Leontaritis (1998) developed a simplified model for prediction of formation damage and productivity decline by asphaltene deposition in under saturated (above bubble-point pressure) asphaltenic oil reservoirs. This model consists of a set of algebraic equations. In this section, the Leontaritis model is presented with some modifications for consistency with the rest of the presentation of this artcle. As schematically shown in this article, for analysis, Leontaritis (1998) considers the portion of the reservoir defined by the radius of drainage of a production well. In this region, the flow is assumed radial. Schematically depicts the variation of the flowing bottom hole pressure during constant rate production, while the external reservoir pressure and the onset of the asphaltene flocculation pressure remain constant. The calculational steps of this model are described briefly in the following.

Step 1. The initiation time for asphaltene precipitation is referred to as zero (i.e., t = 0). Given the well productivity index, PI, the flowing bottom hole pressure, pw [=Q, prior to asphaltene damage is calculated from the definition of the productivity index:

Then, the steady-state radial pressure profile prior to damage is calculated by:

The asphaltene deposition is assumed to occur within the near wellbore region, rw<r<rAF, where the pressure is below the asphaltene flocculation pressure, pAF. The radius of this region, rAF, is determined by Eq. 14-16 for p = pAF. Leontaritis (1998) assumes that the pressure beyond this region (i.e., rAF ≤r≤re) is not influenced by asphaltene deposition in the near wellbore region. The region rw ≤ r ≤ rAF is divided into a number of sections of finite width Δr. Steps 2 and 3 calculations are carried out over each Δr segment for a time increment by Δt, consecutively, as described in the following.

Step 2. Similar to Wojtanowicz et al. (1987, 1988), Leontaritis considers the porous media as a bundle of tortuous flow tubes. Thus, the mean hydraulic diameter is estimated by the ratio of the total pore volume to the total pore surface area of the flow channels according to:

where A, L, and ɸ denote the cross-sectional area, length, and porosity of a core plug, and Ag and Vg are the mean surface area and the mean volume of the porous media grains. If the mean, spherical grain diameter is denoted by d , then Eq. 14-17 can be expressed as:

Next, the tube size distribution function, f(DA), the mole fraction, XA, and molar volume, VA , of the flocculated asphaltenes, and the moles of reservoir fluid, mRF, at the prevailing pressure and temperature conditions within the near wellbore region are determined according to Leontaritis (1997).

Leontaritis (1998) assumes that permeability impairment primarily occurs by pore throat plugging and generalizes the one-third rule-of-thumb of filtration as the particles larger than a certain fraction of the pore size cannot penetrate a filter, and determines the fraction of the particles, ftrap, which are captured and deposited at the pore throats. Thus, the rule-ofthumb for trapment of particles at the pore throats is generalized to estimate the critical particle diameter for plugging as a fraction of the hydraulic tube diameter as:

in which a is as an empirical factor accounting for the smallest particle that can be filtered. Its value is determined to match the model predictions to measured data, instead of setting it to a prescribed value, like 1/3. Then, the fraction of the asphaltene particles that cannot pass through the pore throats and, therefore, are captured at the pore throats is determined by:

Step 3. The incremental moles of asphaltene particles trapped and the incremental flow area closed within the A? time interval are estimated, respectively, by:

where aA is the specific surface of the asphaltene particles retainedin porous media, estimated as:

where DA is the mean diameter of the asphaltene particles retained, p is an empirical factor accounting for the plugging by asphaltene particles. Therefore, combining Eqs. 14-20 through 23 over a number of N consecutive, discrete time steps, Af, the cumulative flow area closed to flow by pore throat plugging is estimated by:

Hence, the area open to flow during damage is given by:

According to Wojtanowicz et al. (1987, 1988), the area open to flow during formation damage by pore throat plugging is linearly related to permeability:

Therefore, the instantaneous permeability is given by:

in which the productivity index is defined by:

Hence, combining Eqs. 14-25 and 30-32 yields the following expression for the productivity ratio:

Note that Eq. 14-33 is different than the equation given by Leontaritis (1998) because Eq. 14-33 is squared. The pressure loss by damage is calculated by Eqs. 14-31 through 33 as:

Step 4. When Steps 2 and 3 over all the Δr segments are completed, the pressure loss by skin and the skin factor are calculated as following. Note that the drawdown pressure is given during damage as:

where s is the van Everdingen-Hurst skin factor. Thus, the loss of the pressure by the skin effect is given by:

Consequently, comparing Eqs. 14-16 and 36 in view of Eq. 14-37 yields:

Once the pressure loss by skin is calculated by Eq. 14-38, the skin factor can then be calculated by Eq. 14-37.

Step 5. Another time increment, Δt, is taken and Steps 2-4 are repeated until either the final time considered for the calculation is reached or the flow rate of production can no longer be kept constant, which is the condition imposed for the above described model. Leontaritis considers that a steady-state is attained when the deposition and erosion rates equal. Then, the asphaltene deposition stops and the area open to flow attains a certain minimum limit value. Because of the lack of a better asphaltene erosion theory, Leontaritis assumes that the area of flow can be empirically expressed as some fraction of the initial area. His equation can be expressed in terms of Eq. 14-31 as:

Based on Eqs. 14-40 and 41, it can be concluded that 0≤a≤l and 0≤b≤l. However, there is no clear evidence of the use of Eqs. 14-39 through 41 in his calculational procedure.

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