Ideal gas in a deformable (fractured) formation

From (9.11) after substituting expressions for density of gas and for permeability of a formation we obtain:

—If——МГН1-;8 apJ41+—apw)5 -=

д. Ш 20j32JL v J 5/3 v wJ

, / ч (9.13)

| l,69Zwp Q2 Г1 П

2л h rw 120(1-h)2[rw re)

Filtration in non-uniform reservoirs

The permeability is not identical at various points of reservoirs. At small-scale chaotic change of filtration characteristics in a reservoir the latter is considered to be of average homogeneous permeability.

The reservoir is called macro non-uniform if its filtration characteristics (permeability, porosity) considerably, stepwise change in different areas.

We shall distinguish the following types of macro non-uniformity:

a) Layered non-uniformity (a multilayered reservoir), i.e. non-uniformity on thickness of a reservoir. In such a case, the reservoir’s thickness is subdivided into several layers or laminae (figure 9.1). Reservoir properties within each lamina are considered to be uniform and different from those in the adjacent laminae. Such reservoirs are also called thickness-non-uniform. The separation boundaries between the laminae of different permeabilities are considered plane. Thus, it is assumed in a layered non-uniform reservoir model that permeability, porosity, etc. change only from one lamina to the next and they are a piecewise-constant functions of the vertical coordinate.

It may be assumed that all laminae are separated by impermeable boundaries (the case of hydraulically isolated laminae). Otherwise crossflows between the laminae must be taken into consideration (the case of hydraulically communicating laminae). In the first case, it is possible to determine the filtration parameters using unidimensional flow schematics. In the second case, to calculate accurate values of filtration flow parameters one has to solve two-dimensional filtration problem.

In hydrodynamic calculations it is convenient to replace the equations of the fluid flow in a non-uniform bed with equations for a uniform bed of the same size with average value of permeability kavg. This average value of permeability may be determined from the condition of both reservoirs’ flow rates equality. It is easy to show that average permeability kavg is determined by the expression:

_ yi kfy

Kavg 2-4

here k? h. are the permeability and the effective thickness of ith lamina, and h is the effective thickness of the whole reservoir.

Figure 9.1. Rectilinear non-uniform flow in a non-uniformly stratified reservoir: 1 -p(x) for liquid, 2 -p(x) for gas

b) Zonal non-uniformity. Ln such a case, filtration properties change in the lamination plane, i.e., the reservoir includes several zones (areas of the reservoir). Reservoir properties within each zone are assumed to be identical and are assumed to change abruptly at the zones’ boundaries (figure 9.2).

Figure 9.2. Rectilinear-parallel flow of a liquid in zonally non-uniform reservoir

According to continuity equation mass flow rate is constant and is equal to:

at rectilinear - parallel flow:

G = Bh

at radial plane flow:

G = 2n: h - Фк~ Фс



here В is the width of a reservoir; li, ri is the length of the ith zone or its external radius (r=rw), i = 1..., n; n is the number of zones.

In practice of great importance is the case of flow to a well with a ring zone around wells’ bottom hole which differs in permeability from the permeability of a reservoir. This may be a result of torpedoing or acid treatment, or installation of gravel-packed filter, claying or wax precipitation at bottom hole zone, etc. Let’s estimate the influence of variation of permeability of bottom hole ring zones from permeability of other part of a reservoir on efficiency of a well. Calculations show that:

  • 1) it is intolerable to forecast future flow rate only on the basis of the data on permeability of the bottom hole zone (it is necessary to use quasi-uniform approximation);
  • 2) the decrease of permeability at the bottom hole zone leads to greater reduction of flow rate compared to the growth of the flow rate caused by the increase in permeability in this zone. If there will be any notable decline of permeability even in the small area of a reservoir adjoining a well then the well flow rate will sharply decrease;
  • 3) in the case of a filtration under Darcy’s law there is no point in increasing permeability at bottom hole zone more than 20 times, because the further increase in permeability practically does not lead to the flow rate growth;
  • 4) violation of Darcy’s law strengthens the positive influence of the increased permeability of the bottom hole zone on wells’ productivity.

Let’s give the main formulas for the case of two-zonal round reservoir. Let’s introduce the following notations: Rc is the reservoir contour radius, rw is the well radius, R1 is the radius of the near-bottom hole zone, k{ is the permeability of the near-bottom hole zone, k2 is the permeability of the rest of reservoir, pc is the pressure at the charge contour, pw is the pressure at the bottom hole of the well, ц is fluid viscosity. The formulas to calculate the main parameters are as follows:

Average permeability is determined by the formula:

k ________1п(Л /4 )_______

1 i


Pressure distribution formula:

in two-zone reservoir is determined by the



xln— Д c

. ( 1 1 л 1 д' A; In - +—fil

P(r) =

ial rw k2 Д

____Pc Pw______

ll Л li Rc

In + In c

К rw ^2 R

Gradient of pressure in two-zone reservoir is determined by the formula:

  • ---—-—-----V X-, Гг
  • 1 1 *1 1 1 r

In 1 + In—

dp(r) dr

A A A)

----Pc~Pw----- xi , R

. , 1 . Rx 1 , r

k, —In 4 + In—


IA rw A A)

Filtration velocity is determined by the formula:

к dp w =—x—

Д r

Flow rate of the well is determined by the formula:

p 2*A Pc-Py p R Al A

Л A ''w

Well’s productivity index is determined by the formula:

Sample diagrams are given below (figures 9.3 - 9.7).

Lecture conclusion

The main issues considered in the lection are: an incompressible liquid in non-deformable formation, ideal gas in non-deformable formation, homogeneous incompressible liquid in deformable (fractured) formation, ideal gas in deformable (fractured) formation, filtration in non-uniform reservoirs. A formula for distribution of pressure in a formation for an incompressible liquid in a non-deformable formation is obtained. For the case of ideal gas flow in non-deformable formation, formulae for well flow rate and for the factors A and В determined in the course of steady-state wells testing are obtained. Also, filtration in non-uniform reservoirs is considered. Layered non-uniformity and zonal non-uniformity are studied.

Pressure gradient Pressure

Figure 9.3. Pressure vs distance from the center of the well

Figure 9.5. Filtration velocity vs distance from the center of the well

Figure 9.6. Flow rate vs kfk2 ratio


Figure 9.7. Map of isobars for two-zonal reservoir

Test questions

  • 1. Write the equation for the flow of an incompressible liquid in non-deformable formation for quadratic filtration law.
  • 2. Write the equation for the flow of an incompressible liquid in fractured deformable formation for quadratic filtration law.
  • 3. Write the equation for flow of the ideal gas in non-deformable formation for quadratic filtration law.
  • 4. Write the formula for volumetric flow rate of filtration of real gas in non-deformable formation for quadratic filtration law.
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