Effect of fluids properties on non-equilibrium capillarity effects: Dynamic pore-network modeling

We have developed a Dynamic Pore-network model for Simulating Two-phase flow in porous media (DYPOSIT). The model is applicable to both drainage and imbibition processes. Employing improved numerical and geometrical features in the model facilitate a physically-based pore-scale simulator. This computational tool is employed to perform several numerical experiments (primary and main drainage, main imbibition) to investigate the current capillarity theory. Traditional two-phase flow formulations state that the pressure difference between the two phase is equal to the capillary pressure, which is assumed to be a function of saturation only. Many theoretical and experimental studies have shown that this assumption is invalid and the pressure difference between the two fluids is not only equal to the capillary pressure but is also related to the variation of saturation with time in the domain; this is referred to as the non-equilibrium capillarity effect. To date, non-equilibrium capillarity effect has been investigated mainly under drainage. In this study, we analyze the non-equilibrium capillarity theory under drainage and imbibition as a function of saturation, viscosity ratio, and effective viscosity. Other aspects of the dynamics of two-phase flow such as trapping and saturation profile are also studied.     

A Mathematical Model for Hysteretic Two-Phase Flow in Porous Media

We develop a mathematical model for hysteretic two-phase flow (of oil and water) in waterwet porous media. To account for relative permeability hysteresis, an irreversible trapping-coalescence process is described. According to this process, oil ganglia are created (during imbibition) and released (during drainage) at different rates, leading to history-dependent saturations of trapped and connected oil. As a result, the relative permeability to oil, modelled as a unique function of the connected oil saturation, is subject to saturation history. A saturation history is reflected by history parameters, that is by both the saturation state (of connected and trapped oil) at the most recent flow reversal and the most recent water saturation at which the flow was a primary drainage. Disregarding capillary diffusion, the flow is described by a hyperbolic equation with the connected oil saturation as unknown. This equation contains functional relationships which depend on the flow mode (drainage or imbibition) and the history parameters. The solution consists of continuous waves (expansion waves and constant states), shock waves (possibly connecting different modes) and stationary discontinuities (connecting different saturation histories). The entropy condition for travelling waves is generalized to include admissible shock waves which coincide with flow reversals. It turns out that saturation history generally has a strong influence on both the type and the speed of the waves from which the solution is constructed.     

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.     

Theoretical Development of a Novel Equation for Dynamic Spontaneous Imbibition with Variable Inlet Saturation and Interfacial Coupling Effects

In oil recovery from fractured reservoirs, dynamic spontaneous imbibition (DSI) plays an important role. Conventional equations used for characterizing dynamic spontaneous imbibition neglect the effects of the driving forces acting across the wetting and non-wetting phases which are flowing in opposite directions. Such effects, defined as interfacial coupling effects (ICE), are known to cause a decrease in the calculated flow rate in drainage processes. Moreover, none of the numerical models have considered a variable inlet saturation (S*) for DSI. A new theoretical model has been developed using generalized transport equations to describe dynamic spontaneous imbibition for immiscible two-phase flow processes. The inclusion of interfacial coupling effects provides a more accurate way to describe dynamic spontaneous imbibition. Furthermore, the addition of variable inlet saturation allows one to establish whether the inlet-face saturation (S*) increases from the initial saturation to 1−Sro, or whether it can remain constant and equal to one minus the residual saturation to the non-wetting phase (1−Sro).     

Percolation theory approach to transport phenomena in porous media

The percolation theory approach to static and dynamic properties of the single- and two-phase fluid flow in porous media is described. Using percolation cluster scaling laws, one can obtain functional relations between the saturation fraction of a given phase and the capillary pressure, the relative permeability, and the dispersion coefficient, in drainage and imbibition processes. In addition, the scale dependency of the transport coefficient is shown to be an outcome of the fractal nature of pore space and of the random flow pattern of the fluids or contaminant.     

Bulk Flow Regimes and Fractional Flow in 2D Porous Media by Numerical Simulations

We investigate a two-dimensional network simulator that models the dynamics of two-phase immiscible bulk flow where film flow can be neglected. We present a method for simulating the detailed dynamical process where the two phases are allowed to break up into bubbles, and bubbles are allowed to merge together. The notions of drainage and imbibition are not adequate to describe this process since there is no clear front between the fluids. In fact, the simulator is constructed so that one can study the behaviour of the system far from inlets and outlets, where the two fluids have been mixed together so much that all initial fronts have broken up. The simulator gives the fractional flow as a function of the saturation of each of the fluids. For the case of two fluids with equal viscosity, we classify flow regimes that are parametrized by the capillary number.     

