New Analytical Solution to Water Content Simulation in Porous Media

Darcy and inertial fluid flow simulations in porous media using the non-orthogonal central moments lattice Boltzmann method

Modelling, simulation, and visualization of porous media have practical applications in the petroleum and chemical industries. Advances in modelling and simulation of porous media can help in the understanding of processes in petroleum rocks. The paper presents new algorithms for modelling and visualizing fluid interfaces within porous media. These algorithms allow the creation and nondestructive testing of porous samples in a virtual laboratory of porous media. The visualization process is one of the main elements in such a laboratory. Results of these simulated processes can be visualized as 3D or 2D images, where colour and shading emphasize fluid saturations and interfaces between two interacting fluids as a function of varying pressure. These algorithms are based on the octree data structure. The idea is to create a porous media sample with a realistic structure and at a level of complexity commensurate with actual samples. Simulations are then run on this model. This method contrasts with traditional porous media modelling where a mathematical model (lattice) is simulated using sets of equations.

The dispersion phenomenon has resulted from the various water flow magnitude and direction in porous media. The dissolved tracer tends to spread due to dispersion and then travel time of tracer through the porous media increases. In unsaturated porous media, dispersion coefficient varies with non-linear Darcy’s velocity and the water content. These nonlinear dispersions were observed in both of the laboratory scale sand and soil columns (20 cm). The unsaturated infiltration column and tracer tests have been used to interpret the relationships between Darcy’s velocity and the water content together with the dispersion coefficient. However, the dispersivity coefficient cannot be measured directly; it has to be determined from advection-dispersion equation (ADE), which can be used to model the tracer transport in unsaturated porous media. The model was used to describe the non-linear functions of water contents and dispersivities for both porous media. The simulations have been verified that the dispersion of tracer through soil is higher than that of sand column and also the travel time of tracer through soil is longer than that of sand column. Even though, soil has very low degree of pore velocity, high dispersivity is observed in the simulations. The water content and tracer concentration profiles reveal that the increase of dispersivity induces the increase of flow path distance and the decrease of pore velocity. The maximum dispersivity was observed when the water content of porous media is relatively low; this leads to the maximum of spreading of tracer.

Fluid mechanics simulation of steady state flow in complex geometries has many applications, from the micro-scale (cell membranes, filters, rocks) to macro-scale (groundwater, hydrocarbon reservoirs, and geothermal) and beyond. Direct simulation of steady state flow in such porous media requires significant computational resources to solve within reasonable timeframes. This study outlines an integrated method combining predictions of fluid flow (fast, limited accuracy) with direct flow simulation (slow, high accuracy) is outlined that reduces computation time by an order of magnitude without loss of accuracy. A convolutional neural network (CNNs) is trained with various configurations on simulations in 2D and 3D porous media to estimate steady state velocity fields. Permeability estimation (as an average of the field) is accurate, but the velocity fields themselves are error prone, unsuitable for further transport studies. This estimate can either be used as an indicative prediction, or as initial conditions in direct simulation to reach a fully accurate result in a fraction of the compute time. Using Deep Learning predictions (or potentially any other approximation method) to accelerate flow simulation to steady state in complex structures shows promise as a technique to push the boundaries fluid flow modelling.

Locally conservative Galerkin and finite volume methods for two-phase flow in porous media

A family of analytical solutions of a nonlinear diffusion–convection equation

Plants grown in porous media are part of a bioregenerative life support system designed for long-duration space missions. Reduced gravity conditions of orbiting spacecraft (microgravity) alter several aspects of liquid flow and distribution within partially saturated porous media. The objectives of this study were to evaluate the suitability of conventional capillary flow theory in simulating water distribution in porous media measured in a microgravity environment. Data from experiments aboard the Russian space station Mir and a U.S. space shuttle were simulated by elimination of the gravitational term from the Richards equation. Qualitative comparisons with media hydraulic parameters measured on Earth suggest narrower pore size distributions and inactive or nonparticipating large pores in microgravity. Evidence of accentuated hysteresis, altered soil-water characteristic, and reduced unsaturated hydraulic conductivity from microgravity simulations may be attributable to a number of proposed secondary mechanisms. These are likely spawned by enhanced and modified paths of interfacial flows and an altered force ratio of capillary to body forces in microgravity.

