Two-phase Displacement Research Articles

One-dimensional model of two-phase fluid displacement in a slot with permeable walls

A one-dimensional model is proposed for transportation of a two-phase fluid (sandcontaining fluid and pure fluid) in the Hele-Shaw cell with permeable walls through which the pure fluid can leak off, causing the growth of the sand concentration. The model describes the process of pure fluid displacement with the emergence of the Saffman–Taylor instability and extends Koval’s model to the case of sand concentration variation owing to pure fluid outflow through the cell walls. The Riemann problem is analyzed. New flow configurations, which are not predicted by Koval’s model, are discovered.

Displacement front behavior of near miscible CO2 flooding in decane saturated synthetic sandstone cores revealed by magnetic resonance imaging

It is of great importance to study the CO2-oil two-phase flow characteristic and displacement front behavior in porous media, for understanding the mechanisms of CO2 enhanced oil recovery. In this work, we carried out near miscible CO2 flooding experiments in decane saturated synthetic sandstone cores to investigate the displacement front characteristic by using magnetic resonance imaging technique. Experiments were done in three consolidated sandstone cores with the permeabilities ranging from 80 to 450mD. The oil saturation maps and the overall oil saturation during CO2 injections were obtained from the intensity of magnetic resonance imaging. Finally the parameters of the piston-like displacement fronts, including the front velocity and the front geometry factor (the length to width ratio) were analyzed. Experimental results showed that the near miscible vertical upward displacement is instable above the minimum miscible pressure in the synthetic sandstone cores. However, low permeability can restrain the instability to some extent.

An extended JBN method of determining unsteady-state two-phase relative permeability

Relative permeability is the reduction of permeability of porous media when subjected to multiphase flow and a key parameter in subsurface hydrology. The JBN method is a well-known method of obtaining relative permeability, which measures the overall pressure drop and the effluent phase ratio versus time during two-phase displacements. By assuming no capillary pressure or gravity, the JBN method obtains the relative permeabilities to both phases at the core outlet. Since data across a range of saturations are acquired in a relatively short time, this method is widely used. This work extends the JBN method by having (1) section-wise pressure drop measurements between the core inlet, four pressure taps on the core and the outlet, (2) local saturation measurements, and (3) local phase fluxes. With these data, the extended JBN method can determine relative permeabilities to both phases at each pressure tap of the core (not just at the core outlet). The JBN extension is shown using a data set where CO2 invades a brine-filled core. From this it is found that the advantages of the extended JBN method over the regular JBN method are: (1) four times more data are obtained, and (2) data are more accurate because the capillary end effect is experimentally avoided. Avoiding the end effect results in tripling the saturation range, and obtaining relative permeabilities that are consistent with steady-state measurements and roughly 40% higher than those from the regular JBN method.

Measurement of specific storage of rock for supercritical carbon dioxide using constant flow pump and constant head permeability techniques

Abstract Advanced laboratory system of rock permeability test associated with constant flow pump, and constant head permeability techniques were developed to measure permeability and specific storage of rock for supercritical CO2. The laboratory system was designed to be capable in reproducing similar physical condition of deep aquifer within high pressure and high temperature where CO2 tends to be in supercritical state. To analyze the result of permeability tests, mathematical models of constant flow pump and constant head permeability techniques were modified to deal with two-phase flow drainage displacement of CO2-water in rock. For the examination of its applicability, experimental tests and numerical analysis were undertaken. The accuracy of the obtained specific storage was validated by employing a ratio of the specific storage of the rock specimen to the storage capacity of the pump used in the permeability test. It was found that the specific storage of low permeability sandstone for storing supercritical CO2 is 1.63 × 10−4 1/Pa, while large permeability sandstone has the specific storage for supercritical CO2 at 1.12 × 10−7 1/Pa. This finding suggested that advanced experimental system of constant flow pump and constant head permeability technique can be used as repeatable, accurate and standardized laboratory test in measuring specific storage of sedimentary rock for CO2 in supercritical state.

