources of the calculation error, and points out the significant effect of the symmetry or uniformity of the node allocation in the node influence domain on the accuracy of generalized difference operators, and the radius of the node influence domain should be small to achieve high calculation accuracy, which is a significant difference between the studied hyperbolic two-phase porous flow problem and the elliptic problems when GFDM is applied. In all, the upwind GFDM with the fully implicit nonlinear solver and related analysis about computational performances given in this work may provide a critical reference for developing a general-purpose meshless numerical simulator for porous flow problems.

Multi-fidelity electrochemical modeling of thermally activated battery cells

Thermally activated batteries undergo a series of coupled physical changes during activation that influence battery performance. These processes include energetic material burning, heat transfer, electrolyte phase change, capillary-driven two-phase porous flow, ion transport, electrochemical reactions, and electrical transport. Several of these processes are strongly coupled and have a significant effect on battery performance, but others have minimal impact or may be suitably represented by reduced-order models. Assessing the relative importance of these phenomena must be based on comparisons to a high-fidelity model including all known processes.In this work, we first present and demonstrate a high-fidelity, multi-physics model of electrochemical performance. This novel multi-physics model enables predictions of how competing physical processes affect battery performance and provides unique insights into the difficult-to-measure processes that happen during battery activation. We introduce four categories of model fidelity that include different physical simplifications, assumptions, and reduced-order models to decouple or remove costly elements of the simulation. Using this approach, we show an order-of-magnitude reduction in computational cost while preserving all design-relevant quantities of interest within 5 percent. The validity of this approach and these model reductions is demonstrated by comparison between results from the full fidelity model and the different reduced models.

Numerical investigation of the POD reduced-order model for fast predictions of two-phase flows in porous media

PurposeThis paper aims to present an efficient IMPES algorithm based on a global model order reduction method, proper orthogonal decomposition (POD), to achieve the fast solution and prediction of two-phase flows in porous media.Design/methodology/approachThe key point of the proposed algorithm is to establish an accurate POD reduced-order model (ROM) for two-phase porous flows. To this end, two projection methods including projecting the original governing equations (Method I) and projecting the discrete form of original governing equations (Method II) are respectively applied to construct the POD-ROM, and their distinctions are compared and analyzed in detail. It is found the POD-ROM established by Method I is inapplicable to multiphase porous flows due to its failed introduction of fluid saturation and permeability that locate on the edge of grid cell, which would lead to unphysical results.FindingsBy using Method II, an efficient IMPES algorithm that can substantially speed up the simulation of two-phase porous flows is developed based on the POD-ROM. The computational efficiency and numerical accuracy of the proposed algorithm are validated through three numerical examples, and simulation results illustrate that the proposed algorithm displays satisfactory computational speed-up (one to two orders of magnitude) without sacrificing numerical accuracy obviously when comparing to the standard IMPES algorithm that without any acceleration technique. In addition, the determination of POD modes number, the relative errors of wetting phase pressure and saturation, and the influence of POD modes number on the overall performances of the proposed algorithm, are investigated.Originality/value1. Two projection methods are applied to establish the POD-ROM for two-phase porous flows and their distinctions are analyzed. The reason why POD-ROM is difficult to be applied to multiphase porous flows is clarified firstly in this study. 2. A highly efficient IMPES algorithm based on the POD-ROM is proposed to accelerate the simulation of two-phase porous flows. 3. Satisfactory computational speed-up (one to two orders of magnitude) and prediction accuracy of the proposed algorithm are observed under different conditions.

Control Volume Approximation of Degenerate Two-Phase Porous Flows

Implicit Euler approximations of the equations governing the porous flow of two immiscible incompressible fluids are shown to be the Euler--Lagrange equations of a convex function. Tools from convex analysis are then used to develop robust fully discrete algorithms for their numerical approximation. Existence and uniqueness of solutions to control volume approximations are established.

On the effect of boundaries in two-phase porous flow

In this paper we study a model of an interface between two fluids in a porous medium. For this model we prove several local and global well-posedness results and study some of its qualitative properties. We also provide numerics.

