-
Chapter 40
Estimation of Oil and Gas Reserves Forrest A. Garb, SPE, H.J.
Grt~y & Assocs. Inc.* Gerry L. Smith ,** H.J. Gruy 6i Asaoca.
Inc.
Estimating Reserves General Discussion Managements decisions are
dictated by the anticipated results from an investment. In the case
of oil and gas, the petroleum engineer compares the estimated costs
in terms of dollars for some investment opportunity vs. the cash
flow resulting from production of barrels of oil or cubic feet of
gas. This analysis may be used in formulat- ing policies for (1)
exploring and developing oil and gas properties; (2) designing and
constructing plants, gather- ing systems, and other surface
facilities; (3) determining the division of ownership in unitized
projects; (4) deter- mining the fair market value of a property to
be bought or sold: (5) determining the collateral value of
producing properties for loans; (6) establishing sales contracts,
rates, and prices; and (7) obtaining Security and Exchange Com-
mission (SEC) or other regulatory body approvals.
Reserve estimates are just what they are called- estimates. As
with any estimate, they can be no better than the available data on
which they are based and are sub- ject to the experience of the
estimator. Unfortunately, reliable reserve figures are most needed
during the early stages of a project, when only a minimum amount of
in- formation is available. Because the information base is
cumulative during the life of a property, the reservoir en- gineer
has an increasing amount of data to work with as a project matures,
and this increase in data not only changes the procedures for
estimating reserves but, cor- respondingly, improves the confidence
in the estimates. Reserves are frequently estimated (1) before
drilling or any subsurface development, (2) during the development
drilling of the field, (3) after some performance data are
available, and (4) after performance trends are well es- tablished.
Fig. 40.1 demonstrates (I) the various periods in the life of an
imaginary oil property, (2) the sequence
of appropriate recovery estimating methods, (3) the im- pact on
the range of recovery estimates that usually re- sults as a
property ages and more data become available, (4) a hypothetical
production profile, and (5) the relative risk in using the recovery
estimates. Time is shown on the horizontal axis. No particular
units are used in this chart, and it is not drawn to any specific
scale. Note that while the ultimate recovery estimates may become
ac- curate at some point in the late life of a reservoir, the
reserve estimate at that time may still have significant risk.
During the last week of production. if one projects a reserve of 1
bbl and 2 bbl are produced, the reserve esti- mate was 100% in
error.
Reserve estimating methods are usually categorized into three
families: analogy, volumetric, and performance techniques. The
performance-technique methods usually are subdivided into
simulation studies, material-balance calculations, and
decline-trend analyses. The relative peri- ods of application for
these techniques are shown in Fig. 40.1. .2 During Period AB,
before any wells are drilled on the property, any recovery
estimates will be of a very general nature based on experience from
similar pools or wells in the same area. Thus, reserve estimates
during this period are established by analogy to other produc- tion
and usually are expressed in barrels per acre.
The second period, Period BC, follows after one or more wells
are drilled and found productive. The well logs provide subsurface
information, which allows an acreage and thickness assignment or a
geologic interpre- tation of the reservoir. The acre-foot volume
considered to hold hydrocarbons, the calculated oil or gas in place
per acre-foot, and a recovery factor allow closer limits for the
recovery estimates than were possible by analogy alone. Data
included in a volumetric analysis may include well logs, core
analysis data, bottomhole sample infor- mation, and subsurface
mapping. Interpretation of these
-
40-2
Fig. 4&l-Range in estimates of ultimate recovery during life
of reservoir.
data. along with observed pressure behavior during ear- ly
production periods, may also indicate the type of producing
mechanism to be expected for the reservoir.
The third period, Period CD, represents the period af- ter
delineation of the reservoir. At this time, performance data
usually are adequate to allow derivation of reserve estimates by
use of numerical simulation model studies. Model studies can yield
very useful reserve estimates for a spectrum of operating options
if sufficient information is available to describe the geometry of
the reservoir, any spatial distribution of the rock and fluid
characteristics, and the reservoir producing mechanism. Because
numer- ical simulators depend on matching history for calibra- tion
to ensure that the model is representative of the actual reservoir,
numerical simulation models performed in the early life of a
reservoir may not be considered to have high confidence.
During Period DE, as performance data mature, the
material-balance method may be implemented to check the previous
estimates of hydrocarbons initially in place. The pressure behavior
studied through the material- balance calculations may also offer
valuable clues regard- ing the type of production mechanism
existent in the reser- voir. Confidence in the material-balance
calculations
PETROLEUM ENGINEERING HANDBOOK
depends on the precision of the reservoir pressures record- ed
for the reservoir and the engineers ability to deter- mine the true
average pressure at the dates of study. Frequent pressure surveys
taken with precision instru- ments have enabled good calculations
after no more than 5 or 6 % of the hydrocarbons in place have been
produced.
Reserve estimates based on extrapolation of established
performance trends, such as during Period DEF, are con- sidered the
estimates of highest confidence.
In reviewing the histories of reserve estimates over an extended
period of time in many different fields, it seems to be a common
experience that the very prolific fields (such as East Texas,
Oklahoma City, Yates, or Redwater) have been generally
underestimated during the early barrels-per-acre-foot period
compared with their later performance, while the poorer ones (such
as West Ed- mond and Spraberry) usually are overestimated during
their early stages.
It should be emphasized that, as in all estimates, the accuracy
of the results cannot be expected to exceed the limitations imposed
by inaccuracies in the available ba- sic data. The better and more
complete the available data, the more reliable will be the end
result. In cases where property values are involved, additional
investment in ac- quiring good basic data during the early stages
pays off later. With good basic data available, the engineer making
the estimate naturally feels more sure of his results and will be
less inclined to the cautious conservatism that often creeps in
when many of the basic parameters are based on guesswork only.
Generally, all possible approaches should be explored in making
reserve estimates and all applicable methods used. In doing this,
the experience and judgment of the evaluator are an intangible
quality, which is of great importance.
The probable error in the total reserves estimated by
experienced engineers for a number of properties dimin- ishes
rapidly as the number of individual properties in- creases. Whereas
substantial differences between independent estimates made by
different estimators for a single property are not uncommon,
chances are that the total of such estimates for a large group of
properties or an entire company will be surprisingly close.
Petroleum Reserves-Definitions and Nomenclature3 Definitions for
three generally recognized reserve categories, proved, probable,
and possible, which are used to reflect degrees of uncertainty in
the reserve estimates, are listed as follows. The proved reserve
definition was developed by a joint committee of the SPE, American
Assn. of Petroleum Geologists (AAPG), and American Petroleum Inst.
(API) members and is consistent with current DOE and SEC
definitions. The joint committees proved reserve definitions,
support- ing discussion, and glossary of terms, are quoted as fol-
lows. The probable and possible reserve definitions enjoy no such
official sanction at the present time but are be- lieved to reflect
current industry usage correctly.
