non-Darcy Flow In Porous Media Research Articles

Homogenization of non-Darcy flow in porous media containing spheroidal impervious inclusions

The aim of the present study is to determine the nonlinear filtration law of a porous medium containing spheroidal impervious inclusions through a homogenization framework. At the local scale, it is assumed that the fluid flow through the porous solid surrounding the inclusions obeys the Forchheimer equation. The macroscopic law is derived in the field of nonlinear variational approach applied to some representative cell, namely a porous spheroid containing a single centered spheroidal inclusion confocal to the outer. Both a static and kinematical approach are developed to bound the overall filtration properties of the heterogeneous solid. Appropriate trial fields are considered to determine closed-form estimates and are obtained from the solution of a linear problem replacing the Forchheimer law in the porous solid at the microscopic scale by the Darcy law. The analytic estimates are calculated through some approximations and lead to accurate solutions of the nonlinear problems. Finally, the analytic models are compared with FFT solutions in the case of a regular array of spheroidal impervious inclusions.

New correlations for non-Darcy flow in porous media

New correlations for non-Darcy flow in porous media

Adoption of Homotopy Perturbation Method (HPM) for non-Darcy Flow in Porous Media

For non-Darcy flow in coarse porous media, an analytical solution based on the Homotopy Perturbation Method (HPM) is proposed. Subsurface water profile data of a laboratory model are used for six different inlet discharges in both rounded and crushed coarse porous media types. The model slope is S = 0.00001 close to the horizon. Different upstream and downstream water level boundary conditions are considered. The results of the analytical solution of non-Darcy flow by the HPM method are compared with experimental data. The normal objective function (NOF) is used for better comparison between the results of analytical solutions and experimental data. Results depict that the HPM method provides acceptable solutions in both rounded and crushed media types. The analytical results of flow rates q = 26.25 lit/s with NOF of 0.000294586 percent in rounded porous media and q = 30 lit/s with NOF of 0.00028660 percent in crushed one are the most consistent with the experimental data. The proposed HPM solution performs very well, particularly at higher flow rates, in both rounded and crushed porous media.

Non-Darcy flow models in porous media via Atangana-Baleanu derivative

Abstract Two fractional Swartzendruber models applying the Atangana-Baleanu (A-B) derivative are proposed to describe non-Darcy flow in porous media. The analytical solutions of models are obtained by using Laplace transform. Fitting curves with the experimental data display the proposed models are suitable to describe non-Darcy flow problems in low and high permeability media. In addition, sensitivity analysis is performed to clarify the influence of relevant parameters on the A-B Swartzendruber models. In the light of the two-scale thermodynamics, the physical explanations of A-B Swartzendruber model I and II are revealed. The proposed A-B Swartzendruber models connect fluid flow of different scales, and provide a unified description of non-Darcy flow in porous media with low and high permeability.

The electroviscous flow of non-Newtonian fluids in microtubes and implications for nonlinear flow in porous media

This paper aims to interpret the low-velocity nonlinear flow occurring in low-permeability reservoirs based on the theories of electrokinetic transport and non-Newtonian rheology of fluids. To achieve this end, we simulate the steady-state electroviscous flow of Bingham-Papanastasiou (BP) fluids in circular microtubes by simultaneously solving the Poisson-Boltzmann and the modified Navier-Stokes equations. The induced electrical field strength Es, velocity profiles, and the transport capacity of the non-Newtonian fluid under the effects of various factors (such as capillary radius R, zeta potential ζ, yield stress τ0, and stress growth index m) were examined. The results show that the generated Es of the BP fluid is highly affected by the fluid rheology, which is quite different from that of the Newtonian liquid. The velocity profiles become lower and flatter as m or τ0 increases, and this is more remarkable in smaller microtubes. The apparent viscosity of non-Newtonian fluid declines monotonically with increasing c∞, yet non-monotonically with R, m, τ0, and ζ. In addition, the low-velocity nonlinear flow in microtubes can be successfully captured when considering the electrokinetic flow of the non-Newtonian fluid rheology. While for the Newtonian fluid, only involving the electroviscous effect fails to generate the nonlinear flow behavior. The contributions of electrokinetic parameters versus rheological properties to the degree of flow nonlinearity are also discussed. The impact of electrokinetic parameters (ζ, c∞) on the flow characteristics is significant at high-pressure gradients and becomes trivial when the pressure gradient is relatively low. In contrast, the fluid rheological parameters (m, τ0) greatly determine the magnitude of the flow nonlinearity occurring at the low-pressure gradients. In sum, the electroviscous flow of BP fluids in microchannels provides a possible explanation of the low-velocity non-Darcy flow in porous media.

