Equations Of Flow In Porous Media Research Articles

Immiscible Viscous Fingering: The Effects of Wettability/Capillarity and Scaling

AbstractRealistic immiscible viscous fingering, showing all of the complex finger structure observed in experiments, has proven to be very difficult to model using direct numerical simulation based on the two-phase flow equations in porous media. Recently, a method was proposed by the authors to solve the viscous-dominated immiscible fingering problem numerically. This method gave realistic complex immiscible fingering patterns and showed very good agreement with a set of viscous unstable 2D water → oil displacement experiments. In addition, the method also gave a very good prediction of the response of the system to tertiary polymer injection. In this paper, we extend our previous work by considering the effect of wettability/capillarity on immiscible viscous fingering, e.g. in a water → oil displacements where viscosity ratio $$\left( {\mu_{{\text{o}}} /\mu_{{\text{w}}} } \right) \gg 1$$ μ o / μ w ≫ 1 . We identify particular wetting states with the form of the corresponding capillary pressure used to simulate that system. It has long been known that the broad effect of capillarity is to act like a nonlinear diffusion term in the two-phase flow equations, denoted here as $$D(S_{w} )$$ D ( S w ) . Therefore, the addition of capillary pressure, $$P_{c} (S_{w} )$$ P c ( S w ) , into the equations acts as a damping or stabilisation term on viscous fingering, where it is the derivative of this quantity that is important, i.e. $$D(S_{w} )\sim\left( {dP_{c} (S_{w} )/dS_{w} } \right)$$ D ( S w ) ∼ d P c ( S w ) / d S w . If this capillary effect is sufficiently large, then we expect that the viscous fingering to be completely damped, and linear stability theory has supported this view. However, no convincing numerical simulations have been presented showing this effect clearly for systems of different wettability, due to the problem of simulating realistic immiscible fingering in the first place (i.e. for the viscous-dominated case where $$P_{c} = 0$$ P c = 0 ). Since we already have a good method for numerically generating complex realistic immiscible fingering for the $$P_{c} = 0$$ P c = 0 case, we are able for the first time to present a study examining both the viscous-dominated limit and the gradual change in the viscous/capillary force balance. This force balance also depends on the physical size of the system as well as on the length scale of the capillary damping. To address these issues, scaling theory is applied, using the classical approach of Rapport (1955), to study this scaling in a systematic manner. In this paper, we show that the effect of wettability/capillarity on immiscible viscous fingering is somewhat more complex and interesting than the (broadly correct) qualitative description above. From a “lab-scale” base case 2D water → oil displacement showing clear immiscible viscous fingering which we have already matched very well using our numerical method, we examine the effects of introducing either a water wet (WW) or an oil wet (OW) capillary pressure, of different “magnitudes”. The characteristics of these two cases (WW and OW) are important in how the value of corresponding $$D(S_{w} )$$ D ( S w ) functions, relate to the (Buckley–Leverett) shock front saturation, $$S_{wf}$$ S wf , of the viscous-dominated ($$P_{c} = 0$$ P c = 0 ) case. By analysing this, and carrying out some confirming calculations, we show clearly why we expect to see much clearer immiscible fingering at the lab scale in oil wet rather than in water wet systems. Indeed, we demonstrate why it is very difficult to see immiscible fingering in WW lab systems. From this finding, one might conclude that since no fingering is observed for the WW lab-scale case, then none would be expected at the larger “field” scale. However, by invoking scaling theory—specifically the viscous/capillary scaling group, $$C_{{{\text{VC1}}}}$$ C VC1 , (and a corresponding “shape group”, $$C_{{{\text{S}}1}}$$ C S 1 ), we demonstrate very clearly that, although the WW viscous fingers do not usually appear at the lab scale, they emerge very distinctly as we “inflate” the system in size in a systematic manner. In contrast, we demonstrate exactly why it is much more likely to observe viscous fingering for the OW (or weakly wetting) case at the lab scale. Finally, to confirm our analysis of the WW and OW immiscible fingering conclusions at the lab scale, we present two experiments in a lab-scale bead pack where $$\left( {\mu_{{\text{o}}} /\mu_{{\text{w}}} } \right) = 100$$ μ o / μ w = 100 ; no fingering is seen in the WW case, whereas clear developed immiscible fingering is observed in the OW case.

