Slow gas expansion in saturated natural porous media by gas injection and partitioning with non-aqueous phase liquids

The partitioning of volatile non-aqueous phase liquid (NAPL) compounds to a discontinuous gas phase can result in the expansion of that gas phase, and the resulting gas flow can significantly affect the mass transfer from NAPL source zones. This recently reported gas flow generated by the spontaneous expansion of a discontinuous gas phase has not been extensively characterized in the literature. This study measured the expansion rate of a single gas cluster in a 1.1 mm sand above a pool of trans-1,2-dichloroethene (tDCE) in small-scale flow cell experiments. To characterize the gas flow, gas injection experiments in three sizes of sand were conducted at very slow injection rates typical of gas flow rates produced by gas expansion due to NAPL partitioning. Gas cluster spontaneous expansion rates above a tDCE pool were found to be 0.34 ± 0.02 and 0.29 ± 0.01 mL/day in duplicate experiments, which is sufficiently slow to result in discontinuous gas flow in porous media with a grain size diameter greater than 0.02 mm. Measured capillary pressures during gas injection showed patterns consistent with discontinuous gas flow, and identified multiple fragmentation events and expansion by coalescence with trapped clusters. The combination of pressure data and light transmission images were used to identify fragmentation and obtain direct measurements of the critical cluster length (i.e. the length at which withdrawal of the gas phase from a pore space occurs) in quasi-two-dimensional porous media for the first time. The measured critical cluster lengths were 1.4–3.6, 3.2–6.0 and 2.8–6.5 cm in 1.1, 0.7 and 0.5 mm sands, respectively. These values agreed well with estimates of the critical cluster length made using previously reported equations, and parameters derived from the medium’s capillary pressure-saturation relationship.     

Experimental analysis of pore-scale flow and transport in porous media

A novel, non-intrusive fluorescence imaging technique has been used to quantitatively measure the pore geometry, fluid velocity, and solute concentration within a saturated, three-dimensional porous medium. Discrete numerical averages of these quantities have been made over a representative volume of the medium and used to estimate macroscopic quantities that appear in conventional continuum models of flow and transport. The approach is meant to illustrate how microscopic information can be measured, averaged, and used to characterize medium-scale processes that are typically approximated constitutively. The experimental system consisted of a clear, cylindrical column packed with clear spherical beads and a refractive index-matched fluid seeded with fluorescent tracer particles and solute dye. By illuminating the fluid within the column with a scanning planar laser beam, details of flow and concentration within the pore spaces can be quantitatively observed, allowing for three-dimensional, dimensional, time dependent information to be obtained at good resolution. In time dependent information to be obtained at good resolution. In the current experiment, volumetrically averaged velocities and void-to-volume ratios are first compared with bulk measurements of fluid flux and medium porosity. Microscopic measurements of concentration are then used to construct cross-sectionally averaged profiles, mean breakthrough curves, and direct measurements of the dispersive flux, velocity variance, and concentration variance. In turn, the dispersive flux measurements are compared with mean concentration gradients to provide a basis for confirming the Fickian dispersion model and estimating dispersion coefficients for the medium. Coefficients determined in this manner are compared with others based upon traditional length-scale arguments, mean breakthrough analyses, and curve fits with numerical simulations.     

Effect of NAPL film stability on the dissolution of residual wetting NAPL in porous media: A pore-scale modeling study

Wettability profoundly affects not only the initial distribution of residual NAPL contaminants in natural soils, but also their subsequent dissolution in a flowing aqueous phase. Under conditions of preferential NAPL wettability, the residual NAPL phase is found within the smaller pores and in the form of continuous corner filaments and thick films on pore walls. Such films expose a much greater interfacial area for mass transfer than would be exposed by the same amount of non-wetting NAPL. Importantly, capillary and hydraulic continuity of NAPL filaments and thick films is essential for sustaining NAPL–water counterflow during the course of NAPL dissolution in flowing groundwater—a mechanism which maintains and even increases the interfacial area for mass transfer. Continued dissolution results in gradual thinning of the NAPL films, which may become unstable and rupture causing disconnection of the residual NAPL in the form of clusters. Using a pore network simulator, we demonstrate that NAPL film instability drastically modifies the microscopic configuration of residual NAPL, and hence the local hydrodynamic conditions and interfacial area for mass transfer, with concomitant effects on macroscopically observable quantities, such as the aqueous effluent concentration and the fractional NAPL recovery with time. These results strongly suggest that the disjoining pressure of NAPL films may exert an important, and hitherto unaccounted, control on the dissolution behaviour of a residual NAPL phase in oil wet systems.     