Proved Reserves Definitions3 The following is reprinted from the
Journal of Petrole- UM Technology (Nov. 1981, Pages 2113-14) proved
reserve definitions, discussion, and glossary of terms.
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ESTIMATION OF OIL AND GAS RESERVES 40-3
Proved Reserves. Proved reserves of crude oil, natural gas, or
natural gas liquids are estimated quantities that geological and
engineering data demonstrate with reasona- ble certainty to be
recoverable in the future from known reservoirs under existing
economic conditions.*
Discussion. Reservoirs are considered proved if economic
producibility is supported by actual production or forma- tion
tests or if core analysis and/or log interpretation dem- onstrates
economic producibility with reasonable certainty. The area of a
reservoir considered proved in- cludes (1) that portion delineated
by drilling and defined by fluid contacts, if any, and (2) the
adjoining portions not yet drilled that can be reasonably judged as
economi- cally productive on the basis of available geological and
engineering data. In the absence of data on fluid contacts, the
lowest known structural occurrence of hydrocarbons controls the
lower proved limit of the reservoir. Proved reserves are estimates
of hydrocarbons to be recovered from a given date forward. They are
expected to be re- vised as hydrocarbons are produced and
additional data become available.
Proved natural gas reserves comprise nonassociated gas and
associated/dissolved gas. An appropriate reduction in gas reserves
is required for the expected removal of natural gas liquids and the
exclusion of nonhydrocarbon
Glossary of Terms Crude Oil Crude oil is defined technically as
a mixture of hydrocar- bons that existed in the liquid phase in
natural underground reservoirs and remains liquid at atmospheric
pressure af- ter passing through surface separating facilities. For
statistical purposes, volumes reported as crude oil include: (1)
liquids technically defined as crude oil; (2) small amounts of
hydrocarbons that existed in the gaseous phase in natural
underground reservoirs but are liquid at at- mospheric pressure
after being recovered from oilwell (casinghead) gas in lease
separators*; and (3) small amounts of nonhydrocarbons produced with
the oil.
Natural Gas Natural gas is a mixture of hydrocarbons and varying
quantities of nonhydrocarbons that exists either in the gaseous
phase or in solution with crude oil in natural underground
reservoirs. Natural gas may be subclassi- fied as follows.
Associated Gas. Natural gas, commonly known as gas- cap gas,
that overlies and is in contact with crude oil in the reservoir.
**
gases if they occur in significant quantities. Reserves that can
be produced economically through
Dissolved Gas. Natural gas that is in solution with crude oil in
the reservoir.
the application of established improved recovery tech-
niques-are included in the proved classification when these
qualifications are met: (1) successful testing by a pilot
Nonassociated Gas. Natural gas in reservoirs that do not
project or the operation of an installed program in that contain
significant quantities of crude oil.
reservoir or one with similar rock and fluid properties pro-
Dissolved gas and associated gas may be produced con-
vides support for the engineering analysis on which the
currently from the same wellbore. In such situations, it
project or program was based, and (2) it is reasonably is not
feasible to measure the production of dissolved gas
certain the project will proceed. and associated gas separately;
therefore, production is
Reserves to be recovered by improved recovery tech- reported
under the heading of associated/dissolved or
niques that have yet to be established through repeated
casinghead gas. Reserves and productive capacity esti-
economically successful applications will be included in mates
for associated and dissolved gas also are reported
the proved category only after successful testing by a pi- as
totals for associated/dissolved gas combined.
lot project or after-the operation of an installed-p&g&~
in the reservoir provides support for the engineering anal- Natural
Gas Liquids ysis on which the project or program was based. Natural
gas liquids (NGLs) are those portions of reser-
Estimates of proved reserves do not include crude oil, voir gas
that are liquefied at the surface in lease separa- natural gas, or
natural gas liquids being held in under- tors, field facilities, or
gas processing plants. NGLs ground storage. include but are not
limited to ethane, propane, butanes,
pentanes, natural gasoline, and condensate.
Proved Developed Reserves. Proved developed reserves are a
subcategory of proved reserves. They are those reserves that can be
expected to be recovered through ex- isting wells (including
reserves behind pipe) with proved equipment and operating methods.
Improved recovery reserves can be considered developed only after
an im- proved recovery project has been installed.
Reservoir A reservoir is a porous and permeable underground for-
mation containing an individual and separate natural ac- cumulation
of producible hydrocarbons (oil and/or gas) that is confined by
impermeable rock and/or water barri- ers and is characterized by a
single natural pressure system.
Proved Undeveloped Reserves. Proved undeveloped reserves are a
subcategory of proved reserves. They are those additional proved
reserves that are expected to be recovered from (I) future drilling
of wells, (2) deepen- ing of existing wells to a different
reservoir, or (3) the installation of an improved recovery
project.
From a technical standpoint, these hqulds are termed condensate,
however, they are commmgled wth Ihe crude stream and it IS
impractical to meawe and report their volumes separately All other
condensate IS reported as either lease condensate or plant
condensate and Included I natural gas l,q,ds
. Where resewar cond,,,ons are such lhat the production of
associated gas does not substantlallv affect the recwerv of crude
011 I the reser~oll. such aas rnav be reclassitledas nonassoclated
gis by a regulatory agency In this w&t, res&es and
producbon are reported I accordance wth the classlficatw used by
the regulatory agency Most reserve,, engmeers add the expression
considering current technology.
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40-4 PETROLEUM ENGINEERING HANDBOOK
OIL-WATER CONTACT -7450
Probable Reserves Probable reserves of crude oil, natural gas,
or natural gas liquids are estimated quantities that geological and
engi- neering data indicate are reasonably probable to be re-
covered in the future from known reservoirs under existing economic
conditions. Probable reserves have a
0 higher degree of uncertainty with regard to extent,
recoverability, or economic viability than do proved reserves.
Possible Reserves Possible reserves of crude oil, natural gas,
or natural gas liquids are estimated quantities that geological and
engi- neering data indicate are reasonably possible to be
recov-
Fig. 40.2-Geological map on top (-) and base (-7) of reservoir.
ered in the future from known reservoirs under existing economic
conditions. Possible reserves have a higher degree of uncertainty
than do proved or probable reserves.
In most situations, reservoirs are classified as oil reser-
voirs or as gas reservoirs by a regulatory agency. In the absence
of a regulatory authority, the classification is based on the
natural occurrence of the hydrocarbon in the reservoir as
determined by the operator.
Computation of Reservoir Volume4
Improved Recovery Improved recovery includes all methods for
supplement- ing natural reservoir forces and energy, or otherwise
in- creasing ultimate recovery from a reservoir. Such recovery
techniques include (1) pressure maintenance, (2) cycling, and (3)
secondary recovery in its original sense (i.e., fluid injection
applied relatively late in the produc- tive history of a reservoir
for the purpose of stimulating production after recovery by primary
methods of flow or artificial lift has approached an economic
limit). Improved recovery also includes the enhanced recovery
methods of thermal, chemical flooding, and the use of miscible and
immiscible displacement fluids.
