Well Test Analysis in Oil Reservoirs With Gas Caps And/or Water Aquifers
Well Test Analysis in Oil Reservoirs With Gas Caps And/or Water Aquifers
Well Test Analysis in Oil Reservoirs With Gas Caps And/or Water Aquifers
SPE 19842
Well Test Analysis in Oil Reservoirs with Gas Caps and/or Water Aquifers
by A. J. AI-Khalifa, ARAMCO, and A. S. Odeh, MOBIL R&D
SPE Members
This paper was prepared for presentation at the 64th Annual Technical Conference and Exhibition of the Society of Petroleum Engineers held in San Antonio, TX, October 8-11, 1989.
This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper,
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Abstract -
This paper proposes a new solution for analyzing well tests run In practice, oil wells are completed quite apart from the gas and
in oil reservoirs underlain by water aquifers or overlain by gas water zones to prevent or at least delay the intrusion of unwanted
caps. fluids into the wellbore. Since oil is a slightly compressible
fluid, it normally takes a short period of time for the voidage to
The presence of a gas cap and/or a water aquifer alters the be felt at the gas-oil or oil-water interface. It is only during this
pressure response of wells penetrating the accompanying oil early time that gravity-segregated reservoirs, producing from the
sands. Only a single phase flows into the wellbore while a oil zone, behave in a manner similar to that of single phase
segregated multiphase flow predominates in the formation. systems. This early pressure response exhibits a short semilog
Neither single-phase nor multiphase conventional pressure straight line which is normally masked by wellbore effects.
transient solutions apply for such an environment. Type curves, Later on, the pressure starts to flatten indicating the support of
published in the literature, were developed assuming constant the gas cap and/or the bottom water. This is illustrated in Figure
pressure at the gas-oil or oil-water interface. For most practical 1 which shows the drawdown response of an oil producer under
cases, this assumption appears to distort the physics of flow and a gas cap support. Since the pressure drop was plotted against
results in non-accurate answers. the same scale, the late response appears to be heading toward a
constant pressure. In other words, the system is trending
In this study, a finely-gridded coning model was used to toward a steady state flow. This only occurs if one of the
simulate more than a hundred drawdown, buildup and injectivity boundaries is at constant pressure. This hypothesis led several
tests. In contrast to the constant-pressure assumption, the investigators (Strelstova-Adamsl,2, Buhidma and Raghavan3,
semilog plots of pressure vs time exhibited straight lines Chu et al4, Kuchuk et al5) to treat the gas-oil or oil-water
reflecting pseudo-radial flow across the entire formation interface as a constant pressure boundary. Such a treatment
thickness. The semilog slopes were then used to yield the total does not allow pseudo..radial flow (late semilog straight line) to
system transmissibility, (kh/J.J.) 1• develop, and more importantly,
the entire formation properties cannot be easily estimated.
This paper also explains a time-lapse technique which uses
successive tests run in the same system to track the location of Here, we investigate the theoretical basis of the constant-
the oil-water or gas-oil contacts with time. The same technique pressure assumption and show that it rarely applies in practice.
may also be used to obtain an estimate of the end point relative We also present a new solution to analyze the transient tests run
permeability in the swept zones.
