Splitting technique for analytical modelling of two-phase multicomponent flow in porous media

In this paper we discuss one-dimensional models for two-phase Enhanced Oil Recovery (EOR) floods (oil displacement by gases, polymers, carbonized water, hot water, etc.). The main result presented here is the splitting of the EOR mathematical model into thermodynamical and hydrodynamical parts. The introduction of a potential associated with one of the conservation laws and its use as a new independent coordinate reduces the number of equations by one. The (n) × (n) conservation law model for two-phase n-component EOR flows in new coordinates is transformed into a reduced (n − 1) × (n − 1) auxiliary system containing just thermodynamical variables (equilibrium fractions of components, sorption isotherms) and one lifting equation containing just hydrodynamical parameters (phase relative permeabilities and viscosities). The algorithm to solve analytically the problem includes solution of the reduced auxiliary problem, solution of one lifting hyperbolic equation and inversion of the coordinate transformation. The splitting allows proving the independence of phase transitions occurring during displacement of phase relative permeabilities and viscosities. For example, the minimum miscibility pressure (MMP) and transitional tie lines are independent of relative permeabilities and phases viscosities. Relative motion of polymer, surfactant and fresh water slugs depends on sorption isotherms only. Therefore, MMP for gasflood or minimum fresh water slug size providing isolation of polymer/surfactant from incompatible formation water for chemical flooding can be calculated from the reduced auxiliary system. Reduction of the number of equations allows the generation of new analytical models for EOR. The analytical model for displacement of oil by a polymer slug with water drive is presented.Adolfo P. Pires, Pavel G. Bedrikovetsky, Alexander A. Shapir

Correction of Basic Equations for Deep Bed Filtration with Dispersion

Copyright © 2006 Elsevier B.V. All rights reserved.Deep bed filtration of particle suspensions in porous media occurs during water injection into oil reservoirs, drilling fluid invasion into reservoir productive zones, fines migration in oil fields, bacteria, virus or contaminant transport in groundwater, industrial filtering, etc. The basic features of the process are advective and dispersive particle transport and particle capture by the porous medium. Particle transport in porous media is determined by advective flow of carrier water and by hydrodynamic dispersion in micro-heterogeneous media. Thus, the particle flux is the sum of advective and dispersive fluxes. Transport of particles in porous media is described by an advection–diffusion equation and by a kinetic equation of particle capture. Conventional models for deep bed filtration take into account hydrodynamic particle dispersion in the mass balance equation but do not consider the effect of dispersive flux on retention kinetics. In the present study, a model for deep bed filtration with particle size exclusion taking into account particle hydrodynamic dispersion in both mass balance and retention kinetics equations is proposed. Analytical solutions are obtained for flows in infinite and semi-infinite reservoirs and in finite porous columns. The physical interpretation of the steady-state flow regimes described by the proposed and the traditional models favours the former.Altoe, J. E., Bedrikovetski, P.G., Siqueira, A. G., de Souza, A. L., Shecaira, F.http://www.elsevier.com/wps/find/journaldescription.cws_home/503345/description#descriptio

The inverse problem of determining the filtration function and permeability reduction in flow of water with particles in porous media

The original publication can be found at www.springerlink.comDeep bed filtration of particle suspensions in porous media occurs during water injection into oil reservoirs, drilling fluid invasion of reservoir production zones, fines migration in oil fields, industrial filtering, bacteria, viruses or contaminants transport in groundwater etc. The basic features of the process are particle capture by the porous medium and consequent permeability reduction. Models for deep bed filtration contain two quantities that represent rock and fluid properties: the filtration function, which is the fraction of particles captured per unit particle path length, and formation damage function, which is the ratio between reduced and initial permeabilities. These quantities cannot be measured directly in the laboratory or in the field; therefore, they must be calculated indirectly by solving inverse problems. The practical petroleum and environmental engineering purpose is to predict injectivity loss and particle penetration depth around wells. Reliable prediction requires precise knowledge of these two coefficients. In this work we determine these quantities from pressure drop and effluent concentration histories measured in one-dimensional laboratory experiments. The recovery method consists of optimizing deviation functionals in appropriate subdomains; if necessary, a Tikhonov regularization term is added to the functional. The filtration function is recovered by optimizing a non-linear functional with box constraints; this functional involves the effluent concentration history. The permeability reduction is recovered likewise, taking into account the filtration function already found, and the functional involves the pressure drop history. In both cases, the functionals are derived from least square formulations of the deviation between experimental data and quantities predicted by the model.Alvarez, A. C., Hime, G., Marchesin, D., Bedrikovetski, P