When sufficient subsurface control is available, the oil- or
gas-bearing net pay volume of a reservoir may be com- puted in
several different ways.
1. From the subsurface data a geological map (Fig. 40.2) is
prepared, contoured on the subsea depth of the top of the sand
(solid lines), and on the subsea depth of the base of the sand
(dashed lines). The total area enclosed by each contour is then
planimetered and plotted as ab- scissa on an acre-feet diagram
(Fig. 40.3) vs. the corre- sponding subsea depth as the ordinate.
Gas/oil contacts (GOCs) and water/oil contacts (WOCs) as determined
from core, log, or test data are shown as horizontal lines.* After
the observed points are connected, the combined gross volume of
oil- and gas-bearing sand may be deter- mined by the following
methods.
lf working I Sl umls, the depths WIII be expressed in meters and
the planlmetered areas enclosed by each contour w,ll be expressed I
hectares The resultant hectare- meter plot can be treated exactly
llke the following acre-foot example to yield reserw~ ~oI!mes m
cubic meters. (1 ha, m = 10,000 m3 )
GROSS GAS BEARING SAND VOLUME:
[(0+8&42lt4(24)] ~2367 ACRE FEET
GAS-OIL CONTACT
GROSS OIL BEARING SAND VOLUME:
y [W-42+ 378 -242)+4(209-1061]=m ACRE FEET
OIL-WATER CONTACT
100 200 300 400 500 600 AREA ENCLOSED BY CONTOUR
Fig. 40.3-Acre-feet diagram
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ESTIMATION OF OIL AND GAS RESERVES 40-5
Fig. 40.4-lsopachous map-gas sand
a. Planimetered from the acre-feet diagram. b. If the number of
contour intervals is even, comput-
ed by Simpsons rule:
So/3[(0+136)+4(24+103)+2(46)]= 12,267 acre-ft.
(The separate calculations of the volume of gross gas- bearing
sand and gross oil-bearing sand by means of Simp- sons rule are
shown in the diagram of Fig. 40.3.)
r. With somewhat less accuracy, computed by the trapezoidal
rule:
SO[%(O+ 136)+(24+46+ 103)] = 12,050 acre-ft.
d. Computed by means of the somewhat more compli- cated
pyramidal rule:
ss[(O+136)+2(24+46+ 103)+J24x88 +m
+d5icEm-m-J]
= 11,963 acre-ft.
e. If the sand is ofuniform thickness, it will oftentimes
suffice to multiply the average gross pay thickness h I by the area
enclosed by the contour 1/2Z fi above the WOC.
J If the area within the top contour is circular (area A, height
Z), then the top volume is QrZ+ %AZ if treated as a segment of a
sphere, and %AZ if treated as a cone.
From a study of the individual well logs or core data, it is
then determined what fraction of the gross sand sec- tion is
expected to carry and to produce hydrocarbons.
Multiplication of this net-pay fraction by the gross sand volume
yields the net-pay volume. If, for example, in the case illustrated
with Figs. 40.2 and 40.3, it is found that 15% of the gross section
consisted of evenly distributed shale or dense impervious streaks,
the net gas- and oil- bearing pay volumes may be computed as,
respectively,
0.85 x2,367=2,012 net acre-ft of gas pay
and
0.85x9,900=8,415 net acre-ft of oil pay.
2. From individual well-log data, separate isopachous maps may
be prepared for the net gas pay (Fig. 40.4) or for the net pay
(Fig. 40.5) and the total net acre-feet of oil- or gas-bearing pay
computed as under It&m la, b, or c.
3. If the nature of the porosity varies substantially from well
to well, and if good log and core-analysis data are
Fig. 40.5-lsopachous map-oil sand
available on many wells, it is sometimes justified to pre- pare
an isopachous map of the number of porosity feet (porosity fraction
times net pay in feet) and compute the total available void space
in the net-pay section from such an isopachous map by the methods
discussed under Item la, b, or c.
Computation of Oil or Gas in Place Volumetric Method If the size
of the reservoir, its lithologic characteristics, and the
properties of the reservoir fluids are known, the amount of oil or
gas initially in place may be calculated with the following
formulas:
Free Gas in Gas Reservoir or Gas Cap (no residual oil present).
For standard cubic feet of free gas,
GFj = 43,5601/,@(1 -Siw)
, . (1) *,
where V, = net pay volume of the free-gas-bearing
portion of a reservoir, acre-ft, 4 = effective porosity,
fraction,
S;, = interstitial water saturation, fraction, B, = gas FVF,
dimensionless, and
43,560 = number of cubic feet per acre-foot.
Values for the gas FVF or the reciprocal gas FVF, l/B,, may be
estimated for various combinations of pres- sure, temperature, and
gas gravity (see section on gas FVF).
Oil in Reservoir (no free gas present in oil-saturated portion).
For stock-tank barrels of oil,
N= 7,758V,4(1 -S,,) , . . . . . . . . . B, (2)
where N = reservoir oil initially in place, STB,
V, = net pay volume of the oil-bearing portion of a reservoir,
acre-ft,
B, = oil FVF, dimensionless, and 7,758 = number of barrels per
acre-foot.
Refer ,oChaps. 20 through 25 for delaled coverage of 011. gas,
condensate and watel properties. and correlalions.
-
40-6
B 0 1.0
1.5
2.0
3.0
PETROLEUM ENGINEERING HANDBOOK
TABLE 40.1--BARRELS OF STOCK-TANK OIL IN PLACE PER ACRE-FT
S iw 0.10 0.20 0.30 0.40 0.50 0.10 0.20 0 30 0.40 0.50 0.10 0 20
0.30 0.40 0.50 0.10 0.20 0.30 0.40 0.50
0.05 0.10 0.15 0.20 0.25 0.30 0.35 349 698 1,047 1,396 1,746
2,095 2,444 310 621 931 1,241 1,552 1,862 2,172 272 543 615 1,066
1.358 1,629 1,901 233 465 698 931 1.164 1,396 1,629 194 388 582 776
970 1,164 1,358 233 465 698 931 1,164 1,396 1,629 206 411 617 822
1,028 1,234 1,439 182 365 547 729 912 1,094 1,276 155 310 465 621
776 931 1,086 128 256 384 512 640 768 896 175 349 524 698 873 1,047
1,222 155 310 465 621 776 931 1,086 136 272 407 543 679 815 950 116
233 349 465 582 698 815
97 194 291 388 485 582 679 116 233 349 465 582 698 815 105 209
314 419 524 628 733
89 178 268 357 446 535 625 78 155 233 310 388 465 543 66 132 198
264 330 396 462
Table 40.1 shows the number of barrels of stock-tank oil per
acre-foot for different values of porosity, 4, intersti- tial water
saturation, S,,,., and the oil FVF, B,,
Solution Gas in Oil Reservoir (no free gas present). For
standard cubic feet of solution gas,
Gs = 7,7581/,@(1 -s,,.)R.,
. . (3) Bo
where G, is the solution gas in place, in standard cubic feet,
and R,T is the solution GOR, in standard cubic feet per stock-tank
barrel.