Constant-Pressure Assumption
Introduction
In order to examine the applicability of the constant-pressure
With the exception of solution gas-drive reservoirs, phases are assumption, one needs to briefly review the coning theory6. A
normally segregated in petroleum reservoirs. Gravity forces drop of water at an oil-water interface is acted upon by two
cause gas to override and water to underride the oil phase. This forces. First is the decline in pressure below the outer-radius
may result in an initial formation of a gas cap and a bottom pressure due to the production sink in the wellbore. Second is
water. In dipping reservoirs, the flow between the wells may the differential hydrostatic head, due to the different densities of
also be segregated, even though a gas cap or a bottom water oil and water, which works against the former force. Whenever
might not be present initially. these two forces equilibrate, the water cone remains in a static
condition below the oil zone where the flow is occurring, Figure
References and illustrations at end of paper. 2. The pressure of the water zone, beneath the well, would then
SfE 19842
be equal to the outer-radius pressure (forming a constant-pressure The index p stands for penetrated, and sP is the skin due to
boundary beneath the oil zone). For such a condition to develop, partial penetration. The sP term can be computed using Odeh's
the well needs to produce below the critical rate? (i.e. the rate empirical relation 12 or Brons-Marting chartsl3. Both methods
beyond which the well starts producing water) for an excessive were derived for single-phase homogeneous systems, where the
period of time. Throughout this long period of time, the term f3 reduces to hpl h 1• Therefore, the term hpl hz in b?th
constant-pressure assumption does not hold. Nor does it hold solutions should be replaced by the term f3 defined by Equation
when the rate changes, until the two forces re-equilibrate and 4.
another static cone develops. In summary, the constant-pressure
assumption may only be valid under very restricted conditions
which may not be observed in the field. Numerical Model
-log[ (~l
0rw2 [{cth)o + {cth)w]j
l
+ 3.23 J (2)
The simulation was repeated three times for three different
discretization schemes in the vertical direction. The first system
was discretized into 20', 10', 10', 10' (top to bottom) for the oil
zone and 70' (one block) for the water zone, (Nz=5). The
second system was discretized into uniform grids of five feet
where s1 is the total wellbore skin. thick. Similarly, the third system was discretized into uniform
grids of only two feet thick. The semilog plots of the shut-in
The skin due to damage can be computed using the following
equation: pressure response vs (th+Llt)/Llt are shown in Figures 4 to 6.
The late response of Figure 6 is drawn on an enlarged scale,
Figure 7. The difference in the unit response slopes (slope/rate)
(3) is due to the significant effect of the different discretization on
the representation of the cone development, i.e. the pressure
where support (khlJ.L) of the water sand.
kh)
( Jl t
=(kh) + (kh) =162.6 X 2000 X 1.09 112888 md-ft
cp
Jl 0 Jl w 3.14
SE.E 1984 ?
This result is in good agreement with the input value of 114370 for the recent test,
md-ft/cp.
Time-Lapse Technigue
[(k:L (k:Ll.. +
8
test+ h [(~L- (~Ll (8)
162.6 qoBo)
(
= slope Recent test
Following, we explain a time-lapse technique which reconciles
the information obtained from a base transient test and a base
saturation log to interpret the results of a recent transient test. Subtracting Equation 7 from Equation 8 yields Equation 6.
The objective is to determine the location of the oil-water or gas-
oil contacts over the entire drainage area, rather than that
determined at the sandface using saturation logs. Or else, to Estimation of k'rw
determine the water endpoint relative permeability, k'rw, which
reflects field-scale heterogeneities, rather than that measured in In order to determine the endpoint relative permeability, one
the laboratories using a small piece of rock, i.e. cores. needs to run a recent saturation log and proceed as follows:
Hereafter we require that h 0 and hw be estimated using 1. Determine hoi and hwi from the base saturation log, also
saturation logs. Prior to logging, the well should be shut-in determine h0 and hw from the recent saturation log.
long enough for sandface saturation to reflect the average h0 and
hw over the entire drainage area This would minimize the effect 2. Use the estimated thickness of Step 1 in the following
of coning on the following procedures. equations:
Tracking Fluid Contacts
k (h)J.1 o,i + kw (h)J.1 w,i = (162.6slopeqoBo lEase
\
test
(9)
In order to determine the location of the fluid contacts, one needs and
to eitlier believe in the laboratory determination of k',w, or
otherwise, determine k'rw using well tests (as will be explained k (h)J.1
0 +
k
w
(h)J.1 _(162.6
w-
qoBo)
slope Recent test
(10)
later). Note that
3. Solve Equations 9 and 10 fork and kw
=1 for virgin aquifers, kw = ko = k
k'rw 4. Evaluate k'rw:
<1 for swept zones,
(11)
Following is a proposed procedure to determine the fluid
contacts:
1. Determine hoi and hwi using the base saturation log Conclusions
(i stands for initial condition).