Smoothed particle hydrodynamics pore-scale simulations of unstable immiscible flow in porous media

We have conducted a series of high-resolution numerical experiments using the Pair-Wise Force Smoothed Particle Hydrodynamics (PF-SPH) multiphase flow model. First, we derived analytical expressions relating parameters in the PF-SPH model to the surface tension and static contact angle. Next, we used the model to study viscous fingering, capillary fingering, and stable displacement of immiscible fluids in porous media for a wide range of capillary numbers and viscosity ratios. We demonstrated that the steady state saturation profiles and the boundaries of viscous fingering, capillary fingering, and stable displacement regions compare favorably with micromodel laboratory experimental results. For a displacing fluid with low viscosity, we observed that the displacement pattern changes from viscous fingering to stable displacement with increasing injection rate. When a high viscosity fluid is injected, transition behavior from capillary fingering to stable displacement occurred as the flow rate was increased. These observations are also in agreement with the results of the micromodel laboratory experiments.     

Micro-computed tomography pore-scale study of flow in porous media: Effect of voxel resolution

A fundamental understanding of flow in porous media at the pore-scale is necessary to be able to upscale average displacement processes from core to reservoir scale. The study of fluid flow in porous media at the pore-scale consists of two key procedures: Imaging - reconstruction of three-dimensional (3D) pore space images; and modelling such as with single and two-phase flow simulations with Lattice-Boltzmann (LB) or Pore-Network (PN) Modelling. Here we analyse pore-scale results to predict petrophysical properties such as porosity, single-phase permeability and multi-phase properties at different length scales. The fundamental issue is to understand the image resolution dependency of transport properties, in order to up-scale the flow physics from pore to core scale. In this work, we use a high resolution micro-computed tomography (micro-CT) scanner to image and reconstruct three dimensional pore-scale images of five sandstones (Bentheimer, Berea, Clashach, Doddington and Stainton) and five complex carbonates (Ketton, Estaillades, Middle Eastern sample 3, Middle Eastern sample 5 and Indiana Limestone 1) at four different voxel resolutions (4.4 µm, 6.2 µm, 8.3 µm and 10.2 µm), scanning the same physical field of view. Implementing three phase segmentation (macro-pore phase, intermediate phase and grain phase) on pore-scale images helps to understand the importance of connected macro-porosity in the fluid flow for the samples studied. We then compute the petrophysical properties for all the samples using PN and LB simulations in order to study the influence of voxel resolution on petrophysical properties. We then introduce a numerical coarsening scheme which is used to coarsen a high voxel resolution image (4.4 µm) to lower resolutions (6.2 µm, 8.3 µm and 10.2 µm) and study the impact of coarsening data on macroscopic and multi-phase properties. Numerical coarsening of high resolution data is found to be superior to using a lower resolution scan because it avoids the problem of partial volume effects and reduces the scaling effect by preserving the pore-space properties influencing the transport properties. This is evidently compared in this study by predicting several pore network properties such as number of pores and throats, average pore and throat radius and coordination number for both scan based analysis and numerical coarsened data.     

Transport upscaling using multi-rate mass transfer in three-dimensional highly heterogeneous porous media

A methodology for transport upscaling of three-dimensional highly heterogeneous formations is developed and demonstrated. The overall approach requires a prior hydraulic conductivity upscaling using an interblock-centered full-tensor Laplacian-with-skin method followed by transport upscaling. The coarse scale transport equation includes a multi-rate mass transfer term to compensate for the loss of heterogeneity inherent to all upscaling processes. The upscaling procedures for flow and transport are described in detail and then applied to a three-dimensional highly heterogeneous synthetic example. The proposed approach not only reproduces flow and transport at the coarse scale, but it also reproduces the uncertainty associated with the predictions as measured by the ensemble variability of the breakthrough curves.     

