Explicit Saturation Research Articles

Unsaturated flow in a packing of swelling particles; a grain-scale model

In this work, a grain-scale modelling technique is introduced for the simulation of unsaturated flow in deforming and swelling granular materials. To do so, a pore-scale model for unsaturated flow is coupled to the discrete element method (DEM). It is assumed that initially a dry packing of particles is quickly invaded by a liquid and becomes fully saturated. Particles start absorbing the liquid and this causes a rearrangement of particles, entrance of air into the packing, and a redistribution of the liquid (i.e. unsaturated flow). Flow was computed using a scheme of implicit pressure solver and explicit saturation update (IMPES), whilst particle movement was modelled using DEM. Simulations are continued until the packing is dry again. This is the first time that such a pore-scale model has been developed. We have used the model to investigate unsaturated flow during drying of a bed of swelling particles. Results indicated that the characteristic time scales of unsaturated flow and water absorption determine the swelling behaviour of the particle packing.

Non-classical flow modeling of spontaneous imbibition in spatially heterogeneous reservoirs

In this study, imbibition process is investigated in heterogeneous hydrocarbon reservoir using the non-classical (effective viscous finger) model equations proposed by Luo et al. (SPE Res. Eval. Engineering, 20(4), 1–16 2017). Counter-current spontaneous imbibition is analyzed in different heterogeneous porous media. The heterogeneity is mainly due to the spatial variation of the porosity and absolute permeability. The proposed mathematical model is solved considering the Implicit Pressure Explicit Saturation Scheme (IMPES) via finite difference method (FDM). Effect of the viscosity ratio, growth rate exponent of the fingering function, and the maximum fingering cross section on the normalized water saturation are analyzed. The present model is verified in the limiting case of classical model equations in homogeneous medium. It is noted that rate and extent of the counter-current imbibition are significantly controlled by the distribution of heterogeneous parameters like porosity and absolute permeability of the reservoir.

Numerical simulation of two-phase flows in 2-D petroleum reservoirs using a very high-order CPR method coupled to the MPFA-D finite volume scheme

The focus of the present work is to model the two-phase flow of oil and water in 2-D anisotropic petroleum reservoirs using a novel appropriate combination of a non-orthodox cell-centered Multipoint Flux Approximation with a Diamond Stencil (MPFA-D) finite volume method to discretize the pressure equation coupled to the very high-resolution Correction Procedure via Reconstruction (CPR) scheme for the discretization of the saturation equation. To suppress numerical oscillations (under/overshoots) near shocks, that are typical in higher-order schemes, and to deliver high accuracy in smooth regions of the solution, a hierarchical Multidimensional Limiting Process (MLP) is used in the reconstruction stage. The integration in time is carried out using a third-order Runge–Kutta method. The coupling of the pressure-saturation set of equations is carried out using a classical Implicit Pressure Explicit Saturation (IMPES) procedure. To properly couple the MPFA-D method with the CPR formulation, it is necessary to obtain an adequate velocity reconstruction throughout the control volumes of the mesh. Because the cell-centered finite volume method naturally delivers fluxes across control surfaces of the primal grid, then, we use a reconstruction operator based on the lowest order Raviart–Thomas interpolation functions and the Piola transformation, to get the complete knowledge of the velocity field throughout the domain. Finally, some representative problems were solved to evaluate the accuracy, efficiency, and shock-capturing capabilities of our new numerical methodology.

Thermodynamically consistent modeling of two-phase incompressible flows in heterogeneous and fractured media

Numerical modeling of two-phase flows in heterogeneous and fractured media is of great interest in petroleum reservoir engineering. The classical model for two-phase flows in porous media is not completely thermodynamically consistent since the energy reconstructed from the capillary pressure does not involve the ideal fluid energy of both phases and attraction effect between two phases. On the other hand, the saturation may be discontinuous in heterogeneous and fractured media, and thus the saturation gradient may be not well defined. Consequently, the classical phase-field models can not be applied due to the use of diffuse interfaces. In this paper, we propose a new thermodynamically consistent energy-based model for two-phase flows in heterogeneous and fractured media, which is free of the gradient energy. Meanwhile, the model inherits the key features of the traditional models of two-phase flows in porous media, including relative permeability, volumetric phase velocity and capillarity effect. To characterize the capillarity effect, a logarithmic energy potential is proposed as the free energy function, which is more realistic than the commonly used double well potential. The model combines with the discrete fracture model to describe two-phase flows in fractured media. The popularly used implicit pressure explicit saturation method is used to simulate the model. Finally, the experimental verification of the model and numerical simulation results are provided.

