CO<Subscript>2</Subscript> injection into saline carbonate aquifer formations I: laboratory investigation

Although there are a number of mathematical modeling studies for carbon dioxide (CO₂) injection into aquifer formations, experimental studies are limited and most studies focus on injection into sandstone reservoirs as opposed to carbonate ones. This study presents the results of computerized tomography (CT) monitored laboratory experiments to analyze permeability and porosity changes as well as to characterize relevant chemical reactions associated with injection and storage of CO₂ in carbonate formations. CT monitored experiments are designed to model fast near well bore flow and slow reservoir flows. Highly heterogeneous cores drilled from a carbonate aquifer formation located in South East Turkey were used during the experiments. Porosity changes along the core plugs and the corresponding permeability changes are reported for different CO₂ injection rates and different salt concentrations of formation water. It was observed that either a permeability increase or a permeability reduction can be obtained. The trend of change in rock properties is very case dependent because it is related to distribution of pores, brine composition and thermodynamic conditions. As the salt concentration decreases, porosity and the permeability decreases are less pronounced. Calcite deposition is mainly influenced by orientation, with horizontal flow resulting in larger calcite deposition compared to vertical flow.     

Use of a gamma ray attenuation technique to study colloid deposition in porous media

An experimental study of colloidal deposition in porous media is presented. The local deposition is determined through a local measurement of porosity variation using a -ray attenuation technique. The basic principle of this technique is described and the accuracy measurement is discussed. An experimental setup was designed using an artificial porous medium flushed with several pore volumes of a latex suspension. The damage to the porous medium was determined from permeability reduction and porosity measurements. A good agreement was obtained for a monolayer deposit. The discrepancy between global and local measurements of multilayer deposition is discussed.     

Effect of Network Topology on Relative Permeability

We consider the role of topology on drainage relative permeabilities derived from network models. We describe the topological properties of rock networks derived from a suite of tomographic images of Fontainbleau sandstone (Lindquist et al., 2000, J. Geophys. Res. 105B, 21508). All rock networks display a broad distribution of coordination number and the presence of long-range topological bonds. We show the importance of accurately reproducing sample topology when deriving relative permeability curves from the model networks. Comparisons between the relative permeability curves for the rock networks and those computed on a regular cubic lattice with identical geometric characteristics (pore and throat size distributions) show poor agreement. Relative permeabilities computed on regular lattices and on diluted lattices with a similar average coordination number to the rock networks also display poor agreement. We find that relative permeability curves computed on stochastic networks which honour the full coordination number distribution of the rock networks produce reasonable agreement with the rock networks. We show that random and regular lattices with the same coordination number distribution produce similar relative permeabilities and that the introduction of longer-range topological bonds has only a small effect. We show that relative permeabilities for networks exhibiting pore–throat size correlations and sizes up to the core-scale still exhibit a significant dependence on network topology. The results show the importance of incorporating realistic 3D topologies in network models for predicting multiphase flow properties.     

Application of X-ray CT investigation of CO<Subscript>2</Subscript>–brine flow in porous media

A clear understanding of two-phase flows in porous media is important for investigating CO₂ geological storage. In this study, we conducted an experiment of CO₂/brine flow process in porous media under sequestration conditions using X-ray CT technique. The flow properties of relative permeability, porosity heterogeneity, and CO₂ saturation were observed in this experiment. The porous media was packed with glass beads having a diameter of 0.2 mm. The porosity distribution along the flow direction is heterogeneous owing to the diameter and shape of glass beads along the flow direction. There is a relationship between CO₂ saturation and porosity distribution, which changes with different flow rates and fractional flows. The heterogeneity of the porous media influences the distribution of CO₂; moreover, gravity, fractional flows, and flow rates influence CO₂ distribution and saturation. The relative permeability curve was constructed using the steady-state method. The results agreed well with the relative permeability curve simulated using pore-network model.     

