Exploring dynamic effects in capillary pressure in multistep outflow experiments

Authors


Abstract

[1] Traditional steady state experiments to measure constitutive relations governing two-phase (organic-aqueous) flow in the subsurface often extend over periods of weeks or months. Alternatively, one-step or multistep outflow (MSO) experiments can be combined with application of a multiphase flow simulator and an optimization algorithm to achieve a more rapid technique for parameter estimation. In this work, MSO experiments were conducted to produce a data set for the estimation of two-phase constitutive parameters using this inverse modeling approach. Examination of experimental results reveals significant discrepancies between observed and simulated outflow data, with simulated curves tending to approach equilibrium at a faster rate than the experimental observations. Similar behavior has been documented by other investigators. Application of alternative equilibrium constitutive models in the multiphase flow optimization simulator failed to improve model fits to observed data. However, when model governing equations were modified to incorporate a dynamic capillary pressure term, there was significant improvement in the agreement between measured and simulated cumulative water outflow and outflow rates. Comparisons of simulated and measured data further suggest that the dynamic capillary pressure constitutive coefficient depends on saturation. Attribution of the observed experimental deviations to dynamic effects in capillarity is also supported by the consistency of the fit equilibrium retention function with an independently measured static retention relation.

1. Introduction

[2] Traditional steady state experiments for the measurement of multiphase flow constitutive relations often have durations of weeks or months. Alternatively, one-step or multistep outflow experiments can be combined with application of a multiphase flow simulator and an optimization algorithm to achieve a more rapid technique for parameter estimation [Kool et al., 1985; Parker et al., 1985; Eching and Hopmans, 1993; Eching et al., 1994; van Dam et al., 1994]. Most one-step outflow (OSO) and multistep outflow (MSO) research has focused on unsaturated flow and the solution of Richards' equation [Parker et al., 1985; van Dam et al., 1992; Eching and Hopmans, 1993; Eching et al., 1994; van Dam et al., 1994]. Early in the development of these techniques, concerns about stability and uniqueness were investigated [Kool et al., 1985; Kool and Parker, 1988; Russo et al., 1991; Toorman et al., 1992; van Dam et al., 1992; Eching and Hopmans, 1993; van Dam et al., 1994]. For MSO experiments it was concluded that use of cumulative water outflow in the objective function leads to a unique estimate of a soil's constitutive relationship parameters [van Dam et al., 1994; Hopmans et al., 2002]. As a result of this pioneering work, OSO and MSO experiments are becoming increasingly common methods to estimate unsaturated flow constitutive relationships.

[3] More recently, MSO experiments have been conducted in nonaqueous phase liquid (NAPL)/water systems to estimate multiphase flow constitutive relationships [Liu et al., 1998; Chen et al., 1999]. In these studies the NAPL phase displaced the water phase in the soil column. For data analysis, Liu et al. [1998] assumed that no NAPL pressure gradient existed in the soil column and only the aqueous phase equation was solved. Chen et al. [1999] improved the conceptual model by including both the NAPL and aqueous phase equations in the multiphase flow simulator. An examination of their model/data comparisons, however, reveals that in simulated cumulative outflow, equilibrium was achieved much faster than observed experimentally and at a different cumulative outflow level. Similar discrepancies between observed and fit cumulative outflow are present in other published studies [Schultz et al., 1999; Hwang and Powers, 2003].

[4] Inconsistencies between observed and fit behavior may be due to the use of inappropriate constitutive models for the relation between capillary pressure and saturation. Under drainage conditions, traditional models assume a unique relationship between capillary pressure and saturation. In two-phase air-water systems, however, some researchers have observed that the equilibrium water saturation under water drainage conditions is also a function of the magnitude of the imposed pressure change [Topp et al., 1967; Smiles et al., 1971; Vachaud et al., 1972]. Larger equilibrium water saturations were observed when one large pressure step was imposed than when a series of smaller pressure steps was used. In the literature this phenomenon, where saturation is observed to depend on both capillary pressure and the rate of saturation change, has been called dynamic or nonequilibrium effects in capillary pressure [Barenblatt and Gil'man, 1987; Hassanizadeh and Gray, 1990; Kalaydjian, 1992]. It has been hypothesized that discrepancies between observed and fit cumulative outflow may also be due to dynamic effects in capillary pressure [Schultz et al., 1999; Wildenschild et al., 2001; Hassanizadeh et al., 2002]. Schultz et al. [1999] found that the use of Richard's equation could not capture the shape of observed outflow data and attributed the differences to neglecting air pressure gradients and also to dynamic effects in capillary pressure. Wildenschild et al. [2001] similarly found that the magnitude of MSO pressure steps led to estimates of distinctly different retention functions.Hollenbeck and Jensen [1998] also found that cumulative outflow experimental results from experiments employing differing pressure step sizes were inconsistent with a unique retention function. Although these researchers observed behavior that may be attributed to dynamic effects in capillary pressure, they did not attempt to quantify the magnitude of these effects.