Material-Balance Method5-8 In the absence of reliable volumetric
data or as an indepen- dent check on volumetric estimates, the
amount of oil or gas in place in a reservoir may sometimes be
computed by the material-balance method.5 This method is based on
the premise that the PV of a reservoir remains con- stant or
changes in a predictable manner with the reser- voir pressure when
oil, gas, and/or water are produced. This makes it possible to
equate the expansion of the reser- voir fluids upon pressure drop
to the reservoir voidage caused by the withdrawal of oil, gas, and
water minus the water influx. Successful application of this method
re- quires an accurate history of the average pressure of the
reservoir, as well as reliable oil-, gas-, and water- production
data and PVT data on the reservoir fluids. Generally, from 5 to 10%
of the oil or gas originally in place must be withdrawn before
significant results can be expected. Without very accurate
performance and PVT data the results from such a computation may be
quite erratic, 6 especially when there are unknowns other than the
amount of oil in place, such as the size of a free-gas cap, or when
a water drive is present.
When the number of available equations exceeds the number of
such unknowns, the solution should prefera- bly be by means of the
method of least squares. Be- cause of the sensitivity of the
material-balance equation
Porositv. d
to small changes in the two-phase FVF, B,, an adjust- ment
procedure, called the Y method, may be used for the pressure range
immediately below the bubblepoint. The method consists of plotting
values of
y= (Ph-PRPoi pR(B,-B,,i) , . . . . . . . . . . . . . . .(4)
where ph = bubblepoint pressure, psia, pR = reservoir pressure,
psia, B, = two-phase FVF for oil, dimensionless, and
Boi = initial oil FVF, dimensionless.
vs. reservoir pressure, PR, and bringing a straight line through
the plotted points, with particular weight given to the more
accurate values away from the bubblepoint. This straight-line
relationship is then used to correct the previous values for Y,
from which the adjusted values for B, are computed. Values of B,
computed with this method for pressures substantially below the
bubblepoint should not be used if differential liberation is
assumed to represent reservoir producing conditions.
When an active water drive is present, the cumulative water
influx, W,, should be expressed in terms of the known pressure/time
history and a water drive constant, thus reducing this term to one
unknown. A completely worked-out example of the use of material
balance that uses this conversion and in which the amount of oil in
place is determined for a partial water drive reservoir where 36
pressure points and equations were available at a time when about 9
% of the oil in place had been pro- duced is given in Ref. 7.
The material-balance equation in its most general form reads
N= N,,[B,+O.l7XIB,(R,~-R,,)I-(W,,-~,,)
B,,, B,q B, rnB + B- -(m+ I) I -
&RR(.,+S,,,,!)
,q, 0, 1 -s,,, II . . . . . . . . . . . . . . . . . . . . . . .
..~.... (5)
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ESTIMATION OF OIL AND GAS RESERVES 40-7
TABLE 40.2-CLASSIFICATION OF MATERIAL-BALANCE EQUATIONS
Reservoir Type
Oil reservoir with gas cap and active water drive
Material-balance Equation Unknowns Equation
Np]B, +0.1781B,(R, -R,,)]-(W, - WP) N= N, W,, m 6
mB,,
Oil reservoir with gas cap; no active water drive (W, = 0)
Initially undersaturated oil reservoir with active water drive
(m =0):
1. Above bubblepoint
2. Below bubblepoint
lnltially undersaturated oil reservoir: no active water drive (m
= 0), (W, = 0):
1. Above bubblepoint
2. Below bubblemint
Gas reservoir with active water drive
Gas reservoir; no active water drive
we =O)
Np[B, +0.1781B,(Rp -I?,,)]+ w, N= N. m 7
ma,,
N,U we-WP
+APpRco) - ~
N= B,, 1 (1 -S,,)
APl(C, +c, -S,&, -c,)l
N= Npl~,+0.f781B,(R,-R,,)1-(W,-W,)
8, -60,
N, W, 8
9 N, W,
N,(l +W&J- F (1 -St,) 01 1
N= N 10 QJDR[c,+c,-S,,(c,-c,)l
N= NJ!3, + O.l781B,(R, -R,,)]+ W,
N 11 6, -go,
G= G,B, -5.615(W, - WP) G W, 12
B, --By
G= G,B, +5.615W,
G 13 6, -B,,
where N,, = R,, = R.,, = w,, = w,, =
Aj?R = B,pi =
III =
f =
c,, =
cumulative oil produced, STB, cumulative GOR, scf/STB. initial
solution GOR. scf/STB, cumulative water influx, bbl, cumulative
water produced, bbl, change in reservoir pressure, psi, initial gas
FVF. res cu ftiscf, ratio of initial reservoir free-gas volume
and initial reservoir-oil volume, compressibility of reservoir
rock, change in
PV per unit PV per psi, and compressibility of interstitial
water, psi -
When a free-gas cap is present, this equation may be simplified
to Eq. 6 of Table 40.2 by neglecting the reser- voir formation
compressibility cf and the interstitial water compressibility
c,,..
When such a reservoir has no active water drive (W,,=O), Eq. 7
results.
For initially undersaturated reservoirs (m = 0) below the
bubblepoint, Eqs. 6 and 7 reduce to Eqs. 9 and I I, de- pending on
whether an active water drive is present.
For initially undersaturated reservoirs (m=O) above the
bubblepoint, no free gas is present (R,) -R,yi =O). while B,
=Bo;+A~~c, (where c, is the compressibility of reservoir oil,
volume per psi), so that general Eq. 5 reduces to Eqs. 8 and 10,
depending on whether an ac- tive water drive is present.
For gas reservoirs the material-balance equation takes the form
of Eq. 12 or 13, depending on whether an ac- tive water drive is
present. The numerator on the right side in each case represents
the net reservoir voidage by production minus water influx, while
the denominator is the gas-expansion factor (BR -B,;) for the
reservoir.
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40-8 PETROLEUM ENGINEERING HANDBOOK
TABLE 40.3-CONDITIONS FOR UNIT-RECOVERY EQUATION. DEPLETION-TYPE
RESERVOIR
Reservoir pressure Interstitial water, @SW, bbllacre-ft Free
gas, &S,, bbllacre-ft Reservoir oil, bbllacre-ft
Initial Conditions
$58 0
7,758 $41 -s,,)
Ultimate Conditions
7pp58 7,758
7,758 $~(l - S,, -S,,)
Stock-tank oil, bbl/acre-ft 1-S
7,758 d2 BO,
7,758 4 1 -s&v -s,, B w
'SubstIMe 10 000 for the 7.758 constanf 11 c"b,c melers per
hectare.mefer IS used.
