Log-normal Permeability Distribution Research Articles

Temporal Markov Processes for Transport in Porous Media: Random Lattice Networks

AbstractMonte Carlo (MC) simulations of transport in random porous networks indicate that for high variances of the lognormal permeability distribution, the transport of a passive tracer is non‐Fickian. Here we model this non‐Fickian dispersion in random porous networks using discrete temporal Markov models. We show that such temporal models capture the spreading behavior accurately. This is true despite the fact that the slow velocities are strongly correlated in time, and some studies have suggested that the persistence of low velocities would render the temporal Markovian model inapplicable. Compared to previously proposed temporal stochastic differential equations with case‐specific drift and diffusion terms, the models presented here require fewer modeling assumptions. Moreover, we show that discrete temporal Markov models can be used to represent dispersion in unstructured networks, which are widely used to model porous media. A new method is proposed to extend the state space of temporal Markov models to improve the model predictions in the presence of extremely low velocities in particle trajectories and extend the applicability of the model to higher temporal resolutions. Finally, it is shown that by combining multiple transitions, temporal models are more efficient for computing particle evolution compared to correlated CTRW with spatial increments that are equal to the lengths of the links in the network.

Effective Hydraulic Conductivity of Three‐Dimensional Heterogeneous Formations of Lognormal Permeability Distribution: The Impact of Connectivity

AbstractUniform mean flow takes place in a 3‐D heterogeneous formation of normal hydraulic logconductivity . The aim of the study is to derive the dependence of the horizontal Kefh and vertical Kefv effective conductivities on the structural parameters of hydraulic conductivity and investigate the impact of departure from multi‐Gaussianity on Kef, by numerical simulations of flow in formations that share the same pdf and covariance of Y but differ in the connectivity of classes of Y. The main result is that for the extreme models of connected and disconnected high Y zones the ratio between the effective conductivities in isotropic media is much smaller than in 2‐D. The dependence of and Kefv upon the logconductivity variance and the anisotropy ratio is compared with existing approximations (first‐order, Landau‐Matheron conjecture, self‐consistent approximation). Besides the theoretical interest, the results offer a basis for empirical relationships to be used in applications.

Impact of heterogeneous permeability distribution on the groundwater flow systems of a small sedimentary basin

Ground water flow systems of shallow sedimentary basins are studied in general by analyzing the fluid dynamics at the real world example of the Thuringian Basin. The impact of the permeability distribution and density differences on the flow velocity pattern, the salt concentration, and the temperature distribution is quantified by means of transient coupled simulations of fluid flow, heat, and mass transport processes. Simulations are performed with different permeabilities in the sedimentary layering and heterogeneous permeability distributions as well as with a non-constant fluid density. Three characteristic numbers are useful to describe the effects of permeability on the development of flow systems and subsurface transport: the relation of permeability between aquiclude and aquifer, the variance, and the correlation length of the log-normal permeability distribution. Density dependent flow due to temperature or concentration gradients is of minor importance for the distribution of the flow systems, but can lead to increased mixing dissolution of salt. Thermal convection is in general not present. The dominant driver of groundwater flow is the topography in combination with the permeability distribution. The results obtained for the Thuringian Basin give general insights into the dynamics of a small sedimentary basin due to the representative character of the basin structure as well as the transferability of the settings to other small sedimentary basins.

Reducing uncertainty associated with CO2 injection and brine production in heterogeneous formations