[5] The physical processes that yield dynamic effects in capillary pressure are not well understood. However, a number of researchers have suggested processes that may contribute to this effect [Kalaydjian, 1992; Wildenschild et al., 2001; Hassanizadeh et al., 2002]. Under certain conditions, as an advancing fluid invades a porous medium the curvature of the fluid/fluid interface in a pore is unable to smoothly change in response to changes in capillary pressure as suggested by traditional capillary pressure/saturation models [Brooks and Corey, 1964; van Genuchten, 1980]. In these instances the interface becomes unstable and the fluid/fluid interface will jump to a new stable location with a corresponding decrease in local capillary pressure and free energy of the system [Morrow, 1970]. However, fluctuations in capillary pressure, due to local instabilities in the fluid/fluid interface, often go unnoticed in traditional equilibrium measurements. As the advancing fluid invades the porous medium, the measured capillary pressure will be larger due to unstable fluid/fluid interfaces and will decrease as the fluids reach equilibrium and minimize the free energy of the system. These pore scale processes, described by Haines jumps, are not captured when upscaling from the pore to the representative elemental volume (REV) scale and may contribute to dynamic effects in capillary pressure [Kalaydjian, 1992; Hassanizadeh et al., 2002]. Following a detailed literature review of studies that have exhibited dynamic effects in capillary pressure, Hassanizadeh et al. [2002] hypothesize that this behavior is also related to the presence of microscale heterogeneities not captured by continuum models. Other explanations for observed dynamic effects in capillary pressure, in air/water systems, include water entrapment, pore water blockage, air entrapment, air-entry effects, and dynamic contact angle effects [Wildenschild et al., 2001].

[6] Both Hassanizadeh and Gray [1990] and Kalaydjian [1992] postulate that dynamic effects in capillary pressure can be explained through the thermodynamic theory defining macroscopic capillary pressure. In their work, dynamic capillary pressure is defined as the difference between the pressures of the nonwetting and wetting fluids flowing in the system. The static capillary pressure is the capillary pressure measured when a change in pressure yields an infinitesimally small change in water saturation.

[7] Their theory suggests that dynamic capillary pressure can be represented as a linear function of the rate of change in water saturation in the porous medium:

equation image

where Pcd is the dynamic capillary pressure, Pcs is the static capillary pressure, τ is a material coefficient, and Sw is water saturation. The material coefficient may not be constant in a given system and may be a function of saturation [Hassanizadeh et al., 2002] or inversely proportional to the total flow rate [Kalaydjian, 1992]. Further work is required to adequately determine an appropriate functional form of the material coefficient.

[8] Using equation (1), Hassanizadeh et al. [2002] estimated an average τ for published studies that exhibited dynamic effects in capillary pressure, where enough experimental data were available. Their estimated difference between static and dynamic capillary pressure was based upon published static and dynamic capillary pressure/saturation curves. They estimated the temporal derivative in saturation based on the change in column saturation over a given time. Estimates of τ ranged between 3 × 104 and 5 × 107 kg m−1 s−1. Similarly, Manthey et al. [2004] estimated average values of τ ranging from 3 × 104 to 105 kg m−1 s−1 for a series of column experiments. In these studies, saturation was assumed to vary linearly in the column, and the rate of saturation change was determined over a finite time interval. Thus τ was not estimated through optimization of experimental fits to multiphase flow simulator predictions. The material coefficient τ has also been estimated theoretically for idealized porous media (5 × 101 to 8 × 102 kg m−1 s−1 for a bundle-of-tubes model [Celia et al., 2004] and 5 × 103 to 4 × 104 kg m−1 s−1 for a pore-scale network model [Gielen et al., 2004]).

[9] In this study, data from a series of water/dense nonaqueous phase liquid (DNAPL) MSO experiments were used to explore whether agreement between observed and simulated MSO could be improved through the inclusion of a dynamic capillary pressure term in the governing equations. This work also examines the sensitivity of outflow predictions to Pc/S and kr/S relationships and investigates whether some of the model discrepancy can be improved by decoupling the kr/S constitutive model from the shape of the Pc/S curve. Data from three equilibrium experiments serve as independent measures of the Pc/S function for the selected sand.

2. Materials and Methods

2.1. Materials

[10] Laboratory grade (99%) tetrachloroethene, PCE (Aldrich Chemical, Milwaukee, Wisconsin), was used as the representative DNAPL. The aqueous phase in all systems was Milli-Q water. Fluid physical properties are presented in Table 1. The porous medium consisted of F35-F50-F70-F110 Ottawa sand (US Silica, Ottawa, Illinois), with a mean grain size of 0.026 cm and a uniformity index of 2.79 [O'Carroll et al., 2005].

