A Pore-Scale Investigation of Low-Salinity Waterflooding in Porous Media: Uniformly Wetted Systems

The potential of low-salinity (LS) water injection as an oil recovery technique has been the source of much recent debate within the petroleum industry. Evidence from both laboratory and field-level studies has indicated significant benefits compared to conventional high-salinity (HS) waterflooding, but many conflicting results have also been reported and, to date, the underlying mechanisms remain poorly understood. In this paper, we aim to address this uncertainty by developing a novel, steady-state pore network model in which LS brine displaces oil from a HS-bearing network. The model allows systematic investigation of the crude oil/brine/rock parameter space, with the goal of identifying features that may be critical to the production of incremental oil following LS brine injection. By coupling the displacement model to a salinity-tracking tracer algorithm, and assuming that a reduction of water salinity within the pore network leads to localised wettability alteration, substantial perturbations to standard pore filling sequences are predicted. The results clearly point to two principal effects of dynamic contact angle modification at the pore scale: a “pore sequence” effect, characterised by an alteration to the distribution of displaced pore sizes, and a “sweep efficiency” effect, demonstrated by a change in the overall fraction of pores invaded. Our study indicates that any LS effect will depend on the relative (scenario-dependent) influence of each mechanism, where factors such as the initial wettability state of the system and the pore size distribution of the underlying network are found to play crucial roles. In addition, we highlight the important role played by end-point capillary pressure in determining LS efficacy.     

Effects of Fines Migration on Low-Salinity Waterflooding: Analytical Modelling

We discuss the governing system for oil–water flow with varying water composition. The model accounts for wettability alteration, which affects the relative permeability, and for salinity-variation-induced fines migration, which reduces the relative permeability of water. The overall ionic strength represents the aqueous phase composition in the model. One-dimensional displacement of oil by high-salinity water followed by low-salinity-slug injection and high-salinity water chase drive allows for exact analytical solution. The solution is derived using the splitting method. The analytical model obtained analyses the effects of wettability alteration and fines migration on oil recovery as two distinct physical mechanisms. For typical reservoir conditions, the significant effects of both mechanisms are observed.     

Shear Rate Determination from Pore-Scale Flow Fields

Aqueous solutions with polymer additives often used to improve the macroscopic sweep efficiency in oil recovery typically exhibit non-Newtonian rheology. In order to predict the Darcy-scale effective viscosity \(\mu _{\mathrm{eff}} \) required for practical applications often, semi-empirical correlations such as the Cannella or Blake–Kozeny correlation are employed. These correlations employ an empirical constant (“C-factor”) that varies over three orders of magnitude with explicit dependency on porosity, permeability, fluid rheology and other parameters. The exact reasons for this dependency are not very well understood. The semi-empirical correlations are derived under the assumption that the porous media can be approximated by a capillary bundle for which exact analytical solutions exist. The effective viscosity \(\mu _{\mathrm{eff}} (v_{\mathrm{Darcy}} )\) as a function of flow velocity is then approximated by a cross-sectional average of the local flow field resulting in a linear relationship between shear rate \(\gamma \) and flow velocity. Only with such a linear relationship, the effective viscosity can be expressed as a function of an average flow rate instead of an average shear rate. The local flow field, however, does in general not exhibit such a linear relationship. Particularly for capillary tubes, the velocity is maximum at the center, while the shear rate is maximum at the tube wall indicating that shear rate and flow velocity are rather anti-correlated. The local flow field for a sphere pack is somewhat more compatible with a linear relationship. However, as hydrodynamic flow simulations (using Newtonian fluids for simplicity) performed directly on pore-scale resolved digital images suggest, flow fields for sandstone rock fall between the two limiting cases of capillary tubes and sphere packs and do in general not exhibit a linear relationship between shear rate and flow velocity. This indicates that some of the shortcomings of the semi-empirical correlations originate from the approximation of the shear rate by a linear relationship with the flow velocity which is not very well compatible with flow fields from direct hydrodynamic calculations. The study also indicates that flow fields in 3D rock are not very well represented by capillary tubes.     

