Porous Flow Equations Research Articles

Surrogate modeling for porous flow using deep neural networks

Neural networks are a potential method to solve partial differential equations (PDEs) without considering equation discretization, linearization, and solving large sparse linear systems. Some data-driven or physics-informed methods are studied for surrogate modeling and uncertainty quantification tasks for PDE systems. Compared with data-driven methods, physics-informed methods reduce the reliance on labeled data and have better generalizations and broader applications. However, the methods have low accuracy for solving non-stationary problems with strong nonlinearities. This paper proposes a physics-informed method to solve a porous flow equation with a single source or sink as the boundary condition. The present network framework consists of two network modules. The first module uses two hidden layers with frozen weights, obtained by pre-training a simple double hidden layer network using boundary conditions and initial conditions as loss functions. The second module is used as a residual module to learn measurable label data to improve the accuracy of the solution on the nonlinear boundary. The numerical results verify the validity of the PDE solution, and the results show that the method performs better in terms of both solution and prediction than existing methods. In addition, the method allows a good inversion of the permeability from the measurable data.

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.

SPH‐based numerical treatment of the interfacial interaction of flow with porous media

SummaryIn this paper, the macroscopic equations of mass and momentum are developed and discretized based on the smoothed particle hydrodynamics (SPH) formulation for the interaction at an interface of flow with porous media. The theoretical background of flow through porous media is investigated to highlight the key constraints that should be satisfied, particularly at the interface between the porous media flow and the overlying free flow. The study aims to investigate the derivation of the porous flow equations, computation of the porosity, and treatment of the interfacial boundary layer. It addresses weak assumptions that are commonly adopted for interfacial flow simulation in particle‐based methods. As support to the theoretical analysis, a two‐dimensional weakly compressible SPH model is developed based on the proposed interfacial treatment. The equations in this model are written in terms of the intrinsic averages and in the Lagrangian form. The effect of particle volume change due to the spatial change of porosity is taken into account, and the extra stress terms in the momentum equation are approximated by using Ergun's equation and the subparticle scale model to represent the drag and turbulence effects, respectively. Four benchmark test cases covering a range of flow scenarios are simulated to examine the influence of the porous boundary on the internal, interface, and external flows. The capacity of the modified SPH model to predict velocity distributions and water surface behavior is fully examined with a focus on the flow conditions at the interfacial boundary between the overlying free flow and the underlying porous media.

Nonlinear Porous Flow Equation Based on Pore Throat Radius Sorting of Porous Media in Low Permeability Reservoirs

ABSTRACT In low-permeability reservoirs, the average pore throat radius is small and the heterogeneity is severe, and the threshold pressure gradient exists obviously. Because of the pore throat radius sorting of porous media, the threshold pressure gradients of the pore throats with different radius are different, which leads to the asynchronism of the fluid participating in flow in the pore throats with different radius. The apparent permeability of the rock is a variable value before it reaches the average permeability of the rock, which makes low-velocity nonlinear porous flow come into being. The essence of nonlinear porous flow is the process of the fluid participating in porous flow gradually according to pore throat radius from large to small in pore throats with different radius under the combined action of threshold pressure gradient and pore throat radius sorting. This is an inevitable stage before the porous media reaches Darcy flow. The porous media keeps in nonlinear porous flow stage before all the pore throats participate in flowing. Based on the pore throat radius sorting of porous media, the nonlinear porous flow equation in low-permeability reservoirs was established using unequal-diameter parallel capillary bundle model with the capillary radius commonly in accordance with normal distribution. The equation can not only reveal low-velocity nonlinear porous flow mechanism but also describe the forming process of nonlinear porous flow. Furthermore, the equation is a general equation. Both Darcy’s law and the pseudo-Darcy equation with the threshold pressure gradient in low-permeability reservoirs are the particular cases.

Evaluation of oscillation-free fluid-porous interface treatments for segregated finite volume flow solvers

The volume-averaged approach to simulate flow in porous media is often used because of its practicality and computational efficiency. Derivation of the volume-averaged porous flow equations introduces additional porous resistance terms to the momentum equation. These porous resistance terms create body force discontinuities at fluid-porous interfaces, which may lead to spurious oscillations if not accounted for properly. A variety of techniques has been proposed to treat this problem, but only recently two approaches were developed by Nordlund et al. [Improved PISO algorithms for modeling density varying flow in conjugate fluid-porous domains. Journal of Computational Physics 2016;306:199–215. http://www.sciencedirect.com/science/article/pii/S0021999115007755. http://dx.doi.org/10.1016/j.jcp.2015.11.035; . Modified Rhie–Chow/PISO algorithm for collocated variable finite porous media flow solvers. No. 40 in ECI Symposium Series; 5th International Conference on Porous Media and Their Applications in Science, Engineering and Industry; 2014], concentrating on the combination of collocated grids and segregated solvers, which are commonly used for industrial applications. In this paper, we compared these two methods for the treatment of fluid-porous interfaces: (i) the Re-Distributed Resistivity (RDR) and (ii) the Face Consistent Pressure (FCP) methods, for a wide range of process conditions and porous media parameters, covering most practical porous media flow applications. The numerical robustness and accuracy of both methods were evaluated by applying a systematic quantitative comparison procedure. Both the RDR and the FCP approaches proved to be suitable for treating discontinuities in the porous resistance terms, yielding smooth solutions around the interface and good convergence with grid refinement. The RDR method was found to be the most robust method, particularly at very low Darcy numbers and on unstructured grids, distinguishing it from the FCP approach.