A locally and globally phase-wise mass conservative numerical algorithm for the two-phase immiscible flow problems in porous media

In this work, we introduce a novel numerical method to solve the problem of two-phase immiscible flow in porous media that is conservative to both phases. In the widely used implicit pressure, explicit saturation (IMPES) scheme, the conservation of mass of both the two phases are summed to form an equation involving the total Darcy’s velocity. In the discretization of such an equation it becomes difficult to enforce the conservation of mass of each phase. To guarantee the conservation of mass of both phases locally and hence globally, we introduce a scheme in which the time discretization of the mass conservation equations is considered separately. Cell-centered finite difference (CCFD) methods are adopted for spatial discretization, where the variables of fluid properties (i.e. relative permeability and mobility) are upwinded separately according to the velocity of each phase and not according to the total velocity. Furthermore, this new scheme updates all phase velocities and uses them to update the corresponding phase saturation. In addition, a two-scale of time-splitting methods are adopted for pressure equation and saturation equations to improve the computational efficiency. For the sake of simplicity, we show a number of examples of two-phase system in two-dimensional geometry solved using the new scheme. It is shown that the new scheme is more embracing the physics and it can be more accurate than traditional IMPES scheme, particularly for the cases in which the phase velocities are in opposite direction, and conventional IMPES schemes fail.

Mathematical modeling and simulation of water-alternating-gas (WAG) process by incorporating capillary pressure and hysteresis effects

Mathematical modeling and simulation of three-phase flow in porous media involves complexities related to the three-phase relative permeability, capillary pressure, and hysteresis effects that are cycle-dependent. Extensive theoretical studies are available in the literature, simulating immiscible and miscible water-alternating-gas (WAG) processes; however, the simulation of recovery behaviors and mechanisms of near-miscible WAG is overlooked. Also, a majority of WAG simulation studies lack the cycle-dependent three-phase hysteresis that appears in the relative permeability and capillary pressure models. In this paper, we study a three-phase flow modeling of near-miscible WAG process for EOR implication, using implicit pressure explicit saturation (IMPES) method. The mathematical model simulates a WAG case study in a strongly water-wet Berea sandstone core, using synthetic oil and brine at 38 °C and 12.7 MPa. Three cycles of water flooding (WF) and gas injection (GI) are used in the WAG operation. The recovery data from our mathematical model is in excellent agreement with the corresponding experimental data. The absolute relative error is less than 1.7%, while estimating the ultimate oil recovery factor in the WF and GI stages of all three cycles. We also study the effects of influential variables such as injection rate, WAG ratio, slug size, crude oil viscosity, and core absolute permeability on the WAG performance. The WAG ratio of 1 is found to be better than 0.5 and 2 cases. Although the absolute permeability changes the breakthrough recovery performance of the individual stages, it does not significantly affect the ultimate recovery of near-miscible WAG. Increasing the injection rate decreases the recovery factor. This effect is more pronounced in the GI cycles. Also, the ultimate recovery factor is reduced by 20% upon an increase in the viscosity of the in-situ oil. The findings of this research work can help to better understand the near-miscible WAG injection at various scenarios/conditions in terms of operational conditions and rock and fluid characteristics for the purposes of WAG design, operation, and optimization.

