Single-phase Flow In Porous Media Research Articles

Stationary Stokes solver for single-phase flow in porous media: A blastingly fast solution based on Algebraic Multigrid Method using GPU

The paper is focused on high efficiency Stokes solver that is applied to the incompressible flow in porous media. Computational domains are represented by binarized 3D computed tomography voxel models. The solution procedure is constructed around a fast algebraic multigrid solver that utilizes the power of graphics processing units (GPUs). In order to minimize memory footprint and accelerate the solver, a simple MAC-type staggered finite difference discretization is used and a coupled Stokes saddle-point type system is solved directly on a GPU. The MAC discretization is discussed, taking particular care of internal solid boundaries around pores. The method includes topological domain analysis on a GPU, which removes isolated volumes with no flow in the domain (based on the connected component labeling), depending on the boundary conditions. We consider various types of boundary conditions and efficient parallel strategies for the GPUs, including fast matrix assembly and residual regularization. Our method is extensively benchmarked both against analytical solutions and applied problems from digital rock physics in terms of computational wall time, precision, approximation order, and convergence. We demonstrate that it takes up to 5–23 s on a modern GPU card to obtain a solution with 1⋅10−6 error residuals for 3D geometries with 300–4503 voxels and a porosity range of 5–37%.

Study of the pore space heterogeneity effect on the absolute permeability tensors calculated under different boundary conditions and driving forces using a “Computational Rock Physics” technology

In this paper, we study the effect of pore space heterogeneity on off-diagonal permeability tensors calculated for four possible combinations between external boundary conditions (non-flow and periodic) and the driving force inducing the fluid flow (pressure drop and body force). The results were obtained using “Computational Rock Physics” technology, which involves numerical experiments in digital models of porous media. As a mathematical model describing a single-phase flow in porous media, the lattice Boltzmann equations in combination with the multi-relaxation time scheme are used. Porous structures with various heterogeneities are generated using a combination of the Monte-Carlo movement algorithm and the Quartet Structure Generation Set method. The results show that for non-flow boundary conditions (BC), the diagonal components calculated for the body force exceed the corresponding values calculated for the pressure drop, but for the periodic BC, the opposite trend was revealed. It has been established that the heterogeneity significantly affects the sensitivity of the permeability components to the type of driving force as follows: an increase in heterogeneity promotes an increase in differences between the permeability components calculated under pressure drop and body force. The sensitivity of the diagonal components to the driving force is higher in the case of non-flow BC. The paper also considers the issue of the asymmetry of the permeability tensor and the influence of heterogeneity, types of boundary conditions, and driving forces on it. It was found that the permeability tensor is absolutely symmetrical only for periodic BC and body force, while for the other three combinations this is not the case. The power of tensor asymmetry increases with increasing pore space heterogeneity. Differences in the permeability components, as well as the power of tensor asymmetry, decrease with the increasing scale of the sample.

Discrimination between Pore and Throat Resistances against Single-Phase Flow in Porous Media

This study investigates the critical agents that cause non-Darrian flow in porous media. Four porous media different in morphology but similar in topology were studied numerically. By varying the throat diameters, the distinct roles of pores and throats in total dissipation were investigated using direct numerical simulation. Forchheimer model was selected to analyze the non-Darcian flow. In our simplified geometry, the ratio KappKD can best be correlated by non-Darcy effect (E). Total dissipation is directly related to the porous medium resistance against fluid flow. The energy dissipated in pores and throats was calculated by summing the dissipation in each computational segment. Pores are more prone to disobey the Darcy model than throats due to irregularity in fluid flow, and they are introduced as the cause of Darcy-model cessation. By increasing the pore-to-throat ratio, the non-Darcian flow in the pores begins sooner. The results show that the energy dissipation due to eddies is negligible. The dissipation in pores and throats was simulated through separate power-law equations, and their exponents were also extracted. The exponent for the pore body is equal to two when the viscous forces are dominant, and it increases by increasing the inertia force. The dissipation due to pore bodies is more apparent when the size of pore and throats are of the same order of magnitude. The relative losses of pore body increase as the velocity increases, in contrast to throats.

