Ordinary Differential Equations
In this section, several examples are given to illustrate the numerical solution of ordinary differential equation models. Specifically, the simplified formation damage and filtration models, developed in previous articles, are solved.
Example 1: Wojtanowicz et al. Fines Migration Model
a. Derive a numerical solution for the following modified Wojtanowicz etal. (1987, 1988) fines migration model
b. Plot c and o versus t using the following data until ɸ = 0 :
A = 1cm2, L = 1cm, ɸ0 =0.20, q = 0.5cm3/min, Cin =0.85gr/cm3, p = 1.00 grl cm3, P= 1.00grlcm3, kr= 0.7mm-1, ke=0.2mm-1
Expanding Eq. 16-1 and then substituting Eqs. 16-2 and 3 and rearranging yields
A simultaneous solution of Eqs. 16-2 and 5 as a function of time, subject to the initial conditions given by Eq. 16-4, can be readily obtained using an appropriate method, such as by the Runge-Kutta-Fehlberg four (five) method available in many ordinary differential equation solving software (IMSL, 1987, for example). Then, the porosity variation is calculated by Eq. 16-3. A typical numerical solution is presented in Figure 16-1.
Example 2: Ceriiansky and Siroky Fines Migration Model
The numerical solution is carried out for T^. =0. Here, the numerical solution approach presented by Cernansky and Siroky (1985) is described. Define the dimensionless time and distance, respectively, by:
Thus, invoking Eqs. 16-6 and 7, Eqs. 10-84 and 91, respectively, become:
Eqs. 16-8 and 9 are a system of hyperbolic partial differential equations, which can be transformed into a system of ordinary differential equations by means of the method of characteristics as:
Applying the condition given by Eq. 16-15, Eq. 16-10 becomes:
The system of ordinary differential equations given by Eqs. 16-10 and 11 are solved by means of the fourth-order Runge-Kutta method, subject to the conditions given by Eqs. 16-15 and 16 along the characteristic represented by Eq. 16-14. Figure 16-2 shows the dimensionless effluent particles concentration as a function of the cumulative volume injected per unit area. Figures 16-3 and 16-4 show typical suspended particle concentration and the particles retained in porous media as a function of distance along the porous media at different times.
Example 3: Civan's Incompressive Cake Filtration without Fines Invasion Model
The equations of Civan's (1998, 1999) incompressive cake filtration model are given in this article. The ordinary differential equations of this model have been solved by the Runge-Kutta- Fehlberg four (five) numerical scheme (Fehlberg, 1969), subject to the initial condition given by Eq. 12-14.
Example 4: Civan's Compressive Cake Filtration Including Fines Invasion Model
The equations of Civan's (1998, 1999) compressive cake filtration including fines invasion model are given in this article. As described in
The ordinary differential equations of this model have been solved by the Runge-Kutta-Fehlberg four (five) numerical scheme (Fehlberg, 1969), subject to the initial condition given by Eq. 12-14.
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