A Computational Approach to Determine Average Reservoir Pressure in a Coalbed Methane (CBM) Well Flowing Under Dominant Matrix Shrinkage Effect

Desorption of gas from coal matrix alters the pore volume of fracture network. Consequently, cleat porosity and permeability of reservoir changes as pressure depletes. The method of standard pressure analysis calculations produces incorrect results in the case of coalbed methane reservoirs producing under dominant matrix shrinkage effect. The change in cleat porosity and permeability due to shrinkage of coal matrix following gas desorption with pressure depletion invalidates the underlying assumptions made in the derivation of diffusivity equation. Consequently, equations of pseudo-steady state commonly used in conventional reservoirs no longer remain valid as the porosity and permeability values change with pressure depletion. In this paper, effort has been made to describe pseudo-steady-state flow in coalbed methane reservoirs in the form of a new equation that accounts for pressure dependency of cleat porosity and permeability due to shrinkage of coal matrix. The concept of Al-Hussainy et al. (1966) has been extended to define a new pseudo-pressure function which assimilates within itself the pressure dependence of porosity and permeability Palmer and Mansoori (1998). Equation has been used to relate the cleat porosity with pressure. The equation-based computational method suggested in this paper finds its usefulness in estimating average reservoir pressure for any known flowing bottom hole pressure and thus reducing the frequency of future pressure buildup tests. The new equation is also useful in predicting reservoir pressure under the situation when coal matrix shrinks below desorption pressure. The equation used in the computational method has been validated with the help of numerical simulator CMG-GEM.     

Tomography-Based Characterization and Optimization Of Fluid Flow Through Porous Media

Numerical simulations to characterize fluid flow through porous media have been carried out using tomography-derived real geometry data that has been manipulated using digital image processing techniques to obtain a wide range of porosities. Two kinds of porous media have been analyzed: (a) a reticulated porous ceramic (RPC) foam and (b) a packed bed of CaCO₃ particles. The porosity of the media is varied via morphological operations between 0.727 and 0.913 in case of the RPC and between 0.329 and 0.824 in case of the packed bed. A mesh generator based on the pore space indicator function is then used to generate unstructured tetrahedral grids from the processed tomography data. Fluid flow simulations are carried out for Reynolds numbers ranging from 0.1 to 200 and the results are used to determine the permeability and the Dupuit?CForchheimer coefficient in each case. The results are then compared with existing analytical models and the applicability of the models is examined. In the RPC case, the Happel?CBrenner (parallel-flow) model predicts the permeability with a normalized root mean square error (NRMSE) of 11.8 % across the porosity range and Modified Ergun (Macdonald et. al) model predicts the Dupuit?CForchheimer coefficient within a NRMSE of 13.5 %. In the packed-bed case, the Brinkman drag model predicts the permeability within a NRMSE of 8.26 % across the porosity range and the Modified Ergun model predicts the Dupuit?CForchheimer coefficient within an NRMSE of 5.94 %. For each material, an adjusted Kozeny constant is determined. For the RPC, the Kozeny constant is evaluated at 7.73 and for the CaCO₃ packed bed, it is found to be 6.10, leading to predictions of the permeability with an NRMSE of 4.16 and 3.37 %, respectively.     

The effect of modeled drag reduction on the wall region

The low-dimensional model derived for the wall region of a turbulent boundary layer (Aubry et al., 1988) is applied to a drag-reduced flow. In agreement with some experimental results, drag reduction is modeled by thickening the wall region, which is achieved by applying stretching transformations to the original flow. By application of a Galerkin projection, a set of ordinary differential equations (ODEs) is obtained whose structure is identical to the set corresponding to the unmodified flow. The coefficients of the ODEs are modified in a nontrivial way. The bifurcation diagrams plotted for different values of the stretching parameter are different in detail but the structure is globally the same. In particular, the intermittent behavior which Aubry et al. identified with the cyclic bursting events experimentally observed is still present. The scenario by which intermittency appears through a subcritical Hopf bifurcation in which a heteroclinic cycle is created and disappears through a bifurcation to traveling waves is identical. These results hold for values of the stretching between 1 and 2.65, the value at which the top of the buffer layer reaches the centerline of the pipe. This is in agreement with experimental results for flows whose drag is reduced but which still display intermittency. The bifurcations occur in the stretched flow at increased levels of dissipation (relative to the unstretched flow), consistent with theoretical pictures of drag reduction, in which the increase of scale is due to stabilization by an increase of dissipation in the turbulent part of the flow. Moreover, this method is a systematic way to perturb the coefficients of the ODEs of Aubry et al. (1988). Under this kind of perturbation, the behavior of the solution (in the part of the bifurcation diagram physically relevant) is found to be extremely robust.     