Empirical Measures of Wettability in Porous Media and the Relationship between Them Derived From Pore-Scale Modelling

The wettability of a crude oil/brine/rock system is of central importance in determining the oil recovery efficiency of water displacement processes in oil reservoirs. Wettability of a rock sample has traditionally been measured using one of two experimental techniques, viz. the United States Bureau of Mines and Amott tests. The former gives the USBM index, I _USBM, and the latter yields the Amott–Harvey index, I _AH. As there is no well-established theoretical basis for either test, any relationship between the two indices remains unclear.Analytical relationships between I _AH and I _USBM for mixed-wet and fractionally-wet media have been based on a number of simplifying assumptions relating to the underlying pore-scale displacement mechanisms. This simple approach provides some guidelines regarding the influence of the distribution of oil-wet surfaces within the porous medium on I _AH and I _USBM. More detailed insight into the relationship between I _AH and I _USBM is provided by modelling the pore-scale displacement processes in a network of interconnected pores. The effects of pore size distribution, interconnectivity, displacement mechanisms, distribution of volume and of oil-wet pores within the pore space have all been investigated by means of the network model.The results of these analytical calculations and network simulations show that I _AH and I _USBM need not be identical. Moreover, the calculated indices and the relationship between them suggest explanations for some of the trends that appear in experimental data when both I _USBM and I _AH have been reported in the literature for tests with comparable fluids and solids. Such calculations should help with the design of more informative wettability tests in the future.     

Pore-by-Pore Modelling,Validation and Prediction of Waterflooding in Oil-Wet Rocks Using Dynamic Synchrotron Data

We predict waterflood displacement on a pore-by-pore basis using pore network modelling. The pore structure is captured by a high-resolution image. We then use an energy balance applied to images of the displacement to assign an average contact angle, and then modify the local pore-scale contact angles in the model about this mean to match the observed displacement sequence. Two waterflooding experiments on oil-wet rocks are analysed where the displacement sequence was imaged using time-resolved synchrotron imaging. In both cases the capillary pressure in the model matches the experimentally obtained values derived from the measured interfacial curvature. We then predict relative permeability for the full saturation range. Using the optimised contact angles distributed randomly in space has little effect on the predicted capillary pressures and relative permeabilities, indicating that spatial correlation in wettability is not significant in these oil-wet samples. The calibrated model can be used to predict properties outside the range of conditions considered in the experiment.

     

Insight into Heterogeneity Effects in Methane Hydrate Dissociation via Pore-Scale Modeling

