Immiscible Displacement in Cross-Bedded Heterogeneous Porous Media

Direct insight into the mechanisms of flow and displacements within small-scale (cm) systems having permeability heterogeneities that are not parallel to the flow direction (cross-bedding and fault zones) have been carried out. In our experiments, we have used visual models with unconsolidated glass bead packs having carefully controlled permeability contrasts to observe the processes with coloured fluids and streamlines. The displacements were followed visually and by video recording for later analysis. The experiments show the significance that heterogeneities have on residual saturations and recovery, as well as the displacement patterns themselves. During a waterflood, high permeability regions can be by-passed due to capillary pressure differences, giving rise to high residual oil saturations in these regions. This study demonstrates the importance of incorporating reservoir heterogeneity into core displacement analysis, but of course the nature of the heterogeneity has to be known. In general, the effects created by the heterogeneities and their unknown boundaries hamper interpretation of flood experiments in heterogeneous real sandstone cores. Our experiments, therefore, offer clear visual information to provide a firmer understanding of the displacement processes during immiscible displacement, to present benchmark data for input to numerical simulators, and to validate the simulator through a comparison with our experimental results for these difficult flow problems.     

Experimental displacement patterns in a 2 × 2 quadrant block with permeability and wettability heterogeneities—problems for numerical modelling

The effect of heterogeneities on miscible and immiscible flood displacements in 2D bead packs in quadrant form, 2 × 2 block heterogeneity, with either a permeability or a wettability contrast is the subject of this paper. The physical processes occurring during miscible and immiscible flow and displacement within permeability and wettability quadrant bead pack models have been studied experimentally. This geometry occurs in a number of situations relevant to hydrocarbon production: particularly faults where adjacent rocks have large permeability contrasts with rapid changes, in the laboratory with core butting, in reservoir simulation where grid blocks have different permeability and in reservoirs having near-wellbore damage problems. The model quadrants 1–4, had 1 and 4 and 2 and 3 with identical properties, either in permeability or wettability. Reported are complete unit mobility miscible displacements, then the effects of viscosity differences (mobility modifiers) and finally immiscible displacements on displacement patterns for initial linear injection. The experiments demonstrate that nodal flow occurs for both miscible and immiscible flow, but for immiscible flow there are boundary effects due to capillary pressure differences created by water saturation changes or wettability contrasts which can leave patches of isolated fluid within a quadrant. The displacement patterns for the different models and fluids change significantly with the viscosity and wettability changes, particularly for the immiscible displacements. This is due to the changing capillary pressure between the quadrant blocks as the water saturation change. These are difficult to address in numerical modelling but should be accounted for. Other effects include coupling of all physical processes governing the flow through the node and creations of microzones of trapped residual oil. Our displacement patterns can therefore be a valuable verification benchmark tool for numerical modelling and a calibration data source for those wishing to simulate the effects of capillary pressure under differing wettability conditions and for those investigating upscaling modelling procedures. However, the possible loss of physical reality when averaging must always be considered.     

A Semi-Analytical Model for Two Phase Immiscible Flow in Porous Media Honouring Capillary Pressure

This article describes a semi-analytical model for two-phase immiscible flow in porous media. The model incorporates the effect of capillary pressure gradient on fluid displacement. It also includes a correction to the capillarity-free Buckley–Leverett saturation profile for the stabilized-zone around the displacement front and the end-effects near the core outlet. The model is valid for both drainage and imbibition oil–water displacements in porous media with different wettability conditions. A stepwise procedure is presented to derive relative permeabilities from coreflood displacements using the proposed semi-analytical model. The procedure can be utilized for both before and after breakthrough data and hence is capable to generate a continuous relative permeability curve unlike other analytical/semi-analytical approaches. The model predictions are compared with numerical simulations and laboratory experiments. The comparison shows that the model predictions for drainage process agree well with the numerical simulations for different capillary numbers, whereas there is mismatch between the relative permeability derived using the Johnson–Bossler–Naumann (JBN) method and the simulations. The coreflood experiments carried out on a Berea sandstone core suggest that the proposed model works better than the JBN method for a drainage process in strongly wet rocks. Both methods give similar results for imbibition processes.     

