# Flow of ideal gas in a non-deformable formation

For the considered case of flow we have *к = const, rf = const, p =p _{st}pl p_{st}* so

*ф =*

^{kPs,}-p^{2}+C trip/After substituting these expressions to the corresponding formulas, we have distribution of pressure:

_{2 2} Лр2 *r _{w}*

P = p„--^{2}=MnA

lnr_{w} *r* (8.17)

A _2 _2

AP_{C} =P_{e}-P_{w}-

It follows from comparison, that in the case of a gas flow, the pressure near the walls of well changes more sharply, than at a filtration of an incompressible liquid (figure 8.6). Piezometric curve for gas has, hence, more flat character (far from a well) on the greater extent, than a curve for an incompressible liquid, however it has a sharper bend at a wall of a well, than the bend of a curve for incompressible liquid. So, in the case of gas flow the pressure changes more sharply in the areas close to the well (figure 8.6) compared to the analogous case for incompressible liquid.

The equation for mass flow rate:

* _{G =} *nhkp

_{st}Ap;

*Л Pst ’*

*Figure 8.6. Distribution ofpressure at filtration of ideal gas (curve 1) and at filtration of incompressible liquid (curve 2)*

Using the previous formula and dividing both parts by *p _{st}* we can obtain an expression for volumetric flow rate at standard pressure:

*n hk* Лр2 *л* in/; ’

(8.18)

So, indicator curve for gas describes linear dependence of volumetric flow rate of the difference of squared pressure at charge contour and squared pressure at well bottom-hole (in distinction to indicator curve for incompressible liquid where we have linear relationship between volumetric flow rate and the difference between the pressures).

Using the following formula

*p ^{2}-p * = 2p p -(bp )^{2}*

we can obtain another expression for volumetric flow rate:

*Q _{st}* =«[2p

_{c}A p

_{c}-(A p

_{c})

^{2}]. (8.19)

*Figure 8.*7. *Indicator curve at a linear filtration of ideal gas*

For the case of filtration of ideal gas under Darcy’s law we have a parabola with an axis parallel to flow rate axis (figure 8.5). The branch of parabola drawn by dashed line has no any physical meaning.

Distribution of pressure gradient:

*dp _* 1 Ap_{c}^{2 }*dr 2r p* In/;

(8.20)

It is evident from (8.20) that pressure gradient grows steeply near bottom-hole zone because of lessening of r and because of pressure drop due to gas compressibility.

It is easy to obtain the following formula for filtration velocity:

- (8-21)
- 2r
*pt]*In/;

It is clear from (8.20) that filtration velocity weakly varies far apart from the center of the well and, like pressure gradient, grows steeply near the bottom-hole zone.

# Real gas filtration in a non-deformable formation

We have to account for real properties of gas when reservoir pressure *p _{res}* >10 MPa and

*p*0,9.

_{w}/p_{c}<We assume that *к = const.* Real gas equation at isothermal flow is as follows:

р =-^— *z(p)*

The potential function looks like:

*ф _{=} ^{к}Р^. v__{+c} , P_{std}*

^{2}n

*z*

where *z = (z _{w}+z_{c}) I* 2;

*rj =*/ 2;

*z*= z(pj,

_{c}*r*

_{c}=rj(p_{w}), z_{c}= z(p_{c}),*V _{c} =^(P_{C})-*

In view of expression for potential function transiting from mass flow rate to volumetric flow rate we shall obtain the equation of inflow:

* _{n} _ nhk* Ap*

(8.22)

*П z* hir_{w}

**Lecture conclusion**

The main issues considered in the lection are: flow of incompressible liquid in a non-deformable formation, flow of incompressible liquid in a fractured deformable formation, flow of ideal gas in a non-deformable formation, filtration of real gas in a non-deformable formation

In the lecture, each type of the flows indicated above is thoroughly analyzed. Indicator curves, pressure distribution, formulas for filtration velocity, volumetric and mass flow rates, description of isobars allocation are presented.

**Test questions**

- 1. Write the equation for flow of an incompressible liquid through non-deformable formation.
- 2. Write the equation for flow of an incompressible liquid in fractured deformable formation.
- 3. Write the equation for flow of the ideal gas through non-deformable formation.
- 4. Write the formula for volumetric flow rate of real gas filtration of real gas in not deformable formation.