Flow of incompressible liquid in a fractured deformable formation

For the case of incompressible liquid, we have p = const, h = const, k = k°_f[l-p* (p_c-p)]³ and:

Ф ⁼~A---O⁷

4*7 Д

[l-)3’(p_c-p)]⁴+C.

After substituting these expressions into corresponding formulas, we obtain the basic dependences.

Distribution of pressure:

Р=Р.-^=Д (8.11)

where

A = 1--тг1п—, a,

Inr r

Figure 8.4. The curves of pressure distributions:

1 — in non-deformable porous formation; 2 - in fractured deformable formation

According to the formula (8.11) the cone of depression for deformable formation is more abrupt, than for non-deformable porous formation (figure 8.4). This confirms that in a deformable fractured formation due to reduction of fractures opening at reduction of rock pressure an additional filtration resistance arises causing sharp pressure decline on rather small distance from a well. The pressure in a formation with big /?* is more sharply reduced.

Depression cone in fractured deformable formation is steeper than in deformable porous formation.

Because of small values of/?* when formation pressure draw down is also small it is possible to accept that:

[1-/3'(р_е-р)]⁴~1-4Д'(р₍-р);

[l-^'(Pe-P.)T “1-4/3'(p_e-P.) •

Volumetric flow rate is equal to:

л hk°_f Q=±, -./-«2 , 2^ In r_c

(8-12)

where the sign before expression in the right part depends on the well type (“-“ is for injection well, “+” is for production well).

Figure 8.5. Indicator curve offiltration of an incompressible liquid in a fractured deformable formation

When /?* = 0, (i.e. the case of non-deformable formation) after evaluation of indeterminate forms in (8.12) we obtain Dupuit’s formula.

From the formula for volumetric flow rate (8.12) it follows, that inflow indicator curve is a parabola of the fourth order with the following coordinates of the top of the curve:

л hk°_f 1

_я./_^{; л}р-=-.

• (8.13)
• 2^ hi r_c p

The parabola passes through the beginning of coordinates, it is symmetric with respect to an axis parallel to flow rate axis; the second branch has no sense (figure 8.5). However, if we take into account real reservoir conditions (full closing of fractures does not occur, the factors connected to the change of characteristics of flow because of the change of fracture opening in the direction of a flow are not taken into account) then it is possible to speak only about the approximate fulfillment of extreme conditions of the curve (8.13).

Aggregate parameter /?* can be determined either by means of diagrams or by means of (8.12). In the latter case it is necessary to calculate the ratio (8.14) at two known values of Q_x and Q₂ and pressure draw down Ap_wl, p_w2

e₂ |-(1-д-_ДРв2)⁴'

(8.14)

After calculating /?* it is possible to determine permeability k*. The gradient of pressure will be determined as follows:

dp _a₂ 1

(815)

Filtration velocity of is equal to:

q/ ^k 1

^U~ /2лг hr~

(8.16)