Capillary Pressure-saturation Function Research Articles

Forced imbibition and uncertainty modeling using the morphological method

The morphological approach is a computationally attractive method for calculating relative permeability and capillary pressure saturation functions. In the corresponding workflow, morphological operations are used to calculate the fluid phase distribution in the pore space of a digital twin. Once the pore space is occupied, the conductivity of the individual fluid phases and thus the relative permeability can be calculated by direct flow simulations. It therefore combines computationally favorable geometric operations with direct flow simulations. In contrast to pore network modeling, all calculations are directly performed on the digital twin without abstraction of the pore space. While the morphological operations conceptually correctly describe primary drainage processes and delivers good results, the method so far failed to describe imbibition processes and the influence of wettability. In this work, we implement contact angle distributions in a deterministic and stochastic way. In this manner, we extend the simulated saturation range from purely spontaneous to forced imbibition, resulting in a full-range imbibition relative permeability. Furthermore, by introducing stochastic contact angle distributions, different fluid phase distributions are obtained, which now allow for an uncertainty analysis. To verify the simulation results, we check (a) whether the simulation results agree with SCAL measurements and (b) compare morphologically and experimentally derived results on the pore scale. With the newly introduced concepts, the imbibition process behaves as physically expected, and shows a good agreement with experimentally derived relative permeability curves and microscopic fluid-phase distributions.

Simultaneous interpretation of SCAL data with different degrees of freedom and uncertainty analysis

Multiphase flow in porous media is relevant in many areas of geoenergy engineering and is governed by relative permeability and capillary pressure saturation functions. These functions are key uncertainties in reservoir engineering and their measurement is demanding and resource intensive. Despite the experimental effort, it is not yet common practice to numerically interpret the data, nor is it common practice to investigate their uncertainty. Furthermore, data interpretation is limited to power-law functions, which is insufficient for describing complex rock types such as microscopically heterogeneous carbonates.We developed a MATLAB-MRST-based simulator for simultaneous interpretation of data sets from different experimental techniques. We discuss the implementation of the common parametrized relative permeability representations and their deficiency to describe data from complex rocks. To overcome this limitation, a point-by-point approach is developed and applied to an extensive carbonate data set. For uncertainty analysis, a Markov Chain Monte Carlo sampling-based workflow is implemented and applied. The uncertainty is discussed in the frame of the individual data set, simultaneously analyzed data sets, and the sample-to-sample variation. The developed method makes the interpretation of relative permeability and capillary pressure saturation functions conclusive, especially in cases of complex line shapes, and is an essential step toward uncertainty-driven stochastic reservoir modeling.

In-situ capillary pressure and wettability in natural porous media: Multi-scale experimentation and automated characterization using X-ray images

HypothesisGeometrical analyses of pore-scale fluid-fluid-rock interfaces have recently been used for in-situ characterization of capillary pressure and wettability in natural porous media. Nevertheless, more robust techniques and multi-scale, well-characterized experimental data are needed to rigorously validate these techniques and enhance their efficacy when applied to saturated porous media. Experiments and Image analysisWe present two new techniques for automated measurements of in-situ capillary pressure and contact angle, which offer several advancements over previous methodologies. These approaches are methodically validated using synthetic data and X-ray images of capillary rise experiments, and subsequently, applied on pore-scale fluid occupancy maps of a miniature Berea sandstone sample obtained during steady-state drainage and imbibition flow experiments. FindingsThe results show encouraging agreement between the image-based capillary pressure-saturation function and its macroscopic counterpart obtained from a porous membrane experiment. However, unlike the macroscopic behavior, the micro-scale measurements demonstrate a nonmonotonic increase with saturation due to the intermittency of the pore-scale displacement events controlling the overall flow behavior. This is further explained using the pertinent micro-scale mechanisms such as Haines jumps. The new methods also enable one to generate in-situ contact angle distributions and distinguish between the advancing and receding values while automatically excluding invalid measurements.