Table 1. Fluid Physical Properties
 WaterPCE
Density, kg/m3999a1623b
Viscosity, N s m−21.12 × 10−3a8.80 × 10−4b
Compressibility, 1/Pa4.4 × 10−10c0.0d

2.2. Multistep Outflow Experiments

[11] The multistep outflow apparatus is presented in Figure 1. Custom-designed aluminum columns (length = 9.62 cm and ID = 5.07 cm) used in the MSO experiments were dry packed with the sand, flushed with carbon dioxide, and then saturated with at least 30 pore volumes of deaired Milli-Q water. The bottom of the column was connected to a custom-designed constant-pressure PCE reservoir (ID = 10.05 cm). The air pressure above the PCE phase in the PCE reservoir was connected to an air pressure/vacuum source. The relatively large diameter of the PCE reservoir, in comparison to that of the soil column, and the use of a constant air pressure source allowed for easy control of the boundary PCE pressure and ensured that the PCE pressure at that location remained relatively constant during a pressure step. The top of the column was connected to a custom-designed water reservoir with an overflow weir (ID = 2.0 cm). The overflow weir ensured a constant water pressure boundary condition. In order to minimize water evaporation, the water reservoir was partially covered with Parafilm® (Pechiney Plastic Packaging, Chicago, Illinois). The water reservoir rested on a balance that measured water outflow weight (Model 4000, Mettler, Columbus, Ohio). The balance was connected directly to a PC computer. A nylon membrane (0.2 μm pore size, Pall Corporation, Ann Arbor, Michigan), emplaced at the top of all columns, acted as a capillary barrier to PCE, allowing only water outflow. A Teflon membrane (0.45 μm pore size, Pall Corporation, Ann Arbor, Michigan), emplaced at the bottom of one column in the study, acted as a capillary barrier to water. The Teflon membrane was emplaced at the bottom of one column and not the other to investigate its effect on multistep outflow results.

Figure 1.

Schematic of multistep outflow experiment.

[12] Fluid flow was induced by imposing a fixed air pressure above the PCE phase. Water and PCE boundary pressures were measured using pressure transducers (MicroSwitch, Freeport, Illinois) which were connected to a CR7 data logger (Campbell Scientific, Logan, Utah). Balance and pressure readings were taken at 30-s intervals. All experiments were conducted at room temperature (22°C). PCE pressure was increased in a series of steps (six to 12) during the outflow experiment. PCE and water phase boundary pressures for the two experiments are presented in Figures 2 and 3. It was assumed that residual water saturation was achieved when water outflow ceased following successive increases in PCE pressure. Each experiment had a target number of pressure steps (eight) with each step yielding approximately equal amounts of water outflow. The sizes of the pressure steps were estimated based on independent static capillary pressure/saturation measurements (see below). However, because of scatter in the static capillary pressure/saturation data and of experimental difficulties associated with precisely adjusting the boundary PCE pressure, precise control of step size and associated outflow was not possible. Tensiometric measurements of the fluid phase pressures inside the columns were not undertaken. In many cases, prior to an increase in PCE boundary phase pressure, water outflow was nearly at equilibrium. This observation suggests that independent tensiometric measurements were not necessary to yield unique capillary pressure/saturation parameter fits [Hopmans et al., 2002].

Figure 2.

Comparison of observed and fit cumulative water outflow for MSOa. Fit parameters and RMSE values are presented in Table 3.

Figure 3.

Comparison of observed and fit cumulative water outflow for MSOb. Fit parameters and RMSE values are presented in Table 3.

2.3. Permeability Measurements

[13] The permeability of the porous medium was quantified at 100% water saturation, prior to the outflow experiment (water flowing), using the constant head method of Klute and Dirksen [1986]. A nylon membrane was then emplaced at one end of the column and the permeability of the column (sand and nylon membrane) was again quantified. Just prior to the outflow experiment a Teflon membrane was emplaced at the other end of the column. At the end of the outflow experiment (at residual water saturation) the nylon membrane was removed and the effective permeability of the column (sand and Teflon membrane) to PCE was quantified. The Teflon membrane was then removed and the effective permeability of the sand to PCE at residual water saturation was quantified. Membrane permeability was determined from the data using Darcy's law for flow in series. For permeability experiments with no membranes present, a stainless steel mesh (0.015 × 0.015 cm pore size) was emplaced at either end of the column to hold the sand in place.

2.4. Equilibrium Capillary Pressure/Saturation Experiments

[14] Equilibrium two-phase (water-PCE) primary drainage capillary pressure/saturation data were measured using a pressure cell system (total volume = 6.37 cm3) based upon the design of Salehzadeh and Demond [1999]. The sand mixtures were dry packed, flushed with carbon dioxide, and then saturated with 200 pore volumes of Milli-Q water. All capillary pressure/saturation experiments began at 100% water saturation [O'Carroll et al., 2005].

[15] Interfacial tension measurements for the fluids, using the axisymmetric drop shape analysis (ADSA) technique [Cheng et al., 1990; Lord et al., 1997], were conducted before and after the capillary pressure/saturation experiments to confirm that no interfacial tension reductions had occurred during the experiment.

2.5. Numerical Model

[16] A one-dimensional, fully implicit, point-centered finite difference multiphase flow simulator was developed to model experimental conditions. In this model, the aqueous phase flow equation is expanded in terms of water pressure:

equation image

where ϕ is porosity, Sw is water saturation, ρw is water density, Pw is water pressure, Pcs is static capillary pressure, λw is water mobility, g is gravity, t is time, and x is the spatial dimension. Matrix compressibility is neglected.