Two-level Galerkin Mixed Finite Element Method for Darcy--Forchheimer Model in Porous Media

A two-level mixed finite element method is developed and analyzed to solve the Darcy--Forchheimer equation modeling non-Darcy flows in porous media. Instead of solving a large nonlinear system of e...

The effect of pore structure on non-Darcy flow in porous media using the lattice Boltzmann method

Non-Darcy flows associated with high Reynolds numbers often occur in the near-wellbore regions of gas reservoirs or hydraulic fractures and thus should not be ignored. However, investigating non-Darcy flow in these porous rocks through laboratory experiments is always expensive and time-consuming. As such, this article sought an alternative method, and a lattice Boltzmann study of non-Darcy flow in various porous models was performed. The applicability of two non-Darcy correlations in porous media and the effect of pore structure on non-Darcy flow were examined. In addition, the reasons for the deviation from the linear Darcy flow and different flow patterns related to inertial effects of the fluid were also studied. The results showed that the characterization of non-Darcy flow in porous media with the cubic law can only be valid in a narrow range of Reynolds number beyond the Darcy regime, outside of which the strong inertia-dominated flow yields to the quadratic correction. On the whole, representing the non-Darcy flows using the quadratic correction is acceptable, especially for porous media with a higher complexity. The features of non-Darcy flow greatly depend on the pore structure of a porous medium, and more heterogeneous pore models always have a faster cessation for Darcy flow and a higher β factor. Furthermore, for simple porous media a small amount of parameters may be adequate for the prediction of the β factor; while the correlations involving more parameters would be needed to determine the β factor for more intricate porous models, although such correlations may not be widely used in various industries. Besides, the non-Darcy flow that occurs in porous media is collectively controlled by different mechanisms. At elevated velocities, the inertial core effect in a large channel will lead the flow to be more homogeneous and less tortuous, while in porous models with complicated pore space, the steady eddy and reversal flow resulting from drag force will make the flow paths more tortuous. As such, it is the hope of this study to provide some new insights into the non-Darcy flow in porous media.

On the well-posedness of generalized Darcy\u2013Forchheimer equation

Under the necessary compatibility condition and some mild regularity assumptions on the interior and the boundary data, we prove the existence, uniqueness, and stability of solution in [L^{m+1}(Omega )]^{d}times(W^{1, frac{m+1}{m}}(Omega)cap L^{2}_{0}(Omega)) for a generalized Darcy–Forchheimer model, governing a non-Darcy flows in porous media with dimension d=2, 3 and min(1, 2].

A Two-Grid Block-Centered Finite Difference Algorithm for Nonlinear Compressible Darcy–Forchheimer Model in Porous Media

In this paper, a block-centered finite difference method is proposed to discretize the compressible Darcy–Forchheimer model which describes the high speed non-Darcy flow in porous media. The discretized nonlinear problem on the fine grid is solved by a two-grid algorithm in two steps: first solving a small nonlinear system on the coarse grid, and then solving a nonlinear problem on the fine grid. On the coarse grid, the coupled term of pressure and velocity is approximated by using the fewest number of node values to construct a nonlinear block-centered finite difference scheme. On the fine grid, the original nonlinear term is modified with a small parameter $$\varepsilon $$ to construct a linear block-centered finite difference scheme. Optimal order error estimates for pressure and velocity are obtained in discrete $$l^\infty (L^2)$$ and $$l^2(L^2)$$ norms, respectively. The two-grid block-centered finite difference scheme is proved to be unconditionally convergent without any time step restriction. Some numerical examples are given to testify the accuracy of the proposed method. The numbers of iterations are reported to illustrate the efficiency of the two-grid algorithm.

Non-Darcy interfacial dynamics of air-water two-phase flow in rough fractures under drainage conditions

Two-phase flow interfacial dynamics in rough fractures is fundamental to understanding fluid transport in fractured media. The Haines jump of non-Darcy flow in porous media has been investigated at pore scales, but its fundamental processes in rough fractures remain unclear. In this study, the micron-scale Haines jump of the air-water interface in rough fractures was investigated under drainage conditions, with the air-water interface tracked using dyed water and an imaging system. The results indicate that the interfacial velocities represent significant Haines jumps when the meniscus passes from a narrow “throat” to a wide “body”, with jump velocities as high as five times the bulk drainage velocity. Locally, each velocity jump corresponds to a fracture aperture variation; statistically, the velocity variations follow an exponential function of the aperture variations at a length scale of ~100 µm to ~100 mm. This spatial-scale-invariant correlation may indicate that the high-speed local velocities during the Haines jump would not average out spatially for a bulk system. The results may help in understanding the origin of interface instabilities and the resulting non-uniform phase distribution, as well as the micron-scale essence of the spatial and temporal instability of two-phase flow in fractured media at the macroscopic scale.