Saturated Depletion-Type Oil Reservoirs-Volumetric Methods
General Discussion Pools without an active water drive that
produce solely as the result of expansion of natural gas liberated
from solution in the oil are said to produce under a depletion
mechanism, also called an internal- or solution-gas drive. When a
free-gas cap is present, this mechanism may be supplemented by an
external or gas-cap drive (Page 40-13). When the reservoir
permeability is sufficiently high and the oil viscosity low, and
when the pay zone has sufficient dip or a high vertical
permeability, the deple- tion mechanism may be followed or
accompanied by gravity segregation (Page 40-14).
When a depletion-type reservoir is first opened to pro- duction,
its pores contain interstitial water and oil with gas in solution
under pressure. No free gas is assumed to be present in the oil
zone. The interstitial water is usual- ly not produced, and its
shrinkage upon pressure reduc- tion is negligible compared with
some of the other factors governing the depletion-type
recovery.
When this reservoir reaches the end of its primary producing
life, and disregarding the possibility of gas-cap drive or gravity
segregation, it will contain the same in- terstitial water as
before, together with residual oil under low pressure. The void
space vacated by the oil produced and by the shrinkage of the
remaining oil is now filled with gas liberated from the oil. During
the depletion proc- ess this gas space has increased gradually to a
maximum value at abandonment time. The amount of gas space thus
created is the key to the estimated ultimate recovery un- der a
depletion mechanism. It is reached when the pro- duced free GOR in
the reservoir, which changes according to the relative permeability
ratio relationship and the vis- cosities of oil and gas involved,
causes exhaustion of the available supply of gas in solution.
Unit-Recovery Equation The unit-recovery factor is the
theoretically possible ul- timate recovery in stock-tank barrels
from a homogene- ous unit volume of 1 acre-t? of pay produced by a
given mechanism under ideal conditions.
The unit-recovery equation for a saturated depletion- type
reservoir is equal to the stock-tank oil initially in place in
barrels per acre-foot at initial pressure pi minus the residual
stock-tank oil under abandonment pressure pi,, as shown in Table
40.3.
By difference, the unit recovery by depletion or solution-gas
drive is, in stock-tank barrels per acre-foot,
1 - S,M - s,, ' .'." B
(14) o(I
where S,, is the residual free-gas saturation under reser- voir
conditions at abandonment time, fraction, and B,, is the oil FVF at
abandonment, dimensionless. The key to the computation of unit
recovery by means of this equa- tion is an estimate of the residual
free-gas saturation S,, at the ultimate time. If a sufficiently
large number of ac- curate determinations of the oil and water
saturation on freshly recovered core samples is available, an
approxi- mation of S,, may be obtained by deducting the average
total saturation of oil plus water from unity. This method is based
on the assumption that the depletion process taking place within
the core on reduction of pressure by bringing it to the surface is
somewhat similar to the actu- al depletion process in the
reservoir. Possible loss of liq- uids from the core before analysis
may cause such a value for S,, to be too high. On the other hand,
the smaller amount of gas in solution in the residual oil left
after flush- ing by mud filtrate has a tendency to reduce the
residual free-gas saturation. Those using this method hope that
these two effects somewhat compensate for each other.
A typical S,, value for average consolidated sand, a medium
solution GOR of 400 to 500 cu ftibbl, and a crude-oil gravity of 30
to 4OAPI is 0.25.
Either a high degree of cementation, a high shale con- tent of
the sand, or a 50% reduction in solution GOR may cut this typical
S,, value by about 0.05, while a complete lack of cementation or
shaliness such as in clean, loose unconsolidated sands or a
doubling of the solution GOR may increase the S,, value by as much
as 0.10.
At the same time, the crude-oil gravity generally in- creases or
decreases the S,, value by about 0.01 for ev- ery 3API gravity.
Example Problem 1. A cemented sandstone reservoir has a porosity
$=0.13, an interstitial water content S,,,.=O.35, a solution GOR at
bubblepoint conditions, /?,I, =300 cu ftibbl, an initial oil FVF
B,,; = 1.20, an oil FVF at abandonment B,, = I .07, and a
stock-tank oil gravity of 40API. Based on the above considerations,
the higher-than-average oil gravity would just about off- set the
effect of the somewhat lower-than-average GOR. and the residual
free-gas saturation S,, after a 0.05 reduc- tion for the
cementation can therefore be estimated at 0.20.
-
ESTIMATION OF OIL AND GAS RESERVES 40-9
Solution. The unit recovery by depletion according to Eq. 14
would be
N,, =(7.758)(0.13) l-0.35 l-O.35 -0.20
1.07 >
= 122 STBiacre-ft [I57 m3/ha.mj.
where N,, is the unit recovery by depletion or solution- gas
drive, STB.
Muskats Method. 9 If the actual relationships between pressure
and oil-FVF B,, gas-FVF B,, gas-solubility in oil (solution GOR) R,
, oil viscosity p,), and gas viscosity ps are available from a PVT
analysis of the reservoir fluids, and if the relationship between
relative permea- bility ratio k,/k, and the total liquid
saturation, S,, is known for the reservoir rock under
consideration, the unit recovery by depletion can be arrived at by
a stepwise com- putation of the desaturation history directly from
the fol- lowing depletion equation in differential form:
As,, -1 APR
B, dR, S,,- +(I -s,, -s,,, )B,L!
d(liB,s) PL,, k,., dB,,
B,, kR -+s,,---
dlR I-,? k,,, BdrR
. ..t... .I..........,......... (15)
where S, = oil or condensate saturation under reservoir
conditions, fraction, PLO = reservoir oil viscosity, cp,
PLK = reservoir gas viscosity, cp, k, = relative permeability to
gas as a fraction of
absolute permeability, and k, = relative permeability to oil as
a fraction of
absolute permeability.
The individual computations are greatly facilitated by computing
and preparing in advance in graphical form the following groups of
terms, which are a function of pressure only,
and the relative permeability ratio k,ik,,, which is a function
of total liquid saturation S, only.
The accuracy of this type of calculation on a desk cal- culator
falls off rapidly if the pressure decrements chos- en are too
large, particularly during the final stages when the GOR is
increasingly rapidly.
With modern electronic computers, however, it is pos- sible to
use pressure decrements of IO psi or smaller, which makes a
satisfactory accuracy possible.