Direct simulations of two-phase flow on micro-CT images of porous media and upscaling of pore-scale forces

Pore-scale forces have a significant effect on the macroscopic behaviour of multiphase flow through porous media. This paper studies the effect of these forces using a new volume-of-fluid based finite volume method developed for simulating two-phase flow directly on micro-CT images of porous media. An analytical analysis of the relationship between the pore-scale forces and the Darcy-scale pressure drops is presented. We use this analysis to propose unambiguous definitions of Darcy-scale viscous pressure drops as the rate of energy dissipation per unit flow rate of each phase, and then use them to obtain the relative permeability curves. We show that this definition is consistent with conventional laboratory/field measurements by comparing our predictions with experimental relative permeability. We present single and two-phase flow simulations for primary oil injection followed by water injection on a sandpack and a Berea sandstone. The two-phase flow simulations are presented at different capillary numbers which cover the transition from capillary fingering at low capillary numbers to a more viscous fingering displacement pattern at higher capillary numbers, and the effect of capillary number on the relative permeability curves is investigated. Overall, this paper presents a new finite volume-based methodology for the detailed analysis of two-phase flow directly on micro-CT images of porous media and upscaling of the results to the Darcy scale.     

Non-Fickian mass transport in fractured porous media

The paper provides an introduction to fundamental concepts of mathematical modeling of mass transport in fractured porous heterogeneous rocks. Keeping aside many important factors that can affect mass transport in subsurface, our main concern is the multi-scale character of the rock formation, which is constituted by porous domains dissected by the network of fractures. Taking into account the well-documented fact that porous rocks can be considered as a fractal medium and assuming that sizes of pores vary significantly (i.e. have different characteristic scales), the fractional-order differential equations that model the anomalous diffusive mass transport in such type of domains are derived and justified analytically. Analytical solutions of some particular problems of anomalous diffusion in the fractal media of various geometries are obtained. Extending this approach to more complex situation when diffusion is accompanied by advection, solute transport in a fractured porous medium is modeled by the advection-dispersion equation with fractional time derivative. In the case of confined fractured porous aquifer, accounting for anomalous non-Fickian diffusion in the surrounding rock mass, the adopted approach leads to introduction of an additional fractional time derivative in the equation for solute transport. The closed-form solutions for concentrations in the aquifer and surrounding rocks are obtained for the arbitrary time-dependent source of contamination located in the inlet of the aquifer. Based on these solutions, different regimes of contamination of the aquifers with different physical properties can be readily modeled and analyzed.     

Influence of mass transfer resistance on detection of nonaqueous phase liquids with partitioning tracer tests

Partitioning interwell tracer tests (PITTs) are a relatively new technique for measuring the amount of nonaqueous phase liquid (NAPL) within saturated porous media. In this work we examined the influence of mass transfer limitations on the accuracy of measured NAPL from PITTs. Two mathematical models were used along with laboratory column experiments to explore the influence of tracer partition coefficient, tracer detection limit, and injected tracer mass on NAPL measurements. When dimensionless mass transfer coefficients were small, NAPL measurement errors decreased with decreasing tracer partition coefficient, decreasing tracer detection limit, and increasing injected tracer mass. Extrapolating breakthrough curves exponentially reduced but did not eliminate systematic errors in NAPL measurement. Although transport in a single stream tube was used in the mathematical models and laboratory experiments, the results from this simplified domain were supported by data taken from a three-dimensional computational experiment, where the NAPL resided as large pool. Based on these results, we suggest guidelines for interpreting tracer breakthrough data to ascertain the importance of mass transfer limitations on NAPL measurements.     