Calculation of relative permeability in reservoir engineering using an interacting triangular tube bundle model

Analytical expressions of relative permeability are derived for an interacting cylindrical tube bundle model. Equations for determining relative permeability curves from both the interacting uniform and interacting serial types of triangular tube bundle models are presented. Model parameters affecting the trend of relative permeability curves are discussed. Interacting triangular tube bundle models are used to history-match laboratory displacement experiments to determine the relative permeability curves of actual core samples. By adjusting model parameters to match the history of oil production and pressure drop, the estimated relative permeability curves provide a connection between the macroscopic flow behavior and the pore-scale characteristics of core samples.     

Relative permeability and capillary pressure functions of porous media as related to the displacement growth pattern

Visualization experiments of the unsteady immiscible displacement of a fluid by another are performed on glass-etched pore networks of well-controlled morphology by varying the fluid system and flow conditions. The measured transient responses of the fluid saturation and pressure drop across the porous medium are introduced into numerical solvers of the macroscopic two-phase flow equations to estimate the non-wetting phase, k_rnw, and wetting phase, k_rw, relative permeability curves and capillary pressure, P_c, curve. The correlation of k_rnw, k_rw, and P_c with the displacement growth pattern is investigated. Except for the capillary number, wettability, and viscosity ratio, the immiscible displacement growth pattern in a porous medium may be governed by the shear-thinning rheology of the injected or displaced fluid, and the porous sample length as compared to the thickness of the frontal region. The imbibition k_rnw increases as the flow pattern changes from compact displacement to viscous fingering or from viscous to capillary fingering. The imbibition k_rw increases as the flow pattern changes from compact displacement or capillary fingering to viscous fingering. As the shear-thinning behaviour of the NWP strengthens and/or the contact angle decreases, then the flow pattern is gradually dominated by irregular interfacial configurations, and the imbibition k_rnw increases. The imbibition P_c is a decreasing function of the capillary number or increasing function of the injected phase viscosity in agreement with the linear thermodynamic theory.     

Three-Dimensional Characterization of Permeability at the Core Scale

In this article, we introduce an integrated method for characterizing permeability heterogeneity at the core scale. It combines the results of laboratory core flooding with already-developed field scale history matching techniques such as gradual deformation and pilot points. Prior to any experiment, X-ray computed tomography (CT) imaging techniques are used to obtain three-dimensional porosity distribution in cores. The samples are submitted to viscous, miscible displacement of water by water–glycerin mixture. The dynamic data collected during injection are the time variations in inlet–outlet pressure drop and three-dimensional CT-scan concentration maps of invading fluid collected at successive times. We develop an inversion or matching process which takes advantage of the available data to characterize the spatial distribution of permeability heterogeneities within core samples. Permeability is assumed to be related to porosity. This matching process involves two successive optimizations. First, an initial permeability guess derived from porosity is modified by varying deterministic parameters until the corresponding simulated pressure answer fits the measured pressure drop. Second, an extended optimization process with both deterministic and stochastic parameters is run to match pressure drop and concentration data. This methodology is applied to a synthetic example for which the permeability–porosity relation is known. It yields a three-dimensional permeability model reproducing the reference pressure and concentration maps. The methodology is also applied to experimental data. In this case, it provides three-dimensional permeability models leading to an improved, but perfectible data match. A major difficulty is the unknown relationship between permeability and porosity.     