A large number (1253) of high-quality streaming potential coefficient (\(C_\mathrm{sp})\) measurements have been carried out on Berea, Boise, Fontainebleau, and Lochaline sandstones (the latter two including both detrital and authigenic overgrowth forms), as a function of pore fluid salinity (\(C_\mathrm{f})\) and rock microstructure. All samples were saturated with fully equilibrated aqueous solutions of NaCl (10\(^{-5}\) and 4.5 mol/dm\(^{3})\) upon which accurate measurements of their electrical conductivity and pH were taken. These \(C_\mathrm{sp}\) measurements represent about a fivefold increase in streaming potential data available in the literature, are consistent with the pre-existing 266 measurements, and have lower experimental uncertainties. The \(C_\mathrm{sp}\) measurements follow a pH-sensitive power law behaviour with respect to \(C_\mathrm{f}\) at medium salinities (\(C_\mathrm{sp} =-\,1.44\times 10^{-9} C_\mathrm{f}^{-\,1.127} \), units: V/Pa and mol/dm\(^{3})\) and show the effect of rock microstructure on the low salinity \(C_\mathrm{sp}\) clearly, producing a smaller decrease in \(C_\mathrm{sp}\) per decade reduction in \(C_\mathrm{f}\) for samples with (i) lower porosity, (ii) larger cementation exponents, (iii) smaller grain sizes (and hence pore and pore throat sizes), and (iv) larger surface conduction. The \(C_\mathrm{sp}\) measurements include 313 made at \(C_\mathrm{f} > 1\) mol/dm\(^{3}\), which confirm the limiting high salinity \(C_\mathrm{sp}\) behaviour noted by Vinogradov et al., which has been ascribed to the attainment of maximum charge density in the electrical double layer occurring when the Debye length approximates to the size of the hydrated metal ion. The zeta potential (\(\zeta \)) was calculated from each \(C_\mathrm{sp}\) measurement. It was found that \(\zeta \) is highly sensitive to pH but not sensitive to rock microstructure. It exhibits a pH-dependent logarithmic behaviour with respect to \(C_\mathrm{f}\) at low to medium salinities (\(\zeta =0.01133 \log _{10} \left( {C_\mathrm{f} } \right) +0.003505\), units: V and mol/dm\(^{3})\) and a limiting zeta potential (zeta potential offset) at high salinities of \({\zeta }_\mathrm{o} = -\,17.36\pm 5.11\) mV in the pH range 6–8, which is also pH dependent. The sensitivity of both \(C_\mathrm{sp}\) and \(\zeta \) to pH and of \(C_\mathrm{sp}\) to rock microstructure indicates that \(C_\mathrm{sp}\) and \(\zeta \) measurements can only be interpreted together with accurate and equilibrated measurements of pore fluid conductivity and pH and supporting microstructural and surface conduction measurements for each sample.     

Pore-Scale Modeling of Electrokinetics in Geomaterials

Pore-scale finite-volume continuum models of electrokinetic processes are used to predict the Debye lengths, velocity, and potential profiles for two-dimensional arrays of circles, ellipses and squares with different orientations. The pore-scale continuum model solves the coupled Navier–Stokes, Poisson, and Nernst–Planck equations to characterize the electro-osmotic pressure and streaming potentials developed on the application of an external voltage and pressure difference, respectively. This model is used to predict the macroscale permeabilities of geomaterials via the widely used Carmen–Kozeny equation and through the electrokinetic coupling coefficients. The permeability results for a two-dimensional X-ray tomography-derived sand microstructure are within the same order of magnitude as the experimentally calculated values. The effect of the particle aspect ratio and orientation on the electrokinetic coupling coefficients and subsequently the electrical and hydraulic tortuosity of the porous media has been determined. These calculations suggest a highly tortuous geomaterial can be efficient for applications like decontamination and desalination.

     

Pore-Scale Flow in Surfactant Flooding

Recovery of oil from the blocks of an initially oil-wet, naturally fractured, reservoir as a result of counter-current flow following introduction of aqueous wettability-altering surfactant into the fracture system is considered, as an example of a practical process in which phenomena acting at the single pore-scale are vital to the economic displacement of oil at the macroscopic scale. A Darcy model for the process is set up, and solutions computed illustrating the recovery rate controlling role of the bulk diffusion of surfactant. A central ingredient of this model is the capillary pressure relation, linking the local values of the pressure difference between the oleic and aqueous phases, the aqueous saturation and the surfactant concentration. Using ideas from single capillary models of oil displacement from oil-wet tubes by wettability-altering surfactant, we speculate that the use of a capillary pressure function, with dependences as assumed, may not adequately represent the Darcy scale consequences of processes acting at the single pore-scale. Multi-scale simulation, resolving both sub-pore and multi-pore flow processes may be necessary to resolve this point. Some general comments are made concerning the issues faced when modelling complex displacement processes in porous media starting from the pore-scale and working upwards.     