This stepwise solution of the depletion equation yields the
reservoir oil saturation S,, as a function of reservoir pressure
pR. The results may be converted into cumula- tive recovery per
acre-foot. In stock-tank barrels per acre-foot,
(16)
The results may be converted into cumulative recovery as a
fraction of the original oil in place (OOIP) by
L+L) (?c), .,....,....... N
(17)
while the GOR history, in standard cubic feet gasistock- tank
barrel, may be computed by
(18)
where R is the instantaneous producing GOR, in standard cubic
feet per stock-tank barrel, and the relative produc- tion rate in
barrels per day by
k o Poi PR . . . (19)
where 90 = kc, =
km = Poi = 40; =
oil-production rate, B/D, effective permeability to oil. md,
initial effective permeability to oil. md, initial reservoir oil
viscosity, cp, and initial oil-production rate, B/D.
It should be stressed that this method is based on the
assumption of uniform oil saturation in the whole reser- voir and
that the solution will therefore break down when there is
appreciable gas segregation in the formation. It is therefore
applicable only when permeabilities are rela- tively low.
Another limitation of this method as well as of the Tarn- er
method, discussed hereafter, is that no condensation of liquids
from the produced gas is assumed to take place in the tubing or in
the surface extraction equipment. It should therefore not be
applied to the high-temperature, high-GOR, and high-FVF volatile
oil reservoirs to be discussed later.
Tarners Method. Babson and Tarner have ad- vanced
trial-and-error-type computation methods for the desaturation
process that require a much smaller number of pressure increments
and can therefore be more readi- ly handled by a desk calculator.
Both methods are based on a simultaneous solution of the
material-balance equa- tion (Eq. 11) and the instantaneous GOR (Eq.
18).
Tarners method is the more straightforward of the two. The
procedure for the stepwise calculation of the cumula- tive oil
produced (N,,)I and the cumulative gas produced (Gp)* for a given
pressure drop from p I to p, is as follows.
-
40-l 0 PETROLEUM ENGINEERING HANDBOOK
TABLE 40.4-COMPUTED DEPLETION RECOVERY IN STBIACRE-FTIPERCENT
POROSITY FOR TYPICAL FORMATIONS
Solution GOR
(cu ftlbbl)
cRsb)
60
200
600
1,000
2,000
Oil Gravity, (OAPI)
-70
;z 50 15 30 50 15 30 50 30 50 50
Unconsolidated
7.2 12.0 19.2
7.0 11.6 19.4
7.6 10.5 15.0 12.3 12.0 10.6
1. Assume that during the pressure drop from p , to pl the
cumulative oil production increases from (N,) , to (N,,)* N, should
be set equal to zero at bubblepoint.
2. Compute the cumulative gas produced (G,,)z at pressure p2 by
means of the material-balance equation (Eq. 111, which for this
purpose-and assuming Wp =0-is rewritten in the following form:
(G,,h =(N,h(R,,):!=N (R.7, -R,\)-5.615
3. Compute the fractional total liquid saturation @,)I at
pressure p2 by means of
(s);=S;~+(l-s;,,J~[l-~]. .., . ..(21)
4. Determine the k,lk,, ratio corresponding to the to- tal
liquid saturation (S,), and compute the instantaneous GOR at p2 by
means of
R* =R,$ +ui15$+. . . . . . . (22) RPK ro
5. Compute the cumulative gas produced at pressure p2 by means
of
RI +R, (G,)2=(Gp)1+ 2 ---[VP)2 -VP) 11, . (23)
in which RI represents the instantaneous GOR comput- ed
previously at pressure p, .
Usually three judicious guesses are made for the value (N,) 2
and the corresponding values of (G,,) 2 computed by both Steps 2
and 5. When the values thus obtained for (G,) 2 are plotted vs. the
assumed values for (N,) 2 , the intersection of the curve
representing the results of Step 2 and the one representing Step 5
then indicates the cu- mulative gas and oil production that will
satisfy both equa- tions. In actual application, the method is
usually simplified further by equating the incremental gas pro-
duction (Gp)z -(G,) I) rather than (G,)Z itself. This
Sand or Sandstone Limestone, Dolomite or (S,, = 0.25) Chert (S,,
=0.15)
Consolidated Highly Cemented Vugular Fractured
4.9 1.4 2.6 0.4 8.5 4.9 6.3 18
13.9 9.5 11.8 5.1 4.6 1.8 2.6 0.5 7.9 4.4 5.8 1.5
13.7 9.2 11.4 4.4 4.8 2.5 3.3 0.9 6.5 3.6 4.7 (1.2) 9.7 5.8 7.2
(2.1) 7.6 4.5 5.4 (1.6) 7.2 4.1 4.8 (1.2) 6.4 4.0 (4.3) (1.5)
equality signifies that at each pressure step the cumula- tive
gas, as determined by the volumetric balance, is the same as the
quantity of gas produced from the reservoir, as controlled by the
relative permeability ratio of the rock, which in turn depends on
the total liquid saturation. Although the Tamer method was
originally designed for graphical interpolation, it also lends
itself well to auto- matic digital computers. The machine then
calculates the quantity of gas produced for increasing oil
withdrawals by both equations and subtracts the results of one from
the other. When the difference becomes negative, the machine stops
and the answer lies between the last and next to last oil
withdrawals.
Tarners method has been used occasionally to com- pute
recoveries of reservoirs with a free-gas cap or to evaluate the
possible results from injection of all or part of the produced gas.
When a free-gas cap is present, or when produced gas is being
reinjected, breakthrough of free gas into the oil-producing section
of the reservoir is likely to occur sooner or later, thus
invalidating the as- sumption of uniform oil saturation throughout
the produc- ing portion of the reservoir, on which the method is
based. Since such a breakthrough of free gas causes the instan-
taneous GOR (Eq. 18) as well as the entire computation method to
break down, the use of Tamers method in its original form for this
type of work is not recommended. It should also be used with
caution when appreciable gas segregation in an otherwise uniform
reservoir is expected.
Computed Depletion-Recovery Factors. Several investi- gators9,
12-14 have used the Muskat and Tarner methods to determine the
effects of different variables on the ulti- mate recovery under a
depletion mechanism. In one such attempt I2 the k,lk, relationships
for five different types of reservoir rock representing a range of
conditions for sands and sandstones and for limestones, dolomites,
and cherts were developed. These five types of reservoir rock were
assumed to be saturated under reservoir conditions with 25 %
interstitial water for sands and sandstones and 15 % for the
limestone group and with 12 synthetic crude- oil/gas mixtures
representing a range of crude-oil gravi- ties from 15 to 5OAPI and
gas solubilities from 60 to 2,000 cu ft/STB. Their production
performance and recovery factors to an abandonment pressure equal
to 10% of the bubblepoint pressure were then computed by means of
depletion (Eq. 15).
-
ESTIMATION OF OIL AND GAS RESERVES
10.0
z 2 1.0 e
= P
0.1
0.01
5 TOT PER
Notes: interstitial water is assumed to be 30% of pore space and
dead-
oil viscosity at reservoir temperature to be 2 cp. Equilibrium
gas saturation is assumed to be 5% of pore space. As here used
ultimate oil recovery is realized when the reser-
voir pressure has declined from the bubblepoint pressure to at-
mospheric pressure.