Étude expérimentale du dépôt de particules colloïdales en milieu poreux : Influence de l'hydrodynamique et de la salinité

This study deals with colloid transport in porous media which applications are found in subsurface water, petroleum engineering or civil engineering. An experimental study of colloidal polystyrene Latex particles deposition in a consolidated porous medium is presented. The influence of ionic strength of the colloid suspension and the flow rate on particle deposition is investigated. We see first that beyond a critical salt concentration, the total collector efficiency increases with the ionic strength. Moreover, such collector efficiency decreases as the flow rate increases according to theory. In other respects, using a γ ray attenuation technique allows us to measure local porosity fluctuation due to particles deposition. By this way deposition kinetics may be followed locally and precisely. Nevertheless when considering the thickness of the adsorbed layer over large scales, obtained results using the γ rays attenuation technique are found in good agreement with those obtained by means of an usual technique especially at latest stages of adsorption process. To cite this article: A. Djehiche et al., C. R. Mecanique 337 (2009).     

The weak shear kinetic phase diagram for nematic polymers

We study the shear problem for nematic polymers as modeled by the molecular kinetic theory of Doi (1981), focusing on the anomalous slow flow regime. We provide the kinetic phase diagram of monodomain (MD) attractors and phase transitions vs normalized nematic concentration (N) and weak normalized shear rate (Peclet number, Pe). We then overlay all rheological features typically reported in experiments: alignment properties, normal stress differences and shear stress. These features play a critical role in the synthesis between theory and experiment for nematic polymers (Larson 1999; Doi and Edwards 1986). MD type is routinely used for rheological shear characterization: cf., flow-aligning 5CB (Mather et al. 1996a), tumbling PBT (Srinivasarao and Berry 1991), and 8CB (Mather et al. 1996b), evidence for a wagging regime (Mewis et al. 1997), out-of-plane kayaking modes (Larson and Ottinger 1991), and evidence for chaotic major director dynamics (Bandyopadhyay et al. 2000). MD transitions correlate with sign changes in normal stresses (Larson and Ottinger 1991; Magda et al. 1991; Kiss and Porter 1978, 1980). Furthermore, structure formation in shear devices appears to be correlated with monodomain precursor dynamics (Tan and Berry 2003; Forest et al. 2002a). In this paper we combine seminal kinetic theory results (Kuzuu and Doi 1983, 1984; Larson 1990; Larson and Ottinger 1991; Faraoni et al. 1999; Grosso et al. 2001), symmetry observations (Forest et al. 2002b), and mesoscopic results on the fate of orientational degeneracy in weak shear (Forest and Wang 2003; Forest et al. 2003a), together with our resolved numerical simulations, to provide the kinetic flow-phase diagram of Doi theory in the weak shear regime, 0<Pe<1, for infinitely thin rods. We report the "birth" of key rheological features at the onset of flow: sign changes and local maxima and minima in normal stress differences (N₁ and N₂) associated with MD transitions. These results serve as the basis for continuation of the kinetic phase diagram to Pe>1 ; as the definitive benchmark for any mesoscopic or continuum model; and experimental data can be compared in order to determine accuracy and limitations of the Doi theory in weak shear.     