Development of a Reservoir Simulator to Model Single-Phase Flow in Porous Media

Any groundwater reservoir requires tools to predict future performance as well as to optimize its operation. It is then necessary to simulate groundwater flow in porous media because of the uncertainty and heterogeneity associated with reservoirs. This study developed a reservoir simulator for modeling a single-phase flow in a porous medium. The development of the simulator consists of the physical and mathematical modeling of the reservoir. A MATLAB code was developed to describe groundwater flow in order to appreciate reservoir hydrodynamic pressure distributions from hydraulic head as a function of radial distance while varying the production flow. The formulation equation obtained was solved by the direct method. Examples of graphical plots generated from the simulator illustrate that before the coordinate point P (r=33.74m; h=286.65m) for any value of production flow, hydraulic head or hydrodynamic pressure of the reservoir increases equally with radial distance. This reflects the same drop in the static pressure of the reservoir. Beyond point P, there is a further increase in the hydraulic head, i.e., the hydrodynamic pressure of the reservoir as the production flow increases with the increase in population. This results in a drop in the static pressure of the reservoir in proportion to the increase in the production flow. The variations of the production flow carried out show that the static pressure of the reservoir decreases when the production flow increases. Finally, the simulator to predict the hydraulic head distributions i.e., the hydrodynamic pressure of the reservoir in single-phase flow during production periods is a springboard towards the implementation of multi-phase fluid flow formulations.

Improved pore network models to simulate single-phase flow in porous media by coupling with lattice Boltzmann method

In this paper, different pore network models to simulate single-phase flow in porous media are built and their accuracy is evaluated. In addition to the conventional pore network model (CPNM) which consists of regular pore bodies and throat bonds, three improved pore network models (IPNMs) are developed allowing to better describing the real pore and throat geometry. The first improved pore network model (IPNM1) replaces the regular throat bond with a throat bond showing the real throat cross section. The second improvement (IPNM2) uses a series of sub-throat bonds with varying cross sections to better describe the real throat geometry, which is firstly proposed in this paper. The third model (IPNM3) extracts the real pore-throat-pore geometry without simplification. The conductance of fluid flow through these more realistic throat bonds is calculated by the lattice Boltzmann method (LBM). The accuracy and computational efficiency of the different pore network models are evaluated taking the LBM simulation over the whole porous medium as reference solution. The global permeability and detailed pressure distributions in the pores for the different pore network models are validated. The results show that the accuracy of the pore network model increases from CPNM to IPNM3, but at the expense of increasing computational cost. This study suggests that IPNM3 can replace a whole-domain LBM simulation with similar accuracy but much lower computational cost. As a first-order approximation the newly proposed IPNM2 is suggested as good compromise between accuracy and computational cost.

On a Second-Order Multipoint Flux Mixed Finite Element Methods on Hybrid Meshes

We consider the numerical approximation of single-phase flow in porous media by a mixed finite element method with mass lumping. Our work extends previous results of Wheeler and Yotov, who showed t...

Separate-phase model and its lattice Boltzmann algorithm for liquid-vapor two-phase flows in porous media.

In this article, we formulate a set of the separate-phase governing equations at the representative-elementary-volume scale and develop its double-distribution function lattice Boltzmann (LB) algorithm to describe liquid-vapor two-phase flows with or without phase change in porous media. Different from those previous studies, the mathematical description in this article involves the Darcy force, viscous force, and pressure gradient, and the resulting LB simulations can well describe two-phase flows and mass transfer throughout porous media under the compounding effects of these forces. The LB algorithm was validated by simulating single-phase flows in porous media. Its results are in good agreement with those available analytical solutions. We also applied it to model water flows through a semi-infinite porous region bounded by a heated solid wall, where liquid-vapor phase change takes place. The numerical simulations recover the previous results in the limit of the zero Darcy number. Significantly, it reveals much richer two-phase flow and mass transfer characteristics in porous media adjacent to solid walls. The separate-phase model and its lattice Boltzmann algorithm in this article are effective means to gain more profound and clearer understandings of complex two-phase transport processes in a porous system.

Numerical study on inertial effects on liquid-vapor flow using lattice Boltzmann method

Liquid-vapor flow in porous media is studied in this article. To fulfill this goal, a double-distribution-function lattice Boltzmann (LB) model is proposed based on the separate-phase governing equations at the representative elementary volume (REV) scale. Importantly, besides the Darcy force and capillary force, which were commonly included in previous studies, the LB model in this article also considers the inertial force characterized by the Forchheimer term. This feature enables the model to offer an effective description of liquid-vapor flow in porous media at low, intermediate and even high flow rates. We validated the LB model by simulating a single-phase flow in porous media driven by a pressure difference and found its results are in good agreement with the available analytical solutions. We then applied the model to study water-vapor flow in a semi-infinite porous region bounded by an impermeable and heated wall. The numerical simulation reveals the flow and mass transfer characteristics under the compounding effects of inertial, Darcy and capillary forces. Through a comparison with the results given by the generalized Darcy’s law, our numerical results directly evidence that the inertial force is a dominating factor when a fluid passes through porous media at an intermediate or high flow rate.