[17] For NAPL phase flow, the organic phase pressure is expanded as the sum of the dynamic capillary and aqueous phase pressures (PoPcd + Pw). Dynamic effects in capillary pressure in the NAPL compressibility term are neglected and equation (1) is incorporated in the Darcy flux term to yield an equation in terms of Pcs and Pw:

equation image

Neglecting dynamic effects in capillary pressure in the NAPL compressibility term is not expected to affect the results since the compressibility of the NAPL is very small. Note also that for this single displacement direction experiment, it is assumed that there is a unique relationship between Pcs and Sw (the equilibrium capillary pressure/saturation relation).

[18] The governing equations (2) and (3) are discretized using a fully implicit centered finite difference formulation with upstream weighting of the phase mobilities. Consistent with the implicit formulation, the nonlinear coefficients are evaluated at the current time level. The temporal capillary pressure derivative in the accumulation term in the NAPL governing equation (3) is expanded using backward finite differences with nonlinear coefficients lagged an iteration. Similarly, the temporal static capillary pressure derivative in the flux term in the NAPL governing equation (3) is also discretized using a backward finite difference expansion with a centered finite difference expansion of the spatial derivatives. The capacity coefficient is estimated using a chord slope approximation

equation image

[19] The discrete, coupled equations are solved directly for the changes in Pw and Pcs over a time step using a block tridiagonal algorithm, and convergence is achieved through Picard (direct) iteration. The complete matrix formulation is presented in Appendix A.

[20] Both the Brooks and Corey [1964] and van Genuchten [1980] models are implemented in the simulator to represent the static capillary pressure as a function of effective water saturation (Table 2). Similarly, the Burdine [1953], Mualem [1976], and Demond and Roberts [1993] models are implemented in the simulator to represent water and organic phase relative permeability as a function of effective water saturation (Table 2). This multiphase flow simulator was verified numerically by comparing selected one-dimensional vertical column simulations with those of the extensively validated multiphase simulator M-VALOR [Abriola et al., 1992]. M-VALOR was not used as a platform for dynamic capillary pressure modifications in this work because it was unable to accommodate the experimental boundary conditions; that is, it does not permit differing type among phase boundary conditions at a boundary (such as a Dirichlet boundary condition for one phase and a Neuman boundary condition for the other phase at the same boundary). In addition, M-VALOR is a two-dimensional, three-phase flow numerical simulator, and its application would thus place an unnecessary computational burden on the nonlinear parameter optimization routine used herein. The multiphase flow numerical simulator developed for this study computes global mass balances for each phase which were monitored to evaluate satisfactory conservation of mass in the system.

Table 2. VGB, BCM, GDM, and DR Constitutive Model Descriptiona
 ReferencesPc/SWater Relative PermeabilityOrganic Relative Permeability
  • a

    Sweff = (SwSwr)/(1 − Swr), where Sw and Swr are the water and residual water saturations, respectively. For the VGB model, m = 1 − (2/n).

VGBvan Genuchten [1980]Burdine [1953]Pc = [(Sweffequation image−1)equation image]/αkrw = (Sweff)2 [1 − (1 − Sweffequation image)m]kro = (1 − Sweff)2 (1 − Sweffequation image)m
BCMBrooks and Corey [1964]Mualem [1976]Pc = PdSweffequation imagekrw = (Sweff)2.5 +equation imagekro = (1 − Sweff)equation image [1 − (Sweff)1 +equation image]2
DRDemond and Roberts [1993] krw = (Sweff)a [1 − (1 − Sweffequation image)b]kro = c(1 − Sweff)d (1 − Sweffequation image)e

[21] The multiphase flow simulator was used in conjunction with the LM-OPT inverse fitting routine of Clausnitzer and Hopmans [1995] to estimate the constitutive model parameters of the porous medium. This history matching routine minimizes the squared difference between observed and simulated outflow data based on the Levenberg-Marquardt algorithm [Levenberg, 1944; Marquardt, 1963].

[22] For experimental simulations, the column domain was discretized with 145 nodes in a variable grid spacing. The maximum grid spacing, at the midpoint of the column, was 10−3 m. To appropriately resolve the membrane influence, the minimum grid spacing, near the column boundaries and membrane/porous medium interfaces, was 8 × 10−6 m. Convergence was assumed if the normalized difference in the primary variables was less than 10−6 over the iteration,

equation image

Additionally, the normalized change in effective water saturation over an iteration was required to be less than 10−6,

equation image

A dynamic time step adjustment algorithm was implemented such that if convergence in these three variables was not achieved in eight iterations, the time step was reduced by a factor of 0.55. If convergence was achieved in two iterations or less, the time step was increased by a factor of 1.05, to a maximum of 10 s. The minimum permissible time step was 10−12 s.