Two-Phase Flow Properties Prediction from Small-Scale Data Using Pore-Network Modeling

The objective of this work is to evaluate the prediction accuracy of network modeling to calculate transport properties of porous media based on the interpretation of mercury invasion capillary pressure curves only. A pore-scale modeling approach is used to model the multi-phase flow and calculate gas/oil relative permeability curves. The characteristics of the 3-D pore-network are defined with the requirement that the network model satisfactorily reproduces the capillary pressure curve (P_c curve), the porosity and the permeability. A sensitivity study on the effect of the input parameters on the prediction of capillary pressure and gas/oil relative permeability curves is presented. The simulations show that different input parameters can lead to similarly good reproductions of the experimental P_c, although the predicted relative permeabilities K_r are somewhat widespread. This means that the information derived from a mercury invasion P_c curve is not sufficient to characterize transport properties of a porous medium. The simulations indicate that more quantitative information on the wall roughness and the node/bond aspect ratio would be necessary to better constrain the problem. There is also evidence that in narrow pore size distributions pore body volume and pore throat radius are correlated while in broad pore size distributions they would be uncorrelated.     

The transverse permeability of disordered fiber arrays: a statistical correlation in terms of the mean nearest interfiber spacing

In the porous media literature, unidirectional fibrous systems are broadly categorized as ordered or disordered. The former class, easily tractable for analysis purposes but limited in its relation to reality, involves square, hexagonal and various staggered arrays. The latter class involves everything else. While the dimensionless hydraulic permeability of ordered fibrous media is known to be a deterministic function of their porosity ϕ, the parameters affecting the permeability of disordered fiber arrays are not very well understood. The objective of this study is to computationally investigate flow across many unidirectional arrays of randomly placed fibers and derive a correlation between K and some measure of their microstructure. In the process, we explain the wide scatter in permeability values observed computationally as well as experimentally. This task is achieved using a parallel implementation of the Boundary Element Method (BEM). Over 600 simulations are carried out in two-dimensional geometries consisting of 576 fiber cross-sections placed within a square unit cell by a Monte Carlo procedure. The porosity varies from 0.45 to 0.90. The computed permeabilities are compared with earlier theoretical results and experimental data. Analysis of the computational results reveals that the permeability of disordered arrays with ϕ < 0.7 is reduced as the non-uniformity of the fiber distribution increases. This reduction can be substantial at low porosities. The key finding of this study is a direct correlation between K and the mean nearest inter-fiber spacing , the latter depending on the microstructure of the fibrous medium.     

Effect of network topology on two-phase imbibition relative permeability

In a previous study Arns et al. (2004, Transport Porous Media 55, 21–46) we considered the role of topology on drainage relative permeability curves computed using network models derived from a suite of tomographic images of Fontainebleau sandstone. The present study extends the analysis to more complex imbibition displacements where the non-wetting fluid can be disconnected by snap-off as a result of swelling of wetting films in the corners of pores and throats. In contrast to the findings for drainage displacements which showed that relative permeabilities are significantly affected by network topology, the present study shows that the effect of topology on imbibition relative permeabilities depends on the level of snap-off. For strongly wetting conditions where snap-off dominates the displacement the effect of network topology is significantly smaller than for weakly wet conditions where snap-off is suppressed. For contact angles sufficiently large to completely suppress snap-off, the effect of topology on imbibition relative permeabilities is similar to that for drainage displacements. The findings are valid for random networks and for networks displaying short-range pore–throat and longer range spatial correlations.     

Equivalent diffusion coefficient and equivalent diffusion accessible porosity of a stratified porous medium

Diffusion is an important transport process in low permeability media, which play an important role in contamination and remediation of natural environments. The calculation of equivalent diffusion parameters has however not been extensively explored. In this paper, expressions of the equivalent diffusion coefficient and the equivalent diffusion accessible porosity normal to the layering in a layered porous medium are derived based on analytical solutions of the diffusion equation. The expressions show that the equivalent diffusion coefficient changes with time. It is equal to the power average with p = −0.5 for small times and converges to the harmonic average for large times. The equivalent diffusion accessible porosity is the harmonic average of the porosities of the individual layers for all times. The expressions are verified numerically for several test cases.     