Pore Network Analysis of Resistivity Index for Water-Wet Porous Media

Pore network analysis is used to investigate the effects of microscopic parameters of the pore structure such as pore geometry, pore-size distribution, pore space topology and fractal roughness porosity on resistivity index curves of strongly water-wet porous media. The pore structure is represented by a three-dimensional network of lamellar capillary tubes with fractal roughness features along their pore-walls. Oil-water drainage (conventional porous plate method) is simulated with a bond percolation-and-fractal roughness model without trapping of wetting fluid. The resistivity index, saturation exponent and capillary pressure are expressed as approximate functions of the pore network parameters by adopting some simplifying assumptions and using effective medium approximation, universal scaling laws of percolation theory and fractal geometry. Some new phenomenological models of resistivity index curves of porous media are derived. Finally, the eventual changes of resistivity index caused by the permanent entrapment of wetting fluid in the pore network are also studied.Resistivity index and saturation exponent are decreasing functions of the degree of correlation between pore volume and pore size as well as the width of the pore size distribution, whereas they are independent on the mean pore size. At low water saturations, the saturation exponent decreases or increases for pore systems of low or high fractal roughness porosity respectively, and obtains finite values only when the wetting fluid is not trapped in the pore network. The dependence of saturation exponent on water saturation weakens for strong correlation between pore volume and pore size, high network connectivity, medium pore-wall roughness porosity and medium width of the pore size distribution. The resistivity index can be described succesfully by generalized 3-parameter power functions of water saturation where the parameter values are related closely with the geometrical, topological and fractal properties of the pore structure.     

Simulation of capillary pressure curves using bond correlated site percolation on a simple cubic network

A stochastic approach to network modelling has been used to simulate quasi-static immiscible displacement in porous media. Both number-based and volume-based network saturation results were obtained. Number-based results include: number-based saturation curves for primary drainage, secondary imbibition and secondary drainage, fluid distribution data, and cluster trapping history. Using pore structure data of porous media, it is possible to convert the number-based curves to capillary pressure — saturation relationships. Pore size distribution functions and pore shapes which are thought to closely represent Berea sandstone samples were used to predict the capillary curves. The physical basis of these calculations is a one-to-one correspondence between the cumulative node and bond index fractions in the network analysis, and the cumulative number-based distributions of pore body and pore throat diameters, respectively. The oil-water capillary pressure curve simulated for primary drainage closely resembles those measured experimentally. The agreement between the simulated and the measured secondary imbition and secondary drainage curves is less satisfactory.     

Effect of dynamic contact angle variation on spontaneous imbibition in porous materials

We investigate the influence of contact angle variations on spontaneous imbibition of moisture in porous materials. While the contact angle is typically assumed constant when modelling the moisture transfer in porous media, experimental findings put this assumption into question. It has been shown that during imbibition the contact angle notably rises with increasing meniscus velocity. This phenomenon resultantly affects the moisture retention curve, the relation linking the local capillary pressure to the local moisture saturation, which in turn impacts the imbibition rate and moisture distribution. This study investigates these dynamic effects via a pore network technique as well as a continuum approach. It is shown that the impacts of pore-scale contact angle variations on the imbibition process can be reproduced at the continuum scale through a modified moisture retention curve including a dynamic term. Complementarily a closed-form equation expressing the dynamic capillary pressure in terms of local saturation and saturation rate is derived. The continuum approach is then finally employed to predict measured moisture saturation profiles for imbibition in Berea sandstone and diatomite found in literature, and a fair agreement between simulated and measured outcomes is observed.

     

A Parametric Model for Predicting Relative Permeability-Saturation-Capillary Pressure Relationships of Oil–Water Systems in Porous Media with Mixed Wettability

A parametric two-phase, oil–water relative permeability/capillary pressure model for petroleum engineering and environmental applications is developed for porous media in which the smaller pores are strongly water-wet and the larger pores tend to be intermediate- or oil-wet. A saturation index, which can vary from 0 to 1, is used to distinguish those pores that are strongly water-wet from those that have intermediate- or oil-wet characteristics. The capillary pressure submodel is capable of describing main-drainage and hysteretic saturation-path saturations for positive and negative oil–water capillary pressures. At high oil–water capillary pressures, an asymptote is approached as the water saturation approaches the residual water saturation. At low oil–water capillary pressures (i.e. negative), another asymptote is approached as the oil saturation approaches the residual oil saturation. Hysteresis in capillary pressure relations, including water entrapment, is modeled. Relative permeabilities are predicted using parameters that describe main-drainage capillary pressure relations and accounting for how water and oil are distributed throughout the pore spaces of a porous medium with mixed wettability. The capillary pressure submodel is tested against published experimental data, and an example of how to use the relative permeability/capillary pressure model for a hypothetical saturation-path scenario involving several imbibition and drainage paths is given. Features of the model are also explained. Results suggest that the proposed model is capable of predicting relative permeability/capillary pressure characteristics of porous media mixed wettability.     

Efficiency of Capillary Imbibition Dominated Displacement of Nonwetting Phase by Wetting Phase in Fractured Porous Media

The critical and optimum injection rates as well as the critical fracture capillary number for an efficient displacement process are determined based on the experimental and numerical modeling of the displacement of nonwetting phase (oil) by wetting phase (water) in fractured porous media. The efficiency of the process is defined in terms of the nonwetting phase displaced from the system per amount of wetting phase injected and per time. Also, the effects of injection rate on capillary imbibition transfer dominated two-phase flow in fractured porous media are clarified by visualizing the experiments. The results reveal that as the injection rate is increased, fracture pattern begins to become an effective parameter on the matrix saturation distribution. As the rate is lowered, however, the system begins to behave like a homogeneous system showing a frontal displacement regardless the fracture configuration.     