膨胀性非牛顿流体多孔介质流动的模拟分析

在表征体元尺度采用格子Boltzmann方法分析膨胀性非牛顿流体在多孔介质中的流动,基于二阶矩模型在演化方程中引入表征介质阻力的作用力项,求解描述渗流模型的广义Navier-Stokes方程．采用局部法计算形变速率张量,通过循环迭代得到非牛顿粘度和松弛时间．对多孔介质的Poiseuille流动进行分析,通过比较发现结果与孔隙尺度的解析解十分吻合,并且收敛较快,表明方法合理有效．分析了渗透率和幂律指数对速度和压力降的影响,研究结果表明,膨胀性流体的多孔介质流动不符合达西规律,压力降的增加幅度小于渗透率的减小幅度．当无量纲渗透率Da小于10-5时,流道中的速度呈现均匀分布,并且速度分布随着幂律指数的减小趋于平滑．压力降随着幂律指数的增加而增加,Da越大幂律指数对压力降的影响越明显．     

Fluid flow simulation in a random elliptical porous medium by using the lattice Boltzmann method

A fluid flow through an isotropic porous medium with randomly arranged elliptical particles is simulated by the lattice Boltzmann method. The dimensionless pressure drop and the dimensionless permeability are evaluated as functions of the Reynolds number. The effect of the aspect ratio of the major to minor semi-axis of the ellipse on the dimensionless permeability is considered for different values of porosity. The pressure drop is thoroughly investigated as a function of fluid viscosity for different values of the aspect ratio and porosity. The influence of various parameters of the problem on the mean tortuosity of the medium is considered.     

Analysis of Thermal Flow in a Rotating Porous U-Turn Duct Using Lattice Boltzmann Method

The lattice Boltzmann method is carried out to investigate the heat transfer enhancement in a U-turn duct which is partially filled with a porous media. The porous layer is inserted at the core of the duct and is modeled using the Brinkman–Forchheimer assumptions. In order to validate the results, first a channel flow problem without any porous layer is compared with available data. Second, the porous Couette flow and partially porous channel flow are successfully compared with the studies of other researchers. Then, fluid flow in a clear U-turn duct is studied looking carefully at the velocity, curvature and rotation effects. Finally, the effects of porous layer thickness on the rate of heat transfer and pressure drop are investigated. Parametric studies are conducted to evaluate the effects of various parameters (i.e., Reynolds number, Darcy number, rotation number), highlighting their influences on the thermo-hydrodynamics behavior of the flow. The optimum values of porous layer thickness are presented for specific flow parameters.     

Transport properties of heterogeneous materials. Combining computerised X-ray micro-tomography and direct numerical simulations

Feasibility of a method for finding flow permeability of porous materials, based on combining computerised X-ray micro-tomography and numerical simulations, is assessed. The permeability is found by solving fluid flow through the complex 3D pore structures obtained by tomography for actual material samples. We estimate overall accuracy of the method and compare numerical and experimental results. Factors contributing to uncertainty of the method include numerical error arising from the finite resolution of tomographic images and the rather small sample size available with the present tomographic techniques. The total uncertainty of computed values of permeability is, however, not essentially larger than that of experimental results. We conclude that the method provides a feasible alternative for finding fluid flow properties of the kind of materials studied. It can be used to estimate all components of permeability tensor and is useful in cases where direct measurements are not achievable. Analogous methods can be applied to other modes of transport, such as diffusion and heat conduction.     

Dispersion Measurements in Highly Heterogeneous Laboratory Scale Porous Media

Laboratory experiments and numerical simulations investigate conservative contaminant transport in a heterogeneous porous medium. The laboratory experiments were performed in cylindrical columns 1m long and 3.5cm inside diameter filled with spherical glass beads. Concentration breakthrough curves are measured at a scale much finer than the size of the heterogeneity. Numerical simulations are based on a random walk in a known constant velocity field. The heterogeneity is a distinct, discontinuous change in the local permeability field. Fluid flow is miscible, flowing in a saturated porous medium. Previous work has shown this to be a very poorly understood phenomenon. The measurements reported here help to better understand how dispersion evolves through and past a heterogeneity.     