Relative Permeability Characterization for Water-Alternating-Gas Injection in Oil Reservoirs

Summary Large quantities of oil usually remain in oil reservoirs after conventional waterfloods. A significant part of this remaining oil can still be economically recovered by water-alternating-gas (WAG) injection. WAG injection involves drainage and imbibition processes taking place sequentially; therefore, the numerical simulation of the WAG process requires reliable knowledge of three-phase relative permeability (kr) accounting for cyclic-hysteresis effects. In this study, the results of a series of unsteady-state two-phase displacements and WAG coreflood experiments were used to investigate the behavior of three-phase kr and hysteresis effects in the WAG process. The experiments were performed on two different cores with different characteristics and wettability conditions. An in-house coreflood simulator was developed to obtain three-phase relative permeability values directly from unsteady-state WAG experiments by history matching the measured recovery and differential-pressure profiles. The results show that three-phase gas relative permeability is reduced in consecutive gas-injection cycles and consequently the gas mobility and injectivity drop significantly with successive gas injections during the WAG process, under different rock conditions. The trend of hysteresis in the relative permeabilty of gas (krg) partly contradicts the existing hysteresis models available in the literature. The three-phase water relative permeability (krw) of the water-wet (WW) core does not exhibit considerable hysteresis effect during different water injections, whereas the mixed-wet (MW) core shows slight cyclic hysteresis. This may indicate a slight increase of the water injectivity in the subsequent water injections in the WAG process under MW conditions. Insignificant hysteresis is observed in the oil relative permeability (kro) during different gas-injection cycles for both WW and MW rocks. However, a considerable cyclic-hysteresis effect in kro is observed during water-injection cycles of WAG, which is attributed to the reduction of the residual oil saturation (ROS) during successive water injections. The kro of the WW core exhibits much-more cyclic-hysteresis effect than that of the MW core. No models currently exist in reservoir simulators that can capture the observed cyclic-hysteresis effect in oil relative permeability for the WAG process. Investigation of relative permeability data obtained from these displacement tests at different rock conditions revealed that there is a significant discrepancy between two-phase and three-phase relative permeability of all fluids. This highlights that not only the three-phase relative permeability of the intermediate phase (oil), but also the three-phase kr of the wetting phase (water) and nonwetting phase (gas) are functions of two independent saturations.

Adaptive Mesh Optimization for Simulation of Immiscible Viscous Fingering

Summary Viscous fingering can be a major concern when waterflooding heavy-oil reservoirs. Most commercial reservoir simulators use low-order finite-volume/-difference methods on structured grids to resolve this phenomenon. However, this approach suffers from a significant numerical-dispersion error because of insufficient mesh resolution, which smears out some important features of the flow. We simulate immiscible incompressible two-phase displacements and propose the use of unstructured control-volume finite-element (CVFE) methods for capturing viscous fingering in porous media. Our approach uses anisotropic mesh adaptation where the mesh resolution is optimized on the basis of the evolving features of flow. The adaptive algorithm uses a metric tensor field dependent on solution-interpolation-error estimates to locally control the size and shape of elements in the metric. The mesh optimization generates an unstructured finer mesh in areas of the domain where flow properties change more quickly and a coarser mesh in other regions where properties do not vary so rapidly. We analyze the computational cost of mesh adaptivity on unstructured mesh and compare its results with those obtained by a commercial reservoir simulator on the basis of the finite-volume methods.

Petrophysical characterization of porous media starting from micro-tomographic images

A quasi-static scheme based on pore space spatial statistics is presented to simulate pore-scale two-phase capillary-dominant displacement processes. The algorithm is coupled with computational fluid dynamics in order to evaluate saturation functions. Wettability heterogeneity in partial and fractional/mixed-wet media is implemented using a contact angle map. The simulation process is pixel-wised and performed directly on binary images. Bypassing and snap-off are tackled as non-wetting phase trapping mechanisms. Post-processing results include residual saturations, effective permeability and capillary pressure curves for drainage and imbibition scenarios. The primary advantages of the proposed workflow are eliminating pore space skeletisation/ discretization, superior time efficiency and minimal numerical drawbacks when compared to other direct or network-based simulation techniques.