Identifying the representative flow unit for capillary dominated two-phase flow in porous media using morphology-based pore-scale modeling

In this paper, we extend pore-morphology-based methods proposed by Hazlett (1995) and Hilpert and Miller (2001) to simulate drainage and imbibition in uniformly wetting porous media and add an (optional) entrapment of the (non-)wetting phase. By improving implementation, this method allows us to identify the statistical representative elementary volume and estimate uncertainty by computing fluid flow properties and saturation distributions of hundreds of subsamples within a reasonable time-frame. The method was utilized to study three different porous medium systems and results demonstrate that morphology-based pore-scale modeling is a viable approach to assess the representative elementary volume with respect to capillary dominated two-phase flow. The focus of this paper is the determination of the representative elementary volume for multiphase-flow properties for a digital representation of a rock.     

Modeling non-equilibrium mass transport in biologically reactive porous media

We develop a one-equation non-equilibrium model to describe the Darcy-scale transport of a solute undergoing biodegradation in porous media. Most of the mathematical models that describe the macroscale transport in such systems have been developed intuitively on the basis of simple conceptual schemes. There are two problems with such a heuristic analysis. First, it is unclear how much information these models are able to capture; that is, it is not clear what the model's domain of validity is. Second, there is no obvious connection between the macroscale effective parameters and the microscopic processes and parameters. As an alternative, a number of upscaling techniques have been developed to derive the appropriate macroscale equations that are used to describe mass transport and reactions in multiphase media. These approaches have been adapted to the problem of biodegradation in porous media with biofilms, but most of the work has focused on systems that are restricted to small concentration gradients at the microscale. This assumption, referred to as the local mass equilibrium approximation, generally has constraints that are overly restrictive. In this article, we devise a model that does not require the assumption of local mass equilibrium to be valid. In this approach, one instead requires only that, at sufficiently long times, anomalous behaviors of the third and higher spatial moments can be neglected; this, in turn, implies that the macroscopic model is well represented by a convection–dispersion–reaction type equation. This strategy is very much in the spirit of the developments for Taylor dispersion presented by Aris (1956). On the basis of our numerical results, we carefully describe the domain of validity of the model and show that the time-asymptotic constraint may be adhered to even for systems that are not at local mass equilibrium.     

A residual flow procedure and application for free surface flow in porous media

A finite element procedure based on the extension of the saturated flow domain into the partially saturated zone above the free surface is proposed. The finite element equations are derived by using a pseudo variational principle which results into a residual or correction load vector. The steady or transient free surface is corrected by using the residual load in an iterative scheme. The proposed procedure uses only one (initial) mesh and does not require modification of the mesh during iterations. It is compared (qualitatively) with other procedures such as variable mesh and variational inequalities. The procedure provides satisfactory comparisons with a number of closed-form solutions and laboratory test results; two applications involving the latter are described in the paper.     

Impact of geometrical properties on permeability and fluid phase distribution in porous media

To predict fluid phase distribution in porous media, the effect of geometric properties on flow processes must be understood. In this study, we analyze the effect of volume, surface, curvature and connectivity (the four Minkowski functionals) on the hydraulic conductivity and the water retention curve. For that purpose, we generated 12 artificial structures with 800³ voxels (the units of a 3D image) and compared them with a scanned sand sample of the same size. The structures were generated with a Boolean model based on a random distribution of overlapping ellipsoids whose size and shape were chosen to fulfill the criteria of the measured functionals. The pore structure of sand material was mapped with X-rays from synchrotrons.     

Phenomenological model of phase transformations of gas hydrates in porous media: Visco-plastic consolidated skeleton

We develop a phenomenological model to describe the behavior of natural porous media saturated with phases that can experience phase transformations which result in changes of the strength, rheology, and transport properties of the medium. A porous medium saturated with gas hydrates cementing the grains of a mineral skeleton is an example of such behavior. On a decrease in pressure or increase in temperature, hydrates in such a medium dissociate into gas and water. The resulting fluid acts as a lubricant between the skeleton grains: the elastic response is changed by the viscous response, and the processes of consolidation and multiphase filtration in the deforming skeleton are initiated.     

