Transport In Porous Media Research Articles

A single mesh approximation for modeling multiphase flow in heterogeneous porous media

The control volume finite element (CVFE) approach is based on the discretization of primary flow and transport unknowns on two different meshes. The element mesh, used to represent the physical properties of the medium, differs from the control volume mesh necessary to ensure a mass conservative solution. The inherent two mesh feature of the CVFE approximation introduces inconsistency in the transport solution due to the averaging of physical and computed quantities between neighboring elements in the mesh. In this work, we present a consistent approach for modeling multiphase flow and transport in heterogeneous porous media. The approach is applied in the CVFE framework by enabling the discretization of primary unknowns on a single mesh. We combine the discretization of an element-wise, discontinuous pressure approximation with a first-order, discontinuous velocity approximation to resolve the elliptic or parabolic flow problem. The effectiveness of the formulation is achieved by exploiting the same finite element mesh as the flow problem, when updating the saturation solution. This direct mapping between the flow and transport mesh is simple yet effective for establishing a consistent solution in addition to circumventing the non-physical mass leakage exhibited in the classical CVFE method. We describe the interface approximations and the discontinuous terms needed for consistent solutions. The method is well suited to model flow in complex geometrical subsurface domains and is shown to be numerically stable while providing locally and globally mass conservative solutions. We apply the approach to several domains with complex geometrical features to emphasize the superiority of the approximation over conventional methods. The analysis shows over two orders of magnitude reduction in solution error compared to the classical CVFE. The new formulation effectively captures accurate transport solutions, even in the presence of varying material properties, without the need for mesh refinement.

Efficient Energy Storage via Methane Production Using Protonic Ceramic Electrochemical Cells

Protonic ceramic electrochemical cells (PCECs) are promising energy storage technologies due to their high performance and ability to convert CO2 into value-added chemicals. However, there are many fundamental aspects of this technology that require investigation to better understand its interactions and opportunities. Thus, this paper focuses on the development of a steady-state PCEC model, encompassing intricate details of reactive porous-media transport, elementary catalytic chemistry, and electrochemistry within unit cells. The model is calibrated and validated using experimental data. The model presents the rate of methane production at different temperatures and revealed that efficient PCEC for co-electrolysis of CO2 and H2O is operated in the temperature range of 420–470 °C and the optimum being 450 °C at the rate of 0.0391 moldm−3 min−1. It provides insights into the effects of overpotentials on cell performance. At a current density of 3000 A/m2 the ohmic, concentration, and activation overpotentials amount to 0.65, 0.004, and 0.19 V, respectively. The results obtained from the model highlight the potential of PCECs as efficient CO2 sinks for decarbonization purposes and as a means of methane production. Furthermore, the model's findings offer valuable guidance for the design, selection of stacks, and choice of building materials in PCEC systems. The potential applications of PCECs as real-life fuel utilization technologies are justified with improved material development to reduce cell overpotentials, opening doors for scale-up and eventual commercialization. These findings contribute to the development of sustainable energy systems and advance the pursuit of decarbonization goals.

A parameter-free and monolithic approach for multiscale simulations of flow, transport, and chemical reactions in porous media

Traditional approaches for transport in porous media rely on empirical hydrodynamic/transport parameters to quantify the interfacial exchange terms between fluid and solid, which leads to considerable uncertainties and also to a narrow application range of each method for realistic investigations of porous media. A parameter-free approach for flow, transport, and chemical reactions in porous media is presented in this paper. This approach adopts a novel, single-domain set of equations that are rigorously derived based on the elementary conservation laws describing the physical, continuous domain. The impact of the solid on nearby fluid being directly quantified in terms of the quantity to be solved, this avoids the introduction of empirical parameters. Various transport problems including external, conjugate, and reactive processes are properly formulated in this context, with the coupling between fluid and porous structures at different scales naturally solved in a monolithic manner. All these equations remain simple in terms of their numerical implementation, with solid structures represented by the porosity field on a background mesh. The application of the present method in both pore-scale and representative elementary volume (REV-scale) simulations are checked by performing a series of test cases, including interfacial momentum/heat/mass transport, conjugate heat/mass transfer, chemical reactions, and two practical applications containing reacting interfaces. An overall first-order convergence rate has been observed in both spatial and temporal accuracy tests.