Modelling the pressure drop of two-phase flow through solid porous media

This contribution discusses a physically meaningful model for the description of the pressure drop of two-phase flow through bi-continuous solid porous media like metal sponges (open-cell metal foams) at high mass fluxes (>25 kg m-2 s-1). Such conditions typically occur during flow boiling in tubes filled with solid sponges as a means to enhance heat transfer. We propose combining the homogeneous model for two-phase pressure drop in empty tubes and the Forchheimer equation describing the pressure drop of single-phase flow in porous media. For known geometrical characteristics of the sponge (i.e. the porosity, the specific surface area, and the window diameter), different prediction methodologies for the Forchheimer coefficient based on models developed for the single-phase pressure drop are compared. The validity range of the model is determined on the basis of all available literature data. If the geometric properties of the sponge are known, the model can predict 80% of all experimental data with a deviation of less than 30%. For experimentally determined single-phase Forchheimer coefficients, 96% of all data are predicted within 30% uncertainty.

A heterogeneous multiscale MPFA method for single-phase flows in porous media with inertial effects

Upscaling of flow from pore to Darcy scale is a long-standing research field within flow in porous media. It is well known that non-linearities can occur in near-well regions and high-porosity or fractured media. At the same time, the upscaled non-linear effects associated with high flow rates are hard to quantify a priori with single-scale models. Advances in pore-scale imaging combined with increased computational have made flow simulations in small pore-scale domain feasible, but computations on domains larger than at most a few centimeters are still elusive. In this work, we present a multiscale simulation framework that automatically adapts to non-linear effects as they arise. We formulate a control volume heterogeneous multiscale method (CVHMM) by coupling of a Darcy-scale control volume method with a constitutive relation that is captured based on the fine-scale physics. While the CVHMM formulation works with arbitrary upscaled laws, we emphasize its ability to be applied in fully discrete multiscale context, in particular when a finite element solver is used for solving Navier-Stokes equations on the fine-scale pore geometry. Previous versions of CVHMM are consistent only when the coarse grid is aligned with the upscaled permeability. Herein, we generalize CVHMM by introducing a new coarse solver, thus significantly improving the applicability of the method. The presented method is applied to study flow in near-well regions, as well as media with fractures and irregular grain shapes. The examples show that the method successfully copes with general grids and pore geometries and handles flows with varying degree of non-linearities even outside the domain of applicability of classical upscaled models. In terms of computational efficiency, the method seamlessly localizes computations to regions where non-linear effects are important.

Macroscopic momentum and mechanical energy equations for incompressible single-phase flow in porous media.

Modeling flow in porous media is usually focused on the governing equations for mass and momentum transport, which yield the velocity and pressure at the pore or Darcy scales. However, in many applications, it is important to determine the work (or power) needed to induce flow in porous media, and this can be achieved when the mechanical energy equation is taken into account. At the macroscopic scale, this equation may be postulated to be the result of the inner product of Darcy's law and the seepage velocity. However, near the porous medium boundaries, this postulate seems questionable due to the spatial variations of the effective properties (velocity, permeability, porosity, etc.). In this work we derive the macroscopic mechanical energy equation using the method of volume averaging for the simple case of incompressible single-phase flow in porous media. Our analysis shows that the result of averaging the pore-scale version of the mechanical energy equation at the Darcy scale is not, in general, the expected product of Darcy's law and the seepage velocity. As a matter of fact, this result is only applicable in the bulk region of the porous medium and, in the derivation of this result, the properties of the permeability tensor are determinant. Furthermore, near the porous medium boundaries, a more novel version of the mechanical energy equation is obtained, which incorporates additional terms that take into account the rapid variations of structural properties taking place in this particular portion of the system. This analysis can be applied to multiphase and compressible flows in porous media and in many other multiscale systems.

Particle velocimetry analysis of immiscible two-phase flow in micromodels

We perform micro-PIV measurements in micromodels using very simple optical equipment combined with efficient image acquisition and processing. The pore-scale velocity distributions are obtained for single-phase flow in porous media with a typical pore size of 5–40 µm at a resolution of 1.8μm×1.8μm vector grid. Because the application of micro-PIV in micromodels is not standard, extensive effort is invested into validation of the experimental technique. The micro-PIV measurements are in very good agreement with numerical simulations of single-phase flows, for which the modeling is well established once the detailed pore geometry is specified and therefore serves as a reference. The experimental setup is then used with confidence to investigate the dynamics of immiscible two-phase flow in micromodels that represent natural complex porous media (e.g., sandstone). For unstable immiscible two-phase flow experiments, micro-PIV measurements indicate that the flow is highly oscillatory long before the arrival of the invading interface. The dynamics are accompanied with abrupt changes of velocity magnitude and flow direction, and interfacial jumps. Following the passage of the front, dissipative events, such as eddies within the aqueous phase, are observed in the micro-PIV results. These observations of complex interface dynamics at the pore scale motivate further measurement of multiphase fluid movement at the sub-pore scale and requisite modeling.