Impact of structured heterogeneities on reactive two-phase porous flow

Two-phase flow through heterogeneous media leads to scale-free distributions of irregularly shaped pockets of one fluid trapped within the other. Although reactions within these fluids are often modeled at the homogeneous continuum scale, there exists no current framework for upscaling from the pore scale that accounts for the complex and scale-free geometry of the bubbles. In this paper, we apply a linear-kinetics reaction-diffusion model to characterize the steady-state chemical environment inside the irregular pockets. Using a combination of theory and invasion-percolation simulations, we derive scaling laws describing the distribution of diffusion times within bubbles. We show that chemical concentrations within the bubbles are determined by the Laplace transform of the entire distribution of diffusion times from each location. This serves as a means to compute average concentrations of reactant within a bubble of unique geometry and size. Furthermore, the overall system size imposes upper bounds on the distribution of bubble sizes, thereby imposing a system-size dependence on the statistics and average concentrations. These conclusions have profound implications for continuum models of porous reactive flow, where kinetic and equilibrium parameters are often chosen from laboratory measurements made at centimeter scales.

Simulation of liquid water breakthrough in a nanotomography reconstruction of a carbon paper gas-diffusion layer

This study reports the feasibility of newly simulating liquid water intrusion into the porous gas diffusion layer (GDL) of a polymer electrolyte fuel cell (PEFC) using X-ray nano-tomography and two-phase lattice Boltzmann (LB) simulation. A digital 3D model of the GDL is reconstructed using X-ray nanotomography while two-phase porous flow is simulated at two different levels of surface wettability by applying a newly-developed numerical LB model. The results show liquid infiltration in a hydrophobic GDL is comparatively lower (pore saturation of 0.11 to 0.90) than that for a hydrophilic GDL (pore saturation of 0.36 to 0.96) over the liquid intrusion range of 1 kPa . 100 kPa. Visualisation of simulated results in three dimensions reveal dissimilar liquid infiltration characteristics for the two levels of wettability considered, yet also reveal a general breakthrough of liquid water at a pressure of 10 kPa due to specific structural features of the GDL.

One-dimensional two-phase porous flow equations

We study initial-boundary value problems for one-dimensional two-phase porous flow equations\(\left\{ \begin{array}{l} V_x = 0, V = - k(x, s)p_x - a(x, s) \\ m(x)s_t = \left\{ {r(x, s)s_x + b(x, s) + Vl(s)} \right\}_x . \\ \end{array} \right.\) The existence and asymptotic behavior of weak solutions are treated, with suitable coefficients and data. The proof is based on regularization and elementary energy inequalities.

Some properties of the generalized solutions of one-dimensional two-phase porous flow problems

In the context of the one-dimensional approximation, some qualitative results are obtained on the behaviour of the generalized solutions of the problem of the displacement of a wetting by a non-wetting fluid in a homogeneous porous medium, where there is monotonic dependence on the initial data, an over-all fluid consumption, and a pressure gradient. The asymptotic behaviour at long and short times is studied.

A solution of the problem of maximizing oil output in conditions of two-phase porous flow

The problem of maximizing oil output in conditions of two-phase porous flow is stated as an optimal control problem. Aspects of the numerical solution of the problem by successive linearization are discussed. An example is given.

A front tracking method applied to Burgers' equation and two-phase porous flow

A method is presented that is capable of following discontinuities in the solution of hyperbolic partial differential equations. At every time step for each cell in the neighborhood of the discontinuity, the fraction of the cell lying behind the discontinuity curve is updated. From this data the front is reconstructed. The method is applied to three scalar differential equations: inviscid Burgers' equation, the Buckley-Leverett equation for immiscible porous flow, and the equation for two-phase miscible flow in a porous medium.

BOUNDARY VALUE PROBLEMS FOR SYSTEMS OF EQUATIONS OF TWO-PHASE POROUS FLOW TYPE; STATEMENT OF THE PROBLEMS, QUESTIONS OF SOLVABILITY, JUSTIFICATION OF APPROXIMATE METHODS

A system of two second order quasilinear equations (one elliptic and one parabolic) is investigated. Such a system is formed by the equations of two-phase porous flow. Boundary value problems are considered for the system, and existence, uniqueness, and stability theorems are proved for the solutions of the problems in the nondegenerate plane case. A justification is given for some approximate methods of constructing the solutions.Bibliography: 10 titles.