3. Results

[23] Results of two multistep outflow experiments MSOa and MSOb are presented in Figures 2 and 3, respectively. An organic-wet membrane was not emplaced at the bottom of the first column, MSOa, but one was emplaced in the second column, MSOb. In both experiments the observed water outflow (solid black line) rapidly increased following an increase in PCE phase boundary pressure and then gradually decreased. Maximum capillary numbers immediately following a PCE phase pressure increase were 7.3 × 10−7 and 1.44 × 10−7 for MSOa and MSOb, respectively. As water outflow decreased and began to plateau, the PCE phase boundary pressure was again increased. Note that observed water outflow did not decrease to zero prior to an increase in PCE phase boundary pressure.

[24] In this study, two representative constitutive relationship models, the van Genuchten/Burdine (VGB) and the Brooks-Corey/Mualem (BCM) models, were used to fit observed cumulative outflow data (Table 2). The VGB model was selected based on its ability to closely reproduce PCE migration in a two-dimensional infiltration experiment [O'Carroll et al., 2004], and the BCM model was selected for comparison purposes. For the VGB model, three parameters were fit (α, n, and residual water saturation). Similarly, three parameters were fit for the BCM model (Pd, λ, and residual water saturation). In these simulations, the dynamic capillary pressure term τ was set to zero. Independently measured sand and membrane permeabilities were input parameters to the model (Table 3). Inspection of Figures 2 and 3 reveals that best fits of both models fail to capture the initial outflow increase observed experimentally. Fit VGB and BCM parameters are presented in Table 3. As time increases, in a given pressure step, the simulations tend to approach equilibrium at a rate faster than the experimental observations, reaching zero outflow more rapidly, except at the very end of the experiment. In addition, the simulations often yielded cumulative outflow plateaus that differed from experimental results. Examination of published work suggests that discrepancies of this nature are common [Chen et al., 1999; Schultz et al., 1999; Hwang and Powers, 2003].

Table 3. Measured Sand and Membrane Permeabilities, VGB and BCM Constitutive Model Parameters Fit to Multistep Outflow Experiments as Well as Cumulative Outflow and Pc/S RMSE Values
  MSOaMSOb
Measured sand permeability at Sw = 1.0, m2 1.58 × 10−111.26 × 10−11
Measured nylon membrane permeability at Sw = 1.0, m2 5.72 × 10−155.71 × 10−15
Measured Teflon membrane permeability at Swr, m2 not emplaced9.95 × 10−15
VGBα (cm H2O)−13.79 × 10−23.23 × 10−2
 n5.565.77
 SWR0.1330.0
 RMSE for fit cumulative outflow1.991.41
 RMSE for predicted Pc/S6.46 × 10−27.17 × 10−2
BCMPd (cm H2O)21.3224.49
 λ2.072.20
 SWR0.0870.0
 RMSE for fit cumulative outflow2.313.15
 RMSE for predicted Pc/S8.37 × 10−28.57 × 10−2

[25] The root-mean-square error (RMSE) was used to quantitatively evaluate the quality of simulated cumulative water outflow fits. Observed and simulated cumulative water outflow were compared at a given time. In both MSO experiments the outflow fits employing the VGB model yield lower RMSE values when compared with the outflow fits employing the BCM model (Table 3). The capillary pressure/saturation curves generated with the fit parameters (equations (2) and (3)) were also compared with independent data from capillary pressure/saturation experiments, conducted in modified Tempe cell experiments [O'Carroll et al., 2005], using an RMSE analysis. Here the RMSE was calculated using differences in saturation, rather than capillary pressure, to reduce the importance of data at high and low effective water saturations and increase the importance of data in the intermediate effective water saturation range. Both VGB and BCM constitutive parameters, generated by fitting equations (2) and (3) to the experimental data from experiments MSOa and MSOb, yield retention functions that are consistent with the three independent data sets (Table 3 and Figure 4). Similar to results of the outflow fits, the VGB constitutive model yields the lowest RMSE values (Table 3).

Figure 4.

Comparison of observed capillary pressure/saturation data to retention functions derived from MSO data fits. Fit parameters and RMSE values are presented in Table 3.

[26] Although cumulative water outflow simulations and capillary pressure/saturation curves using the VGB model yielded lower RMSE values than the BCM model, neither constitutive relation model yielded outflow rates that were consistent with observed behavior. A sensitivity analysis was conducted to explore the relative importance of retention parameter selection on predicted cumulative water outflow. In this analysis, capillary parameters were systematically varied within their 95% confidence intervals, based upon the independent capillary pressure/saturation measurements. Here the van Genuchten constitutive model was used. Observed and fit capillary pressure/saturation data are presented in Figure 5 along with a 95% confidence interval envelope. Fitting parameters along with their 95% confidence intervals are presented in Table 4. Figure 6 presents the range of predicted cumulative outflows for MSOb achieved by varying capillary parameters within their 95% confidence intervals. Similar to the optimal fits presented in Figure 3, these model predictions exhibit an initial outflow rate much higher than observed experimentally, followed by an early plateau of cumulative outflow. The outflow curves also exhibit considerable vertical separation at a specific time. Other researchers have also observed significant changes in cumulative outflow predictions following relatively minor changes in the constitutive relationship model [Hwang and Powers, 2003]. Examination of simulations in Figure 6 that used the same relative permeability model (same n) but α values at the upper and lower 95% confidence limits indicates that much of the vertical spread in the predicted outflow curves is attributable to differences in the entry pressure of the retention functions. This also suggests an insensitivity of predictions to the relative permeability, which depends only on n.