Effects of intermediate wettability on entry capillary pressure in angular pores

Entry capillary pressure is one of the most important factors controlling drainage and remobilization of the capillary-trapped phases as it is the limiting factor against the two-phase displacement. It is known that the entry capillary pressure is rate dependent such that the inertia forces would enhance entry of the non-wetting phase into the pores. More importantly the entry capillary pressure is wettability dependent. However, while the movement of a meniscus into a strongly water-wet pore is well-defined, the invasion of a meniscus into a weak or intermediate water-wet pore especially in the case of angular pores is ambiguous. In this study using OpenFOAM software, high-resolution direct two-phase flow simulations of movement of a meniscus in a single capillary channel are performed. Interface dynamics in angular pores under drainage conditions have been simulated under constant flow rate boundary condition at different wettability conditions. Our results shows that the relation between the half corner angle of pores and contact angle controls the temporal evolution of capillary pressure during the invasion of a pore. By deviating from pure water-wet conditions, a dip in the temporal evolution of capillary pressure can be observed which will be pronounced in irregular angular cross sections. That enhances the pore invasion with a smaller differential pressure. The interplay between the contact angle and pore geometry can have significant implications for enhanced remobilization of ganglia in intermediate contact angles in real porous media morphologies, where pores are very heterogeneous with small shape factors.

Effects of shear-thinning fluids on residual oil formation in microfluidic pore networks

Two-phase immiscible displacement in porous media is controlled by capillary and viscous forces when gravitational effects are negligible. The relative importance of these forces is quantified through the dimensionless capillary number Ca and the viscosity ratio M between fluid phases. When the displacing fluid is Newtonian, the effects of Ca and M on the displacement patterns can be evaluated independently. However, when the injecting fluids exhibit shear-thinning viscosity behaviour the values of M and Ca are interdependent. Under these conditions, the effects on phase entrapment and the general displacement dynamics cannot be dissociated. In the particular case of shear-thinning aqueous polymer solutions, the degree of interdependence between M and Ca is determined by the polymer concentration. In this work, two-phase immiscible displacement experiments were performed in micromodels, using shear-thinning aqueous polymer solutions as displacing fluids, to investigate the effect of polymer concentration on the relationship between Ca and M, the recovery efficiency, and the size distribution of the trapped non-wetting fluid. Our results show that the differences in terms of magnitude and distribution of the trapped phase are related to the polymer concentration which influences the values of Ca and M.

Influence of Initial Conditions on Propagation, Growth and Decay of Saturation Overshoot

A sequence of drainage and imbibition shocks within the traditional theory of two-phase immiscible displacement can give rise to shallow non-monotone saturation profiles as shown in Hilfer and Steinle (Eur Phys J Spec Top 223:2323, 2014). This phenomenon depends sensitively on model parameters and initial conditions. The dependence of saturation overshoot on initial conditions is investigated more systematically in this article. The results allow to determine regions in the parameter space for the observation of saturation overshoot and to explore limitations of the underlying idealized hysteresis model. Numerical solutions of the nonlinear partial differential equations of motion reveal a strong dependence of the overshoot phenomenon on the boundary and initial conditions. Overshoot solutions with experimentally detectable height are shown to exist numerically. Extensive parameter studies reveal different classes of initial conditions for which the width of the overshoot region can decrease, increase or remain constant.