Impact of the capillary pressure-saturation pore-size distribution parameter on geological carbon sequestration estimates

Cost estimates for geologic carbon sequestration (GCS) are vital for policy and decision makers evaluating carbon capture and storage strategies. Numerical models are often used in feasibility studies for the different stages of carbon injection and redistribution. Knowledge of the capillary pressure-saturation function for a selected storage rock unit is essential in applications used for simulating multiphase fluid flow and transport. However, the parameters describing these functions (e.g. the van Genuchten m pore size distribution parameter) are often not measured or neglected compared to other physical properties such as porosity and intrinsic permeability. In addition, the use of average instead of point estimates of m for numerical simulations of flow and transport can result in significant errors, especially in the case of coarse-grained sediments and fractured rocks. Such erroneous predictions can pose great risks and challenges to decision-making. We present a comparison of numerical simulation results based on average and point estimates of the van Genuchten m parameter for different porous media. Forward numerical simulations using the STOMP code were employed to illustrate the magnitudes of the differences in carbon sequestration predictions resulting from the use of height-averaged instead of point parameters. The model predictions were converted into cost estimates and the results indicate that varying m values in GCS modeling can cause cost differences of up to hundreds of millions dollars.

Non-uniqueness and uncertainty quantification of relative permeability measurements by inverse modelling

Relative permeability is traditionally obtained in core flooding experiments from production data and pressure drop either using analytical methods or inverse modelling. In most cases the inverse modelling is performed manually, which does, however, not provide a realistic estimate of the resulting uncertainty. Here we provide a framework for a consistent uncertainty assessment of relative permeability measurements. We find that extracting relative permeability from experimental core flooding measurements by inverse modelling with the numerical solution of the 2-phase Darcy equations is an ill-posed problem for which the non-uniqueness, correlation of matching parameters and non-Gaussian errors are observed. We also show that the assisted history matching strategy where the two-phase immiscible displacement modelling of the 2-phase Darcy equations in 1D is coupled with a Levenberg-Marquardt based optimization scheme and using an adequate representation of relative permeability and capillary pressure-saturation functions leads to a realistic description of the data and the associated uncertainty which is – depending on conditions – significantly smaller than error of a manual interpretation with tabulated relative permeability. These findings suggest that the resulting does depend to some extent on the choice of relative permeability parameterization which are either too restrictive or have too many (correlated) parameters.

Linking continuum-scale state of wetting to pore-scale contact angles in porous media

HypothesisWetting phenomena play a key role in flows through porous media. Relative permeability and capillary pressure-saturation functions show a high sensitivity to wettability, which has different definitions at the continuum- and pore-scale. We hypothesize that the wetting state of a porous medium can be described in terms of topological arguments that constrain the morphological state of immiscible fluids, which provides a direct link between the continuum-scale metrics of wettability and pore-scale contact angles. ExperimentsWe perform primary drainage and imbibition experiments on Bentheimer sandstone using air and brine. Topological properties, such as Euler characteristic and interfacial curvature are measured utilizing X-ray micro-computed tomography at irreducible air saturation. We also present measurements for the United States Bureau of Mines (USBM) index, capillary pressure and pore-scale contact angles. Additional studies are performed using two-phase Lattice Boltzmann simulations to test a wider range of wetting conditions. FindingsWe demonstrate that contact angle distributions for a porous multiphase system can be predicted within a few percent difference of directly measured pore-scale contact angles using the presented method. This provides a general framework on how continuum-scale data can be used to describe the geometrical state of fluids within porous media.

Dynamic measurements of drainage capillary pressure curves in carbonate rocks

The heterogeneity of rocks represents a challenge for interpreting and using outcomes from multiphase-flow experiments carried out on laboratory samples. While the capillary pressure–saturation function, pc(S), is known to vary spatially and cause local saturation development during immiscible displacements, its variation remains difficult to measure. This is particularly challenging for rocks with complex fabrics, such as carbonates. Here, we present a workflow for the dynamic measurement of core- and subcore-scale drainage pc(S) curves in heterogeneous porous media. Multi-rate, two-phase core-flooding tests are conducted on three carbonate rocks with direct observations of local saturation data. The interpretation of the experiments is done by fitting the parameters of the pc(S) curve, while describing both steady-state saturation and pressure profiles with a detailed one-dimensional model that accounts for the variation of subcore-scale properties in the direction of displacement. Workflow validation is achieved by means of synthetic data, thereby demonstrating the uniqueness of the solution of the resulting multi-objective optimisation problem. The model reproduces accurately experimental data on the three rocks and enables computing the effective core-scale pc(S) curve in the limit of zero velocity, as it would be expected during a porous-plate experiment. The output of the proposed technique is however much richer and includes the relativepc(S) curve that is universal and independent of the specific pattern of heterogeneity, in addition to a set of scaling factors. The latter describe the distribution of the pc(S) curves at the subcore-scale due to heterogeneity and form the statistical basis needed for upscaling studies.