DIGITAL СORE: TEMPERATURE FIELD INFLUENCE ON TWO-PHASE FILTRATION OF FLUIDS IN ROCKS

Link for citation: Katanov Yu.E., Yagafarov A.K., Aristov A.I. Digital сore: temperature field influence on two-phase filtration of fluids in rocks. Bulletin of the Tomsk Polytechnic University. Geo Аssets Engineering, 2023, vol. 334, no. 10, рр. 108-118. In Rus. Relevance. Computational fluid dynamics is a powerful tool for studying geological processes. Fluid flow equations in porous media can be solved directly with the resulting three-dimensional X-ray and computer tomographic images. Numerous studies of the geological features of rocks depend on heat transfer in porous materials. These processes include heat conduction in the rock material, convection between matter and the surrounding gas, electrical conductivity, viscous dissipation of fluids, kinetics of chemical reactions, and interfacial heat transfer at the solid/liquid interface. While energy transfer in porous media under conditions of local thermal equilibrium is well studied on the scale of Darcy's law, it is less often considered in porous models. Heat transfer can be accounted for by creating a «heat network» based on a porous network model representing a set of fluid-conducting channels with an appropriate solution to the unified energy balance equation. It is also necessary to take into account convective transfer of thermal energy by the flow of liquid phases (oil, water), each of which is represented by a separate network. Knowing the distribution of the temperature field in the texture of each lithological type of rock, it is possible to determine the probabilistic displacement of the fluid displacement front in the corresponding fluid-conducting space of the rock. Objective: neural network modeling of the effect of the temperature field on the filtration characteristics of hydrocarbons in the fluid-conducting space of rocks. Objects: polymictic sandstones of the Tyumen Formation. Methods. Digital reconstruction of rock texture was performed using artificial intelligence methods and neural networks; neural network algorithms of probabilistic displacement front of two-phase flow (oil, water) were developed in Python programming language; methodical approach to the study of thermal field propagation in rock texture and its influence on two-phase fluid flow was developed using Fourier law, Navier–Stokes equations and similarity criteria of hydrocarbon systems. Results. We developed algorithms for neural network modeling of the temperature field in the fluid-conducting space of a rock (polymictic sandstone) as well as neural network algorithms to estimate the displacement of two-phase filtration front under the corresponding influence of the temperature distribution in the texture of a digital core. The basic mathematical models of these algorithms are outlined. The source code of the algorithms is written in the Python programming language with additional use of non-commercial libraries. The results of neural network modeling have a high degree of reliability, confirmed by experiments with the data obtained in the laboratory of core studies (University of Tyumen, Tyumen), the laboratory of the V.I. Shpilman Research and Analytical Centre for the Rational Use of the Subsoil (Khanty-Mansiysk), laboratory of digital research in oil and gas in the framework of the technological project «Digital core» (Industrial University of Tyumen, Tyumen).

Effect of homogeneous chemical reaction on the dispersion of a solute in a three – dimensional flow of a Newtonian liquid through a porous medium

All-Time dispersion of a reactive solute in a Newtonian fluid flow through a Darcy-Brinkman-Forchheimer Porous medium is carried out using the approach of Gill - Sankarasubramanian (1970) and Doshi et al. (1978). The velocity profile of the non - linear porous medium flow equation is solved using the Maclaurin series of two variables. The three - dimensional model brings into the focus the convective and dispersion coefficients. The chemical reaction is assumed to be homogeneous. The chemical reaction is shown to increase the value of the convective coefficient while it decreases with increase in the value of the reaction rate parameter. The effect of the presence of the porous medium is to decrease the flow and hence the convective coefficient. Similar effect of the porous parameter is seen on the dispersion coefficient. The reaction rate parameter and the porous parameter have opposite effect on the mean concentration distribution.

