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 R_c and on pressure from p up to p_c, we shall obtain the formula for distribution of pressure in a formation:

On , R_c 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 r_w (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[kpr^2Sl'

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

p² = p² + (L In—+-^p^-^-Q^{2 w} 7t kh r_w 17t²h²4k

(9-6)

Integrating the equation (9.5) in limits from p_c up to p_w and from R_c up to r_w, and neglecting 1/R_c in comparison to l/r_w we shall obtain the expression:

p² - p² = ^Pst QIn — + —Q²

• (9.7)
• 7t kh r_w 2n²h²r_wylk

or in the standard form:

Pe-p²_w=AQ_st+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-/n_z)&_z ’

l_bl 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}¹ ’ where

Фг = (^f-dp + C.

^J TJ

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

A ^G 1 ^rc G² Г 1 О

** 2nh г,⁺120») (1-т_/)(2ягй)Ч^г. ^rj

Using the formula for permeability in fractured formations we obtain:

2tj/3 G n hk°_f

In—

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