Mixed methods using standard conforming finite elements

We investigate the mixed finite element method (MFEM) for solving a second order elliptic problem with a lowest order term, as might arise in the simulation of single-phase flow in porous media. We find that traditional mixed finite element spaces are not necessary when a positive lowest order (i.e., reaction) term is present. Hence, we propose to use standard conforming finite elements Q k × ( Q k ) d on rectangles or P k × ( P k ) d on simplices to solve for both the pressure and velocity field in d dimensions. The price we pay is that we have only sub-optimal order error estimates. With a delicate superconvergence analysis, we find some improvement for the simplest pair Q k × ( Q k ) d with any k ⩾ 1 , or for P 1 × ( P 1 ) d , when the mesh is uniform and the solution has one extra order of regularity. We also prove similar results for both parabolic and second order hyperbolic problems. Numerical results using Q 1 × ( Q 1 ) 2 and P 1 × ( P 1 ) 2 are presented in support of our analysis. These observations allow us to simplify the implementation of the MFEM, especially for higher order approximations, as might arise in an hp-adaptive procedure.

A space-time discontinuous Galerkin method applied to single-phase flow in porous media

A space-time discontinuous Galerkin finite element method is proposed and applied to a convection-dominant single-phase flow problem in porous media. The numerical scheme is based on a coupled space-time finite element discretization allowing for discontinuous approximations in space and in time. The continuities on the element interfaces are weakly enforced by the flux treatments, so that no extra penalty factor has to be determined. The resulting space-time formulation possesses the advantage of capturing the steep concentration front with sharp gradients efficiently. The stability and reliability of the proposed approach is demonstrated by numerical experiments.

Fractured Gas-Reservoir Performance

Abstract Gas production from fractured reservoirs can be very prolific despite the extremely low matrix permeability that may be encountered. In order to quantify production, the interaction between matrix and fracture must be well understood. Knowing the limitations to production is important for operators who drill into very tight matrix reservoirs with varying degrees of natural fractures. This paper describes a laboratory protocol for assessing matrix-fracture interaction and discusses field implications. The well-known pulse decay technique was used for determining matrix-fracture interaction by fitting parameters from a simple numerical model to experimental data. On this basis, the values of four parameters were regressed: fracture permeability, matrix permeability, mass transfer from matrix to fracture and the fraction of the porous media that was fracture. A full discussion of the mathematical approach, optimization technique and experimental protocol is included. It was concluded that the technique has significant non-unique convergence characteristics which make the interpretation of core parameters dependent on the shape of the experimental production history and, therefore, introduces some subjectivity. Results indicate that fracture permeability is relatively insensitive to water saturation whereas matrix permeability and the matrix-fracture transfer function are extremely sensitive to increasing water saturation; the latter two parameters become rate-limiting once early-time fracture production is complete. Background The impetus for this work is experimental. The authors ’ objective was to investigate an experimental approach, known as pulse decay, in order to try and deduce parameters associated with gas flow from fractured porous media and to do so quickly and accurately. This would involve gas flow in the absence of any liquid phase, as well as gas flow in the presence of immobile water saturation. These measurements are important to operators of dry gas fractured reservoirs, especially in the presence of mobile water either naturally occurring or induced during drilling or hydraulic fracturing. Quantifying phase interference effects is routinely done empirically, but the state-of-the-art leak-off tests are time consuming and can be expensive. The authors wanted to investigate the merit of being able to obtain a series of relevant parameters through a simple, inexpensive approach such as pulse decay. Single-phase flow in porous media is well known. For gas flow in porous media, potential areas for obfuscation include gas-solid slip at the wall and turbulence effects. The Klinkenberg protocol is a very effective means of removing the effects of gas-slippage which causes deviation from the assumptions inherent in using the Darcy equation (laminar flow with no flow at the wall). Figure 1 shows a standard Klinkenberg plot done for a sample analyzed in this work. The significance of this correction is that if the pore pressure is high enough, the gas will act more like a liquid and, at low rates, will behave in a laminar manner which requires the development of a quadratic velocity profile in the flow conduits and a no-slip condition at the solid surfaces. Figure 1, with its associated trend-line, shows that by extrapolating to infinite pore pressure where any gas would condense, the effective permeability for this sample is 41.2 mD.