A Numerical Approach to Investigate the Impact of Acid-Asphaltene Sludge Formation on Wormholing During Carbonate Acidizing

AbstractCarbonate acidization is the process of creating wormholes by injecting acid to increase reservoir permeability and oil production. Nevertheless, some reservoir oils are problematic with low asphaltene stability, which affects the wormholing process. The interactions between acid, rock, and asphaltene lead to acid-asphaltene sludge formation, which reduces oil productivity and acid injectivity. Neglecting this sludge formation leads to over predicting the depth of the wormhole penetration. Therefore, a numerical model was developed in this study to provide a better understanding of acid-asphaltene sludge formation effect on wormhole creation and propagation in carbonates. A one-dimensional radial model was developed by coupling fluid flow equations in porous media with asphaltene deposition and acid-asphaltene reactions. Then, the developed model was validated and utilized to investigate the effects of different parameters on wormholing including asphaltene presence, acid injection volume and concentration, formation temperature and porosity, and asphaltene concentration. Results showed that acid injection in carbonates with asphaltenic oils reduce wormhole penetration from 40% to total pore blockage as opposed to reservoirs without asphaltene deposition. The findings also highlighted that shallow wormhole penetration is more pronounced with a high volume of acid injection, high porous formations, less diluted acid, and a high concentration of asphaltene. In addition, there is an optimum acid injection volume at which wormhole penetration is high and its infiltration is deep into the formation. This is the first work to discuss modeling of acid-asphaltene sludge formation and subsequent wormhole development in carbonates, which is particularly important for problematic crude oils.

Application of Implicit Pressure-Explicit Saturation Method to Predict Filtrated Mud Saturation Impact on the Hydrocarbon Reservoirs Formation Damage

Hydrocarbon reservoirs’ formation damage is one of the essential issues in petroleum industries that is caused by drilling and production operations and completion procedures. Ineffective implementation of drilling fluid during the drilling operations led to large volumes of filtrated mud penetrating into the reservoir formation. Therefore, pore throats and spaces would be filled, and hydrocarbon mobilization reduced due to the porosity and permeability reduction. In this paper, a developed model was proposed to predict the filtrated mud saturation impact on the formation damage. First, the physics of the fluids were examined, and the governing equations were defined by the combination of general mass transfer equations. The drilling mud penetration in the core on the one direction and the removal of oil from the core, in the other direction, requires the simultaneous dissolution of water and oil flow. As both fluids enter and exit from the same core, it is necessary to derive the equations of drilling mud and oil flow in a one-dimensional process. Finally, due to the complexity of mass balance and fluid flow equations in porous media, the implicit pressure-explicit saturation method was used to solve the equations simultaneously. Four crucial parameters of oil viscosity, water saturation, permeability, and porosity were sensitivity-analyzed in this model to predict the filtrated mud saturation. According to the results of the sensitivity analysis for the crucial parameters, at a lower porosity (porosity = 0.2), permeability (permeability = 2 mD), and water saturation (saturation = 0.1), the filtrated mud saturation had decreased. This resulted in the lower capillary forces, which were induced to penetrate the drilling fluid to the formation. Therefore, formation damage reduced at lower porosity, permeability and water saturation. Furthermore, at higher oil viscosities, due to the increased mobilization of oil through the porous media, filtrated mud saturation penetration through the core length would be increased slightly. Consequently, at the oil viscosity of 3 cP, the decrease rate of filtrated mud saturation is slower than other oil viscosities which indicated increased invasion of filtrated mud into the formation.

Investigation of thermal model effects on geothermal and oil reservoir well testing behavior during water and steam injection

Global warming and reducing fossil fuel resources have increased the interest in using renewable resources such as geothermal energy. In this paper, in the first step, heat transfer equations have been presented for reservoir during water (steam) injection by considering heat loss to adjacent formations. According to radius of thermal front, the reservoir is partitioned into two regions with different fluid physical properties. The heat transfer model is coupled with a fluid flow model which is used to calculate the reservoir pressure or fluid flow rates. Then by calculating outer radius of heated region and using radial composite reservoir model, the fluid flow equations in porous media are solved. Using pressure derivative plot in regions with different thermal conductivity coefficients, a type curve plot is presented. The reservoir and adjacent formation thermal conductivity coefficients can be calculated by matching the observed pressure data on the thermal composite type curve. Additionally, the interference test in composite geothermal reservoir is discussed. In the composite reservoir model, parameters such as diffusivity coefficient, conductivity ratio and the distance to the radial discontinuity are considered. New type curves are provided to introduce the effect of diffusivity/conductivity contrast ratios on temperature behavior. Improving interpretations, and performing fast computations and fast sensitivity analysis are the benefits of the presented solutions.

