Since j2 < 1-0, the function / ( 4 ^ L)JAx6 is calculated from Eq.
3-53
with y yi.
From Eq. 3^9
Using Eq. 3-47
From Eq. 3-45, with A1 = 2yeh and A' = 0.750
A comparison of the horizontal well productivity obtained by three
methods is shown below:
Method 1 Jh = 1.644 stb/(day/psi)
Method 2 Jh = 1.638 stb/(day/psi)
Method 3 Jh = 1.472 stb/(day/psi)
3.3 Horizontal Oil Well Performance
During Transient State
Refs. 6, 13, 14 have presented the solution for a horizontal well in an
infinite reservoir (see Figure 3-16). The equation is given below:
(3-59)
where
(3-60)
(3-61)
(3-62)
(3-63)
(3-64)
Horizontal well
Figure 3-16. Horizontal well model.
(3-65)
(3-66)
(3-67)
zw = vertical distance measured from the bottom boundary of the pay
zone to the well.
Uniform-flux solution.
Infinite-conductivity solution.
Uniform-flux solutions with wellbore pressure averaging.
For an infinite-conductivity wellbore, Figure 3-17 is a log-log plot of PWD
versus tD with a dimensionless wellbore length, L 0 , as a parameter. Figure
3-17 shows that:
Bottom curve represents the pressure response of a vertical well with a
fully penetrating infinite-conductivity fracture.
Time between dashed lines AA and BB represents transitional flow
period from early-time radialflow(vertical radialflow)to pseudo-radial
flow.
Once the pseudo-radial flow starts, horizontal well solution for LD > 10
is practically the same as the vertically fractured well solution.
Dimensionless pressure, PWD
Start of pseudoradial
flow
End of initial radial
flow period
Tw 0 =IO" 4 , z WD = 0.5
Vertical fracture
solution Infinite-conductivity
Dimensionless time, tD
Figure 3-17. Pressure response of horizontal well.
3.4 Transient Well Testing Techniques
in Horizontal Oil Wells
Horizontal well testing is complex and on many occasions it is difficult
to interpret. In this chapter the limitations and use of horizontal well
testing are outlined. There are four transient flow regimes that are theore-
tically possible with a buildup or drawdown test in a horizontal well, which
are as follows.
Early-Time Radial Flow
The flow is radial and is equivalent to that of a fully penetrating vertical
well in an infinite reservoir. (See Figure 3-18.)
Intermediate-Time Linear Flow
A horizontal well will generally be long compared to the formation
thickness; a period of linear flow may develop once the pressure transient
reaches the upper and lower boundaries. (See Figure 3-19.)
Late-Time Radial Flow
If the horizontal well length is sufficiently small as compared to the
reservoir size, a second radial flow known as a pseudo-radial flow will
develop at late times. (See Figure 3-20.)
Horizontal Well
The pressure transient is
moving radially from the
wellbore and has not
encountered any boundaries
Figure 3-18. Early-time radial flow.
Horizontal Well
The duration of this second
major flow regime is directly
related to the effective length of
the horizontal well
Figure 3-19. Intermediate-time linear flow.
Pressure transient becomes
effectively radial in nature after
a long enough time.
Figure 3-20. Late-time radial flow (pseudo-radial flow).
Late-Time Linear Flow
This flow period occurs when the pressure transient reaches the lateral
extremities of the reservoir. This second and final linear flow period develops
only for reservoir of finite width. The identification of these flow regimes is
critical to the proper interpretation of a horizontal well test. (See Figure 3-21.)
Possible Flow Regimes and Analytical Solutions
Figures 3-18 through 3-21 show four possible transient flow regimes
depending on the well length relative to the reservoir thickness and drainage
area.5'12 Under certain circumstances, permeability, k, anisotropy, and skin
factors can be estimated by analyzing these transient flow pressure data.
Time and pressure response equations relating to each of theflowregimes to
solve specific reservoir parameters for drawdown and buildup tests can
This is the last major flow
regime; it is not commonly
seen in tests.
Figure 3-21. Late-time linear flow (pseudo-steady state).
be found in the next sections. Figure 3-22 shows transient flow regime and
analytical solutions.