FVF units are reservoir barrels per barrel of residual oil.
Solution GOR units are standard cubic feet per barrel of
residual
oil. Example 1: Required: Ultimate recovery from a system
-having a bub-
blepoint pressure = 2,250 psia, FVF = 1.6, and a solution GOR.
Procedure: Starting at the left side of the chart, proceed
horizontally along the 2,250-psi line to FVF = 1.6. Now rise
verti- cally 10 the 1,300-scflbbl line. Then go horizontally and
read an ultimate recovery of 23.8%. Example 2: F)eqoired: Convert
the recovery figure determined in Exam-
ple 1 to tank oil recovered. Data requirements: Differential
liberation data given in Exam-
ple 1. Flash liberation data: bubblepoint pressure = 2,250 psia,
FVF = 1.485, FVF at atmospheric pressure = 1.080 for both flash and
differential liberation.
FORMATION VOLUME FACTOR
Procedure: Calculate the oil saturation at atmospheric pres-
sure by substituting differential liberation data in the equation
as follows:
Oil saturation at atmospheric pressure = 0.360. Next, substitute
the calculated value of oil saturation and the
flash liberation data into the previous equation and calculate
the ultimate oil recovery as a percentage of tank oil originally in
place.
N,, (ultimate oil recovery)=29.3% of tank oil originally in
place.
Fig. 40.6-Chart for estimating ultimate recovery from solution
gas-drive reservoirs.
These theoretical depletion-recovery factors, expressed as
barrels of stock-tank oil per percent porosity, will be found in
Table 40.4 for the different types of reservoir rocks, oil
gravities, and solution GORs assumed.
In cases where no detailed data are available concern- ing the
physical characteristics of the reservoir rock and its fluid
content, Table 40.4 has been found helpful in es- timating the
possible range of depletion-recovery factors. It may be noted that
the k,lk, relationship of the reser- voir rock is apparently the
most important single factor governing the recovery factor.
Unconsolidated intergranu- lar material seems to be the most
favorable, while in- creased cementation or consolidation tends to
affect recoveries unfavorably. Next in importance is crude-oil
gravity with viscosity as its corollary. Higher oil gravi-
ties and lower viscosities appear to improve the recov- ery. The
effect of GOR on recovery is less pronounced and shows no
consistent pattern. Apparently the benefi- cial effects of lower
viscosity and more effective gas sweep with higher GOR is in most
cases offset by the higher oil FVFs.
In general, these data seem to indicate a recovery range from
the poorest combinations of 1 to 2 bbl/acre-fi for each percent
porosity to the best combinations of 19 to 20 bbllacre-Mpercent
porosity. An overall average seems to be around 10
bbliacre-ftlpercent porosity.
It is also of interest to note that when the reservoir is about
two-thirds depleted, the pressure has usually dropped to about
one-half the value at bubblepoint.
-
40-12 PETROLEUM ENGINEERING HANDBOOK
In another attempt nine nomographs were developed, each for a
given combination of the k, lk ,.(, curve, dead- oil viscosity, and
interstitial water content. The nomo- graph for an average k,lk,
relationship, an interstitial water content of 0.30. and a dead-oil
viscosity of 2 cp is reproduced as Fig. 40.6. Instructions for its
use are shown opposite the figure.
The authors also introduced an interesting empirical
relationship between the relative permeability ratio k,/k,, the
equilibrium gas saturation S,,., the intersti- tial water
saturation S,,., and the oil saturation S,:
k ri: = i(O.0435 +0.4556E), k
. (24) t-0
where t;=(l -S,,.-S,, -S,)/(S, -0.25). A similar correlation I5
for sandstones that show a linear relation- ship between lip,
(where p,.=critical pressure) and saturation is
k rg (1 -S*)I[ 1 -@*)I] -= k ro
(s*)4 , . (25)
where effective saturation S*=S,I(l -Si,). This equa- tion
represents a useful expression for calculating rela- tive
permeability ratios in sandstone reservoirs for which an average
water saturation has been obtained by either electrical log or core
analysis.
API Estimation of Oil and Gas Reserves In a statistical study of
the actual performance of 80 so- lution gas-drive reservoirs, the
API Subcommittee on Recovery Efficiency I6 developed the following
equation for unit recovery (N,,) below the bubblepoint for solu-
tion gas-drive reservoirs, in stock-tank barrels per
acre-foot*:
N,, =3,*44 [ 44;,y 1.6 x (2-J o.0979
( > 0.1741
x(s, ,)O.3722x !k IM . . . . (26) Pa
where k = absolute permeability, darcies,
B ob = oil FVF at bubblepoint, RBLSTB, P,~ = oil viscosity at
bubblepoint, cp, Pa = abandonment pressure, psig, and pb =
bubblepoint pressure, psig.
The permeability distribution in most reservoirs is usually
sufficiently nonuniform in vertical and horizon- tal directions to
cause the foregoing depletion calculations on average material to
be fairly representative.
However, when distinct layers of high and low perme- ability,
separated by impervious strata, are known to be present, the
depletion process may advance more rapidly in high-permeability
strata than in low-permeability zones. In such cases separate
performance calculations should
be made for each permeability bank that is known to be
continuous and the results converted into rate/time curves for each
by combining Eqs. 16 and 19. The estimated ul- timate recovery will
then be based on a superposition of such rate/time curves for the
different zones.
If there is a wide divergence in permeabilities, one may find
that at a time when the combined rate for all zones has reached the
economic limit the more permeable banks will be depleted and have
yielded their full unit recovery while the pressure depletion and
the recovery from the tighter zones are still incomplete.
Undersaturated Oil Reservoirs Without Water Drive Above the
Bubblepoint- Volumetric Method t7-19 With progressively deeper
drilling, a number of oil reser- voirs have been encountered that,
while lacking an ac- tive water drive, are in undersaturated
condition. Because of the expansion of the reservoir fluids and the
compac- tion of the reservoir rock upon pressure reduction, sub-
stantial recoveries may sometimes be obtained before the
bubblepoint pressure pb is reached and normal depletion sets in.
Such recoveries may be computed as follows.
The oil initially in place in stock-tank barrels per acre- foot
at pressure pi is according to Eq. 2,
. . 73758x4i(1-Siw) .
Boi
where 4; is initial porosity. By combining this expres- sion
with the material-balance equation (Eq. 10). the recovery factor
above the bubblepoint in stock-tank bar- rels per acre-foot may be
expressed as
Np= 7375Wi(Pi-Pb)[Co +Cf-Siw(cc~-~w)l
Boi[lfco(Pi-Pb)l
I (27)
where c,,, is the compressibility of interstitial water in
volume per volume per psi.