Upscaling Capillary Pressure–Saturation Functions Using Different Reference Pressure Elevations

Core Ideas The BC‐vG Upscaler predicts the drainage of porous media of different heights. The default reference pressure elevation (RPE) is the middle of the porous medium. Predicted drainage functions deviate from observed functions as height increases. A top RPE improves accuracy of predicted drainage functions regardless of height. Choice of reference pressure elevation (RPE) on average effective saturation–capillary pressure functions, 〈〉(ψ), was investigated for monotonic drainage of homogeneous porous media. Nine columns of Flint sand with heights ranging from 4.3 to 55.0 cm were prepared. Measured 〈〉(ψ) functions were determined gravimetrically using the hanging water column method. Predicted 〈〉(ψ) functions were obtained by upscaling point function parameters determined by neutron radiographic imaging of a single drainage experiment. Bottom and midpoint RPEs resulted in the inaccurate parameterization of 〈〉(ψ) functions for tall columns. A top RPE produced accurate upscaled functions for all column heights. To evaluate the overall performance of this RPE, observed effective saturations for the nine columns were linearly regressed against predicted values. The resulting best‐fit model (slope = 0.98; intercept = 0.03; R2 = 0.98) corresponded closely to a 1:1 line. The van Genuchten (vG) α and n parameters for the observed and predicted 〈〉(ψ) functions decreased with increasing column height. A power model explained >95% of the variance in the predicted vG parameters and between 69 and 78% of the variance in the observed vG parameters. The lower R2 values for the observed parameter models were attributed to experimental variation among the nine columns, whereas the predicted parameter models were upscaled from a single column. Despite these differences, the magnitudes and height dependencies of the observed and predicted average vG parameters were similar. For tall columns, the RPE should be established at the top for drainage experiments and at the bottom for wetting experiments.

Impact of the capillary pressure-saturation pore-size distribution parameter on geological carbon sequestration estimates

Abstract Cost estimates for geologic carbon sequestration (GCS) are vital for policy and decision makers evaluating carbon capture and storage strategies. Numerical models are often used in feasibility studies for the different stages of carbon injection and redistribution. Knowledge of the capillary pressure-saturation function for a selected storage rock unit is essential in applications used for simulating multiphase fluid flow and transport. However, the parameters describing these functions (e.g. the van Genuchten m pore size distribution parameter) are often not measured or neglected compared to other physical properties such as porosity and intrinsic permeability. In addition, the use of average instead of point estimates of m for numerical simulations of flow and transport can result in significant errors, especially in the case of coarse-grained sediments and fractured rocks. Such erroneous predictions can pose great risks and challenges to decision-making. We present a comparison of numerical simulation results based on average and point estimates of the van Genuchten m parameter for different porous media. Forward numerical simulations using the STOMP code were employed to illustrate the magnitudes of the differences in carbon sequestration predictions resulting from the use of height-averaged instead of point parameters. The model predictions were converted into cost estimates and the results indicate that varying m values in GCS modeling can cause cost differences of up to hundreds of millions dollars.

Displacement and Mass Transfer of CO2/Brine in Sandstone

The displacement and fluid/fluid mass transfer between CO2 and brine in Berea sandstone have been investigated by unsteady-state core-flood experiments combined with x-ray computed tomography. Relative permeability and capillary pressure saturation functions of primary drainage for mutually saturated fluid phases have been determined from production data and the saturation profiles by history matching and have been benchmarked against standard SCAL. The displacement stability of the CO2/brine drainage process has been investigated by numerical modeling, and the consequences for experimental procedures and for geological storage of CO2 are discussed. Aspects of mass transfer during drainage and imbibition that are relevant for nonequilibrated fluid phases were studied by core flooding with unsaturated CO2 and brine phases.