Splitting-based domain decomposition methods for two-phase flow with different rock types

In this paper, we are concerned with the global pressure formulation of immiscible incompressible two-phase flow between different rock types. We develop for this problem two robust schemes based on domain decomposition (DD) methods and operator-splitting techniques. The first scheme follows a sequential procedure in which the (global) pressure, the saturation-advection and the saturation-diffusion problems are fully decoupled. In this scheme, each problem is treated individually using various DD approaches and specialized numerical methods. The coupling between the different problems is explicit and the time-marching is with no iterations. To adapt to different time scales of problem components and different rock types, the novel scheme uses a multirate time stepping strategy, by taking multiple finer time steps for saturation-advection within one coarse time step for saturation-diffusion and pressure, and permits independent time steps for the advection step in the different rocks. In the second scheme, we review the classical Implicit Pressure–Explicit Saturation (IMPES) method (by decoupling only pressure and saturation) in the context of multirate coupling schemes and nonconforming-in-time DD approaches. For the discretization, the saturation-advection problem is approximated with the explicit Euler method in time, and in space with the cell-centered finite volume method of first order of Godunov type. The saturation-diffusion problem is approximated in time with the implicit Euler method and in space with the mixed finite element method, as in the pressure problem. Finally, in a series of numerical experiments, we investigate the practicality of the proposed schemes, the accuracy-in-time of the multirate and nonconforming time strategies, and compare the convergence of various DD methods within each approach.

Mathematical modeling of formation damage effect during underbalanced drilling operations

Formation damage during underbalanced drilling is created due to the spontaneous counter-current imbibition when drilling mud is in contact with saturated formations. Therefore, it should be appropriately recognized and investigated. The objective of this study is to develop a numerical model to simulate the penetration rate of drilling mud in the formation during underbalanced drilling. Modeling of this phenomenon is done by implementing its formulation through implicit pressure–explicit saturation (henceforth, IMPES) method. To validate the results of the developed model with a pilot study in this field, it is demonstrated that the model performs effectively in determining the penetration range of muds and finding the distribution of drilling mud saturation across the core. Consequently, the degree of damage to the formation through laboratory studies is usually time-consuming and therefore, the developed model can be employed for initial studies. Increasing the unbalanced pressure difference, oil viscosity and porosity will reduce the amount of drilling mud penetration in the formation during underbalanced drilling operations.

An analytical approach to investigate formation damage caused by underbalanced drilling performances

Underbalanced drilling performances have always considered as one of the proper solutions to prevent the formation damage problems during the drilling operational performances. To numerically model the spontaneous counter-current imbibition during the underbalanced drilling operational performances, different imbibition mechanism in the porous media should be taken into consideration. In this paper, a developed numerical approach was proposed to evaluate the various parameters which significantly affected the formation damage caused by underbalanced drilling performances. To verify the accuracy of the proposed model, which is analytically modeled by implicit pressure explicit saturation (henceforth; IMPES), a set of experimental CT scan data from a laboratory pilot was administered. Subsequently, the proposed analytical model is in a good agreement with CT scan laboratory data which clearly indicated the reliability of the model to evaluate filtrated mud saturation.

Iterative Numerical Scheme for Non-Isothermal Two-Phase Flow in Heterogeneous Porous Media

In the current paper, an iterative algorithm is developed to simulate the problem of two-phase flow with heat transfer in porous media. The convective body force caused by heat transfer is described by Boussinesq approximation throughout with the governing equations, namely, pressure, saturation, and energy. The two coupled equations of pressure and saturation are solved using the implicit pressure-explicit saturation (IMPES) scheme, while the energy equation is treated implicitly, and the scheme is called iterative implicit pressure, explicit saturation, implicit temperature (I-IMPES-IMT). In order to calculate the pressure implicitly, the equations of pressure and saturation are coupled by linearizing the capillary pressure which is a function of saturation. After that, the equation of saturation is solved explicitly. Then, the velocity is computed which is used in the energy equation to calculate the temperature implicitly. The cell-centered finite difference (CCFD) method is utilized for spatial discretization. Furthermore, a relaxation factor along is used with the Courant–Friedrichs–Lewy (CFL) condition. Finally, in order to illustrate the efficiency of the developed algorithm, error estimates for saturation and temperature for different values of time steps and number of iterations are presented. Moreover, numerical examples of different physical scenarios of heterogamous media are presented.