Effects of Initial Injection Condition on Colloid Retention

AbstractThis study investigates the impact of initial injection conditions on colloid transport and retention in porous media. Employing both uniform and flux‐weighted distributions for the initial colloid locations, the research explores diverse flow scenarios, ranging from simple Poiseuille flow to more complex geometries. The results underscore the pivotal role the injection mode plays on the shape of colloid retention profiles (RPs), particularly those that display anomalous non‐exponential decay with distance. Broadly, uniform injection yields multi‐exponential profiles, while flux‐weighted injection can lead to nonmonotonic profiles in certain conditions. The study identifies preferential flow paths as a key factor in producing nonmonotonic RPs. Notably, variations in fluid velocity, colloid size, and ionic strength affect attachment rates near the inlet but do not significantly alter the qualitative transition between multi‐exponential and nonmonotonic profiles. The study emphasizes that the chosen injection mode dictates retention profile shapes, highlighting its crucial role in porous media colloid transport. These insights provide a possible partial explanation of previously observed anomalous transport behaviors, urging consideration of injection conditions in interpretations of experiments, where they can be difficult to accurately control and measure with high precision.

In Situ Quantification of Colloidal Kaolinite Transport and Attachment in Porous Media Using Positron Emission Tomography

In Situ Quantification of Colloidal Kaolinite Transport and Attachment in Porous Media Using Positron Emission Tomography

A Mathematical Modelling of Solid Particles Transport in Two-Zone Porous Media with Non-Linear Kinetics of Deposition

A Mathematical Modelling of Solid Particles Transport in Two-Zone Porous Media with Non-Linear Kinetics of Deposition

Nanobubble transport in porous media: Towards agro- and environmental applications

Nanobubbles have been increasingly used in various applications involving porous media, such as groundwater remediation and irrigation. However, the fundamental scientific knowledge regarding the interactions between nanobubbles and the media is still limited. The interactions can be repulsive, attractive, or inert, and can involve reversible or irreversible attachment as well as destructive mechanisms. Specifically, the stability and mobility of nanobubbles in porous media is expected to be dependent on the dynamic conditions and the physicochemical properties of the porous media, solutions, and nanobubbles themselves. In this study, we investigated how changes in solution chemistry (pH, ionic strength, and valence) and media characteristics (size and wettability) affect the size and concentration of nanobubbles under dynamic conditions using column experiments. Quartz crystal microbalance with dissipation monitoring provided a deeper understanding of irreversible and elastic nanobubbles’ interactions with silica-coated surfaces. Our findings suggest that nanobubbles are less mobile in solutions of higher ionic strength and valence, acidic pH and smaller porous media sizes, while the wettability of porous media has a negligible influence on the retention of nanobubbles. Overall, our findings provide insights into the underlying mechanisms of nanobubble interactions and suggest potential strategies to optimize their delivery in various applications.

Effect of particle density on microplastics transport in artificial and natural porous media

The occurrence and persistence of microplastics (MPs) in natural environments are of increasing concern. Along with this, the transport of MPs in sediments has been investigated mainly focusing on the effect of plastic size and shape, media size effect, and solution chemistry. Yet, the influence of particle density is only partially understood. Therefore, column experiments on the transport of variably buoyant MPs in saturated natural sediments and glass beads were conducted, and transport parameters were quantified using a two-site kinetic transport model with a depth-dependent blocking function (the amount of retained MPs does not decrease at a constant rate with increasing depth, the majority of MPs were retained near the column inlet). Neutral, sinking, and buoyant MPs within the same size range were selected, with stable water isotope applied as conservative tracer to explore water and MP movement in the tested sediments. The results showed that 95.5 ± 1.4% of sinking MPs remained in columns packed with gravel, followed by buoyant and neutral MPs, thus indicating that particle density does affect MP mobility. Similar recovered amounts of MPs were found in columns packed with glass beads, indicating that tested sediment types do not affect the deposition behavior of MPs. The breakthrough curves of MPs were accurately described by the selected model. However, the simulated retention profiles overestimated the observed MP amount in layers closest to the column inlet. The coupled experimental and modeled results suggest an enhanced retention of sinking MPs, while neutrally and buoyant MPs exhibit a higher mobility in comparison. Thus, neutral or buoyant MPs can potentially pose a higher contamination risk to subsurface porous media environments compared to sinking MPs. Discrepancies between observed and simulated retention profiles indicate that future model development is needed for advancing the MP deposition as affected by particle density.

Colloid Transport in Bicontinuous Nanoporous Media.