Investigation of Barree-Conway non-Darcy flow effects on coalbed methane production

Coalbed gas non-Darcy flow has been observed in high permeable fracture systems, and some mathematical and numerical models have been proposed to study the effects of non-Darcy flow using Forchheimer non-Darcy model. However, experimental results show that the assumption of a constant Forchheimer factor may cause some limitations in using Forchheimer model to describe non-Darcy flow in porous media. In order to investigate the effects of non-Darcy flow on coalbed methane production, this work presents a more general coalbed gas non-Darcy flow model according to Barree-Conway equation, which could describe the entire range of relationships between flow velocity and pressure gradient from low to high flow velocity. An expanded mixed finite element method is introduced to solve the coalbed gas non-Darcy flow model, in which the gas pressure and velocity can be approximated simultaneously. Error estimate results indicate that pressure and velocity could achieve first-order convergence rate. Non-Darcy simulation results indicate that the non-Darcy effect is significant in the zone near the wellbore, and with the distance from the wellbore increasing, the non-Darcy effect becomes weak gradually. From simulation results, we have also found that the non-Darcy effect is more significant at a lower bottom-hole pressure, and the gas production from non-Darcy flow is lower than the production from Darcy flow under the same permeable condition.

A block-centered finite difference method for general non-Darcy flow in porous media

A block-centered finite difference scheme is introduced to solve the nonlinear equation of non-Darcy model, in which it can maintain local mass conservation and the velocity and pressure can be approximated simultaneously. The second-order error estimates in discrete $l^2$ norm for both pressure and velocity are established on a nonuniform rectangular grid. Numerical experiments using the scheme show the consistency of the convergence rates of our method with the theoretical analysis.

A Two-Grid Block-Centered Finite Difference Method For Darcy--Forchheimer Flow in Porous Media

A two-grid block-centered finite difference method is proposed for solving the two-dimensional Darcy--Forchheimer model describing non-Darcy flow in porous media. To construct the two-grid method we modify the original nonlinear elliptic operator of Darcy--Forchheimer flow to a twice continuously differentiable one by introducing a small and positive parameter $\varepsilon$. By using the two-grid method, solving a nonlinear equation on a fine grid is reduced to solving a nonlinear equation on a coarse grid together with solving a linear equation on a fine grid. Optimal order error estimates for pressure and velocity in discrete $L^2$ norms are obtained. Some numerical examples are given to show the accuracy and efficiency of the presented method.

Analysis of Multiphase Non-Darcy Flow in Porous Media

Recent laboratory studies and analyses (Lai et al. Presented at the 2009 Rocky Mountain Petroleum Technology Conference, 14–16 April, Denver, CO, 2009) have shown that the Barree and Conway model is able to describe the entire range of relationships between flow rate and potential gradient from low- to high-flow rates through porous media. A Buckley and Leverett type analytical solution is derived for non-Darcy displacement of immiscible fluids in porous media, in which non-Darcy flow is described using the Barree and Conway model. The comparison between Forchheimer and Barree and Conway non-Darcy models is discussed. We also present a general mathematical and numerical model for incorporating the Barree and Conway model in a general reservoir simulator to simulate multiphase non-Darcy flow in porous media. As an application example, we use the analytical solution to verify the numerical solution for and to obtain some insight into one-dimensional non-Darcy displacement of two immiscible fluids with the Barree and Conway model. The results show how non-Darcy displacement is controlled not only by relative permeability, but also by non-Darcy coefficients, characteristic length, and injection rates. Overall, this study provides an analysis approach for modeling multiphase non-Darcy flow in reservoirs according to the Barree and Conway model.

Lattice Boltzmann Simulation of non-Darcy Flow in Porous Media

Flow through porous media at low Reynolds numbers has been studied in detail with the Lattice Boltzmann Method (LBM) for applications such as groundwater flow, pollution transport or adsorption processes. In contrast to that, medium to high Reynolds number flow through porous media, which occurs in many areas of industrial engineering, has not yet widely been investigated on a microscopic level by detailed numerical simulations.In this paper, we focus on air flow through a porous medium, because our far goal entails the simulation of acoustic excitations from the turbulent flow leaving the porous medium. We validate the LBM at Reynolds numbers beyond the limit of Darcy's law, and compare the results of direct numerical simulation with those achieved by applying a Smagorinsky-type large eddy turbulence model. For this, we performed flow simulations through a generic (periodic) porous medium at a variety of resolutions to investigate the effect of LES modelling at lower mesh sizes, where the subgrid scale effects become important.

