# Two-Phase Wellbore Mud Invasion and Filter Cake Formation Model

The following assumptions are made:

1 oil/water system,

2 aqueous phase density varies by the salt content,

3 radial and horizontal flow, and

4 homogeneous formation.

Civan (1994) and Ramakrishnan and Wilkinson (1997) modeled the radial invasion of water-based drilling muds into near-wellbore formation in oil reservoirs. Ramakrishnan and Wilkinson (1997) neglected the effects of the fluid compressibility, capillary pressure, gravity, salt dispersion, the porosity variation, and fine particle invasion. Therefore, the simplified model obtained under these conditions can be solved conveniently by the method of characteristics. Civan's (1994) model includes all of these effects. Therefore, it is much more complicated and requires a much more complicated numerical solution scheme. In the following, the formulation of the

filtrate invasion problem is presented by combining the features of the formulations given by Civan (1994) and Ramakrishnan and Wilkinson (1997). Ramakrishnan and Wilkinson (1997) consider a two-phase fluid system in the near wellbore formation. The aqueous phase forms the wetting phase, denoted by W, and the oleic phase forms the nonwetting phase, denoted by N. Neglecting the change of volume by mixing, they express the density of the aqueous phase as a volumetrically weighted average of the densities of the aqueous phase at two extreme cases, namely, the density of the saturated solution, Psw,, and the density of pure water, Pow. Hence,

in which a denotes the volume fraction of the saturated solution in the aqueous mixture. All dissolved ions are lumped together into a "salt" pseudo-component. Then, the mass concentration of the salt in the aqueous phase is given by:

The initial condition is given by:

Neglecting the molecular diffusion, the dispersion coefficient is given by:

where α and β are some empirical constants, and SWc and SNr denote the irreducible saturations of the aqueous and oleic phases, respectively. Thus, assuming that p^ is constant at reservoir conditions and substituting Eq. 18-35 into Eq. 18-36 yields:

Substituting Eq. 18-34 into 42 and applying Eq. 18-41 yields a volumetric water phase balance equation as:

The boundary conditions are given by:

The mass balance equation of the oleic phase is given by:

Assuming the nonwetting phase density is constant, Eq. 18-47 is simplified and then added to Eq. 18-43 to obtain a total volumetric balance equation as:

For convenience, the volumetric flux of the aqueous phase can be expressed in terms of the fractional flow function according to Buckley and Leverett as (Collins, 1961):

where the fractional flow function of the aqueous phase is given by (Richardson, 1961), neglecting the gravity term for horizontal, radial flow:

in which Fw is the zero capillary pressure and zero gravity fractional flow term, given by:

Substituting Eq. 18-49 for the total volumetric flux, and Darcy's equations for the aqueous and oleic phases into Eq. 18-48, results in the following equation for the pressure of the aqueous phase:

The rate of invasion and the pressure of the filtrate at the formation face can be estimated as following using a filtercake buildup model according to Civan (1994, 1998, 1999). Assuming an incompressible filtercake, the cake radius is given by (Civan, 1994, 1998, 1999):

In the field, usually the mud pressure, Pmud, is maintained constant. Thus, the filtrate invasion rate varies. Note that Donaldson and Chernoglazov (1987) and Civan and Engler (1994) used empirical correlations for the filtrate invasion rate. Whereas, the filtrate invasion rate can be estimated by means of Darcy's law assuming incompressible filtercake and constant viscosity filtrate, as following:

Application. Ramakrishnan and Wilkinson (1997) neglected the porosity variation and the capillary and gravity effects, and defined dimensionless distance and time, and normalized saturated solution volume fraction and saturation, respectively, as:

where Q(t) is the cumulative filtrate volume, rw is the wellbore radius, and σc and σf denote the saturated solution volume fractions of the connate and filtrate aqueous phases. In addition, a normalized saturation can be defined as:

Therefore, neglecting the capillary and gravity terms in Eq. 18-52 and applying Eqs. 18-51 through 57 into Eqs. 18-43 and 41, respectively, yields the following aqueous phase saturation and saturated solution concentration equations:

Ramakrishnan and Wilkinson (1997, 1999) have solved Eqs. 18-64 through 67 by applying the relative permeability functions given by Ramakrishnan and Wasan (1986). They considered that the filtrate invasion rate, q(t), is an unknown function, but it can be uniquely determined by an inverse problem approach if the resistivity of the near wellbore formation is measured as a function of the radial distance. However, the filtrate invasion rate can be directly predicted by applying Civan's (1994) formulation as following. Applying the same simplifying assumptions used in deriving Eqs. 18- 64 and 65 to Eq. 18-54 and Eq. 18-60 and PW =PN=P, yields the following simplified pressure equation:

Eqs. 18-68 through 71 can be solved by the finite difference method to obtain the near wellbore pressure profile and the filtrate invasion rate.

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