Bounding of Flow and Transport Analysis in Heterogeneous Saturated Porous Media: A Minimum Energy Dissipation Principle for the Bounding and Scale-Up

We apply minimum kinetic energy principles from classic mechanics to heterogeneous porous media flow equations to derive and evaluate rotational flow components to determine bounding homogenous representations. Kelvin characterized irrotational motions in terms of energy dissipation and showed that minimum dynamic energy dissipation occurs if the motion is irrotational; i.e., a homogeneous flow system. For porous media flow, reductions in rotational flow represent heterogeneity reductions. At the limit, a homogeneous system, flow is irrotational. Using these principles, we can find a homogenous system that bounds a more complex heterogeneous system. We present mathematics for using the minimum energy principle to describe flow in heterogeneous porous media along with reduced special cases with the necessary bounding and associated scale-up equations. The first, simple derivation involves no boundary differences and gives results based on direct Kelvin-type minimum energy principles. It provides bounding criteria, but yields only a single ultimate scale-up. We present an extended derivation that considers differing boundaries, which may occur between scale-up elements. This approach enables a piecewise less heterogeneous representation to bound the more heterogeneous system. It provides scale-up flexibility for individual model elements with differing sizes, and shapes and supports a more accurate representation of material properties. We include a case study to illustrate bounding with a single direct scale-up. The case study demonstrates rigorous bounding and provides insight on using bounding flow to help understand heterogeneous systems. This work provides a theoretical basis for developing bounding models of flow systems. This provides a means to justify bounding conditions and results.

Porous media acidizing simulation: New two-phase two-scale continuum modeling approach

In Recent years some experimental studies have been conducted to investigate the effects of presence of a non-aqueous phase during carbonate acidizing process. Narrower and longer wormholes were observed to be formed when acid is injected into an oil-saturated porous medium, instead of a water-saturated one; which can provide breakthrough earlier with less injected pore volume. However, the mechanisms responsible for these observations had not been comprehensively studied. In this research, a two-phase two-scale continuum (TPTSC) model for simulation of dissolution process during acid injection into carbonate rocks is presented. This model couples two-phase fluid flow equations in porous media and the previously developed two-scale continuum (TSC) model equations for carbonate acidizing. A synthetic case is simulated and the effect of the restricted flow due to various relative permeability values, viscous fingering effects on injected pore volumes to breakthrough; and dissolution patterns are investigated. Our simulation results demonstrate positive effects of two-phase flow and viscous fingering on formation of narrower and longer wormholes. It is observed that pore volumes to breakthrough for an initially high viscosity oil-saturated core is significantly less than pore volumes to breakthrough for an initially water-saturated core. Flow restriction around wormhole due to less relative permeability values and viscous fingering phenomenon can be responsible for gaining lower injected acid volumes to breakthrough. Thus, especially for high viscosity oil reservoir conditions, direct injection of acid to an oil-saturated medium, instead of injecting a pre-flush to push back the saturating oil, can be beneficial.

Identification of diffusion parameters in a non-linear convection–diffusion equation using adaptive homotopy perturbation method

This paper is concerned with the numerical identification of diffusion parameters in a non-linear convection–diffusion equation, which arises as the saturation equation in the fractional flow formulation of the two-phase porous media flow equations. In order to overcome the defect of the local convergence of traditional methods, an adaptive homotopy perturbation method is applied to solve this parameter identification inverse problem. The adaptive homotopy perturbation method provides a simple way to adapt computational refinement to the choice of the homotopy parameter. Numerical simulations illustrate that the proposed algorithm is globally convergent and computationallyefficient.

Monotone nonlinear finite‐volume method for nonisothermal two‐phase two‐component flow in porous media

SummaryThis article presents a new nonlinear finite‐volume scheme for the nonisothermal two‐phase two‐component flow equations in porous media. The face fluxes are approximated by a nonlinear two‐point flux approximation, where transmissibilities nonlinearly depend on primary variables. Thereby, we mainly follow the ideas proposed by Le Potier combined with a harmonic averaging point interpolation strategy for the approximation of arbitrary heterogeneous permeability fields on polygonal grids. The behavior of this interpolation strategy is analyzed, and its limitation for highly anisotropic permeability tensors is demonstrated. Moreover, the condition numbers of occurring matrices are compared with linear finite‐volume schemes. Additionally, the convergence behavior of iterative solvers is investigated. Finally, it is shown that the nonlinear scheme is more efficient than its linear counterpart. Copyright © 2016 John Wiley & Sons, Ltd.