On the calculation of wall temperatures in the post dryout heat transfer region

A non-equilibrium post dryout heat transfer model for calculating the wall temperature distribution in vertical upflows is presented in this study. The model is based upon the three path heat transfer formulation developed by MIT researchers (Laverty & Rohsenow 1964, Forslund & Rohsenow 1968, Hynek et al. 1969 and Plummer et al. 1974) that involves heat transfer from wall to vapor, from wall to droplets in contact with the wall and from vapor to liquid droplets in the vapor core. Downstream gradients for the bulk vapor temperature, vapor quality, droplet size and vapor velocities are identical to those used by Hynek et al. (1969) and Plummer et al. (1974). Conditions at the dryout location are calculated using a modified version of a technique developed by Hynek et al. (1969).A procedure for determining an average droplet diameter based on a size distribution is introduced. Migration of droplets through the boundary layer and droplet deposition flux are predicted with the model of Gani? & Rohsenow (1979). Heat transfer from the wall to the impinging liquid droplets is calculated with a correlation by Holman & McGinnis (1969). Mechanisms contributing to wall to droplet heat transfer are identified as (a) droplet-wall contact, (b) intensive droplet evaporation inside the boundary layer, and (c) destruction of the boundary layer due to droplet migration to, and rebound from, the hot surface. The significance of the average droplet size and size distribution is demonstrated through its control over the free stream evaporation and droplet deposition rates.Predicted uniform heat flux wall temperature profiles for water, nitrogen and freon 12 are in good agreement with the data of Era et al. (1966), Bennett et al. (1967), Forslund & Rohsenow (1968), Ling et al. (1971), Groeneveld (1972) and Janssen & Kervinen (1975).     

Permeability of Pore Networks Under Compaction

The characteristic pore length fixes the scale of permeability of a porous medium. For pore networks undergoing strong random compaction, this length becomes singular at transition porosities, revealing a change in the microstructure of the porespace controlling the transport. Nodal balances and lattice Boltzmann simulations of flow in pore networks under compaction show that the scaling between permeability and porosity changes near the transition porosities. Simulation results are compared with experimental permeability data from transparent two-dimensional micromodels of networks decorated with the same pore size distribution. Permeability?Cporosity data of media undergoing smooth compaction is well described by a single power law. Under strong compaction, however, the scaling between permeability and porosity is possible by traits only, the scaling exponent changes notably at given transition porosities. These behaviors are common to a wealth of permeability?Cporosity data thus far unexplained.     

Foam Mobility in Heterogeneous Porous Media

It is shown experimentally that in situ generation of foam is an effective method for achieving gas mobility control and diverting injected fluid to low permeability strata within heterogeneous porous media. The experimental system is composed of a 0.395 porosity, 5.35 µm² synthetic sandstone and a 0.244 porosity, 0.686 µm² natural sandstone. The cores are arranged in parallel and communicate through common injection and production conditions. Nitrogen is the gas phase and alpha-olefin sulfonate (AOS 1416) in brine is the foamer. Three types of experiments were conducted. First, gas alone was injected into the system after presaturation with the foamer solution. Second, gas and foamer solution were coinjected at an overall gas fraction of 90% into cores presaturated with surfactant. Each core accepted a portion of the injected gas and liquid according to the mobility within the core. Lastly, gas and foamer solution were coinjected into the individual, isolated porous media in order to establish baseline behavior. The results are striking. It is possible to achieve total diversion of gas injection to the low permeability medium in some cases. The results also confirm previous predictions that foamed gas can be more mobile in lower permeability porous media.     

On the Validation of a Compositional Model for the Simulation of $$\text {CO}_2$$ Injection into Saline Aquifers