Figure 5.

Observed and fit retention function as well as 95% confidence interval bounds. Fit parameters are presented in Table 4.

Figure 6.

Comparison of cumulative outflow predictions for MSOb for the parameters fit to independent retention data assuming Swr = 0.

Table 4. The 95% Confidence Intervals for the van Genuchten Model Fit to Independent Retention Data
 Lower 95% Confidence Confidence IntervalEstimateUpper 95% Confidence Interval
α (cm H2O)−13.15 × 10−23.32 × 10−23.49 × 10−2
n5.816.587.34

[27] Traditional predictive relative permeability/saturation models are based on the pore size distribution of the medium [Burdine, 1953; Mualem, 1976], which is usually estimated from the shape of the capillary pressure/saturation curve. Consequently, the capillary pressure and relative permeability relationships are usually coupled through the shape of the capillary pressure/saturation curve (e.g., constants n and λ in the VG and BC models, respectively). However, one-dimensional steady state flow experiments by Demond and Roberts [1993] suggest that common predictive models for kr/S may not adequately reproduce nonwetting phase relative permeability. In particular, these investigators found that nonwetting phase relative permeability does not approach a value of 1 as the residual water saturation is approached. To further investigate the sensitivity of outflow predictions to the selected relative permeability model, a series of optimization simulations was undertaken in which the relative permeability relationships were decoupled from the capillary pressure/saturation relationship. In this approach, the VG Pc/S model and the Demond-Roberts (DR) kr/S model parameters (Table 2) were fit to the MSOb outflow data. Note that the DR parameter c is equivalent to the organic end point relative permeability at the end of primary drainage. This end point permeability was measured at residual water saturation following the multistep outflow experiment (kro = 0.52 at Swr). Thus c was fixed at 0.52 and six parameters were fit to the outflow data (α, n, a, b, d, and e) (Table 2). Consistent with the fit of the three-parameter model to MSOb, Swr was set to zero. Optimal parameter fits are presented in Table 5. Comparison of Tables 3 and 5 reveals that the parameters α and n are nearly identical for both the three- and six-parameter fits. For the tortuosity parameters, a and d, the optimal fit values (0.5) were at the lower bound of the search range, consistent with the Mualem model. The quality of the six-parameter fit to the cumulative outflow data was nearly identical to the three-parameter MSOb VGB fit (RMSE = 1.39). Also, similar to that obtained for the three-parameter fit, the simulated outflow curve rose too steeply, following a pressure step (Figure 7). In summary, these simulation results indicate that under these experimental conditions, outflow predictions are extremely sensitive to the retention function but not to the selected relative permeability model. In addition, these results indicate that changes in the constitutive model do not yield a better representation of measured cumulative water outflow.

Figure 7.

Comparison of observed outflow as well as the three- and six-parameter fits to cumulative water outflow for MSOb assuming Swr = 0. Fit parameters are presented in Tables 3 and 5, respectively.

Table 5. VG Capillary Pressure/Saturation and DR Relative Permeability Parameters fit to MSOba
 Fitted Value
  • a

    VG and DR models are presented in Table 2.

α (cm H2O)−13.28 × 10−2
n5.76
a0.5
b0.373
d0.5
e1.35

[28] Differences in observed and fit cumulative outflow, such as those observed in this study, may be attributed to dynamic effects in capillary pressure that are not accounted for in traditional multiphase flow simulators [Schultz et al., 1999; Hassanizadeh et al., 2002]. To explore this hypothesis, a four-parameter model, employing the VGB relationship and the dynamic capillary pressure term (equation (1)), was fit to the MSO outflow experiments. In this approach the parameters α, n, Swr, and τ were fit (Table 6). Here the material coefficient τ was assumed to be a constant, independent of saturation. Potential dynamic effects within the column membranes were neglected based upon the fact that these water and organic-wet membranes were initially water and PCE saturated, respectively, and no change in saturation occurred during the experiment. Inclusion of a constant material coefficient τ in the parameter optimization routine resulted in fits that were only slightly better than the three-parameter fits and did not satisfactorily capture observed outflow rates (see Figures 8 and 9) .

Figure 8.

Comparison of observed and fitted cumulative water outflow for MSOa. The fits incorporate dynamic effects in capillary pressure. Fit parameters and RMSE values are presented in Table 6.

Figure 9.

Comparison of observed and fitted cumulative water outflow for MSOb. The fits incorporate dynamic effects in capillary pressure. Fit parameters and RMSE values are presented in Table 6.