On a form of the system of porous media flow equations convenient for the construction of parallel algorithms

A geometric approach and the concept of streamlines are used to reformulate the equations governing the multiphase multicomponent flow through porous media with allowance for phase transitions. With the use of the streamline simulation technology, the three-dimensional problem is split into a set of onedimensional subproblems, for which efficient parallel algorithms can be constructed. A new geometric formulation of the streamline simulation technology is presented, which is in a sense rigorous. It is demonstrated how the accuracy of the method can be estimated with the help of this formulation.

Application of discrete wavelet transform to the solution of boundary value problems for quasi-linear parabolic equations

The wavelet method for solving the linear and quasi-linear parabolic equations under initial and boundary conditions is set out. By applying regular multi-resolution analysis and received formula for differentiating wavelet decompositions of functions of many variables the problem is reduced to a finite set of linear and accordingly nonlinear algebraic equations for the wavelet coefficients of the problem solution. The general scheme for finite-dimensional approximation in the regularization method is combined with the discrepancy principle. For quasi-linear parabolic equations the convergence rate of an approximate weak solution to a classical one is estimated.The proposed method is used for constructing stable approximate wavelet decompositions of weak solutions to boundary value problems for the unsteady porous-medium flow equation with discontinuous coefficients and inexact data.

Leaking gas from a snow-covered pipe: empirical evidence and modeling of preferential flow paths

The potential dangers associated with exposed fuel gas pipes in areas of significant snowfall are examined. There have been numerous incidents in the past two decades where snow loading or impact on fuel gas equipment has been identified as one step in the sequence of events leading to fires or explosions. This study experimentally and numerically examines the possibility of snow cover creating a preferential pathway for gas flow into a nearby structure from a leaking pipe. Simple experiments show that when a suspended pipe is covered with snow, a gap often forms directly underneath the pipe, which has the potential to serve as a low-resistance flow path for fugitive fuel gas. Using a simplified geometry, the fraction of gas entering a nearby structure from a leaking, snow-covered pipe is examined using finite differences to solve the flow equations in porous media. The snow cover is shown to force a large portion of the fugitive gas toward the structure under certain circumstances and has the potential to create additional hazards that would not be present without the snow layer. Using sensitivity analysis, the major parameters governing fugitive gas flow are determined, which include internal structure pressure, gap size, snow permeability, and distance of the leak from the structure.

Quasi-Newton acceleration of ILU preconditioners for nonlinear two-phase flow equations in porous media

In this work, preconditioners for the iterative solution by Krylov methods of the linear systems arising at each Newton iteration are studied. The preconditioner is defined by means of a Broyden-type rank-one update of a given initial preconditioner, at each nonlinear iteration, as described in [5] where convergence properties of the scheme are theoretically proved. This acceleration is employed in the solution of the nonlinear system of algebraic equations arising from the finite element discretization of two-phase flow model in porous media. We report numerical results of the application of this approach when the initial preconditioner is chosen to be the incomplete LU decomposition of the Jacobian matrix at the initial nonlinear stage. It is shown that the proposed acceleration reduces the number of linear iterations needed to achieve convergence. Also, the cost of computing the preconditioner is reduced as this operation is made only once at the beginning of the Newton iteration.