Fluid Flow Across Capillary Heterogeneities,Experiments and Simulations

A comparative study of numerical modelling and laboratory experiments of two-phase immiscible displacements in a 33 cm × 10 × 3 cm thick cross-bedded reservoir model is reported. Dynamic two-dimensional fluid saturation development was obtained from experiments by use of a nuclear tracer imaging technique and compared to numerical predictions using a full-field black oil simulator.The laboratory cross-bedded reservoir model was a sandpack consisting of two strongly waterwet sands of different grain sizes, packed in sequential layers. The inlet and outlet sand consisted of low permeable, high capillary, sand while the central crosslayer with a dip angle of 30° was a high permeable, low capillary, sand. Results on moderate contrasts in permeability and capillary heterogeneities in the cross-bedded reservoir model at different mobility ratios and capillary number floods temporarily showed a bypass of oil, resulting in a prolonged two-phase production. The final remaining oil saturations, however, were as for isolated samples. Hence, permanently trapped oil was not observed.Simulations of waterfloods, using a commercial software package, displayed correct water breakthrough at low flow rate and unity viscosity ratio, but failed in predicting local saturation development in detail, probably due to numerical diffusion.The simulator was used to test several cases of heterogeneity contrasts, and influence from different relative permeability curves. Further, by altering the capillary pressure at the outlet, the end effects were proven important.     

Unsteady flow of a dusty Bingham fluid through a porous medium in a circular pipe

A time-varying flow through a porous medium of a dusty viscous incompressible Bingham fluid in a circular pipe is studied. A constant pressure gradient is applied in the axial direction, whereas the particle phase is assumed to behave as a viscous fluid. The effect of the medium porosity, the non-Newtonian fluid characteristics, and the particle phase viscosity on the transient behavior of the velocity, volumetric flow rates, and skin friction coefficients of both the fluid and particle phases is investigated. A numerical solution is obtained for the governing nonlinear momentum equations by using the method of finite differences.     

Determination of Permeability Distributions Using NMR Velocity Imaging

This study develops a methodology for determining the absolute permeability distribution in a porous media sample using velocity data obtained from NMR imaging experiments. An objective function describing the discrepancy between observed and simulated data is reduced by iteratively updating the permeability. This parameter estimation scheme is based on an iterative method which uses optimal control theory to refine the estimates. Although this theory is developed for both isotropic and anisotropic porous media, the permeability reconstructions examined in this paper are restricted to the isotropic case. Synthetic data are used to investigate the impact of varying the noise in the experimental data, the degree of parameterization, the relative weighting of the regularization term in the objective function, and the amount and type of data required to obtain a satisfactory permeability reconstruction. These synthetic data are extracted from the solution of numerical experiments that have utilized an assumed permeability distribution. The methodology is also applied to data gathered in laboratory experiments for water flow in a sandstone sample.     

On shear thinning fluid flow induced by continuous mass injection in porous media with variable conductivity

A new formulation is proposed to examine the propagation of the pressure disturbance induced by the injection of a time-variable mass of a weakly compressible shear thinning fluid in a porous domain with generalized geometry (plane, radial, or spherical). Medium heterogeneity along the flow direction is conceptualized as a monotonic power-law permeability variation. The resulting nonlinear differential problem admits a similarity solution in dimensionless form which provides the velocity of the pressure front and describes the pressure field within the domain as a function of geometry, fluid flow behavior index, injection rate, and exponent of the permeability variation. The problem has a closed-form solution for an instantaneous injection, generalizing earlier results for constant permeability. A parameter-dependent upper bound to the permeability increase in the flow direction needs to be imposed for the expression of the front velocity to retain its physical meaning. An example application to the radial injection of a remediation agent in a subsurface environment demonstrates the impact of permeability spatial variations and of their interplay with uncertainties in flow behavior index on model predictions.     

Metal Foam Heat Exchangers for Heat Transfer Augmentation from a Cylinder in Cross-Flow

A numerical study has been conducted to examine the heat transfer from a metal foam-wrapped solid cylinder in cross-flow. Effects of the key parameters including the free stream velocity and characteristics of metal foam such as porosity, permeability, and form drag coefficient on heat and fluid flow are examined. Being a determining factor in pressure drop and heat transfer increment, the porous layer thickness is changed systematically to observe that there is an optimum layer thickness beyond which the heat transfer does not improve while the pressure drop continues to increase. This has been verified by the application of Bejan’s Intersection of Asymptotes method. Results have been compared to those of a finned-tube heat exchanger to observe much higher heat transfer rate with reasonable excess pressure drop leading to a higher area goodness factor for metal foam-wrapped cylinder.     