Comparison of Average and Point Capillary Pressure–Saturation Functions Determined by Steady‐State Centrifugation

The capillary pressure–saturation function can be determined from centrifuge drainage experiments. In soil physics, the data resulting from such experiments are usually analyzed by the “averaging method.” In this approach, average relative saturation, 〈S〉, is expressed as a function of average capillary pressure, 〈ψ〉, i.e., 〈S〉(〈ψ〉). In contrast, the capillary pressure–saturation function at a physical point, i.e., S(ψ), has been extracted from similar experiments in petrophysics using the “integral method.” The purpose of this study was to introduce the integral method applied to centrifuge experiments to a soil physics audience and to compare S(ψ) and 〈S〉(〈ψ〉) functions, as parameterized by the Brooks–Corey and van Genuchten equations, for 18 samples drawn from a range of porous media (i.e., Berea sandstone, glass beads, and Hanford sediments). Steady‐state centrifuge experiments were performed on preconsolidated samples with a URC‐628 Ultra‐Rock Core centrifuge. The angular velocity and outflow data sets were then analyzed using both the averaging and integral methods. The results show that the averaging method smoothes out the drainage process, yielding less steep capillary pressure–saturation functions relative to the corresponding point‐based curves. Maximum deviations in saturation between the two methods ranged from 0.08 to 0.28 and generally occurred at low suctions. These discrepancies can lead to inaccurate predictions of other hydraulic properties such as the relative permeability function. Therefore, we strongly recommend use of the integral method instead of the averaging method when determining the capillary pressure–saturation function by steady‐state centrifugation. This method can be successfully implemented using either the van Genuchten or Brooks–Corey functions, although the latter provides a more physically precise description of air entry at a physical point.

Effective properties of two‐phase flow in heterogeneous aquifers

The spectral perturbation approach is employed to predict the field‐scale relative permeabilities and the variance of capillary pressure and saturation under steady two‐phase flow conditions. The theoretical analysis treats stationary three‐dimensional variations in pressure and flow produced as a result of spatial variations of permeability and two‐phase flow characteristics. The analysis is developed in a general form that is applied to different commonly used two‐phase flow characterizations, and results are illustrated for two‐phase flow characteristics corresponding to those observed at the Borden site in Canada. For gravity‐driven flow in horizontally layered aquifers, both nonwetting phase flow (dense nonaqueous phase liquid (DNAPL) flow) and wetting phase flow (water, unsaturated flow) show effective vertical relative permeabilities that are substantially decreased in comparison with those corresponding to a homogeneous system. The resulting saturation‐dependent anisotropy of effective permeability is substantially larger for DNAPL flow than for the case of unsaturated flow. The Leverett scaling characterization frequently used in Monte Carlo simulations is shown to underestimate system anisotropy owing to omission of variability of the parameter governing the slope in the capillary pressure–saturation function. Application of the Brooks‐Corey characterization of the relative permeability resulted in significantly greater anisotropy than with the van Genuchten characterization. In contrast to the strong influence of heterogeneity predicted in the case of effective permeabilities, the effective capillary pressure characteristic (mean capillary pressure versus mean saturation curve) is only weakly influenced by heterogeneity.

The influence of field-scale heterogeneity on the infiltration and entrapment of dense nonaqueous phase liquids in saturated formations

Numerical simulations are used for the systematic exploration of the migration and entrapment of dense nonaqueous phase liquids (DNAPLs) in heterogeneous formations. Ensembles of realizations of random, spatially correlated permeability fields are generated and employed in model simulations of a spill event. Statistical techniques are then used to quantify the sensitivity of model predictions to input parameters, thereby identifying the parameters or processes that may be of primary importance in the determination of organic liquid distributions in heterogeneous systems. Results of the study indicate that the most critical factors in modeling organic entrapment include the spill release rate, reliable estimates of the mean, variance, and vertical correlation scale of the formation permeability, and an accurate representation of the correlation between the capillary pressure–saturation function and the permeability. In contrast, the hydraulic gradient and cross-correlation of residual saturations with permeabilities are found to have only minor influence on organic liquid distributions in such heterogeneous formations.