Development of an approximate model for unsaturated flow with root water uptake under rectangular water content profiles assumption

Stability analysis of traveling wave solution for gravity-driven flow

Using scaling methods, a single solution of Richards' equation (RE) will suffice for numerous specific cases of water flow in unsaturated soils. In this study, a new method is developed to scale RE for the soil water redistribution process. Two similarity conditions are required: similarity in the shape of the soil water content profiles as well as of the water flux density curves. An advantage of this method is that it is not restricted to a specific soil hydraulic model - hence, all such models can be applied to RE. To evaluate the proposed method, various soil textures and initial conditions were considered. After the RE was solved numerically using the HYDRUS-1D model, the solutions were scaled. The scaled soil water content profiles were nearly invariant for medium- and fine-textured soils when the soil profile was not deeply wetted. The textural range of the soils in which the similarity conditions are held decreases as the initial conditions deal with a deeply wetted profile. Thus, the scaling performance was poor in such a condition. This limitation was more pronounced in the coarse-textured soils. Based on the scaling method, a procedure is suggested by which the solution of RE for a specific case can be used to approximate solutions for many other cases. Such a procedure reduces complicated numerical calculations and provides additional opportunities for solving the highly nonlinear RE as in the case of unsaturated water flow in soils.

A 6M digital twin for modeling and simulation in subsurface reservoirs

Numerical solution of Richards’ equation of water flow by generalized finite differences

Estimating contaminant migration in the context of waste disposal and/or environmental remediation of polluted soils requires a complete understanding of the underlying transport processes. In unsaturated porous media, water content is one of the most determining parameters to describe solute migration because it impacts directly on solute pore velocity. However, numerous studies are satisfied with only a global or a partial spatial distribution of water content within the studied porous media. Therefore, distribution of water content in porous media must be precisely achieved to optimize transport processes modeling. Tracer experiments with downward flow were performed on the BEETI experimental device equipped with a sand column. Water content and concentration profiles of tracer (KI) were measured along the column during experiment. The relative dispersion of water content, calculated along the column, gives an idea of influence of this parameter on transport properties. A relationship between pore velocity, Darcy flow velocity and water content is proposed.

Simulation of unsaturated water movement in porous media has conventionally been based on the Richards equation (RE), coupled with hydraulic conductivity functions that account solely for capillary-driven liquid flow. This approach, however, overlooks the presence of thin water films adsorbed on solid surfaces, which may contribute appreciably to transport processes under moderately dry to dry conditions. Recent advances, particularly the Peters–Durner–Iden (PDI) framework, enable a physically consistent representation of film flow and isothermal vapor diffusion within formulations of unsaturated hydraulic conductivity.In this work, we introduce the Virtual Soil Simulator, a finite-volume, OpenFOAM-based implementation of the RE augmented with the PDI model to explicitly represent capillary, film, and vapor transport processes. Model performance was assessed using a suite of benchmark tests with analytical or well-established numerical reference solutions, including one-dimensional infiltration, infiltration under steep hydraulic gradients, and two-dimensional nonlinear infiltration scenarios. The results demonstrate high numerical accuracy and robust mass conservation.The applicability of the model is further demonstrated through two case studies. In the first, inverse simulation of a 12-day soil core drying experiment showed that the classical RE formulation reproduced measurements only during the early, wet stage, whereas the PDI-enhanced model remained consistent with observations over the entire drying period and accurately represented regimes dominated by film and vapor flow. In the second case, a synthetic desaturation analysis conducted across 467 soil types indicated that film flow markedly accelerates drainage, with significant effects persisting even at comparatively high pressure heads (−10 m). These findings indicate that neglecting film flow leads to systematic underestimation of unsaturated hydraulic conductivity and distorted predictions of drying and drainage behavior. Moreover, simulations at very low pressure heads emphasize that reliable representation of transport processes requires the combined consideration of both film and vapor fluxes.AcknowledgmentsThis research was founded by the National Science Centre within contract 2021/43/B/ST10/03143.

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.