Displacement of a Newtonian fluid by a non-Newtonian fluid in a porous medium

This paper presents an analytical Buckley-Leverett-type solution for one-dimensibnal immiscible displacement of a Newtonian fluid by a non-Newtonian fluid in porous media. The non-Newtonian fluid viscosity is assumed to be a function of the flow potential gradient and the non-Newtonian phase saturation. To apply this method to field problems a practical procedure has been developed which is based on the analytical solution and is similar to the graphic technique of Welge. Our solution can be regarded as an extension of the Buckley-Leverett method to Non-Newtonian fluids. The analytical result reveals how the saturation profile and the displacement efficiency are controlled not only by the relative permeabilities, as in the Buckley-Leverett solution, but also by the inherent complexities of the non-Newtonian fluid. Two examples of the application of the solution are given. One application is the verification of a numerical model, which has been developed for simulation of flow of immiscible non-Newtonian and Newtonian fluids in porous media. Excellent agreement between the numerical and analytical results has been obtained using a power-law non-Newtonian fluid. Another application is to examine the effects of non-Newtonian behavior on immiscible displacement of a Newtonian fluid by a power-law non-Newtonian fluid.     

Effects of Temperature-Dependent Viscosity on Forced Convection Inside a Porous Medium

Considering the exponential viscosity–temperature relation, effect of temperature-dependent viscosity on forced convection of a liquid through a porous medium, bounded by isoflux parallel plates, is investigated numerically based on the general model of momentum transfer. Local effects of viscosity variation on the distribution of velocity and temperature are analyzed. Moreover, global aspects of the problem are investigated where corrections are proposed for total pressure drop and the fully developed Nusselt number, in the form of out/in viscosity ratio. Results are obtained over a wide range of permeabilities from clear (of solid material) fluid to very low permeability, where for constant properties one expects a nearly slug flow.     

Mathematical model of micropolar fluid in two-phase immiscible fluid flow through porous channel

This paper is concerned with the flow of two immiscible fluids through a porous horizontal channel. The fluid in the upper region is the micropolar fluid/the Eringen fluid, and the fluid in the lower region is the Newtonian viscous fluid. The flow is driven by a constant pressure gradient. The presence of micropolar fluids introduces additional rotational parameters. Also, the porous material considered in both regions has two different permeabilities. A direct method is used to obtain the analytical solution of the concerned problem. In the present problem, the effects of the couple stress, the micropolarity parameter, the viscosity ratio, and the permeability on the velocity profile and the microrotational velocity are discussed. It is found that all the physical parameters play an important role in controlling the translational velocity profile and the microrotational velocity. In addition, numerical values of the different flow parameters are computed. The effects of the different flow parameters on the flow rate and the wall shear stress are also discussed graphically.     

Effective Rheology of Two-Phase Flow in Three-Dimensional Porous Media: Experiment and Simulation

We present an experimental and numerical study of immiscible two-phase flow of Newtonian fluids in three-dimensional (3D) porous media to find the relationship between the volumetric flow rate (Q) and the total pressure difference (\(\Delta P\)) in the steady state. We show that in the regime where capillary forces compete with the viscous forces, the distribution of capillary barriers at the interfaces effectively creates a yield threshold (\(P_t\)), making the fluids reminiscent of a Bingham viscoplastic fluid in the porous medium. In this regime, Q depends quadratically on an excess pressure drop (\(\Delta P-P_t\)). While increasing the flow rate, there is a transition, beyond which the overall flow is Newtonian and the relationship is linear. In our experiments, we build a model porous medium using a column of glass beads transporting two fluids, deionized water and air. For the numerical study, reconstructed 3D pore networks from real core samples are considered and the transport of wetting and non-wetting fluids through the network is modeled by tracking the fluid interfaces with time. We find agreement between our numerical and experimental results. Our results match with the mean-field results reported earlier.