A Lax–Wendroff-IMPES scheme for a two-phase flow in porous media using interior penalty discontinuous Galerkin method

An immiscible two-phase flow numerical model in the porous media is developed using the interior penalty Discontinuous Galerkin (DG) finite element method. As a novel approach, the implicit pressure explicit saturation (IMPES) in conjunction with the second-order Lax–Wendroff based on the Taylor expansion was used. Further, the exchanging numerical fluxes were stabilized using projection of the velocity field in the H (div) vectorial interpolation space for improvement of results at heterogeneities. At the end of each time step, nonphysical oscillations were removed using the modified Chavent–Jaffre slope limiter and the calculated saturation values were modified. In order to assess the performance of this novel approach, the Buckley–Leverett and Mc Worther benchmark problems were considered. The model efficiency and capabilities have also been evaluated using two test cases for high heterogeneous porous media.

Fully mass-conservative IMPES schemes for incompressible two-phase flow in porous media

In this paper we consider fully mass-conservative numerical schemes for the simulation of incompressible and immiscible two-phase flow in porous media with capillary pressure. Compared with two kinds of conventional IMplicit Pressure Explicit Saturation (IMPES) schemes, the new schemes deserve a merit that the conservation of mass of both phases can be obtained. The total conservation equation is obtained by the summation of the discretized conservation equation for each phase. This approach is quite different from the conventional IMPES schemes. We present two kinds of fully mass-conservative IMPES schemes to solve the coupled systems for pressure, auxiliary velocity and saturation of each phase. The upwind mixed finite element methods are used to solve the pressure–velocity systems which can be decoupled, and the problems in the decoupled systems can be proved to be well-posed. Moreover, the new schemes are unbiased and the saturation of each phase can be proved to be bounds-preserving if the time step size is smaller than a certain value. The new schemes can also be applied to approximate the incompressible and immiscible two-phase flow in heterogeneous porous media with different capillarity pressures. Several interesting examples of incompressible and immiscible two-phase flow in porous media are presented to demonstrate the robustness of the new algorithms.

EFFECT OF A WAXY CRUDE OIL'S YIELD STRESS ON THE CONING PHENOMENON: A NUMERICAL STUDY

The effect of a waxy crude oil's yield stress is numerically investigated on the "water coning phenomenon" using the Bingham rheological model. After modifying Darcy's law for purely viscous non-Newtonian fluids, it was used to relate the flow rate to the pressure gradient. Having derived the governing equations for the axisymmetric case, the implicit pressure explicit saturation (IMPES) method was used to numerically solve the governing equations. To verify the code, a comparison was made with published data for the water coning of Newtonian/Newtonian pair. The effect of yield stress and plastic viscosity on the critical rate, the breakthrough time, and the cone shape was obtained numerically. It is shown that by an increase in the yield stress of the displaced oil, the time needed by the water to cone upward and break into the wellbore is decreased. That is to say, the yield stress of the displaced fluid has a negative effect on the water coning phenomenon. On the other hand, the yield stress induced in the displacing fluid by polymeric additives is predicted to delay the coning phenomenon.

An element-free Galerkin method using vertically integrated multiphase flow model for carbon sequestration

In this paper, the element-free Galerkin (EFG) method is applied to the study of vertically integrated two-phase flow of CO2 and brine in a deep saline aquifer. The methodology comprises of solving two simultaneous partial differential equations (PDEs) involving pressure and saturation terms, which are obtained by combining mass and momentum conservation equations. The obtained pressure and the average CO2 saturation resulting from the model will help to study the spatial and temporal distribution of the aquifer, bottom pressure buildup and the CO2 storage capacity of the aquifer respectively. The temporal analysis is solved using an IMplicit Pressure and Explicit Saturation (IMPES) solution strategy. A local refinement zone is introduced to capture the asymptotic pressure solution near the injection wells. Furthermore, the region surrounding the well is modelled as a non-homogeneous interface using Functionally Graded Materials (FGMs) analogy. Various case studies such as the horizontal aquifer, the sloped aquifer, the non-homogenous aquifer and, the horizontal aquifer with multiple wells have been presented to show the efficiency of the proposed model. The performance of the proposed EFG method is compared with that of FEM and XFEM numerical solutions. It is observed that the EFG method can capture average CO2 saturation (height of CO2) more effectively than the FEM and as effectively as the XFEM at the injection well. Moreover, it is easier to use the EFG method to solve complex aquifer problems, having non-homogeneity and multiple wells by considering a local nodal refinement around the well with ease. The simplicity of the proposed EFG method is demonstrated in this paper.