1. The oil-water or gas-oil interface might behave as a constant-
2. Determine k using the semilog slope of the base pressure boundary under very restricted conditions which
transient test in the following equation: may not be observed in the field.
12. Odeh, A. S., "An Equation for Calculating Skin Factor Due
to Restricted Entry", Journal of Petroleum Technoloc:y,
(June 1980), pp. 964-965.
TABLE-2
Sw krw kro
0.1 o· 0.0 1. 0
TABLE-3
ho = 50'
hw = 70'
0 = 0.30
kh = 1000 md
kv = 1000 · md
rw = 0.24
Q = 15.6 X 10·6
Cw = 2.3 x 10·6
Pi = 2490 psi at the oil-water contact
tt = 9 hours
N
..::r
-
cc
Q\
FIGURE -1
L&.l
ei DRAWDOWN TEST, OIL GAS SYSTEM, hg/ h 0 = 0.162
2997
.
2996 -a
.I
l:l
2995-
·-D..
U)
. a·
ft
iD. 2994-
B
1:1
.
a
2993- El
1:]
. a .m
lEI lEI lEI lEI
2992 I
I I I I I . I I I I I I I I I - I I I I I I I I I I I I
.01 .1 1 10
t, DAYS
C\1
4'
co FIGURE- 2
...
a-
A STATIC CONE BENEATH AN OIL PRODUCER
.L&J
5J
I
I
I
I
I.
l I
. OIL
STATIC
CONE
CONSTANT PRESSURE
WATER
. .
N FIGURE- 3
~
co
: LATE RESPONSE OF FIGURE 1, ENLARGED SCALE .
Ul
3i 2993.2...-----------------------.
1!1
·c;; 2992.8
D.
iD.
2992.6
2992.4
2992.2 -1----..--....--,r---...---...---..---r-r-r------,.--~-r---r---,--.---r--r-t
.1 1 10
t, DAYS
C\1
FIGURE- 4
..:r
co
...
0'
UJ
THE SIMULATED BUILDUP, USING COARSE GRIDS
IN THE VERTICAL DIRECTION .
OJ
cn. 2478.6-.-----------------------~
w
§ 2478.0
en
en
w
a: 2477.8
D.
2477.6
1!1
2477.4
2477.2 -L--.-----.r----..---r--~--~---~.-----------.
10 1
.th + ~t
~t
N
FIGURE- 5
~
Q)
...
0\ THE SIMULATED BUILDUP, USING 5' UNIFORM. GRID
THICKNESS IN THE VERTICAL DIRECTION
51 2473.2-.----------------------------.
~ 2472.8
~
(/)
(/)
w
·a:
D. 2472.6
2472.4
2472.2 -t--...----.-......-----.~---.---...----.--------......-------.....iol
10 1
th + L\t .
.L\t
N FIGURE· 6
-4"
(()
...
o-
THE SIMULATED BUILDUP, USING 2' UNIFORM GRID
LLI
DJ. THICKNESS IN THE VERTICAL DIRECTION
2480,---------------------------------------------~
2380
[!]
2360 I I I I I I I I I I I I I I I I I I II I I I I I I I I I I I I I I I I I
LLI
THE SIMULATED BUILDUP, USING .2' UNIFORM GRID
THICKNESS IN THE VERTICAL DIRECTION, ENLARGED SCALE
»J
2472.0-.------------------------..
w 2471.2
a:
:l
(/)
(/)
w
~ 2470.8
2470.4
1!1
2470.0 L---....-..--~-~~-.,r------,r-----..,..--------t
10 1
th+~t
~t