3.5 Flow Time Equations and Solutions
These sets of equations are presented here for estimating the various flow
regimes based on the concepts of Goode and Thambynayagam,5 Odeh and
Babu,12and Joshi.18
Method 1 - Goode and Thambynayagam's Equations5
Early-Time Radial Flow
The early-time radial flow period ends at
(3-68)
Intermediate-Time Linear Flow
Intermediate-time linear flow is estimated to end at
(3-69)
The intermediate-time linear flow may not develop if the time estimated
from Eq. 3-69 is less than the time calculated for the early-time radial flow to
end (Eq. 3-68).
Tra nsie ntflow reg imes
and analytical solutions
Early-time radial Time to end the Early-time Pseudo-radial
flow early radial flow linear flow flow period
This flow regime This flow period If horizontal well If the well length is
can be short and ends when the is long enough sufficiently short to
may be difficult to effect of the top or compared to the reservior size, the
identify in field bottom boundary formation pseudo-steady state
applications is felt thickness, a will develop at late
period of linear times. The flow period
(Figure 3-18)
flow may develop, ends when the
once the pressure pressure transient
transient reaches reaches one of the
Buildup tests Drawdown tests the upper and outer boundaries
(P) (Pi)-(Pw/) lower boundaries (see Figure 3-20)
versus versus
log(tp +At)ZAt logt
Buildup tests Drawdown tests
(P) (Pi)-(P wf)
versus
\og(tp + Af)/At versus
logt
From slope, estimate (kjcy)05 and 5.
Parameter L(IcJc)0'5 can be estimated, if
reservoir isotropic kx=k =kh; then
Kff=(hKf;5-
One can estimate producing well length L, From slope estimate
if producing well length L is known from kh = (kjc/-5mds.
well logging Extrapolate initial reservior pressure, pt
Figure 3-22. Flow regimes and horizontal wellbore pressure responses during
flow period.
Late-Time Radial Flow or Pseudo-Radial Flow
If late-time radial flow or pseudo-radial flow develops, it will begin at
approximately
(3-70)
Ref. 5 suggested the following equation to determine the beginning of
pseudo-radial flow:
(3-71)
For a reservoir of finite width, this would end at
(3-72)
where
dz = distance from the upper reservoir boundary to the center of the
horizontal well, ft
kv = permeability in vertical direction, mD
L effective length of horizontal well, ft
kx permeability in x-direction, mD
Lx\ = distance in x-direction to beginning of horizontal wellbore, ft
Lxd = distance in x-direction to end of horizontal wellbore, ft.
This radialflowperiod will not develop if the estimated time at the end of
late-time radial flow (Eq. 3-72) is less than that calculated at the beginning
of pseudo-radial flow (Eq. 3-70). It also means that the reservoir is smaller
than anticipated. A plot of pwf versus yft can be used to calculate Lx\ and
Lxd in Eq. 3-72.
Method 2 - Odeh and Babu's Equations12
Early-Time Radial Flow
The duration of this period may be approximated by the minimum of the
following two terms:
(3-73)
(3-74)
Intermediate-Time Linear Flow
Time durations for the start and end of linear flow can be found by
(3-75)
and
(3-76)
Late-Time Radial Flow
This flow period starts at
(3-77)
and ends at minimum of
(3-78)
(3-79)
Late-Time Linear Flow
The flow ends at the maximum of
(3-80)
(3-81)
where
dz = the shortest distance between the well and the z-boundary, ft
D2= h dz; the longest distance between the well and the z-boundary, ft,
and h is the reservoir height
ky permeability in j-direction, mD
dx the shortest distance between the well and the x-boundary, ft
Dx the longest distance between the well and the x-boundary, ft.
Method 3 - Ozkan etal.'s Equations14
Early-Time Radial Flow
(3-82)
(3-83)
Assuming isotropic reservoir, i.e., kx ky, Eq. 3-83 reduces to
(3-84)
After estimating the value of LD and rwD, and using Figure 3-17, one can
find tD (dashed line AA) and duration of the early-time radial flow and is
given by
(3-85)
where
LD = dimensionless length
h = reservoir thickness, ft
I*WD = dimensionless radius
tD dimensionless time.
Late-Time Radial Flow
Start of this radial (pseudo-radial) flow can be calculated by using
Eqs. 3-82 and 3-83. Find tD from Figure 3-17 (dashed line BB) and then
substitute in Eq. 3-85.