Analyzing Finite Volume for Single-Phase Flow in Porous Media

Two-point finite volume methods (2P-FVMs) are extensively used for understanding porous media flow because these methods are fast and simple. In this article, we present numerical analysis of the two-point finite volume discretization of a pressure equation of a single-phase flowing in porous media. We present numerical problems with discontinuous permeability and diagonal permeability, together with Neumann and Dirichlet boundary conditions. An analysis of the effect of boundary conditions on the conditioning of the discrete systems is presented. We also analyze convergence of the 2P-FVM in various norms (L2 convergence for pressure and Darcy velocity and L∞ convergence for pressure) for problems with regularity H1+γ, for γ = 0.1, 0.2, ..., 0.9, 0.99.

Simple Models for Permeability Impairment in Reservoir Rocks Caused by Asphaltene Deposition

This study formulates two models for estimating permeability impairment caused by asphaltene deposition. The first model is based on a mass balance and is capable of predicting permeability reduction in either vertical or horizontal wells. The model accounts for both the effects of position in the reservoir and time. The second model is formulated on the basis of the capillary tube theory. It accounts for the rock pore-size distribution but does not account for the time dependence. Both models are formulated in integral form to allow for analytic solutions when symbolic formulation exists for the parameters affecting the deposition process. Both modeling approaches account implicitly for the effects of pressure, temperature, and fluid composition, by including the asphaltene radius distribution. Even though the mass-balance model is one-dimensional and is valid for only single-phase flow in porous media, it can be used to estimate the permeability reduction for both homogeneous and heterogeneous formations. The capillary tube model requires less input variables than the mass-balance model and does not depend on any tuning parameters. However, it is neither sensitive to the position in the reservoir nor to the well type penetrating the reservoir. These reasons make the capillary tube model useful only for a qualitative description and thus capable of only giving a first-order estimate of the permeability reduction caused by asphaltene deposition.

A Cellular Automaton Model for Flow in a Heterogeneous Reservoir

ABSTRACT This paper reports on the development of a 2-dimensional cellular automaton (CA) model to simulate fluid flow in porous media. In CA, simple rules of particle interactions at a lattice are used to simulate complex flow phenomena. Since the numerical operations involved are largely bit manipulation, CA can be potentially more efficient in memory usage than conventional methods, such as finite difference or finite element methods. While other research on the use of CA to model porous media flow primarily focuses on Stokes flow at the pore level, with the intent of understanding flow at the core scale, this work concentrates on modeling Darcy flow in large heterogeneous fields. Permeability variations and anisotropy are modeled by a distribution of "soft scatterers" and "directional scatterers". Collision of incoming particles with scatterers is a stochastic process. Anisotropy is modeled via a preferred direction of collision between the directional scatterers and incoming particles The CA model is verified through the study of single-phase flow in porous media. Simulation results obtained from this CA model show that particle flow rates are linearly proportional to pressure gradient (pressure is modeled through particle density in CA), and permeability is linearly correlated with scatterer density over a certain domain. Therefore, the CA model can be used to predict Darcy-like flow. It is shown that heterogeneity is modeled properly by varying the density of scatterers. In addition to the single fluid model, a model with two fluids with identical physical properties has also been developed to study dispersion characteristics of a CA model. In the numerical simulation results, the transverse diffusivity converges to the theoretical constant, but the longitudinal diffusivity increases with time. Finally, extension of the current model to a more general case (i.e., three-dimensional, multiphase flow) is also discussed.

Multiphase Flow In Porous Media

The field of two-phase flow through porous media is currently undergoing intense development following many years of relatively sluggish activity. This makes it difficult to present a definitive account at this time. Even classical concepts such as relative permeabilities [the cornerstones of which were laid down by M. Muskat and coworkers long ago (cf. Section 3)] are now seriously questioned. Owing to this provisional state, we adopt a conservative attitude in organizing this review by adhering closely to chronological developments in the field. Historically, global macro scale conservation laws and consti­ tutive equations were developed merely by analogy with comparable enti­ ties already existing for single-phase flow in porous media. As such, these relations may be regarded as semiempirical in nature. For example, per­ tinent quantities such as relative permeabilities were invariably measured experimentally rather than calculated or estimated from basic physical principles. This reflected the generally poor predictive power of such governing laws. This strictly phenomenological focus endured from roughly the early 1930s until the 1960s, when attention shifted to local, micro scale phenom­ ena. These proved difficult to analyze owing to both the tortuous geometry of porous media and the influence of capillarity. Since pore dimensions