A practical model for mobile, residual, and entrapped NAPL in water-wet porous media

Flow of nonvolatile nonaqueous phase liquid (NAPL) and aqueous phases that account for mobile, entrapped, and residual NAPL in variably saturated water-wet porous media is modeled and compared against results from detailed laboratory experiments. Residual saturation formation in the vadose zone is a process that is often ignored in multifluid flow simulators, which might cause an overestimation of the volume of NAPL that reaches the ground water. Mobile NAPL is defined as being continuous in the pore space and flows under a pressure gradient or gravitational body force. Entrapped NAPL is defined as being occluded by the aqueous phase, occurring as immobile ganglia surrounded by aqueous phase in the pore space and formed when NAPL is replaced by the aqueous phase. Residual NAPL is defined as immobile, nonwater entrapped NAPL that does not drain from the pore spaces and is conceptualized as being either continuous or discontinuous. Free NAPL comprises mobile and residual NAPL. The numerical model is formulated on mass conservation equations for oil and water, transported via NAPL and aqueous phases through variably saturated porous media. To account for phase transitions, a primary variable switching scheme is implemented for the oil-mass conservation equation over three phase conditions: (1) aqueous or aqueous-gas with dissolved oil, (2) aqueous or aqueous-gas with entrapped NAPL, and (3) aqueous or aqueous gas with free NAPL. Two laboratory-scale column experiments are modeled to verify the numerical model. Comparisons between the numerical simulations and experiments demonstrate the necessity to include the residual NAPL formation process in multifluid flow simulators.     

A new approach for phase transitions in miscible multi-phase flow in porous media

The tightly coupled, strongly nonlinear nature of non-isothermal multi-phase flow in porous media poses a tough challenge for numerical simulation. This trait is even more pronounced, if miscibility is also considered. A primary reason why inclusion of miscibility tends to be problematic are the difficulties stemming from phase transitions: on the one hand, phase transitions need to be included since the presence or absence of fluid phases has a major impact on the flow behavior; on the other hand, convergence of the nonlinear solver may be severely affected if they are not handled robustly.In this work, we present a mathematically sound approach to include phase transitions in the nonlinear system of equations: first, the transition conditions are formulated as a set of local inequality constraints, which are then directly integrated into the nonlinear solver using a nonlinear complementarity function. Under this scheme, Newton-Raphson solvers exhibit considerably more robust convergence behaviour compared to some previous approaches, which is then illustrated by several numerical examples.     

Transport in chemically and mechanically heterogeneous porous media IV: large-scale mass equilibrium for solute transport with adsorption

In this article we consider the transport of an adsorbing solute in a two-region model of a chemically and mechanically heterogeneous porous medium when the condition of large-scale mechanical equilibrium is valid. Under these circumstances, a one-equation model can be used to predict the large-scale averaged velocity, but a two-equation model may be required to predict the regional velocities that are needed to accurately describe the solute transport process. If the condition of large-scale mass equilibrium is valid, the solute transport process can be represented in terms of a one-equation model and the analysis is simplified greatly. The constraints associated with the condition of large-scale mass equilibrium are developed, and when these constraints are satisfied the mass transport process can be described in terms of the large-scale average velocity, an average adsorption isotherm, and a single large-scale dispersion tensor. When the condition of large-scale mass equilibrium is not valid, two equations are required to describe the mass transfer process, and these two equations contain two adsorption isotherms, two dispersion tensors, and an exchange coefficient. The extension of the analysis to multi-region models is straight forward but tedious.     

Estimation of time dependent virus inactivation rates by geostatistical and resampling techniques: application to virus transport in porous media

A methodology is developed for estimating temporally variable virus inactivation rate coefficients from experimental virus inactivation data. The methodology consists of a technique for slope estimation of normalized virus inactivation data in conjunction with a resampling parameter estimation procedure. The slope estimation technique is based on a relatively flexible geostatistical method known as universal kriging. Drift coefficients are obtained by nonlinear fitting of bootstrap samples and the corresponding confidence intervals are obtained by bootstrap percentiles. The proposed methodology yields more accurate time dependent virus inactivation rate coefficients than those estimated by fitting virus inactivation data to a first-order inactivation model. The methodology is successfully applied to a set of poliovirus batch inactivation data. Furthermore, the importance of accurate inactivation rate coefficient determination on virus transport in water saturated porous media is demonstrated with model simulations.