In this work we apply a recently proposed Bayesian Markov chain Monte Carlo framework (Akbarabadi et al. in Comput Geosci 19(6):1231–1250, 2015) to quantify uncertainty in the three-dimensional permeability field of a rock core. This process establishes the credibility of a compositional two-phase flow model to describe the displacement of brine by \(\text {CO}_2\) and \(\text {CO}_2\) storage in saline aquifers. We investigate the predictive capabilities of the compositional model in the context of an unsteady-state \(\text {CO}_2\)-brine drainage experiment at the laboratory scale, performed at field-scale aquifer conditions. We employ forward models consisting of a system of discretized partial differential equations along with relative permeability curves obtained by a curve fitting of experimental measurements. We consider a forward model to be validated when: (1) numerical simulations reveal that the Bayesian framework has accurately characterized the core’s permeability and (2) Monte Carlo predictions show excellent agreement between measured and simulated data. A large set of numerical studies with an accurate compositional simulator shows that forward models have been successfully validated. For such models, our numerical results show that we are able to capture all the dominant features and general trends of the \(\text {CO}_2\) saturation fields observed in the core. Our study is consistent with the design and findings of real experiments. Fluid properties, relative permeability data, measured porosity field, physical dimensions, and thermodynamic conditions are the same as those reported in Akbarabadi and Piri (Adv Water Resour 52:190–206, 2013). However, the measured saturation data are from flow experiments different from those reported in Akbarabadi and Piri (2013), and will be presented here.     

One- and Two-Phase Permeabilities of Vugular Porous Media

The single and double phase macroscopic permeabilities of bimodal reconstructed porous media have been studied. The structure of these bimodal media is characterized by the micro and macroporosities (vug system) and by the micro and macrocorrelation lengths l _p and l _v. For a single phase, if the vugular system does not percolate, it is shown that the absolute permeability K mainly depends on l _p and very little on the other parameters. However, when the vugs percolate, K is also influenced by the density of vugs. For double phase calculations (in strong wettability conditions), it is shown that a vuggy percolating system affects mainly the nonwetting phase permeability. Moreover, the relative permeabilities for a nonpercolating vuggy system are only slightly influenced by the porosity distribution. These predictions are in good agreement with some experimental data obtained with limestones.     

On the skin friction coefficient in viscoelastic wall-bounded flows

Analysis of the skin friction coefficient for wall bounded viscoelastic flows is performed by utilizing available direct numerical simulation (DNS) results for viscoelastic turbulent channel flow. The Oldroyd-B, FENE-P and Giesekus constitutive models are used. First, we analyze the friction coefficient in viscous, viscoelastic and inertial stress contributions, as these arise from suitable momentum balances, for the flow in channels and pipes. Following Fukagata et al. (Phys. Fluids, 14, p. L73, 2002) and Yu et al. (Int. J. Heat. Fluid Flow, 25, p. 961, 2004) these three contributions are evaluated averaging available numerical results, and presented for selected values of flow and rheological parameters. Second, based on DNS results, we develop a universal function for the relative drag reduction as a function of the friction Weissenberg number. This leads to a closed-form approximate expression for the inverse of the square root of the skin friction coefficient for viscoelastic turbulent pipe flow as a function of the friction Reynolds number involving two primary material parameters, and a secondary one which also depends on the flow. The primary parameters are the zero shear-rate elasticity number, El₀, and the limiting value for the drag reduction at high Weissenberg number, LDR, while the secondary one is the relative wall viscosity, μ_w. The predictions reproduce both types A and B of drag reduction, as first introduced by Virk (Nature, 253, p. 109, 1975), corresponding to partially and fully extended polymer molecules, respectively. Comparison of the results for the skin friction coefficient against experimental data shows good agreement for low and moderate drag reduction which is the region covered by the simulations.     

Influence of phase behavior on chemical flood transport phenomena

We propose that there are two classes of temporal development in the degradation of permeability of porous media due to deposition of fines: (1) Deposition, and therefore permeability degradation, is localized to bands growing orthogonally to the average local flow direction, and (2) permeability degradation occurs in stripes parallel to the local flow direction. These latter stripes do not influence total permeability much as they develop. When these stripes are allowed to develop, they coalesce and worm holes form. We discuss how imposing different flow conditions such as constant flow and constant pressure influence the deposition process. Our conclusion is that constant pressure conditions typically lead to a slower permeability degradation compared to constant flow conditions as a direct consequence of the formation of low-permeability bands. We test our ideas by numerical simulations on a simple model for fines migration and deposition in porous media.