Multiple-Mixing-Cell Method for Three-Hydrocarbon-Phase Displacements

Summary Low-temperature oil displacements by carbon dioxide involve complex phase behavior, in which three hydrocarbon phases can coexist. Reliable design of miscible gasflooding requires knowledge of the minimum miscibility pressure (MMP), which is the pressure required for 100% recovery in the absence of dispersion or as defined by slimtube experiments as the “knee” in the recovery curve with pressure in which displacement efficiency is greater than 90%. There are currently no analytical methods to estimate the MMP for multicomponent mixtures exhibiting three hydrocarbon phases. Also, the use of compositional simulators to estimate MMP is not always reliable. These challenges include robustness issues of three-phase equilibrium calculations, inaccurate three-phase relative permeability models, and phase identification and labeling problems that can cause significant discontinuities and failures in the simulation results. How miscibility is developed, or not developed, for a three-phase displacement is not well-known. We developed a new three-phase multiple-mixing-cell method that gives a relatively easy and robust way to determine the pressure for miscibility or, more importantly, the pressure for high-displacement efficiency. The procedure that moves fluid from cell to cell is robust because it is independent of phase labeling (i.e., vapor or liquid), has a robust way to provide good initial guesses for three-phase flash calculations, and is also not dependent on three-phase relative permeability (fractional flow). These three aspects give the mixing-cell approach significant advantages over the use of compositional simulation to estimate MMP or to understand miscibility development. One can integrate the approach with previously developed two-phase multiple-mixing-cell models because it uses the tie-line lengths from the boundaries of tie triangles to recognize when the MMP or pressure for high-displacement efficiency is obtained. Application of the mixing-cell algorithm shows that, unlike most two-phase displacements, the dispersion-free MMP may not exist for three-phase displacements, but rather a pressure is reached in which the dispersion-free displacement efficiency is maximized. The authors believe that this is the first paper to examine a multiple-mixing-cell model in which two- and three-hydrocarbon phases occur and to calculate the MMP and/or pressure required for high displacement efficiency for such systems.

Displacement flows under elastic membranes. Part 1. Experiments and direct numerical simulations

The injection of fluid into the narrow liquid-filled gap between a rigid plate and an elastic membrane drives a displacement flow that is controlled by the competition between elastic and viscous forces. We study such flows using the canonical set-up of an elastic-walled Hele-Shaw cell whose upper boundary is formed by an elastic sheet. We investigate both single- and two-phase displacement flows in which the localised injection of fluid at a constant flow rate is accommodated by the inflation of the sheet and the outward propagation of an axisymmetric front beyond which the cell remains approximately undeformed. We perform a direct comparison between quantitative experiments and numerical simulations of two theoretical models. The models couple the Föppl–von Kármán equations, which describe the deformation of the thin elastic membrane, to the equations describing the flow, which we model by (i) the Navier–Stokes equations or (ii) lubrication theory. We identify the dominant physical effects that control the behaviour of the system and critically assess modelling assumptions that were made in previous studies. The insight gained from these studies is then used in Part 2 of this work, where we formulate an improved lubrication model and develop an asymptotic description of the key phenomena.

Numerical simulation of displacement characteristics of CO2 injected in pore-scale porous media

Pore structure of porous media, including pore size and topology, is rather complex. In immiscible two-phase displacement process, the capillary force affected by pore size dominates the two-phase flow in the porous media, affecting displacement results. Direct observation of the flow patterns in the porous media is difficult, and therefore knowledge about the two-phase displacement flow is insufficient. In this paper, a two-dimensional (2D) pore structure was extracted from a sandstone sample, and the flow process that CO2 displaces resident brine in the extracted pore structure was simulated using the Navier–Stokes equation combined with the conservative level set method. The simulation results reveal that the pore throat is a crucial factor for determining CO2 displacement process in the porous media. The two-phase meniscuses in each pore throat were in a self-adjusting process. In the displacement process, CO2 preferentially broke through the maximum pore throat. Before breaking through the maximum pore throat, the pressure of CO2 continually increased, and the curvature and position of two-phase interfaces in the other pore throats adjusted accordingly. Once the maximum pore throat was broken through by the CO2, the capillary force in the other pore throats released accordingly; subsequently, the interfaces withdrew under the effect of capillary fore, preparing for breaking through the next pore throat. Therefore, the two-phase displacement in CO2 injection is accompanied by the breaking through and adjusting of the two-phase interfaces.

