# Single-Phase Formation Damage by Fines Migration and Clay Swelling

The majority of the formation damage models were developed for single phase fluid systems. This assumption is valid only for very specific cases such as the production of particles with oil flow and for special core tests. Nevertheless, it is instructive to understand these models before looking into the multi-phase effects. Therefore, the various processes involving single-phase formation damage are discussed and the selected models available are presented along with some modifications and critical evaluation as to their practical applicability and limitations. The methodology for determination of the model parameters are presented. The parameters that can be measured directly are identified. The rest of the parameters are determined by means of a history matching technique. The applications of the models and the parameter estimation method are demonstrated using several examples.

An evaluation and comparison of six selected models bearing direct relevance to formation damage prediction for petroleum reservoirs are carried out. The modeling approaches and assumptions are identified, interpreted, and compared. These models are applicable for special cases involving single-phase fluid systems in laboratory core tests.

Porous media is considered in two parts:

1 the flowing phase, denoted by the subscript /, consists of a suspension of fine particles flowing through and

2 the stationary phase, denoted by the subscript s, consists of the porous matrix and the particles retrained.

### The Thin Slice Algebraic Model

#### Model Formulation

Wojtanowicz et al. (1987, 1988) considered a thin slice of a porous material and analyzed the various formation damage mechanisms assuming one distinct mechanism dominates at a certain condition. Porous medium is visualized as having tortuous pathways represented by Nh tubes of the same mean hydraulic equivalent diameter, Dh, located between the inlet and outlet ports of the core as depicted. The crosssectional area of the core is A and the length is L. The tortuosity factor for the tubes is defined as the ratio of the actual tube length to the length of the core.

in which Cl is an empirical shape factor that incorporates the effect of deviation of the actual perimeter from a circular perimeter. As a suspension of fine particles flows through the porous media, tubes having narrow constrictions are plugged and put out of service. If the number of nonplugged tubes at any given time is denoted by Nnp and the plugged tubes by Np, then the total number of tubes is given by:

The Darcy and Hagen-Poiseuille equations given respectively by

are considered as two alternative forms of the porous media momentum equations, q is the flowrate of the flowing phase and Ap is the pressure differential across the thin core slice. Thus, equating Eqs. 10-5 and 10-6 and using Eqs. 10-1 and 10-2 the relationship between permeability, K, and open flow area, A is obtained as:

The permeability damage in porous media is assumed to occur by three basic mechanisms:

1 gradual pore reduction (pore narrowing, pore lining) by surface deposition,

2 single pore blocking by screening (pore throat plugging) and

3 pore volume filling by straining (internal filter cake formation by the snowball effect).

Gradual pore reduction is assumed to occur by deposition of particles smaller than pore throats on the pore surface to reduce the cross-sectional
area, A, of the flow tubes gradually as depicted in this article. the number of tubes open for flow, Nnp, at any time remains the same as the total number of tubes, Nh, available. Hence,

Then, using Eq. 10-9 and eliminating A between Eqs. 10-4 and 10-7 leads to the following equation for the permeability to open flow area relationship during the surface deposition of particles:

Single pore blocking is assumed to occur by elimination of flow tubes from service by plugging of a pore throat or constriction, that may exist somewhere along the tube, by a single particle to stop the flow through that particular tube. Therefore, the cross-sectional areas of the individual tubes, Ah, do not change. But, the number of tubes, Nnp, open for the flow is reduced as depicted in Figure 10-3. The area of the tubes eliminated from service is given by:

The number of tubes plugged is estimated by the ratio of the total volume of pore throat blocking particles to the volume of a single particle of the
critical size.

The critical particle size, d, is defined as the average size of the critical pore constrictions in the core. fd is the volume fraction of particles in the flowing phase, having sizes comparable or greater than d. pp is the particle grain density. pp/ is the mass concentration of particles in the flowing suspension of particles. Because Ah is a constant, Eq. 10-7 leads to the following permeability to open flow area relationship:

Pore filling occurs near the inlet face of the core when a suspension of high concentration of particles in sizes larger than the size of the pore throats is injected into the core as depicted in this article. The permeability, Kc, of the particle invaded region decreases by accumulation of particles. But, in the uninvaded core region near the outlet, the permeability of the matrix, Km, remains unchanged. The harmonic mean permeability, K, of a core section (neglecting the cake at the inlet face) can be expressed in terms of the permeability, Kc, of the Lc long pore filling region and the permeability, Km, of the Lm long uninvaded region as

Rc(t) and Rm are the resistances of the pore filling and uninvaded regions defined by

The rate of increase of the filtration resistance of the pore filling particles is assumed proportional to the particle mass flux of the flowing phase according to:

kc is the pore filling particles resistance rate constant.The instantaneous porosity of a given cross-sectional area is given by:

(|)0 and (() denote the initial and instantaneous porosity values, e is the fractional bulk volume of porous media occupied by the deposited particles, given by

mp is the mass of particles retained per unit volume of porous media and pp is the particle grain density. For convenience, these quantities can be expressed in terms of initial and instantaneous open flow areas, Afo and Ay> and the area covered by the particle deposits, A , as

Substituting Eqs. 10-25 through 10-27, Eqs. 10-23 and 24 become, respectively

at themlet and outlet of the core. Wojtanowicz et al. (1987, 1988) omitted the accumulation of particles in the thin core slice and simplified Eq. 10-30 to express the concentration of particles leaving a thin section by:

The rate of particle retention on the pore surface is assumed proportional to the particle mass concentrations in the flowing phase according to:

The rate of entrainment of the surface deposited particles by the flowing phase is assumed proportional to the mass of particles available on the pore surface according to:

Then, the net rate of deposition is given as the difference between the retention and entrainment rates as:

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