Table 6. VGB Parameters Fit to MSO Experiments, Incorporating Dynamic Effects in Capillary Pressure
 MSOaMSOb
VGB Parameters Fit to MSO Experiments Assuming τ = Constant
α (cm H2O)−14.11 × 10−23.29 × 10−2
n5.105.77
τ, kg m−1 s−12.56 × 1061.06 × 106
SWR0.120.0
RMSE – outflow data1.971.37
 
VGB Parameters Fit to MSO Experiments With τ a Linear Function of Effective Water Saturation(τ = {−A × Sweff + A})
α (cm H2O)−13.82 × 10−23.33 × 10−2
n6.395.78
SWR0.1490.0
A (kg/(m·s))5.64 × 1071.99 × 107
RMSE – outflow data1.341.12
RMSE – Pc/S data1.07 × 10−16.57 × 10−2

[29] A number of researchers have speculated that τ may be a function of saturation [Kalaydjian, 1992; Hassanizadeh et al., 2002; Celia et al., 2004; Gielen et al., 2004; Manthey et al., 2004]. In an attempt to improve the fit of the dynamic model to measured outflow data, a second four-parameter model was fit to experiments MSOa and MSOb where a linear functional form τ = {−A × Sweff + A} was assumed (A is a constant). In these simulations, τ is temporally and spatially variable over the pressure step. Thus equations (2) and (3) were fit to observed data by varying Swr, VGB α and n, as well as A. This four-parameter model resulted in fits that were better than the three-parameter fit simulations. Cumulative outflow RMSE values were 30% less for MSOa and 18% less for MSOb (Table 6). In general observed and fit outflow rates were similar (Figure 8 and 9). For the MSOb fit, PCE infiltration was not sufficiently retarded for the sixth pressure step (cumulative outflow = 14 and 25 g) to match observed data. Otherwise modeled cumulative outflow approached equilibrium at a slower rate than in previous simulations. The retention curves fit to the MSO experiments were also in good agreement with the independent retention data (Table 6 and Figure 10).

Figure 10.

Comparison of observed capillary pressure/saturation data to four-parameter model fits to MSO data. Fit parameters and RMSE values are presented in Table 6.

4. Discussion

[30] The fit material coefficient τ increases to a maximum 5.64 × 107 kg m−1 s−1 in MSOa and 1.99 × 107 kg m−1 s−1 in MSOb (Table 6 and Figure 11). The magnitudes of the optimal fit linear function for τ are at the upper range of the average τ values reported by Hassanizadeh et al. [2002] (3 × 104 to 5 × 107 kg m−1 s−1) for the studies they examined but are considerably larger than those for the experimental data presented by Manthey et al. [2004] (3 × 104 to 105 kg m−1 s−1). The reason for the large discrepancy between the values reported by Manthey et al. [2004] and those presented in this study is not obvious.

Figure 11.

The τ = f(Sweff) from the four-parameter fit (Swr, VGB α and n, τ = {−A × Swapp + A}) for MSOa and MSOb. Fit parameters are presented in Table 6.

[31] Stauffer [1978] suggested the following empirical relationship for the magnitude of τ:

equation image

where ɛ is porosity, μ is viscosity, k is the instrinsic permeability, ρ is density, g is gravity, and λ and Pe are the Brooks-Corey capillary pressure/saturation model parameters. This empirical relationship suggests that τ would be larger in the study of Manthey et al. [2004] since they used a finer sand (permeability contrast = 4.6); however, this trend is not consistent with their reported values. Hassanizadeh et al. [2002] qualitatively compared relation (4) to results of experimental studies and similarly found that fine-grained media with a high entry pressure and low permeability do not consistently exhibit larger dynamic effects when compared to coarse-grained media. Scaling issues are not likely to be a significant factor since the column volume used by Manthey et al. [2004] (84.8 cm3) was of the same order of magnitude as that used in the present study (188.9 cm3). Finally, Manthey et al. [2004] estimated a column-averaged τ whereas no averaging was required in this study. To explore if the differing techniques for determining τ were the source of the discrepancy in the magnitude of τ, a sample calculation was performed to determine a column averaged τ for MSOa at 50% effective water saturation. Here τ was calculated immediately following a PCE boundary pressure increase using a linearized form of equation (1). For this calculation, it was assumed that the dynamic capillary pressure (Pcd) was equivalent to the difference between the imposed PCE boundary pressure at the bottom of the column and the imposed water boundary pressure at the top of the column. The static capillary pressure (Pcs) was estimated from the VGB fits to the outflow data, assuming an effective saturation equivalent to the average value within the column immediately before the PCE boundary pressure increase. The rate of saturation change in the column was then estimated as the change in the average column saturation over the 5-min interval immediately following a PCE boundary pressure increase. On the basis of this calculation, τ (1.63 × 107 kg m−1 s−1) is consistent with the value determined at 50% effective water saturation (2.82 × 107 kg m−1 s−1) through parameter optimization in this study and orders of magnitude larger than the values reported by Manthey et al. [2004]. This analysis suggests that the technique for estimating τ is not the source of the discrepancy and that further work is required to determine the reasons for the large differences in the estimated values of τ between studies.

