A Stochastic Model for Deep Bed Filtration and Well Impairment

Abstract

Abstract Injectivity decline during sea/produced water flooding is a wide spread phenomenon in offshore waterflood projects. The injectivity decreases due to solid and liquid particles that are present in the injected water. Capture of particles by porous rock results in permeability damage. In this work, a new pore level model for deep bed filtration of water with particles in porous media is developed. The model accounts for particles and pore size distribution, as well as size exclusion mechanism (a particle is captured by a smaller size pore). The system of equations derived consists of the continuity particle number equations and of the kinetics of particle entrapment and pore plugging. The particle velocity reduction and the particle accessibility, due to restriction for particles to move via large pores only, are incorporated in the model. An analytical solution of the pore level stochastic system is obtained for a medium with small pore size variation. It allows derivation of the averaged concentration equations which differs from the traditional ones. Finally, the modification of the classical deep bed filtration model that accounts for velocity reduction and particle accessibility effects is proposed. Introduction Deep bed filtration of water with particles occurs in several industrial and environmental processes like water filtration and soil contamination. In petroleum industry, deep bed filtration happens near to injectors during the injection of seawater causing injectivity reduction. The particles capture in porous media can be caused by different physical mechanisms:size exclusion (large particles are captured in thin pores)electrical forces (London - Van der Walls, Double Layer)cloggingbridging In the current paper, the size exclusion mechanism is discussed. A phenomenological model for the particle-capture and permeability-damage process was proposed by Iwasaki1 and used for filtration theory2 and for rock permeability decline3,4. The model assumes linear kinetics of particle deposition, and exhibits a good agreement with laboratory data. So, the model can be used for prediction purposes, like forecast of well injectivity decline based on laboratory coreflood test. Nevertheless, the model does not distinguish between different mechanisms of formation damage. Therefore, the model cannot be used for diagnostic purposes, like determination of the dominant capture mechanism from well data. In case of size exclusion mechanism, the larger are the particles and the smaller are the pores, the more intensive is the capture and the larger is the formation damage. Nevertheless, several attempts to correlate the formation damage with sizes of particles and pores were unsuccessful5,6. It could mean that either size exclusion mechanisms never dominate, or the phenomenological model for average concentrations is not general/universal enough. One of ways out of this contradiction is micro scale modelling of each capture mechanism. Different network micro models have been developed by Sahimi et al.7,8, Imdakm et al.9,10, Payatakes et al.11,12, Rege and Fogler13,14 (see Khilar and Fogler15), Siqueira et al.16. Different physical mechanisms of particle retention are included in these models. Sharma and Yortsos17,18,19 derived basic population balance equations for transport of particulate suspensions in porous media. The model accounts for particle and pore size distribution variation due to different particle capture mechanisms. In the mentioned papers it is assumed that the particle population moves with the average flow velocity of the carrier water. In the case of uniform porous media, this assumption results in independent deep bed filtration of different size particles. Nevertheless, during the deep bed filtration with particle size exclusion mechanism, a particle can pass via large pores only, and thin pores capture it. So, depending on its size, the particle can either pass the uniform pore size medium without being captured, or does not enter the porous media at all.