Three-Phase Flow Modelling Using Pore-Scale Capillary Pressures and Relative Permeabilities for Mixed-Wet Media at the Continuum-Scale

When regions of three-phase flow arise in an oil reservoir, each of the flow parameters, i.e. capillary pressures and relative permeabilities, are generally functions of two phase saturations and depend on the wettability state. The idea of this work is to generate consistent pore-scale based three-phase capillary pressures and relative permeabilities. These are then used as input to a 1-D continuum core- or reservoir-scale simulator. The pore-scale model comprises a bundle of cylindrical capillary tubes, which has a distribution of radii and a prescribed wettability state. Contrary to a full pore-network model, the bundle model allows us to obtain the flow functions for the saturations produced at the continuum-scale iteratively. Hence, the complex dependencies of relative permeability and capillary pressure on saturation are directly taken care of. Simulations of gas injection are performed for different initial water and oil saturations, with and without capillary pressures, to demonstrate how the wettability state, incorporated in the pore-scale based flow functions, affects the continuum-scale displacement patterns and saturation profiles. In general, wettability has a major impact on the displacements, even when capillary pressure is suppressed. Moreover, displacement paths produced at the pore-scale and at the continuum-scale models are similar, but they never completely coincide.     

Accurate Modelling of Pore-Scale Films and Layers for Three-Phase Flow Processes in Clastic and Carbonate Rocks with Arbitrary Wettability

Three-phase flow is a key process occurring in subsurface reservoirs, for example, during $\text{ CO }_2$ sequestration and enhanced oil recovery techniques such as water alternating gas (WAG) injection. Predicting three-phase flow processes, for example, the increase in oil recovery during WAG, requires a sound understanding of the fundamental flow physics in water- to oil-wet rocks to derive physically robust flow functions, i.e. relative permeability and capillary pressure. In this study, we use pore-network modelling, a reliable and physically based simulation tool, to predict the flow functions. We have developed a new pore-scale network model for rocks with variable wettability, from water- to oil-wet. It comprises a constrained set of parameters that mimic the wetting state of a reservoir. Unlike other models, it combines three main features: (1) A novel thermodynamic criterion for formation and collapse of oil layers. The new model hence captures wetting film and layer flow of oil adequately, which affects the oil relative permeability at low oil saturation and leads to accurate prediction of residual oil. (2) Multiple displacement chains, where injection of one phase at the inlet triggers a chain of interface displacements throughout the network. This allows for the accurate modelling of the mobilisation of many disconnected phase clusters that arise, in particular, during higher order WAG floods. (3) The model takes realistic 3D pore-networks extracted from pore-space reconstruction methods and CT images as input, preserving both topology and pore shape of the sample. For water-wet systems, we have validated our model with available experimental data from core floods. For oil-wet systems, we validated our network model by comparing 2D network simulations with published data from WAG floods in oil-wet micromodels. This demonstrates the importance of film and layer flow for the continuity of the various phases during subsequent WAG cycles and for the residual oil saturations. A sensitivity analysis has been carried out with the full 3D model to predict three-phase relative permeabilities and residual oil saturations for WAG cycles under various wetting conditions with different flood end-points.     

Upscaling of Anomalous Pore-Scale Dispersion

Transport in Porous Media - We study the upscaling of advective pore-scale dispersion in terms of the Eulerian velocity distribution and advective tortuosity, both flow attributes, and of the...     

Anomalous Dispersion in Pore-Scale Simulations of Two-Phase Flow

Transport in Porous Media - We investigate anomalous dispersion in steady-state two-phase flow though a random, artificial porous domain. A natural distribution of trapped wetting-phase fluid was...     

A Dynamic Two-Phase Pore-Scale Model of Imbibition

We present a dynamic pore-scale network model of imbibition, capable of calculating residual oil saturation for any given capillary number, viscosity ratio, contact angle, and aspect ratio. Our goal is not to predict the outcome of core floods, but rather to perform a sensitivity analysis of the above-mentioned parameters, except from the viscosity ratio. We find that contact angle, aspect ratio, and capillary number all have a significant influence on the competition between piston-like advance, leading to high recovery, and snap-off, causing oil entrapment. Due to significant CPU-time requirements we did not incorporate long-range correlations among pore and throat sizes in our network, but were limited to small-range correlations. Consequently, the gradual suppression of snap-off occurs within one order of magnitude of the capillary number. At capillary numbers around 10⁸ - 10⁷ snap-off has been entirely inhibited, in agreement with results obtained by Blunt (1997) who used a quasi-static model. For higher aspect ratios, the effect of rate and contact angle is more pronounced.     