Colloid transport and retention in porous media are critical processes influencing various Earth science applications, from groundwater remediation to enhanced oil recovery. These phenomena become particularly complex in the confined spaces of nanoporous media, where strong boundary layer effects and nanoconfinement significantly alter colloid behavior. In this work, we use particle dynamics models to simulate colloid transport and retention processes in bicontinuous nanoporous (BNP) media under pressure gradients. By utilizing particle-based models, we track the movement of each colloid and elucidate the underlying colloid retention mechanisms. Under unfavorable attachment conditions, the results reveal two colloid retention mechanisms: physical straining and trapping in low-flow zone. Furthermore, we investigate the effects of critical factors including colloid volume fraction, d, pressure difference, ΔP, interaction between colloids and BNP media, Ec-p, and among colloids, Ec-c, on colloid transport. Analysis of breakthrough curves and colloid displacements demonstrates that higher values of d, lower values of ΔP, and strong Ec-p attractions significantly increase colloid retention, which further lead to colloid clogging and jamming. In contrast, Ec-c has minimal impact on colloid transport due to the limited colloid-colloid interaction in nanoporous channels. This work provides critical insights into the fundamental factors governing colloid transport and retention within stochastic nanoporous materials.

Transport and deposition of nanoparticles in porous media at the pore scale using an Eulerian-Lagrangian method

BackgroundNanoparticle deposition in a subsurface environment decreases the amount of material in the bulk fluid, increases surface roughness, and affects nanofluid flow. Nanoparticle transport in porous media is a complex, multiscale phenomenon. Its macroscopic behavior is determined by phenomena at the pore-scale. Hence, it is necessary and of interest to study the transport and deposition of nanoparticles in porous media using a pore-scale approach. MethodsA computational solver is developed using an Eulerian-Lagrangian approach within the OpenFOAM library. A Berea sandstone and three open-cell metal foams with porosities of 0.8, 0.85, and 0.9, constructed by Laguerre-Voronoi tessellation, were analyzed to study nanoparticle behavior in water. Significant findingsNanoparticle deposition on solid surfaces is influenced by porosity, double layer thickness (NDL), surface potential magnitude (NE1), and nanoparticle diameter. Brownian motion that has a significant impact on nanoparticle transport and deposition is also examined. For nanoparticles with 20 nm diameter, the fraction of particles deposited in foam with 0.8 porosity is about 17 % more than the foam with 0.9 porosity. This difference decreases with nanoparticle diameter. For the particles with 100 nm diameter and more, the effects of Brownian motion on nanoparticle deposition becomes negligible.

Evaluating nickel removal efficacy of Filtralite under laboratory conditions: Implications for sustainable urban drainage systems

Our study deals with the critical issue of heavy metal pollution in urban runoff; we focused particularly on the threat posed by heavy metals such as nickel to living organisms and water resources owing to their toxicity at low concentrations and non-biodegradability. In this study, we evaluated the efficacy of Filtralite, a filtration medium with excellent hydraulic properties, in removing nickel from water through laboratory experiments and advanced simulation tools. We analysed the geochemical processes involved in nickel removal, including precipitation and adsorption. Importantly, we implemented an innovative approach for numerical modelling using HP1, by combining HYDRUS-1D and PHREEQC for modelling solute transport in porous media and geochemical reactions, respectively. The experimental results revealed that the Filtralite-laden column achieved a removal efficiency of 93.5 % for injected nickel. We observed a significant increase in the effluent pH from 7.0 to 11.5 during the interaction with Filtralite, affecting the solubility of the nickel phases. The model showed the highest nickel concentration in the initial column layers because of rapid chemical precipitation and adsorption upon contact with Filtralite. High-resolution scanning electron microscopy images confirmed the presence of amorphous precipitates around the soil particles, aligning with the changes in pH. This study contributes significantly to environmental engineering and urban water management by introducing an innovative numerical model in HP1 to accurately simulate the removal of nickel using Filtralite. These findings validate the efficiency of Filtralite in extracting nickel from aqueous solutions. Thus, Filtralite is a strong candidate for inclusion in sustainable urban drainage systems.

