The PORO software is based on principles similar to those reported by Abbaszadeh-Dehghani and Brigham [35, 36], but includes no regular and unbalanced injection patterns, anisotropic permeabilities and faults effects. The strategy is based on the computation of the streamlines and the numerical evaluation of the convection diffusion equations on each stream tube. In this process, the time is converted to frequency, employing Fourier transforms. All the numerical computations are made using FORTRAN.

It is considered that, in a mature secondary recovery project, only water is flowing and that the stationary state has been reached. Therefore, if there is a horizontal, homogeneous and non-isotropic layer, then the Darcy velocity components are [39]:

where

where

is the well index; is the number of wells; are the spatial coordinates;

Equation (60) was obtained from the non-isotropic version of the Laplace equation:

where X and Y are the spatial coordinates related to the principal axis.

When sealing faults are present, additional ‘image wells’ are included in Eq. (60) for confining the flux. The streamlines are computed from the Eq. (60), by solving:

(61)

where Ф is the porosity. The boundary conditions at the beginning of each streamline are: and, at the end of each streamline:

rwp=Vc^^WP^+c^-rWP)7, *=i,2,-5nsi (63)

In Eq. (62),

FIG. 103. Streamlines generated by PORO.

is the starting angle of the streamline (i), Nsl is the number of streamlines, and xwI and ywI are the spatial coordinates of the injection well. In Eq. (63), rwP is the radius of the production well, and xwP and ywP are its spatial coordinates.

For illustrating the results, Fig. 103 shows the obtained streamlines of a five spot pattern for an isotropic, balanced case (above), non-isotropic balanced case (half) and sealing fault case (under).

IV2.3.1. Tracer transport

On each streamline, the tracer transport problem can be considered as one dimensional. Hence, a new spatial coordinate, s, (along each streamline) must be defined, satisfying:

with the boundary condition: si(0) = 0.

As in the Abbaszadeh-Dehghani and Brigham model, the convection diffusion equation governs the tracer transport along each streamline: where aL is the longitudinal dispersivity.

By considering that C(s, m) is the Fourier transform of C(s, t):

(66)

and taking into account the fundamental property:

(67)

it is possible to write the Eq. 67 as:

Converting the spatial variable, s, in a discrete form:

Ct = C(tAs, w) ~ C(s, w)

dC(5, t) 1 л 1 1 1 (3C’ __ — — C(s + ^t)-—C(^і) ~ —C,+1 -~C, — I I (70)
and the second spatial derivative:
(71)
a*) (d? C+2 ~~Ds[ C+1+lb C J-«s) (£C+1 — iC> — iwf Ct — 0
(72)
and:
C (2v(s)a + v(s)Ds) ■ )+ — av(s) ■ Ci+2 ‘ av(s) + v(s)Ds — iwfDs 2 The input boundary condition (in the Fourier domain) is: C(0 , f( , V2Coe-mT/2 . wT C „ ) C(0,w) — f (w) — 0 sin— = C — f (w) w-jn 2 where C0 is the pulse height and T its lifetime. At the output, a ‘flow’ condition was imposed:
(73)
(74)
dC(L, w) 1 — kC( T 1 C 1 C 1 — kC ds — a C(L, w)~ DsCn Ds Cn-1 — a Cn-1
(75)
where N is the greatest value taken by the index, i.

a

By solving these equations, it is possible to obtain the C(L, ra) for each streamline. After this, by composing the individual responses and returning to the time domain, the program obtains the complete tracer response.

To illustrate the final result, Fig.104 shows the tracer records (expressed as daily fractional recovery) of a five spot pattern (for an isotropic balanced case). It can be seen how the layer thickness controls the breakthrough, the peak position and the broadness of the tracer records.

Dispersivity controls the breakthrough, the broadness and, to a slight extent, the peak position of the tracer records. The effect of anisotropy (for wells along the direction of Kmax) is opposite to that of the dispersivity. While increased dispersivity results in greater peak width, increased anisotropy reduces the peak width. Figure 105 illustrates the tracer records for an isotropic case and a non-isotropic case.

FIG 105. Daily fractional recovery of a five spot pattern (for a non-isotropic balanced case). Influence of Kma/Kmin-

Additionally, the presence of a sealing fault enhances the injection water support in the producers located in the same fault block as the injector, especially in the wells along the fault (e. g. P-1 in Fig. 106).

The effect on the daily tracer recovery is similar to that caused by anisotropy (on well P-1), giving earlier breakthrough and peak position, but reducing the peak width. Finally, sometimes the lack of tracer (or the scarce production of tracer) in a producer, may be the consequence of actions outside the involved injectors. As a consequence, the tracer daily fractional production in well P-1 is strongly reduced (Fig. 106).