Spatial heterogeneity and variability across many orders of magnitude are two properties of formation permeability that are challenging to effectively capture in subsurface flow models. With regard to geological CO2 storage, heterogeneity affects storage capacity, plume migration, pressure build-up around injection wells and, if utilized, the possibility of early breakthrough at brine extraction wells. Faced with under-characterized heterogeneity and an information-poor environment, the challenge is to make as much use as possible of the available data in reducing uncertainty associated with model prediction of a CO2 injection operation. The two types of data used here to inform models are point permeability measurements from each well and interference tests between wells; these are both obtained from a synthetic permeability distribution approximating a hypothetical CO2 storage formation. These data are used to reduce uncertainty in predictions of CO2 injection, brine and CO2 production rates, and CO2 breakthrough times obtained from simulations of five-spot CO2 injection/brine production performed on >600 conditional permeability realizations. First, assuming the underlying permeability field exhibits long-range correlation and a log-normal permeability distribution, a Bayesian analysis was undertaken using the point permeability measurements to reduce the likely magnitude of permeability heterogeneity. Comparison of realizations drawn from the prior and posterior indicate that the data were sufficient to exclude extreme scenarios with very rapid or very slow CO2 transport and breakthrough at brine production wells. Thirty-year prediction envelopes for net CO2 retention, brine and CO2 production were reduced by 40, 45 and 72%, respectively. In the second stage, simulated interference tests for each of the posterior realizations were compared against interference tests simulated for the synthetic “true” permeability field. Weighting functions derived from these comparisons are used to further refine predictions, providing additional reductions of uncertainty by 38 and 69% for brine and CO2 production rates, respectively. Predictions of CO2 breakthrough are especially sensitive to analysis of interference tests, with a good estimate of the correct time obtained in three out of four cases.

Solver-based vs. grid-based multilevel Monte Carlo for two phase flow and transport in random heterogeneous porous media

We consider two phase flow and transport in heterogeneous porous media with uncertain permeability distribution. The resulting transport uncertainty is assessed by means of multilevel Monte Carlo (MLMC). In contrast to the Monte Carlo (MC) method, which operates on one specific numerical grid with one numerical solver, MLMC samples from a hierarchy of grids or numerical solvers. In this work, the MLMC performance resulting from a hierarchy consisting of a finite volume transport solver and a streamline-based solver is compared to a purely grid-based hierarchy. Unlike the established grid-based MLMC method, our solver-based MLMC method operates on the same numerical grid and therefore avoids difficulties related to the upscaling of permeability fields or boundary conditions on coarser grids. For a two dimensional test case with log-normal permeability distribution, both MLMC approaches are compared to a MC reference run. At equivalent accuracy, significant speedups of MLMC with respect to MC are achieved.

The Modification of the Dykstra-Parsons Method for Inclined Stratified Reservoirs

Summary The Dykstra-Parsons method (Dykstra and Parsons 1950) is used to predict the performance of waterflooding in noncommunicating stratified reservoirs. Much interest has been shown recently in the application of the method to chemical flooding, particularly for the case of polymer injection used for mobility control. The original method assumes that the reservoir layers are horizontal; however, most oil reservoirs exhibit a dip angle, with water being injected in the updip direction. Therefore, it is important to account for the effect of inclination on the performance of the method. A modification of the Dykstra-Parsons equations is obtained to account for reservoir inclination. The developed model includes a dimensionless gravity number that accounts for the effect of the dip angle and the density difference between the displacing and displaced fluids. The derived equation that governs the relative locations of the displacement fronts in different layers is nonlinear, includes a logarithmic term, and requires an iterative numerical solution. This solution is used to estimate the fractional oil recovery, the water cut, the injected pore volume, and the injectivity ratio at the time of water breakthrough in successive layers. Solutions for stratified systems with log-normal permeability distribution were obtained and compared with horizontal systems. The effects of the gravity number, the mobility ratio, and the Dykstra-Parsons permeability-variation coefficient (VDP) on the performance were investigated. Cases of updip and downdip injection are discussed. It was found that for a positive gravity number (updip water injection), performance is enhanced in terms of delayed water breakthrough, increased fractional oil recovery, and decreased water cut as compared with horizontal layers. This occurs for both favorable and unfavorable mobility ratios but is more evident in unfavorable mobility ratios and more-heterogeneous cases. For the case of a negative gravity number (downdip water injection or updip gas injection), the opposite behavior was observed. The results were also compared with the performance of inclined communicating reservoirs with complete crossflow. The effect of communication between layers was found to improve fractional oil recovery for favorable and unit mobility ratios and decrease recovery for unfavorable mobility ratio.