Upscaling the reaction rates in porous media from pore- to Darcy-scale

Modeling and prediction of reactive transport in porous media are challenging due to the complexity of the characterization of reactions across spatial and time scales. The application of reaction rates obtained from batch experiments to porous media is not valid because it disregards the influences of flow pathways, distribution of residence time, transport limitations within the porous media, as well as neglecting the impact of surface mineralogy heterogeneity. In this study, we presented how the pore-scale modeling can help to improve the Darcy-scale simulations. First, the reactive transport was simulated in a pore-network model and was validated against the experiments. Then, the pore-network results were averaged to obtain the effective reaction rates as a function of pore velocity, the inlet concentration, and the resident concentration. The upscaled function was incorporated into the Darcy-scale model and showed a significant improvement in predicting the concentration profiles. Notably, the use of batch reaction rates in the Darcy model (ADRE-batch) led to significant underestimation, up to 93%, of the required pore volume injection for complete contaminant removal from the simulation domain.

Analysis of discharge performance and thermo-electric conversion efficiency in thermally regenerative ammonia-based flow battery with foam copper electrode

This paper established a numerical model of mass transport and electrochemical reaction coupling in porous media of thermally regenerative ammonia-based flow battery with foam copper electrode (TRAFB-FCE) for the first time. The effects of the structural parameters of the foam copper electrode on the power density, energy density, discharge capacity, and active species distribution of the battery were comprehensively studied based on the coupled numerical model. Special attention was paid to the selection strategy of the initial reactant concentration for the battery under different discharge conditions. The results suggest that increasing the porosity or thickness of the foam copper electrode causes the voltage and power density of the battery to first increase and then decrease, while the discharge capacity and energy density monotonically increase. Moreover, the battery should avoid discharging at current densities exceeding 720 A·m−2. When the discharge current density is less than 200 A·m−2, it is preferable to set the initial concentration of Cu2+ to 0.4 mol/L, while the discharge current density is in the range of 200–720 A·m−2, the initial concentration of Cu2+ is preferable to be 0.3 mol/L. When it discharges at about 5 % of its maximum power density, the Carnot-relative efficiency of this battery is comparable to the highest value currently reported in the field of liquid-based thermo-electric conversion achieved by the Thermal Regenerative Electrochemical Cycle (TREC) technology, and the power density of the former is 8.87 times that of the latter.

Microscopic insights of phase-transition-induced vapor transport enhancement in porous media

Vapor transport in porous media, often associated with liquid-vapor phase change, is an fundamental process in many emerging underground energy storage and extraction processes (i.e., seasonal solar thermal aquifer storage, geothermal extraction, extraterrestrial in-situ water extraction). By jointly using experimental imaging and numerical modeling at the micro-scale, we conduct mechanistic pore scale investigation of capillarity-dominated phase change dynamics and its influence on vapor transport in partially saturated porous rock micromodel. Strongly linked to surface roughness and wettability condition, the capillarity hinders water vaporization from rock surface micro/nano-structures as observed under the environmental scanning electron microscope. By varying the contact angle of 0°, 60°, and 120°, the lattice Boltzmann simulation shows the wettability-dependent vaporization process of capillary-hold water, where pores with hydrophilic surfaces contains significantly more liquid water than that of the hydrophobic ones under the same temperature. When saturated vapor flows through rock porous patterns, capillarity further induces water condensation on the strongly water-wet surfaces. Water condensation, yet forming water bridges/islands and causing the blockage of vapor diffusion, enhances the vapor diffusion ability counterintuitively. The reduction of diffusion path is revealed as the main reason by assessing the local vapor pressure distribution before and after the pore filling by condensate. The findings support the debatable enhancement mechanisms postulated by Philip and de Vries. This work offers the insightful interfacial hydrodynamics of vapor transport in porous media and potentially provides operational guidance for geothermal applications and beyond.

Lattice Boltzmann model for incompressible flows through porous media with time-fractional effects

Anomalous transport has been commonly observed in the fluid flow through a complex porous medium, where the evolution exhibits a complex memory-like behavior. Classical integer-order models fail to depict anomalous transport phenomena, while fractional calculus has been proved effective in describing such behavior due to its ability to characterize long memory processes. In this paper, the time-fractional generalized Navier-Stokes (N-S) equations are formulated to model anomalous transport in porous media at the representative elementary volume (REV) scale by incorporating the Caputo fractional derivative of time into the widely employed generalized N-S equations. An innovative lattice Boltzmann (LB) model is presented for solving the time-fractional generalized N-S equations. We validate the proposed LB model by conducting a numerical example with analytical solutions, and observe a good agreement between the LB results and the analytical solutions. The proposed LB model is utilized for simulating Poiseuille flow, Couette flow, and cavity flow in porous media. Unlike the previous studies, which focus on the steady state of these flows, the present work focuses on the unsteady process from the initial state to the final steady state. It is found that time-fractional effects play a significant role in the unsteady processes of these flows. As the fractional order decreases, the flows undergo a slower evolution process and the impact of Darcy number on the unsteady process becomes increasingly noticeable. The Reynolds number has a significant impact on the velocity profile in the steady state, but it has relatively little effect on the duration of the unsteady process.