[32] The material coefficient τ is generally assumed to be a property of the porous medium and fluids by those who have made theoretical arguments for equation (1) [Gray and Hassanizadeh, 1991; Kalaydjian, 1992; Hassanizadeh et al., 2002]. The outflow data/simulation comparisons presented here, however, suggest that additional factors may influence the magnitude of τ. Although τ increased with decreasing water saturation, for both columns, linear fits differed, even though both columns were packed with the same sand material and static capillary pressure curves were consistent. Further work will be required to investigate the factors that affect the magnitude of τ and to develop information on its functional dependence. Similar to the results presented here, other investigators have suggested that the dynamic constitutive capillary parameter τ varies with saturation [Kalaydjian, 1992; Celia et al., 2004; Gielen et al., 2004; Manthey et al., 2004]. The wetting fluid entrapment process, another potential contributing factor of dynamic effects in capillary pressure [Wildenschild et al., 2001], could also result in a saturation-dependent dynamic constitutive capillary parameter. This analysis demonstrates that the assumption of a linear dependence of τ on effective saturation improves the agreement between simulated and measured outflow, in comparison with all other optimization approaches attempted in this study.

[33] In the analysis of experimental data in a number of previous studies, there has been an implicit averaging of saturation and capillary pressure values along the entire length of the column [Hassanizadeh et al., 2002]. In the numerical simulations presented in this work, however, no assumptions were required about the linearity of saturation or capillary pressure profiles along the column length. Although an additional level of averaging is not present in this study, a dynamic capillary pressure term was still necessary to accurately capture observed outflow data. Thus upscaling does not appear to contribute to the dynamic effects in capillary pressure observed in these data.

[34] A number of NAPL infiltration studies have reported greater NAPL capillary spreading in numerical simulations when compared with experimental observations [Oostrom and Lenhard, 1998; Schroth et al., 1998; Rathfelder et al., 2003; O'Carroll et al., 2004]. Discrepancies are particularly large when utilizing the van Genuchten retention function in numerical simulations, because of the large capillary driving force present at high water saturations. The inclusion of dynamic effects in capillary pressure in these numerical simulations would serve to decrease the rate of lateral spreading, when the rate of saturation change is large, and ultimately reduce the extent of simulated NAPL spreading following the termination of active NAPL infiltration. Following vertical penetration into a system, DNAPL ganglia saturations reach a steady state value behind the front, provided the boundary conditions remain unchanged. At these times the rate of saturation change in the ganglia will therefore be negligible and vertical DNAPL migration in ganglia will not be retarded due to dynamic effects in capillary pressure. Since the rate of saturation change will be nonzero in the lateral direction, DNAPL spreading will continue to be retarded. Therefore the inclusion of dynamic effects in capillary pressure may improve predictions of NAPL migration in these studies.

5. Summary and Conclusions

[35] Multiphase flow model simulations of cumulative outflow data from two MSO experiments conducted in this study indicate that traditional two-phase flow constitutive models fail to adequately capture observed outflow rates. Furthermore, a sensitivity analysis of simulated outflow, varying retention functions within 95% confidence intervals of independently measured data, indicate that simulated results are very sensitive to the retention function but not very sensitive to the kr/S constitutive relationships. This point was further illustrated by decoupling the relative permeability constitutive relationship from the shape of the retention function. A relative permeability model with four independent parameters did not improve outflow simulations in comparison with the coupled Pc/kr/S constitutive relationship. This lack of sensitivity to relative permeability parameters suggests that two-phase (liquid) relative permeability parameters estimated using traditional MSO fitting procedures should be used with caution. The inclusion of a dynamic capillary pressure term that varies with saturation, in the numerical simulator, however, significantly improved the outflow simulations, and the fit retention function was in good agreement with independently measured static retention data.

[36] Further work will be required to quantify the magnitude of the τ for a variety of media types, to determine a thermodynamic basis for the relationship between τ and saturation, and to investigate if the magnitude of τ can be determined a priori as suggested by Kalaydjian [1992].

Appendix A: Discretized Governing Equations

[37] The block tridiagonal algorithm solves the temporal change in water and static capillary pressure over a time step.

[38] Water phase governing equation

equation image

where ϕ is porosity, Sw is water saturation, ρw is water density, ΔPw is the change in water pressure over the time step, ΔPcs is the change in static capillary pressure over the time step, Pw is the water pressure at the previous time step, Pcs is the static capillary pressure at the previous time step, λ is mobility, g is gravity, t is time, x is the spatial dimension and the subscript j denotes the iteration level. Δx and Δx+ denote the distance between the i − 1 and i nodes and the i and i + 1 nodes, respectively.

[39] NAPL phase governing equation

equation image

Acknowledgments

[40] This research was supported in full by grant DE-FG07-96ER14702, Environmental Management Science Program, Office of Science and Technology, Office of Environmental Management, U.S. Department of Energy (DOE). Any opinions, findings, conclusions, or recommendations expressed herein are those of the authors and do not necessarily reflect the views of DOE.

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