A Predictive Pore-Scale Model for Non-Darcy Flow in Porous Media

Summary Non-Darcy flow is often observed near wells and in hydraulic fractures where relatively high velocities occur. In these regions, an empirical model, Forchheimer's equation, is used in place of Darcy's Law. It includes a quadratic correction to the linear model and has been adequately fit to many experimental data sets, while found to be insufficient in others. Furthermore, a number of numerical and theoretical approaches have shown limitations of the Forchheimer model in the laminar flow regime. It is important to understand the applicability of Forchheimer's equation and to be able to obtain good predictions of macroscopic properties so that nonlinear flow can be properly modeled in reservoir simulators. In this work, non-Darcy flow of an incompressible fluid is modeled using a physically representative (Bryant et al. 1993) pore-scale network model. Quantitative and predictive results are obtained using both computer-generated porous media and real sandstones digitized through x-ray computed microtomography (XMT). A new friction factor correlation is developed for laminar flow in converging/diverging ducts using the numerical solution to the Navier-Stokes equations. The new equation is used to describe flow in pore throats of the network model. The permeability and non-Darcy coefficient, β, are determined for these isotropic and anisotropic media in which Forchheimer's equation is applicable. The numerical model is compared to existing experimental data and appears to be at least as successful as correlations for predicting the non-Darcy coefficient in isotropic media. Furthermore, limitations to Forchheimer's equation at both low and high velocities are determined and discussed.

Multiphase Non-Darcy Flow in Proppant Packs

Summary This paper builds on a model for single-phase, non-Darcy flow in porous media presented in by Barree and Conway (2004). In this work, the generalized equation for single-phase non-Darcy flow is extended to multiphase-flow conditions. Laboratory measurements of gas/water flow were conducted in several flow cells to determine relative permeability curves and the nonlinear relationship between individual-phase flow rate and induced potential gradient over a wide range of input fractional-flow conditions and over a range of Reynolds numbers (Re). A model is presented to predict multiphase non-Darcy flow in porous media that is applicable to high-deliverability reservoirs and hydraulically fractured wells.

Geometric framework for modeling nonlinear flows in porous media, and its applications in engineering

Abstract Our work is focused on certain theoretical aspects of non-linear non-Darcy flows in porous media, and their application in reservoir and hydraulic engineering. The goal of this paper is to develop a mathematically rigorous framework to study the dynamical processes associated to nonlinear Forchheimer flows for slightly compressible fluids. Using fundamental geometric methods, we have proved the existence of a nonlinear scaling operator which relates constant mean curvature surfaces and time invariant pressure distribution graphs constrained by the Darcy–Forchheimer law. The hereby obtained properties of fast flows and their geometric interpretation can be used as analytical tools to evaluate important technological parameters in reservoir engineering.

Applicability of the Forchheimer Equation for Non-Darcy Flow in Porous Media

Summary The subject of non-Darcy flow in hydraulically fractured wells has generated intense debates recently. One aspect of the discussion concerns the inertia resistance factor, or the so-called beta factor, β, in the Forchheimer equation, and whether the beta factor β of a proppant pack should be constant over the range of flow rates of practical interests. The problem was highlighted in a recent discussion by van Batenburg and Milton-Tayler (2005) and the reply by Barree and Conway (2005) regarding paper SPE 89325 (Barree and Conway 2004) in the August 2005 JPT. This discussion in essence revolves around the applicability of the Forchheimer equation and whether the Forchheimer equation is adequate to describe the experimental results of high rate flow in proppant packs. In order to properly assess the arguments in this debate, and to get a better understanding of the state-of-the-art on non-Darcy flow in porous media in general, literature concerning the theoretical basis of the Forchheimer equation and experimental work on the identification of flow regimes is reviewed. These areas of work provide insights into the applicability of the Forchheimer equation to conventional oilfield flow tests for proppant packs. Models for flow beyond the Forchheimer regime are also suggested.

A Criterion for Non-Darcy Flow in Porous Media

Non-Darcy behavior is important for describing fluid flow in porous media in situations where high velocity occurs. A criterion to identify the beginning of non-Darcy flow is needed. Two types of criteria, the Reynolds number and the Forchheimer number, have been used in the past for identifying the beginning of non-Darcy flow. Because each of these criteria has different versions of definitions, consistent results cannot be achieved. Based on a review of previous work, the Forchheimer number is revised and recommended here as a criterion for identifying non-Darcy flow in porous media. Physically, this revised Forchheimer number has the advantage of clear meaning and wide applicability. It equals the ratio of pressure drop caused by liquid–solid interactions to that by viscous resistance. It is directly related to the non-Darcy effect. Forchheimer numbers are experimentally determined for nitrogen flow in Dakota sandstone, Indiana limestone and Berea sandstone at flowrates varying four orders of magnitude. These results indicate that superficial velocity in the rocks increases non-linearly with the Forchheimer number. The critical Forchheimer number for non-Darcy flow is expressed in terms of the critical non-Darcy effect. Considering a 10% non-Darcy effect, the critical Forchheimer number would be 0.11.