Flow in temporally and spatially varying porous media: a model for transport of interstitial fluid in the brain.

Flow in a porous medium can be driven by the deformations of the boundaries of the porous domain. Such boundary deformations locally change the volume fraction accessible by the fluid, creating non-uniform porosity and permeability throughout the medium. In this work, we construct a deformation-driven porous medium transport model with spatially and temporally varying porosity and permeability that are dependent on the boundary deformations imposed on the medium. We use this model to study the transport of interstitial fluid along the basement membranes in the arterial walls of the brain. The basement membrane is modeled as a deforming annular porous channel with the compressible pore space filled with an incompressible, Newtonian fluid. The role of a forward propagating peristaltic heart pulse wave and a reverse smooth muscle contraction wave on the flow within the basement membranes is investigated. Our results identify combinations of wave amplitudes that can induce either forward or reverse transport along these transport pathways in the brain. The magnitude and direction of fluid transport predicted by our model can help in understanding the clearance of fluids and solutes along the Intramural Periarterial Drainage route and the pathology of cerebral amyloid angiopathy.

Geo-guided deep learning for spatial downscaling of solute transport in heterogeneous porous media

Resolving solute transport in heterogeneous porous media is a complex task, because of the sparse experimental data and the high computational cost of numerical simulations. This work proposes a unique two-stage deep learning architecture comprising a dual-branch autoencoder and a geo-guided super-resolution generative adversarial network (Gg-SRGAN) to address this dual challenge. The dual-branch autoencoder addresses the issue of sparsity by constructing a continuous, but coarse representation of concentration and pressure profiles from a sparse, discontinuous profile with up to 85% missing data points. The Gg-SRGAN is then employed to generate a finer representation of field variables from the outputs generated by the dual-branch autoencoder (i.e., downscaling). We train and test our framework using six solute transport cases with varying levels of heterogeneity and compare the results with standalone methods, namely the vanilla autoencoder and vanilla SRGAN, in addition to ground truth profiles generated by the finite element method (FEM). The comparisons are performed based on several statistical metrics, such as absolute point error (APE), mean squared error (MSE), peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), and learned perceptual image patch similarity (LPIPS). The first four cases are used for training, evaluation, and testing. The last two cases are utilized for blind testing to determine the generalizability of the framework. Our results show that the dual-branch autoencoder outperforms the vanilla autoencoder, and the Gg-SRGAN outperforms the SRGAN during both the training and evaluation phases. Moreover, the proposed framework can successfully construct the fine representation of concentration profiles, compared to FEM, using the coarse representation of the pressure, concentration, and domain permeability fields. When tested using the two blind test cases, the proposed dual-branch autoencoder and Gg-SRGAN exhibit superior performance compared to their counterparts in terms of all evaluation metrics.

Shannon entropy of transport self-organization due to dissolution–precipitation reaction at varying Peclet numbers in initially homogeneous porous media