Multi-frame elastography using a handheld force-controlled ultrasound probe.

We propose a new method for strain field estimation in quasi-static ultrasound elastography based on matching RF data frames of compressed tissues. The method benefits from using a handheld force-controlled ultrasound probe, which provides the contact force magnitude and therefore improves repeatability of displacement field estimation. The displacement field is estimated in a two-phase manner using triplets of RF data frames consisting of a pre-compression image and two post-compression images obtained with lower and higher compression ratios. First, a reliable displacement field estimate is calculated for the first post-compression frame. Second, we use this displacement estimate to warp the second post-compression frame while using linear elasticity to obtain an initial approximation. Final displacement estimation is refined using the warped image. The two-phase displacement estimation allows for higher compression ratios, thus increasing the practical resolution of the strain estimates. The strain field is computed from a displacement field using a smoothness- regularized energy functional, which takes into consideration local displacement estimation quality. The minimization is performed using an efficient primal-dual hybrid gradient algorithm, which can leverage the architecture of a graphical processing unit. The method is quantitatively evaluated using finite element simulations. We compute strain estimates for tissue-mimicking phantoms with known elastic properties and finally perform a qualitative validation using in vivo patient data.

Two-phase flow properties of a horizontal fracture: The effect of aperture distribution

A systematic numerical method has been presented to investigate the constitutive relationships between two-phase flow properties of horizontal fractures and aperture distributions. Based on fractal geometry, single rough-walled fractures are generated numerically by modified successive random addition (SRA) method and then aperture distributions with truncated Gaussian distribution are formed by shear displacement between lower and upper surfaces. (The truncated Gaussian distribution is used to describe aperture evolution under different normal stresses.) According to the assumption of two-dimensional porous media and local parallel plate model, invasion percolation approach is employed to model the two-phase flow displacement (imbibition) in generated horizontal fractures, in which capillary forces are dominant over viscous and gravity forces. For truncated Gaussian distributions, constitutive relationships from numerical simulation are compared to closed-form relationships and a good agreement is obtained. The simulation results indicate strong phase interference with the sum of two phase relative permeability values being less than one in the intermediate saturations. It is found that fracture properties related to residual saturations depend on spatial correlation of aperture distributions. Based on the simulation results, we proposed an empirical relationship between the fracture residual-saturation-rated parameters and the corresponding aperture distributions.

Design and Simulation Analysis of Two-Phase Grating-Type Displacement Sensors with Unequal-Pitch

With the development of grating displacement sensors towards higher precision and higher speed of dynamic responses, the traditional design structure is hard to solve the processing difficulty of the increasing number of poles for time grating sensors. In order to realize the analysis on the internal magnetic field distribution of high precision displacement sensor, the unequal pitch grating displacement sensor with two phase model is established using Ansoft Maxwell simulation environment. The feasibility of the two unequal pitch grating displacement sensor and magnetic field simulation are performed based on the proposed sensor model. In addition the detailed synthetic method is proposed for the simulation data. The analysis show the feasibility of the unequal pitch grating displacement sensor with two phase. The proposed displacement sensor decreases the difficulties of machining process technology and increase the pole number. As the result, the measuring precision is improved.

CO2injectivity in saline aquifers: The impact of non-Darcy flow, phase miscibility, and gas compressibility

A key aspect of CO2 storage is the injection rate into the subsurface, which is limited by the pressure at which formation starts to fracture. Hence, it is vital to assess all of the relevant processes that may contribute to the pressure increase in the aquifer during CO2 injection. Building on an existing analytical solution for immiscible and spatially varying non-Darcy flow, this paper presents a mathematical model that accounts for combined effects of non-Darcy flow, phase miscibility, and gas compressibility in radial two-phase displacements. Results show that in low-permeability formations when CO2 is injected at high rates, non-Darcy simulations forecast better displacement efficiency compared to flow under Darcy conditions. This will have a positive effect on the formation CO2 storage capacity. This, however, comes at the cost of increased well pressures. More favorable estimations of the pressure buildup are obtained when CO2 compressibility is taken into account because reservoir pressures are reduced due to the change in the gas phase properties. Also, non-Darcy flow results in a significant reduction in halite precipitation in the near-well region, with a positive effect on CO2 injectivity. In the examples shown, non-Darcy flow conditions may lead to significantly different pressure and saturation distributions in the near-well region, with potentially important implications for CO2 injectivity.