Relative permeabilities from two- and three-dimensional pore-scale network modelling

We present a computer study of two-phase flow in a porous medium. The porous medium is represented by an isotropic network of up to 80 000 randomly placed nodes connected by thin tubes. We then simulate two-fluid displacements in this network and are able to demonstrate the effects of viscous and capillary forces. We use the local average flow rates and pressures to calculate effective saturation dependent relative pemeabilities, fractional flows and capillary pressures. Using a radial Buckley-Leverett theory, the mean saturation profile can be inferred from the solution of the fractional flow equation, which is consistent with the computed saturation. We show that the relative permeability may be a function of both viscosity ratio and capillary number.     

Matrix–Fracture Transfer through Countercurrent Imbibition in Presence of Fracture Fluid Flow

Although a lot of research has been done in modeling the oil recovery from fractured reservoirs by countercurrent imbibition, less attention has been paid to the effect of the fracture fluid velocity upon the rate of oil recovery. Experiments are conducted to determine the effect of fracture flow rate upon countercurrent imbibition. A droplet detachment model is proposed to derive the effective water saturation in a thin boundary layer at the matrix–fracture interface. This effective boundary water saturation is a function of fluid properties, fluid velocity in the fracture and fracture width. For a highly water–wet porous medium, this model predicts an increase in the boundary water saturation with increase in fracture fluid velocity. The increase in boundary water saturation, in turn, increases the oil recovery rate from the matrix, which is consistent with the experimental results. The model also predicts that the oil recovery rate does not vary linearly with the boundary water saturation.     

Lattice-Boltzmann simulations of flow through Fontainebleau sandstone

We report preliminary results from simulations of single-phase and two-phase flow through three-dimensional tomographic reconstructions of Fontainebleau sandstone. The simulations are performed with the lattice-Boltzmann method, a variant of lattice-gas cellular-automation models of fluid mechanics. Simulations of single-phase flow on a sample of linear size 0.2 cm yield a calculated permeability in the range 1.0–1.5 darcys, depending on direction, which compares qualitatively well with a laboratory measurement of 1.3 darcys on a sample approximately an order of magnitude larger. The sensitivity of permeability calculations to sample size, grid resolution, and choice of model parameters is quantified empirically. We also present a qualitative study of immiscible two-phase flow in a sample of linear size 0.05 cm; simulations of both drainage and imbibition are presented.     

Numerical Solution of Equation for Dynamic,Spontaneous Imbibition with Variable Inlet Saturation and Interfacial Coupling Effects

In the oil industry, dynamic spontaneous imbibition plays an important role in several flow processes in porous media. A numerical approach is developed to simulate dynamic spontaneous imbibition with variable inlet saturation and interfacial coupling. The inclusion of interfacial coupling effects invalidates the assumption that the interfaces (fluid/fluid and fluid/solid) act in the same way. The one-dimensional numerical simulation model is developed using a Lagrangian formulation discretized in time and saturation. The solution of the partial differential equations utilizes an iteration process that includes two material balance criteria to ensure the validity of the variable inlet saturation. Furthermore, an error analysis, the validation of the model and a sensitivity study on the optimal number of time steps and saturation grid cells are undertaken. The numerical simulation solution represents an accurate approach to investigate the effect of fluid and rock properties on dynamic spontaneous imbibition.     

Mathematical model of two-phase fluid nonlinear flow in low-permeability porous media with applications

IntroductionItisasuccessfulexampleinadevelopmentstoryofscienceandtechnologyformechanicsoffluidsinporousmediatocombinewithengineeringtechnology .Fieldsinfluencedbythemechanicsinvolveddevelopmentofoil_gasandgroundwaterresources,controlonseawaterintrusionandsubsidenceandgeologichazards,geotechnicalengineeringandbioengineering ,andairlineindustry[1～ 7].Aproblemonnonlinearflowinlow_permeabilityporousmediaisbutonlyabasiconeindifferentkindsofengineeringfields,butalsooneoffrontlineresearchfieldsofmod…     

On Barenblatt's Model of Spontaneous Countercurrent Imbibition

Water imbibition is a critical mechanism of secondary oil recovery from fractured reservoirs. Spontaneous imbibition also plays a significant role in storage of liquid waste by controlling the extent of rock invasion. In the present paper, we extend a model of countercurrent imbibition based on Barenblatt's theory of non-equilibrium two-phase flow by allowing the model's relaxation time to be a function of the wetting fluid saturation. We obtain two asymptotic self-similar solutions, valid at early and late times, respectively. At a very early stage, the time scale characterizing the cumulative volume of imbibed (and expelled) fluid is a power function with exponent between 1.5 and 1. At a later stage, the time scaling for this volume approaches asymptotically classical square root of time, whereas the saturation profile asymptotically converges to Ryzhik's self-similar solution. Our conclusions are verified against experiments. By fitting the laboratory data, we estimate the characteristic relaxation times for different pairs of liquids.