Abstract. Dissolution and precipitation processes in reactive transport in porous media are ubiquitous in a multitude of contexts within the field of Earth sciences. In particular, the dynamic interaction between the reactive dissolution and precipitation processes and the solute transport is of interest as it is capable of giving rise to the emergence of preferential flow paths in the porous host matrix. It has been shown that the emergence of preferential flow paths can be considered to be a manifestation of transport self-organization in porous media as these create spatial gradients that distance the system from the state of perfect mixing and allow for a faster and more efficient fluid transport through the host matrix. To investigate the dynamic feedback between the transport and the reactive processes in the field and its influence on the emergence of transport self-organization, we consider a two-dimensional Darcy-scale formulation of a reactive-transport setup, where the precipitation and dissolution of the host matrix are driven by the injection of an acid compound, establishing local equilibrium with the resident fluid and an initially homogeneous porous matrix, composed of a calcite mineral. The coupled reactive process is simulated in a series of computational analyses employing the Lagrangian particle-tracking (LPT) approach, capable of capturing the subtleties of the multiple-scale heterogeneity phenomena. We employ the Shannon entropy to quantify the emergence of self-organization in the field, which we define as a relative reduction in entropy compared to its maximum value. Scalability of the parameters, which characterize the evolution of the reactive process, with the Peclet number in an initially homogeneous field is derived using a simple one-dimensional ADRE model with a linear adsorption reaction term and is then confirmed through numerical simulations, with the global reaction rate, the mean value, and the variance of the hydraulic-conductivity distribution in the field all exhibiting dependency on the reciprocal of the Peclet number. Our findings show that transport self-organization in an initially homogeneous field increases with time, along with the emergence of the field heterogeneity due to the interaction between the transport and reactive processes. By studying the influence of the Peclet number on the reactive process, we arrive at a conclusion that self-organization is more pronounced in diffusion-dominated flows, characterized by small Peclet values. The self-organization of the breakthrough curve exhibits the opposite tendencies, which are observed from the perspective of a thermodynamic analogy. The hydraulic power, required to maintain the driving head pressure difference between the inlet and outlet of the field, was shown to increase with the increasing variance, as well as with the increasing mean value of the hydraulic-conductivity distribution in the field, using a simple analytic model. This was confirmed by numerical experiments. This increase in power, supplied to the flow in the field, results in an increase in the level of transport self-organization. Employing a thermodynamic framework to investigate the dynamic reaction–transport interaction in porous media may prove to be beneficial whenever the need exists to establish relations between the intensification of the preferential flow path phenomenon, represented by a decline in the Shannon entropy of the transport, with the amount of reaction that occurred in the porous medium and the change in its heterogeneity.

BioReactPy: An open-source software for simulation of microbial-mediated reactive processes in porous media

This paper provides a new open-source software, named BioReactPy, for simulation of microbial-mediated coupled processes of flow and reactive transport in porous media. The software is based on the micro-continuum approach, and geochemistry is handled in a fully coupled manner with biomass-nutrient growth treated with Monod equation in a single integrated framework, without dependencies on third party packages. The distinguishing features of the software, its design principles, and formulation of multiphysics problems and discretizations are discussed. Validation of the Python implementation using several established benchmarks for flow, reactive transport, and biomass growth is presented. The flexibility of the framework is then illustrated by simulations of highly non-linearly coupled flow and microbial reactive transport at conditions relevant to carbon mineralization for CO2 storage. All results can be reproduced by openly available simulation scripts.

Probabilistic assessment of scalar transport under hydrodynamically unstable flows in heterogeneous porous media

Quantitative predictions of scalar transport in natural porous media is a nontrivial task given the presence of multi-scale spatial heterogeneity in the permeability field. Due to data scarcity, the structural map of the permeability field is subject to uncertainty and therefore, model predictions are uncertain. For such reasons, probabilistic models of flow and transport in natural porous media are required in risk assessment and to provide reliable decision making under uncertainty. Further complexities arise when the viscosity of the injected solute differs from that of the ambient fluid. Under the presence of viscosity contrast, hydrodynamic instabilities give rise to viscous fingering, which induces additional disorder in both velocity and solute concentration fields. This work examines the combined role of viscous fingering and permeability heterogeneity in the probabilistic description of transport predictions. In particular, we focus on metrics that are important for risk analysis, such as the solute plume’s early arrival times and the maximum concentration observed at a given location. We propose to use the Projection Pursuit Adaptation (PPA) method in the Polynomial Chaos Expansion (PCE) framework to quantify uncertainty in transport model predictions. The PPA method is a data-driven approach that optimally represents a given quantity of interest in a low-dimensional manifold. Unlike other dimension reduction techniques in uncertainty quantification, the PPA method utilizes non-linear information of the quantity of interest to identify the low-dimensional manifold, thereby increasing the likelihood of finding a more accurate lower-dimensional space. Moreover, the PPA model converges to the physical solution in a mean squared sense with respect to the polynomial order, enabling the construction of a converged model even with limited available data. Then, the PPA results are compared to Monte Carlo simulations using the same amount of data. This comparison illustrates that while Monte Carlo simulations are able to capture low-order statistics, they struggle to represent more detailed probability density functions. Our results show how the combined effect of permeability heterogeneity and viscosity contrast can enhance the mobility of the solute plume.