A fully coupled porous flow and geomechanics model for fluid driven cracks: a peridynamics approach

A state-based non-local peridynamic formulation is presented for simulating fluid driven fractures in an arbitrary heterogeneous poroelastic medium. A recently developed peridynamic formulation of porous flow has been coupled with the existing peridynamic formulation of solid and fracture mechanics resulting in a peridynamic model that for the first time simulates poroelasticity and fluid-driven fracture propagation. This coupling is achieved by modeling the role of pore pressure on the deformation of porous media and vice versa through porosity variation with medium deformation, pore pressure and total mean stress. The poroelastic model is verified by simulating the one-dimensional consolidation of fluid saturated rock. An additional porous flow equation with material permeability dependent on fracture width is solved to simulate fluid flow in the fractured region. Finally, single fluid-driven fracture propagation with a two-dimensional plane strain assumption is simulated and verified against the corresponding classical analytical solution.

Multiscale Linear Solvers for Very Large Systems Derived from PDES

We present a novel linear solver that works well for large systems obtained from discretizing PDEs. It is robust and, for the examples we studied, the computational effort scales linearly with the number of equations. The algorithm is based on a wavelength decomposition that combines conjugate gradient, multiscaling, and iterative splitting methods into a single approach. On the surface, the algorithm is a simple preconditioned conjugate gradient with all the sophistication of the algorithm in the choice of the preconditioning matrix. The preconditioner is a very good approximate inverse of the linear operator. It is constructed from the inverse of the coarse grained linear operator and from smoothing operators that are based on an operator splitting on the fine grid. The coarse graining captures the long wavelength behavior of the inverse operator, while the smoothing operator captures the short wavelength behavior. The conjugate gradient iteration accounts for the coupling between long and short wavelengths. The coarse grained operator corresponds to a lower resolution approximation to the PDEs. While the coarse grained inverse is not known explicitly, the algorithm requires only that the preconditioner can be a applied to a vector. The coarse inverse applied to a vector can be obtained as the solution of another preconditioned conjugate gradient solver that applies the same algorithm to the smaller problem. Thus, the method is naturally recursive. The recursion ends when the matrix is sufficiently small for a solution to be obtained efficiently with a standard solver. The local feedback provided by the conjugate gradient step at every level makes the algorithm very robust. In spite of the effort required for the coarse inverse, the algorithm is efficient because the increased quality of the approximate inverse greatly reduces the number of times the preconditioner needs to be evaluated. A feature of the algorithm is that the transition between coarse grids is determined dynamically by the accuracy requirement of the conjugate gradient solver at each level. Typically, later iterations on the finer scales need fewer iterations on the coarser scales and the computational effort is proportional to N rather than N log N, where N is the number of equations. We have tested our solver on the porous flow equation. On a workstation we have solved problems on grids ranging in dimension over three orders of magnitude, from 103 to 106, and found that the linear scaling holds. The algorithm works well, even when the permeability tensor has spatial variations exceeding a factor of 109.

Rheological Study of Phenolic-Urethane-Amine Process

The flow dynamics of phenolic-urethane-amine process in a core box of an inverted "U" configuration was studied experimentally using a Laempe LI Core Shooter. The effects of reduction in area of the vents were evaluated by manual reduction of the active vent areas. In addition, experimental data on the pressure drop and friction factor for the vents were obtained as a function of the amounts of sand deposited in vent area. The experimental data are compared with the predictions of the Blake-Kozeny and Ergun porous flow equations and good agreement was found.

An Inverse Problem for the Hydraulic Properties of Porous Media

An inverse problem is formulated to determine the two coefficients in the pressure head formulation of the porous flow equation from a simple hydraulic experiment. Integral identities are derived which relate changes in the coefficients to changes in measured outputs. These identities are used to precisely define the sense in which the experimental data are able to distinguish between different porous media. It is also shown that the mapping associating input coefficient values to output data values is explicitly invertible and that there is a related output least squares problem whose solution is the solution of the inverse problem.

On the one-dimensional steady and unsteady porous flow equations

Porous flow in coarse granular media is discussed theoretically with special concern given to the variation of the flow resistance with the porosity. For steady state flow, the Navier-Stokes equation is applied as a basis for the derivations. A turbulent flow equation is suggested. Alternative derivations based on dimensional analysis and a pipe analogy, respectively, are discussed. For non-steady state flow, the derivations are based on a cylinder/sphere analogy leading to a virtual mass coefficient. For the fully turbulent flow regime, existing experimental data values of the quadratic flow resistance coefficients are presented. Moreover, a simple formula for estimation of the turbulent flow coefficient is given. Virtual mass coefficients based on existing data are presented, however, no definite conclusions can be given due to the scarce data available.

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.

The effect of temperature on the permeability of a porous material

The effect of temperature on the flow of compressible fluids through consolidated porous media has been studied experimentally. The work covers a temperature range of 255–1520 K at a pressure of 1 atm. The porous medium was aluminum oxide ceramic with a porosity range of 38.1–49.8%. The gases studied were air, nitrogen, helium and argon. In particular the effect of the ‘slip’ correction term used to explain the change in permeability in the porous flow equations was studied. This permeability correction term depends on the mean free path of the gas and would be expected to increase with temperature. Also the observed permeability should be different for each gas, due to the molecular diameter dependency of the mean free path according to kinetic theory. Both of these expectations were found consistent with the microscopic assumptions. Plots are given showing the relationship of permeability and temperature for the four gases. Helium behavior departs from that of the other three gases. Also shown is how the porous material parameters vary with temperature.