An Experimental Study of Three-Phase Trapping in Sand Packs

The trapped saturations of oil and gas are measured as functions of initial oil and gas saturation in water-wet sand packs. Analogue fluids—water, octane and air—are used at ambient conditions. Starting with a sand-pack column which has been saturated with brine, oil (octane) is injected with the column horizontal until irreducible water saturation is reached. The column is then positioned vertically and air is allowed to enter from the top of the column, while oil is allowed to drain under gravity for varying lengths of time. At this point, the column may be sliced and the fluids analyzed by gas chromatography to obtain the initial saturations. Alternatively, brine is injected through the bottom of the vertical column to trap oil and gas, before slicing the columns and measuring the trapped or residual saturations by gas chromatography and mass balance. The experiments show that in three-phase flow, the total trapped saturations of oil and gas are considerably higher than the trapped saturations reported in the literature for two-phase systems. It is found that the residual saturation of oil and gas combined could be as high as 23 %, as opposed to a maximum two-phase residual of only 14 %. For very high initial gas saturations, the residual gas saturation, up to 17 %, was also higher than for two-phase displacement. These observations are explained in terms of the competition between piston-like displacement and snap-off. It is also observed that less oil is always trapped in three-phase flow than in two-phase displacement, and the difference depends on the amount of gas present. For low and intermediate initial gas saturations, the trapped gas saturation rises linearly with initial saturation, followed by a constant residual, as seen in two-phase displacements. However, at very high initial gas saturations, the residual saturation rises again.

Analytical model of CO2 storage efficiency in saline aquifer with vertical heterogeneity

CO2 injection into saline aquifers for storage is the most effective way to reduce greenhouse gas emissions. However, the storage layers are heterogeneous because of millions of years of tectono-sedimentary evolution. A multilayer supercritical CO2–brine two-phase displacement model with vertical heterogeneity was established to make clear the impact of heterogeneity on storage efficiency. The analytical solutions were obtained after simplifying the mathematical model. The results show that, the percolating resistance increased with the increase of injection rate and radial extent, with the decrease of heterogeneity coefficient and porosity. Larger heterogeneity coefficient or higher injection rate will result in lower sweep efficiency, and is not conducive to CO2 storage. Larger radial extent or larger porosity will result in higher sweep efficiency, and is conducive to CO2 storage. Calculation showed that heterogeneity coefficient has much more significant effect on storage efficiency than other influencing factors such as injection rate, radial extent and porosity. The analytical solutions of supercritical CO2–brine two-phase displacement model considering vertical heterogeneity provide an easy and accurate method for the evaluation of CO2 storage.

A comparison between simulation and experiment for hysteretic phenomena during two-phase immiscible displacement

[1] The paper compares a theory for immiscible displacement based on distinguishing percolating and nonpercolating fluid parts with experimental observations from multistep outflow experiments. The theory was published in 2006 in Physica A, volume 371, pages 209–225; the experiments were published in 1991 in Water Resources Research, volume 27, pages 2113. The present paper focuses on hysteretic phenomena resulting from repeated cycling between drainage and imbibition processes in multistep pressure experiments. Taking into account, the hydraulic differences between percolating and nonpercolating fluid parts provides a physical basis to predict quantitatively the hysteretic phenomena observed in the experiment. While standard hysteretic extensions of the traditional theory are nonlocal in time the theory used in this paper is local in time. Instead of storing the pressure and saturation history, it requires only the current state of the system to reach the same quantitative agreement.