Parameter estimation of two-fluid capillary pressure–saturation and permeability functions

Accurate characterization and modeling of subsurface flow in multi-fluid soil systems require development of a rapid, reproducible experimental method that yields the information necessary to determine the parameters of the capillary pressure–saturation and permeability functions. Previous work has demonstrated that parameter optimization using inverse modeling is a powerful approach to indirectly determine these constitutive relationships for air–water systems. We consider extension of the inverse parameter estimation method to the modified multi-step outflow method for two-fluid (air–water, air–oil and oil–water) flow systems. The wellposedness of the proposed parameter estimation problem is evaluated by its accuracy, uniqueness and parameter uncertainty. Seven different parametric models for the capillary pressure–saturation and permeability functions were tested in their ability to fit the multi-step outflow experimental data; these included the van Genuchten–Mualem (VGM) model, van Genuchten–Burdine (VGB), Brooks–Corey–Mualem (BCM), Brooks–Corey–Burdine (BCB), Brutsaert–Burdine (BRB), Gardner–Mualem (GDM) and Lognormal Distribution–Mualem (LNM) models. The VGB, BCM and BCB models fitted the multi-step outflow data poorly, and resulted in relatively large values for the root-mean-squared residuals. It was concluded that the remaining VGM, LNM, BRB and GDM models successfully characterized the multi-step experimental data for two-fluid flow systems. Although having one parameter less than the other models, the GDM model's performance was excellent. Finally, we conclude that optimized capillary pressure–saturation functions for the oil–water and air–oil could be predicted from the respective air–water function using interfacial tension ratios.

Flow and entrapment of dense nonaqueous phase liquids in physically and chemically heterogeneous aquifer formations

The migration and entrapment of dense nonaqueous phase liquids (DNAPLs) in aquifer formations is typically believed to be controlled by physical heterogeneities. This belief is based upon the assumption that permeability and capillary properties are determined by the soil texture. Capillarity and relative permeability, however, will also depend on porous medium wettability characteristics. This wettability may vary spatially in a formation due to variations in aqueous phase chemistry, contaminant aging, and/or variations in mineralogy and organic matter distributions. In this work, a two-dimensional multiphase flow simulator is modified to simulate coupled physical and chemical formation heterogeneity. To model physical heterogeneity, a spatially correlated permeability field is generated, and then related to the capillary pressure-saturation function according to Leverett scaling. Spatial variability of porous medium wettability is assumed to be correlated with the natural logarithm of the intrinsic permeability. The influence of wettability on the hysteretic hydraulic property relations is also modeled. The simulator is then employed to investigate the potential influence of coupled physical and chemical heterogeneity on DNAPL flow and entrapment. For reasonable ranges of wettability characteristics, simulations demonstrate that spatial variations in wettability can have a dramatic impact on DNAPL distributions. Higher organic saturations, increased lateral spreading, and decreased depth of infiltration were predicted when the contact angle was varied spatially. When chemical heterogeneity was defined by spatial variation of organic-wet solid fractions (fractional wettability porous media), however, the resultant organic saturation distributions were more similar to those for perfectly water-wet media, due to saturation dependent wettability effects on the hydraulic property relations.