CHEMO-hydrodynamic coupling between forced advection in porous media and self-sustained chemical waves

Autocatalytic reaction fronts between two reacting species in the absence of fluid flow, propagate as solitary waves. The coupling between autocatalytic reaction front and forced simple hydrodynamic flows leads to stationary fronts whose velocity and shape depend on the underlying flow field. We address the issue of the chemico-hydrodynamic coupling between forced advection in porous media and self-sustained chemical waves. Towards that purpose, we perform experiments over a wide range of flow velocities with the well characterized iodate arsenious acid and chlorite-tetrathionate autocatalytic reactions in transparent packed beads porous media. The characteristics of these porous media such as their porosity, tortuosity, and hydrodynamics dispersion are determined. In a pack of beads, the characteristic pore size and the velocity field correlation length are of the order of the bead size. In order to address these two length scales separately, we perform lattice Boltzmann numerical simulations in a stochastic porous medium, which takes into account the log-normal permeability distribution and the spatial correlation of the permeability field. In both experiments and numerical simulations, we observe stationary fronts propagating at a constant velocity with an almost constant front width. Experiments without flow in packed bead porous media with different bead sizes show that the front propagation depends on the tortuous nature of diffusion in the pore space. We observe microscopic effects when the pores are of the size of the chemical front width. We address both supportive co-current and adverse flows with respect to the direction of propagation of the chemical reaction. For supportive flows, experiments and simulations allow observation of two flow regimes. For adverse flow, we observe upstream and downstream front motion as well as static front behaviors over a wide range of flow rates. In order to understand better these observed static state fronts, flow experiments around a single obstacle were used to delineate the range of steady state behavior. A model using the "eikonal thin front limit" explains the observed steady states.

Interpreting tracer tests in the enhanced geothermal system (EGS) of Soultz-sous-Forêts using the equivalent stratified medium approach

A new interpretation of tracer tests performed in 2005 in the French EGS located at Soultz-sous-Forêts is presented. A previous analysis of fluorescein tests revealed significant differences in terms of connectivity between the injection well (GPK3) and the two production wells (GPK2 and GPK4). The results suggested the existence of at least three fluid circulation loops. Two of them (loops 1 and 2) connecting GPK3 to GPK2, and the third one (loop 3) connecting GPK3 to GPK4. The present analysis is based on the equivalent stratified porous medium model, in which each circulation loop corresponds to a flow path through the rock formation and is regarded as a stratified porous medium having a log-normal permeability distribution. The heterogeneity of this equivalent porous medium is quantified by the stratification factor defined as the standard deviation to the mean permeability ratio. A very accurate fit of the concentration curves for both production wells was obtained. The stratification factors show that the three circulation loops have different levels of heterogeneity. The wellhead pressure recorded during the tests was used to calculate the mean permeability of the loops. Loop 1 appears to be very permeable and narrow while loops 2 and 3 are less permeable and wider. This tracer test model is compared to a new deterministic 3D model of the fracture zones observed in the Soultz granitic reservoir. A well-known major fault in the Soultz Graben intersecting GPK2 and GPK3 could explain the rapid tracer circulation in loop 1.

Waterflooding Performance in Inclined Communicating Stratified Reservoirs

Summary A mathematical model is developed for performance prediction of waterflooding performance in communicating stratified reservoirs with a dip angle from the horizontal. The effect of the gravitational force is reflected by a dimensionless gravity number in the fractional flow formula. The gravity number accounts for the dip angle and the density difference between the displacing and displaced fluids. The developed fractional flow formula is used to estimate the fractional oil recovery, the dimensionless time, and the injectivity ratio at times of water breakthrough in the successive layers. The developed model allows for each layer to have its own porosity, endpoint saturations, and endpoint relative permeabilities. Solutions for the waterflooding performance in inclined communicating stratified systems with log-normal permeability distribution were obtained and compared with that of the horizontal systems. The effects of the gravity number, the mobility ratio, and the Dykstra-Parsons permeability-variation coefficient VDP on the performance were investigated. The obtained results showed that the gravity effect of the dip angle enhances the performance in terms of delayed water breakthrough, higher fractional oil recovery, and lower water cut. This improved performance is more significant in the cases of unfavorable mobility ratio and of highly heterogeneous reservoirs. Reservoir dipping does not affect the pseudorelative permeability functions but results in a decrease in the injectivity ratio.