Evaluating transport in irregular pore networks

A general approach for investigating transport phenomena in porous media is presented. This approach has the capacity to represent various kinds of irregularity in porous media without the need for excessive detail or computational effort. The overall method combines a generalized effective medium approximation (EMA) with a macroscopic continuum model in order to derive a transport equation with explicit analytical expressions for the transport coefficients. The proposed form of the EMA is an anisotropic and heterogeneous extension of Kirkpatrick's EMA which allows the overall model to account for microscopic alterations in connectivity (with the locations of the pores and the orientation and length of the throat) as well as macroscopic variations in transport properties. A comparison to numerical results for randomly generated networks with different properties is given, indicating the potential for this methodology to handle cases that would pose significant difficulties to many other analytical models

Permeability hysteresis in gravity counterflow segregation

Hysteresis effects in two-phase flow in porous media are important in applications such as waterflooding or gas storage in sand aquifers. In this paper, we develop a numerical scheme for such a flow where the permeability hysteresis is modeled by a family of reversible scanning curves enclosed by irreversible imbibition and drainage permeability curves. The scheme is based on associated local Riemann solutions and can be viewed as a modification of the classical Godunov method. The Riemann solutions necessary for the scheme are presented, as well as the criteria that guarantee the well-posedness ofthe Riemann problem with respect to perturbations of left and right states. The numerical and analytical results show strong influence of the permeability hysteresis on the flow. In addition, the numerical scheme accurately reproduces the available experimental data once hysteresis is taken into account in the model.Shaerer, C., Sarkis, M., Marchesin, D. and Bedrikovetski, P

Joule-Thomson cooling of CO2 injected into aquifer under heat exchange with adjacent formations by Newtons law- 1D exact solution

This paper discusses axi-symmetric flow during CO2 injection into a non-adiabatic reservoir accounting for Joule-Thomson cooling and steady-state heat exchange between the reservoir and the adjacent layers by Newtons law. An exact solution for this 1D problem is derived and a new method for model validation by comparison with quasi 2D analytical heat-conductivity solution is developed. The temperature profile obtained by the analytical solution shows a temperature decrease to a minimum value, followed by a sharp increase to initial reservoir temperature. The analytical model exhibits stabilization of the temperature profile and cooled zone. When accounting for heat gain from adjacent layers, the minimum temperature is higher compared to the case with no heat exchange. The analytical model can be used to evaluate the risks of hydrate formation and/or rock integrity during CO2 storage in depleted reservoirs.Comment: 38 pages, 15 figures, 2 table

Exact Solution for Long-Term Size Exclusion Suspension-Colloidal Transport in Porous Media

Long-term deep bed filtration in porous media with size exclusion particle capture mechanism is studied. For monodispersed suspension and transport in porous media with distributed pore sizes, the microstochastic model allows for upscaling and the exact solution is derived for the obtained macroscale equation system. Results show that transient pore size distribution and nonlinear relation between the filtration coefficient and captured particle concentration during suspension filtration and retention are the main features of long-term deep bed filtration, which generalises the classical deep bed filtration model and its latter modifications. Furthermore, the exact solution demonstrates earlier breakthrough and lower breakthrough concentration for larger particles. Among all the pores with different sizes, the ones with intermediate sizes (between the minimum pore size and the particle size) vanish first. Total concentration of all the pores smaller than the particles turns to zero asymptotically when time tends to infinity, which corresponds to complete plugging of smaller pores