A Novel Framework for Integration of Random-Walk Particle-Tracking Simulation in Subsurface Multi-phase Flow Modeling

Coarse-scale models are generally preferred in the numerical simulation of multi-phase flow due to computational constraints. However, capturing the effects of fine-scale heterogeneity on flow and isolating the impacts of numerical (artificial) dispersion, which increases with scale, are not trivial. In this paper, a particle-tracking method is devised and integrated in a scale-up workflow to estimate the conditional probability distributions of multi-phase flow functions, which can be considered as inputs in coarse-scale simulations with existing commercial packages. First, a novel particle-tracking method is developed to solve the saturation transport equation. The transport calculation is coupled with a velocity update, following the implicit pressure, explicit saturation framework, to solve the governing equations of two-phase immiscible flow. Each phase particle is advanced in a deterministic convection step according to the phase velocity, as well as in a stochastic dispersion step based on the random Brownian motion. A kernel-based formulation is proposed for computation of fluid saturation in accordance with the phase particle distribution. A novel aspect is that this method employs the kernel approach to construct saturation from phase particle distribution, which is an important improvement to the conventional box method that necessitates a large number of particles per grid cell for consistent saturation interpolation. The model is validated against various analytical solutions. Finally, the validated model is integrated in a statistical scale-up procedure to calibrate effective, or “pseudo,” multi-phase flow functions (e.g., relative permeability functions). The proposed scale-up framework does not impose any length scale requirement regarding the distribution of sub-grid heterogeneities.

Waterflooding of Surfactant and Polymer Solutions in a Porous Media Micromodel

In this study, we examine microscale waterflooding in a randomly close-packed porous medium. Three different porosities were prepared in a microfluidic platform and saturated with silicone oil. Optical video fluorescence microscopy was used to track the water front as it flowed through the porous packed bed. The degree of water saturation was compared to water containing two different types of chemical modifiers, sodium dodecyl sulfate (SDS) and polyvinylpyrrolidone (PVP), with water in the absence of a surfactant used as a control. Image analysis of our video data yielded saturation curves and calculated fractal dimension, which we used to identify how morphology changed the way in which an invading water phase moved through the porous media. An inverse analysis based on the implicit pressure explicit saturation (IMPES) simulation technique used mobility ratio as an adjustable parameter to fit our experimental saturation curves. The results from our inverse analysis combined with our image analysis show that this platform can be used to evaluate the effectiveness of surfactants or polymers as additives for enhancing the transport of water through an oil-saturated porous medium.

Dynamic Pore-Scale Model of Drainage in Granular Porous Media: The Pore-Unit Assembly Method.

Dynamics of drainage is analyzed for packings of spheres, using numerical experiments. For this purpose, a dynamic pore‐scale model was developed to simulate water flow during drainage. The pore space inside a packing of spheres was extracted using regular triangulation, resulting in an assembly of grain‐based tetrahedra. Then, pore units were constructed by identifying and merging tetrahedra that belong to the same pore, resulting in an assembly of pore units. Each pore unit was approximated by a volume‐equivalent regular shape (e.g., cube and octahedron), for which a local capillary pressure‐saturation relationship was obtained. To simulate unsaturated flow, a pore‐scale version of IMPES (implicit pressure solver and explicit saturation update) was employed in order to calculate pressure and saturation distributions as a function of time for the assembly of pore units. To test the dynamic model, it was used on a packing of spheres to reproduce the corresponding measured quasi‐static capillary pressure‐saturation curve for a sand packing. Calculations were done for a packing of spheres with the same grain size distribution and porosity as the sand. We obtained good agreement, which confirmed the ability of the dynamic code to accurately describe drainage under low flow rates. Simulations of dynamic drainage revealed that drainage occurred in the form of finger‐like infiltration of air into the pore space, caused by heterogeneities in the pore structure. During the finger‐like infiltration, the pressure difference between air and water was found to be significantly higher than the capillary pressure. Furthermore, we tested the effects of the averaging, boundary conditions, domain size, and viscosity on the dynamic flow behavior. Finally, the dynamic coefficient was determined and compared to experimental data.