A Study of Gravity Counterflow Segregation

Abstract It has been customary, in predicting saturation changes, to use the Leverett "fractional flow formula", obtained by eliminating the unknown pressure gradient from the generalized Darcy equations for the separate phases. The formula presents difficulties in the case of counterflow, since the "fractional" flow may be negative, greater than unity, or, in the case of a closed system, infinite. Recently, it has been shown by several authors that the corresponding equations (with capillary pressure and gravity terms) for actual flow of the phase may be used just as well. These equations are in agreement with Pirson's statement that, if the two mobilities differ considerably from each other in a closed system, the flow is largely governed by the lower value. The present study was undertaken because of an apparent lack of experimental data on gravity counterflow with which to test the theory. A 4-ft sandpacked tube in a vertical position was employed. Electrodes for determining saturations by resistivity were spaced along the tube, one phase being always an aqueous salt solution. Air, heptane, naphtha, or Bradford crude oil was used for the other phase. A reasonably uniform initial saturation was set up by pumping the phases through the system, after which the tube was shut in and saturation profiles obtained at definite intervals. Cumulative flows over certain horizontal levels were obtained by integration of the distributions; hence, differentiation of the cumulative flows with respect to time gave instantaneous flow rates. To compare experimental and theoretical flow values, capillary pressures were assumed given by the final saturation-distribution curve. The upper part corresponds to the "drainage" region and the lower part to the "imbibition" region, where trapping of the nonwetting phase occurred. While calculations indicated that the capillary pressure saturation function and, probably, the relative permeability saturation functions changed during the segregation, the relation of the measured rates to saturation distributions are in general accord with the frontal-advance equation. It appears that the Darcy equations, as modified for the separate phases, are generally valid for counterflow due to density differences. The usual method of predicting saturation changes, which involves a continuity equation and the elimination of the unknown pressure gradient from the flow equations, should therefore be applicable. However, the need for advance knowledge of drainage and imbibition "capillary pressures" and relative permeabilities during various stages presents difficulties. Introduction The present study was undertaken because of a seeming lack of experimental data relating to vertical counterflow of fluids of different densities in porous media. In particular, it was desired to determine whether data obtained from these laboratory tests were in accordance with certain mathematical treatments of counterflow which have been proposed. The gravity "correction" has been incorporated into the flow equations (and, hence, into displacement theory) nearly as long as both have been used. Field and laboratory data have generally borne out the validity of the theory as applied, for instance, to downward displacement by gas, with all fluids moving downward. However, the modifications for counterflow have only recently been pointed out. It has been customary to use fractional flow rates instead of actual flow rates in displacement calculations. In the case of counterflow, this results in negative values, values greater than unity and, when rates are equal and opposite, in infinite values. As pointed out by Sheldon, et al, and by Fayers and Sheldon, actual flow rates may be used just as well. The fact that these may be of opposite signs for the two fluids does not present any difficulty. SPEJ P. 185^

Experiments on the Capillary Properties of Porous Solids

Abstract A report is made of experimental work performed on the capillary retentionof water within porous solid systems, the displacement being accomplished withair and various organic liquids. A portion of the experiments were designed tomeasure the lowering of vapor pressure of water within a porous solid withsubsequent conversion of such vapor pressure data to capillary pressure values.The form of the capillary pressure curve at high capillary pressures has beenelucidated from this data. Certain theoretical approaches are presented toindicate the correlation between work done by previous investigators. Thepresent work is correlated also with theory. Surface area values are calculatedfor the various core systems studied. Introduction Leverettt in 1941 presented a paper which gave the essential concepts ofcapillary behavior in porous solids. His presentation included both theoreticaland experimental aspects of the problem. He defined the term "capillarypressure" and applied it to an ideal porous system. His work included thedefinition of a dimensionless quantity which was a function of fluid saturationand which could be correlated with porosity and permeability for clean, unconsolidated sands. His experimental work was performed by the drainage ofwater from packed columns of unconsolidated sands. Subsequently, other authorshave presented experimental techniques which permit the determination ofcapillary pressure data on small core samples. Data have been presented onporous systems other than unconsolidated sand and with the use of liquids otherthan water. Notable among these are the works of Hassler, Brunner and Deahl, Bruce and Welge, Amyx and Yuster, and Purcell. Others have indicated theapplicability of such data to field problems without regard to the theoreticalaspects of the capillary pressure saturation function. Within the recentpublished literature there have also been reported studies of the correlations between capillary pressure data and otherfundamental properties of porous solids. It was the intent of the presentreported work to investigate the behavior noted by water and various otherliquids within porous systems in an attempt to amplify existing correlationsbetween capillary properties, surface properties, and other fundamentalcharacteristics of porous systems. The experimental work described here wasperformed as two different investigations but due to its nature it is reportedas separate phases of the same general topic. T.P. 2640