Letter to the Editor: Comments on the Paper “Inertia Effects in High-Rate Flow Through Heterogeneous Porous Media” by M. Fourar, R. Lenormand, M. Karimi-Fard, and R. Horne, Transport in Porous Media, DOI 10.1007/s11242-004-6800-6, 2005

In a recent article, Fourar et al. (Transp Porous Med, 2005, doi:10.1007/s11242-004-6800-6) analyzed the effect of heterogeneity in the permeability distribution on Forchheimer flow in porous media. They derived expressions to calculate the effective inertial coefficient in serial layers, parallel layers, and two-dimensional correlated media. Here, we highlight an inconsistency in their first-order expression for serial layers and extend their findings by providing closed-form expressions for the effective inertial coefficient in the case of a lognormal permeability distribution.

Immiscible Displacement of Non-Newtonian Fluids in Communicating Stratified Reservoirs

Summary The displacement of non-Newtonian power-law fluids in communicating stratified reservoirs with a log-normal permeability distribution is studied. Equations are derived for fractional oil recovery, water cut, injectivity ratio, and pseudorelative permeability functions, and the performance is compared with that for Newtonian fluids. Constant-injection-rate and constant-total-pressure-drop cases are studied. The effects of the following factors on performance are investigated: the flow-behavior indices, the apparent mobility ratio, the Dykstra-Parsons variation coefficient, and the flow rate. It was found that fractional oil recovery increases for nw>no and decreases for nw<no, as compared with Newtonian fluids. For the same ratio of nw/no, oil recovery increases as the apparent mobility ratio decreases. The effect of reservoir heterogeneity in decreasing oil recovery is more apparent for the case of nw>no. Increasing the total injection rate increases the recovery for nw>no, and the opposite is true for nw<no. It also was found that the fractional oil recovery for the displacement at constant total pressure drop is lower than that for the displacement at constant injection rate, with the effect being more significant when nw<no.

Towards a unified framework for anomalous transport in heterogeneous media

Abstract We develop a unified framework to model the anomalous transport of tracers in highly heterogeneous media. While the framework is general, our working media in this study are geological formations. The basis of our approach takes into account the different levels of uncertainty, often associated with spatial scale, in characterizing these formations. The effects on the transport of smaller spatial scale heterogeneities are treated probabilistically with a model based on a continuous time random walk (CTRW), while the larger scale variations are included deterministically. The CTRW formulation derives from the ensemble average of a disordered system, in which the transport in each realization is described by a Master Equation. A generic example of such a system – a 3D discrete fracture network (DFN) – is treated in detail with the CTRW formalism. The key step in our approach is the derivation of a physically based ψ( s ,t) , the joint probability density for a displacement s with an event-time t. We relate the ψ( s ,t) to the velocity spectrum Φ(ξ) (|ξ|=1/v, ξ = v ) of the steady flow-field in a fluid-saturated DFN. Heterogeneous porous media are often characterized by a log-normal permeability distribution; the Φ(ξ) we use in this case is an analytic form approximating the velocity spectrum derived from this distribution. The common approximation of ψ( s ,t)∼p( s )t −1−β with a constant β, is evaluated in these cases. For the former case it is necessary to include s −t coupling while the latter case points to the presence of an effective t-dependent β. The full range of these features can be included in the CTRW solution but, as is shown, not in the fractional-time derivative equation (FDE) formulation of CTRW. Finally, the methods used for the unified framework are critically examined.