Analytical Upgridding Method To Preserve Dynamic Flow Behavior

Summary A geocellular model contains millions of gridblocks and needs to be upscaled before the model can be used as an input for flow simulation. Available techniques for upgridding vary from simple methods such as proportional fractioning to more complicated methods such as maintaining heterogeneities through variance calculations. All these methods are independent of the flow process for which simulation is going to be used, and are independent of well configuration. We propose a new upgridding method that preserves the pressure profile at the upscaled level. It is well established that the more complex the flow process, the more detailed the level of heterogeneity needed in the simulation model. In general, ideal upscaling is the process that preserves the "pressure profile" from the fine-scale model under the applicable flow process. In our method, we upgrid the geological model using simple flow equations in porous media. However, it should be remembered that to obtain a better match between fine scale and coarse scale, we also need to use appropriate upscaling of the reservoir properties. The new method is currently developed for single-phase flow; however, we used it for both single-phase and two-phase flows for 2D and 3D cases. The method differs fundamentally from the other methods that try to preserve heterogeneities. In those methods, gridblocks are combined that have similar velocities (or other properties) by assuming constant pressure drop across the blocks. Instead, we combine the gridblocks that have similar pressure profiles, although to release some of our assumptions such as having constant velocities in gridblocks, we balance our equation with the K2 term. The procedure is analytical and, hence, very efficient, but preserves the pressure profile in the reservoir. The gridblocks (or layers) are combined in a way so that the difference between fine- and coarse-scale pressure profiles is minimized. In addition, we also propose two new criteria that allow us to choose the optimum number of layers more accurately so that a critical level of heterogeneity is preserved. These criteria provide insight into the overall level of heterogeneity in the reservoir and the effectiveness of the layering design. We compare the results of our method with proportional layering and the King et al. method (King et al. 2006) and show that, for the same number of layers, the proposed method captures the results of the fine-scale model better. We show that the layer merging not only depends on the variation in the permeability between the gridblocks (K2 term), but also on the relative magnitude of the permeability values by combining 1/K2 and K2 terms.

Two-phase flow equations with outflow boundary conditions in the hydrophobic–hydrophilic case

We introduce an approximation procedure and provide existence results for two-phase flow equations in porous media. The medium can have hydrophobic and hydrophilic components such that the capillary pressure function is degenerate for extreme saturations. Our main interest is the outflow boundary condition which models an interface with open space. The approximate system introduces standard boundary conditions and can be used in numerical schemes. It allows the derivation of maximum principles. This is the basis for the derivation of the limiting system in the form of a variational inequality.

Wavelet method for solving the unsteady porous-medium flow problem with discontinuous coefficients

A method is proposed for constructing stable approximate wavelet decompositions of weak solutions to boundary value problems for the unsteady porous-medium flow equation with discontinuous coefficients and inexact data. The method is based on the general scheme for finite-dimensional approximation in Tikhonov regularization and on multiresolution analysis with basis functions defined as the product of one-dimensional Daubechies wavelets.

Electroosmosis law via homogenization of electrolyte flow equations in porous media

By the homogenization approach we justify a two-scale model of ion transport in porous media for one-dimensional horizontal steady flows driven by a pressure gradient and an external horizontal electrical field. By up-scaling, the electroosmotic flow equations in horizontal nanoslits separated by thin solid layers are approximated by a homogenized system of macroscale equations in the form of the Poisson equation for induced vertical electrical field and Onsager's reciprocity relations between global fluxes (hydrodynamic and electric) and forces (horizontal pressure gradient and external electrical field). In addition, the two-scale approach provides macroscopic mobility coefficients in the Onsager relations.

Transient‐state groundwater flow in various geometries with wells at arbitrary positions: analytical methods

AbstractWe have developed a method for analytically solving the porous medium flow equation in many different geometries for horizontal (two‐dimensional), homogeneous and isotropic aquifers containing impermeable boundaries and any number of pumping or injection wells located at arbitrary positions within the system. Solutions and results are presented for rectangular and circular aquifers but the method presented here is easily extendible to many geometries. Results are also presented for systems where constant head boundary conditions can be emulated internal to the aquifer boundary. Recommendations for extensions of the present work are briefly discussed. Copyright © 2003 John Wiley & Sons, Ltd.

Hydrologic Contaminant Transport Modeling: A Novel Analytical and Computational Approach Title

We have developed a method for modeling contaminant transport in aquifers with rectangular boundaries utilizing an analytical solution to the porous medium flow equation and a finite difference solution to the advection-dispersion equation. Any number of wells may be placed within the aquifer, as well as any number of non-interacting contaminants. Constant head boundaries that simulate rivers may be included by means of a source term. With realistic parameters we are able to successfully model and predict contamination transport in an on campus well site used for both undergraduate pedagogy and research.