 # FILTRATION AT NONLINEAR LAWS

Lecture Outline

• 9.1. Incompressible liquid in a non-deformable formation
• 9.2. Ideal gas in a non-deformable formation
• 9.3. Homogeneous incompressible liquid in a deformable (fractured) formation
• 9.4. Ideal gas in a deformable (fractured) formation
• 9.5. Filtration in non-uniform reservoirs

In the region of Darcy’s law violation, filtration is described by the equation:

— = —u+bu2 (9.1)

dr к V f

where Ь = ррЦк.

## INCOMPRESSIBLE LIQUID IN A NON-DEFORMABLE FORMATION

Let’s express filtration velocity for the case of radial-plane flow with flow rate Q

u=Q / (2л r h).

In view of this expression the equation (9.1) will be written in the form:

dp _ Л Q ' K Q2 ---—---h Ь-------— . dr к 2 л rh (2nrh)

Separating variables and integrating on radius from r up to Rc and on pressure from p up to pc, we shall obtain the formula for distribution of pressure in a formation:

On , Rc Q2b fl H

p = p„--—In—---I---I

2nkh r (2nh) Л)

• (9-2)
• (93)

The curve of pressure distribution (9.2) is a hyperbole.

To express flow rate of the well we shall write:

On , R, Q2b ( 1 1 'I

c Pw 2M rw (2nh) w Rj

Flow rate is a positive root of quadratic equation (9.3).

## Ideal gas in a non-deformable formation

Let’s express velocity through the volumetric flow rate at standard pressure:

.. _ _ PstQst _ QstPst ZQ A

Pst

Substituting expression (9.4) in (9.1), we obtain:

— = Pst O + PstPstP 01 (Q О

dr Inkhpr^ 4„2 h2y[kpr2Sl'

Separating variables and integrating in the limits from p up to pw and from r up to rw, we have:

p2 = p2 + (L In—+-p^-^-Q2 w 7t kh rw 17t2h24k

(9-6)

Integrating the equation (9.5) in limits from pc up to pw and from Rc up to rw, and neglecting 1/Rc in comparison to l/rw we shall obtain the expression:

p2 - p2 = Pst QIn — + —Q2

• (9.7)
• 7t kh rw 2n2h2rwylk

or in the standard form:

Pe-p2w=AQst+BQ2. (9.8)

Factors A and В are determined according to wells testing at the steadystate regimes.

## Homogeneous incompressible liquid in a deformable (fractured) formation

For fractured mediums quadratic filtration law can be written in the form: ^- = ar1u + bpu2, (9.9)

where; r

a= 1/ ? /,=_______

/'kf’ 120(l-/nz)&z

lbl is the average linear size of block.

Multiplying (9.9) by density p after some algebra we shall obtain:

__G , 1,69/,, G2

dr In hr 12(h) (l-m/)(2^ hr}1where

Фг = (^f-dp + C.

J TJ

After separating variables and integrating (9.10) in the limits from rw up to rc and from up to фс, we shall obtain:

A G 1 rc G2 Г 1 О

** 2nh г,+120») (1-т/)(2ягй)Чг. rj

Using the formula for permeability in fractured formations we obtain:

2tj/3 G n hk°f

In—

• 1,69)3 lblp G2 Г 1 120^(1-/^)^ h)2[rw
• -1. (9-12) 