Waterflooding Performance of Communicating Stratified Reservoirs With Log-Normal Permeability Distribution

Summary An analytical solution is developed for waterflooding performance of layered reservoirs with a log-normal permeability distribution with complete crossflow between layers. The permeability distribution is characterized by the Dykstra-Parsons (DP) variation coefficient VDP or the standard deviation of the distribution σk. The performance is expressed in terms of vertical coverage as function of the producing water-oil ratio. Also an expression for the dimensionless time (pore volumes of injected water) at a given water-oil ratio is derived. Expressions are also derived for pseudorelative permeability functions and fractional flow curves that can be used in reservoir simulation. Correlation charts are also presented to enable graphical determination of the performance. The variables are combined in such a way that a single chart is constructed for the entire range of water-oil ratio, mobility ratio and permeability variation. Analogy to the Buckley-Leverett (BL) multiple-valued saturation profile is found to occur at low mobility ratios (M<1) where a multiple-valued displacement front is formed. A procedure similar to the BL discontinuity is suggested to handle this situation. Successive layers with different permeabilities are allowed to move with the same velocity resulting in a single-valued profile with a discontinuity. No such behavior is observed for mobility ratios greater than unity. A criterion for the minimum mobility ratio at which this behavior occurs is presented as a function of the variation coefficient VDP.

On the presence of double log-normal permeability distributions in ultrafiltration membranes

The possibility of a double log-normal (bimodal) permeability distribution for ultrafiltration membranes is discussed and the existence demonstrated by plotting the number of water flux observations against the logarithm of the pure water fluxes. Ultrafiltration membrane water fluxes were also simulated using the Monte Carlo method assuming double log-normal distributions of the pore size. Some explanations of the phenomenon of double log-normal distributions are proposed, based on the presence of two independent distributions: one of the spacing between the subunits, and one of the pores within the microporous subunits that make up the material, for instance an ultrafiltration membrane. The importance of the membrane thickness and its variation is also pointed out. A thick membrane has lower permeability than a thin at a given pore size distribution.

Groundwater Flux, Travel Time, and Radionuclide Transport

AbstractInflow measurements at Stripa and in other underground openings in Sweden, as well as observations elsewhere in mines and tunnels, reveal that there is generally an extremely broad distribution of groundwater flux in fractured rock. Non-sorbing and sorbing tracer tests typically show similar variability in groundwater travel time (GWTT) and tracer transport.In the U.S. Nuclear Waste Program, Nuclear Regulatory Commission regulations require the GWTT from the disturbed zone to the accessible environment to exceed 1000 years. The regulations seem to envision a rather uniform and narrow distribution of travel time, with perhaps a few identifiable “fast pathways” contained within the rock mass surrounding a potential repository. The premise is that most of these features could be mapped during site characterization, and that regions of the potential repository host rock containing such features could be avoided during waste emplacement.However, both field experience and theoretical studies in recent years provide strong evidence that groundwater flux, GWTT, and aqueous transport of dissolved substances exhibit extremely heterogeneous behavior, even in intact porous media and in fractured rock regions between major features. These phenomena are all dominated by the spatial distribution of permeability within the rock mass of interest. The permeability distribution is often approximately log-normal, with a natural log standard deviation, σ. For unfractured porous rock, σ typically ranges from about 0.6 to about 1.2 for field-scale investigations, and for fractured permeable media, it frequently exceeds 2. Values of σ smaller than 0.6 may be observed in small field-scale projects when the macroscopic flow regime is essentially linear within very uniform sediments and in laboratory displacement experitments.With some additional assumptions, a log-normal permeability distribution implies that groundwater flux, GWTT, and the transport of radionuclides from a potential repository are also log-normal. To first order, the appropriate value of σ describing these distributions is the same as the value for the permeability distribution. This allows σ to be estimated from a large number of hydraulic or pneumatic packer tests within the fractured rock mass of interest.We define a groundwater transport function (GWTF) for the rate of radioactivity release to the accessible environment (AE) at time t resulting from the release of a pulse of unit activity at time 0. The GWTF depends on the mean groundwater travel time, tw, and σ, as well as the retardation factor and decay constant. As σ increases from 0 (a hypothetical completely homogeneous system), the radioactivity breakthrough at early time increases from 0 to 100%. This behavior is consistent with our intuitive notions of “fast transport pathways” in heterogeneous systems, and σ is thus seen to be a parameter for quantifying the effects of heterogeneity.Convolution of the GWTF with a time-dependent release function for the Engineered Barrier System (EBS) is easily performed numerically, resulting in the rate of release to the AE as a function of time, which can then be integrated numerically to calculate the cumulative release as a function of time. The convolution approach clearly separates the effects of uncertainty and heterogeneity on repository performance and is extremely useful for sensitivity analyses. An example calculation shows the combinations of σ and tw required for compliance with total system release standards.Since the effect of heterogeneity is captured by a single parameter in a deterministic calculation, uncertainty can be investigated separately by Monte Carlo sampling from distributions of such parameters as σ, tw and source term strength, allowing (in the future) specific and scientifically meaningful goals to be defined for both site characterization and design.Finally, we emphasize that this approach, in its present form, does not include thermal effects. These effects may dominate both the EBS failure rate and hydrogeochemical behavior, including radionuclide transport, for most of the compliance period and beyond. It cannot be used directly to support any particular thermal loading strategy.