Predictive Pore-Scale Modeling of Single and Multiphase Flow

We show how to predict flow properties for a variety of rocks using pore-scale modeling with geologically realistic networks. The pore space is represented by a topologically disordered lattice of pores connected by throats that have angular cross-sections. We successfully predict single-phase non-Newtonian rheology, and two and three-phase relative permeability for water-wet media. The pore size distribution of the network can be tuned to match capillary pressure data when a network representation of the system of interest is unavailable. The aim of this work is not simply to match experiments, but to use easily acquired data to estimate difficult to measure properties and to predict trends in data for different rock types or displacement sequences.     

Microtomography and Pore-Scale Modeling of Two-Phase Fluid Distribution

Synchrotron-based X-ray microtomography (micro CT) at the Advanced Light Source (ALS) line 8.3.2 at the Lawrence Berkeley National Laboratory produces three-dimensional micron-scale-resolution digital images of the pore space of the reservoir rock along with the spacial distribution of the fluids. Pore-scale visualization of carbon dioxide flooding experiments performed at a reservoir pressure demonstrates that the injected gas fills some pores and pore clusters, and entirely bypasses the others. Using 3D digital images of the pore space as input data, the method of maximal inscribed spheres (MIS) predicts two-phase fluid distribution in capillary equilibrium. Verification against the tomography images shows a good agreement between the computed fluid distribution in the pores and the experimental data. The model-predicted capillary pressure curves and tomography-based porosimetry distributions compared favorably with the mercury injection data. Thus, micro CT in combination with modeling based on the MIS is a viable approach to study the pore-scale mechanisms of CO₂ injection into an aquifer, as well as more general multi-phase flows.     

Grain Shape Effects on Permeability,Formation Factor,and Capillary Pressure from Pore-Scale Modeling

We invoke pore-scale models to evaluate grain shape effects on petrophysical properties of three-dimensional (3D) images from micro-CT scans and consolidated grain packs. Four sets of grain-packs are constructed on the basis of a new sedimentary algorithm with the following shapes: exact angular grain shapes identified from micro-CT scans, ellipsoids fitted to angular grains, and spheres with volume and surface-to-volume ratio equal to original angular grains on a grain-by-grain basis. Subsequently, a geometry-based cementation algorithm implements pore space alteration due to diagenesis. Eight micro-CT scans and 144 grain-pack images with $500 \times 500 \times 500$ voxels (the resolution units of 3D images) are analyzed in this study. Absolute permeability, formation factor, and capillary pressure are calculated for each 3D image using numerical methods and compared to available core measurements. Angular grain packs give rise to the best agreement with experimental measurements. Cement volume and its spatial distribution in the pore space significantly affect all calculated petrophysical properties. Available empirical permeability correlations for non-spherical grains underestimate permeability between 30 and 70 % for the analyzed samples. Kozeny–Carman’s predictions agree with modeled permeability for spherical grain packs but overestimate permeability for micro-CT images and non-spherical grain packs when volume-based radii are used to calculate the average grain size in a pack. We identify surface-to-volume ratio and grain shape as fundamental physical parameters that control fluid distribution and flow in porous media for equivalent porosity samples.     

DNS and Modelling of Passive Scalar Transport in Turbulent Channel Flow with a Focus on Scalar Dissipation Rate Modelling

A direct numerical simulation of turbulent channel flow with an imposed mean scalar gradient is analyzed with a focus on passive scalar flux modelling and in particular the treatment of the passive scalar dissipation equation. The Prandtl number is 0.71 and the Reynolds number based on the wall friction velocity and the channel half width is 265. Budgets are presented for the passive scalar variance and its dissipation rate, as well as for the individual scalar flux components. These form a basis for a discussion of modelling issues related to explicit algebraic scalar flux modelling. This revised version was published online in July 2006 with corrections to the Cover Date.