Stochastic porous media modeling and high-resolution schemes for numerical simulation of subsurface immiscible fluid flow transport

This paper proposes stochastic petroleum porous media modeling for immiscible fluid flow simulation using Dykstra–Parson coefficient (VDP) and autocorrelation lengths to generate 2D stochastic permeability values which were also used to generate porosity fields through a linear interpolation technique based on Carman–Kozeny equation. The proposed method of permeability field generation in this study was compared to turning bands method (TBM) and uniform sampling randomization method (USRM). On the other hand, many studies have also reported that, upstream mobility weighting schemes, commonly used in conventional numerical reservoir simulators do not accurately capture immiscible displacement shocks and discontinuities through stochastically generated porous media. This can be attributed to high level of numerical smearing in first-order schemes, oftentimes misinterpreted as subsurface geological features. Therefore, this work employs high-resolution schemes of SUPERBEE flux limiter, weighted essentially non-oscillatory scheme (WENO), and monotone upstream-centered schemes for conservation laws (MUSCL) to accurately capture immiscible fluid flow transport in stochastic porous media. The high-order schemes results match well with Buckley Leverett (BL) analytical solution without any non-oscillatory solutions. The governing fluid flow equations were solved numerically using simultaneous solution (SS) technique, sequential solution (SEQ) technique and iterative implicit pressure and explicit saturation (IMPES) technique which produce acceptable numerical stability and convergence rate. A comparative and numerical examples study of flow transport through the proposed method, TBM and USRM permeability fields revealed detailed subsurface instabilities with their corresponding ultimate recovery factors. Also, the impact of autocorrelation lengths on immiscible fluid flow transport were analyzed and quantified. A finite number of lines used in the TBM resulted into visual artifact banding phenomenon unlike the proposed method and USRM. In all, the proposed permeability and porosity fields generation coupled with the numerical simulator developed will aid in developing efficient mobility control schemes to improve on poor volumetric sweep efficiency in porous media.

Enriched Galerkin methods for two-phase flow in porous media with capillary pressure

In this paper, we propose an enriched Galerkin (EG) approximation for a two-phase pressure saturation system with capillary pressure in heterogeneous porous media. The EG methods are locally conservative, have fewer degrees of freedom compared to discontinuous Galerkin (DG), and have an efficient pressure solver. To avoid non-physical oscillations, an entropy viscosity stabilization method is employed for high order saturation approximations. Entropy residuals are applied for dynamic mesh adaptivity to reduce the computational cost for larger computational domains. The iterative and sequential IMplicit Pressure and Explicit Saturation (IMPES) algorithms are treated in time. Numerical examples with different relative permeabilities and capillary pressures are included to verify and to demonstrate the capabilities of EG.

On the efficiency of the IMPES method for two phase flow problems in porous media

In this work, we show a method to efficiently solve the multiphase flow through a porous media in the near wellbore region. The numerical discretisation is based on the IMplicit Pressure Explicit Saturation (IMPES) approach.While most of the works in the literature rely on Fully Implicit Methods (FIM) to simulate the reservoir, this is not suited for the near wellbore region, where smaller computational and physical times are required, therefore parametric models with history matching are used in this zone. However, parametric models present several uncertainties that can affect the estimation of the pressure drop. An accurate and fast simulation of that region is therefore required.Here, we propose a new and robust implementation of the IMPES scheme, aiming to reduce the large computational cost of the (implicit) pressure system of equations and increase the robustness and reliability of the scheme. The method takes advantage of the short physical time steps between iterations, observed in the near wellbore region, to produce a new pressure field at the cost of a numerical explicit scheme. Two and three dimensional numerical experiments are carried out to show the validity of the proposed discretisation. We show that the cost of the method is reduced in a factor of 7–19 when compared to the classical IMPES.