Tracer- and Pressure-Test Analysis for Characterization of Areally Heterogeneous Reservoirs

Summary This study compares simulated well-to-well tracer and transient pressure tests in a quadrant of a repeated five-spot with spatial variations in permeability. Single-layer heterogeneous media with an autocorrelated and log-normal permeability distribution are generated with a stochastic moving-average method. Finite-difference simulation of pressure behavior in these systems indicates that the geometric mean of effective permeabilities around injection and production wells is a good approximation for the steady-state interwell permeability. A dimensionless permeability difference, defined in terms of these quantities, correlates with a heterogeneity index, defined as the product of permeability variance and a dimensionless correlation length scale. Tracer flow is simulated with a particle-tracking procedure. Simulations show that when the heterogeneity index is small, tracer response can be matched with solutions of the convection-dispersion equation with a constant dispersivity that is proportional to the heterogeneity index. For larger values of this index, preferential flow paths are created that cause tracer-breakthrough curves to behave as though they result from flow in a layered system. A method is proposed to estimate the heterogeneity index from pressure-test data and thus to predict the nature of the tracer response qualitatively.

Prediction of Waterflood Behavior in a Stratified System

Abstract A general model which approximates the effect of cross-flow has been developed to give a practical method for predicting the water flood behavior of a stratified reservoir. The model is based on a modification of Dietz's theory and allows for variations in both the permeability and hydrocarbon pore volume.in the particular cases considered, it is assumed that the permeability can be characterized by a log- normal distribution and the hydrocarbon pore volume by a normal distribution. A simple graphical method which enables the practicing engineer to predict the behavior of a stratified system is presented The results obtained by the proposed method are compared with those obtained by the Dykstra-Parsons method As a result of this study the following conclusions have been drawn: - The effect of cross-flow in a stratified system can be appreciable, particularly at very favorable or very unfavorable mobility ratios; - Under normal conditions, the effect of variations in the hydrocarbon pore volume can be neglected; and - The failure to use all of the available permeability data can lead to large errors in the prediction of the behavior of a stratified reservoir. Introduction Various methods have been proposed to characterize and to predict the waterflood behavior of a stratified reservoir. Most of the methods assume that the reservoir is composed of discrete homogeneous continuous layers. With this model, the degree of stratification can be measured by several parameters based on core-analysis data. Among these are the Lorenz coefficient and the variation of a log-normal permeability distribution. The behavior of stratified systems is usually predicted by the Stiles, Dykstra-Parsons method or some modification of these. In both of these methods the reservoir is divided into discrete homogeneous layers with no cross-flow between layers. In the Stiles method the mobility ratio is assumed to be equal to unity while in the Dykstra-Parsons method it is allowed to take on any value. Other predictive methods have been proposed; Hiatt's method allows for cross-flow between the beds and a method by Schmalz and Rahme correlates the recovery directly with the Lorenz coefficient. In this new approach, which is essentially a continuous analog of Hiatt's method, the effects of both mobility ratio and cross-flow between the beds have been included. It is assumed that the permeability can be represented by a log-normal distribution and the hydrocarbon pore volume or porosity by a normal distribution. These types of distributions have been observed by several authors, and most field data seem to confirm their observations; e.g., if these distributions are assumed and there is a 1:1 correspondence between porosity and permeability samples, the commonly encountered linear relationship between porosity and the log of permeability is obtained.