Example Problem 2. Zone D-7 in the Ventura Avenue field,
described by E.V. Watts, is an example of an undersaturated oil
reservoir without water drive. Its reser- voir characteristics
are
pi = 8,300 psig at 9,200 ft, pb = 3,500 psig, #Ii = 0.17,
s 1M = 0.40, B oh = 1.45, B o(1 = 1.15,
70 = 32 to 33API, CO = 13x10-6, c w = 2.7~10-~, Cf =
1.4x10-6,
S,, = 0.22, and Rsb = 900 cu ft/bbl.
Solution. On the basis of these data, Watts computes the
recovery by expansion above the bubblepoint at 47 bbliacre-ft and
by a depletion mechanism below the bub- blepoint at 110 bbl/acre-ft
(see Ref. 19 for details).
-
ESTIMATION OF OIL AND GAS RESERVES 40-13
Volatile Oil Reservoirs- Volumetric Methods20-25 Deeper
drilling, with accompanying increases in reser- voir temperatures
and pressures, has also revealed a class of reservoir fluids with a
phase behavior between that of ordinary black oil and that of gas
or gas condensate. These intermediate fluids are referred to as
high- shrinkage or volatile crude oils because of their rela-
tively large percentage of ethane through decane compo- nents and
resultant high volatility. Volatile-oil reservoirs are
characterized by high formation temperatures (above 200F) and
abnormally high solution GOR and FVF (above 2). The stock-tank
gravity of these volatile crudes generally exceeds 45 API.
The inherent differences in phase behavior of volatile oils are
sufficiently significant to invalidate certain premises implicit in
the conventional material-balance methods. In such conventional
material-balance work it is assumed that all produced gas, whether
solution gas or free gas, will remain in the vapor phase during the
depletion process, with no liquid condensation on passage through
the surface separation facilities. Furthermore, the produced oil
and gas are treated as separate independent fluids, even though
they are at all times in compositional equilibrium. Although these
basic assumptions simplify the conventional material-balance
calculations, highly in- accurate predictions of reservoir
performance may result if they are applied to volatile-oil
reservoirs.
In highly volatile reservoirs, the stock-tank liquids re-
covered by condensation from the gaseous phase may ac- tually equal
or even exceed those from the associated liquid phase. This rather
surprising occurrence is exem- plified in a paper by Woods,24 in
which the case histo- ry of an almost depleted volatile-oil
reservoir is presented.
Example Problem 3. Woods reservoir data for this volatile-oil
reservoir were
pi = 5,000 psig, pb = 3,940 psig, TR = 250F,
c$ = 0.198. k = 75 md,
Sib,, = 0.25, R,,, = 3,200 scf/bbl, yoi = 44API, You = 62API,
and B oh = 3.23.
Solution. At 80% depletion when pR = 1,450 psig and R =23,000
scf/bbl, the percentage recovery was 2 1% of which 5% was from
expansion above the bubblepoint, 9% from the depletion mechanism,
and 7% from liquids con- densed out of the gas phase by
conventional field separa- tion equipment (see Ref. 24 for
details).
In view of the increasing number and importance of volatile-oil
reservoirs in recent years, appropriate tech- niques have been
developed to provide realistic predic- tions of the anticipated
production performance of these reservoirs. 2o-z5 The depletion
processes are simulated by an incremental computation method, using
multicompo- nent flash calculations and relative-permeability data,
as indicated in the following stepwise sequence for a cho- sen
pressure decrement:
1. The change in composition of the in-place oil and gas is
determined by a flash calculation.
2. The total volume of fluids produced at bottomhole conditions
is determined by a volumetric material balance.
3. The relative volumes of oil and gas produced at bot- tomhole
conditions are determined by a trial-and-error procedure that
involves simultaneously satisfying the volu- metric material
balance and the relative-permeability rela- tionship.
4. This total well-stream fluid is then flashed to actual
surface conditions to obtain the producing GOR and the volume of
stock-tank liquid corresponding to the select- ed pressure
decrement.
When this calculation procedure is repeated for succes- sive
pressure decrements, the resultant tabulations rep- resent the
entire reservoir depletion and recovery processes. Since these
stepwise calculations are rather tedious and time-consuming, the
use of digital computers is recommended.
This method of reservoir analysis provides composi- tional data
on all fluid phases, including the total well- stream. This
information is then readily available for sepa- rator,
crude-stabilization, gasoline-plant, or related studies at any
desired stage of depletion.
In the case of small reservoirs with relatively limited
reserves, such lengthy laboratory work and phase- behavior
calculations may not be justified. An empirical correlation was
developed24 for prediction of the ultimate recovery in such cases,
based only on the initial produc- ing GOR, R, the reservoir
temperature, TR, and the ini- tial stock-tank oil gravity, yO;.
143.50 N,, = -0.070719+-
R +O.O001208OT,
+O.O011807y~i, . . . . (28)
where N,, =ultimate oil production from saturation pres- sure ph
to 500 psi, in stock-tank volume per reservoir volume of
hydrocarbon pore space.
It is claimed that this correlation will give values with- in
10% of those calculated by the more rigorous proce- dure previously
outlined.
Oil Reservoirs With Gas-Cap Drive- Volumetric Unit Recovery
Computed by Frontal-Drive MethodZ628 The Buckley-Leverett
frontal-drive method may be used in calculating oil recovery when
the pressure is kept con- stant by injection of gas in a gas cap
but is also applica- ble to a gas-cap drive mechanism without gas
injection when the pressure variation is relatively small so that
changes in gas density, solubility, or the reservoir volume factor
may be neglected. A reservoir with a very large gas-cap volume as
compared with the oil volume can sometimes be considered to meet
these qualifications even though no gas is being injected.
The two basic equations, Eqs. 29a and b, refer to a linear
reservoir under constant pressure with a constant cross-sectional
area exposed to fluid flow and with the free gas moving in at one
end of the reservoir and fluids being produced at a constant rate
at the other end. Inter- stitial water is considered as an immobile
phase.
-
40-14 PETROLEUM ENGINEERING HANDBOOK
s? I I I lbfil I VE A
0 I I I -Al !I --i
0= I I I 0 0.10 0.20 0.30
&O 0.50 0.60 0.70
S&GAS SATURATION, FRACTION OF HYDROCARBON FILLED PORE
SPACE
Fig. 40.7-Frontal-drive method in gas-cap drive
If the capillary-pressure forces are neglected. the
fractional-flow equation of gas is
(294
E= k sin @A@,--pR)
. . . 36%.,qr
(29b)
where fX = fractional flow of gas, E = parameter, 8 = dip angle,
degrees, A = area of cross-section normal to bedding
plane, sq ft, PO = density of reservoir oil, g/cm3,
ph = density of reservoir gas, g/cm3. and q, = total flow rate,
reservoir cu ft/D.
Since the ratio of k,lk, is a function of gas satura- tion, and
all other factors are constant, j$ can be deter- mined by Eq. 29a
as a function of gas saturation (see Fig. 40.7, Curve A).
The rate-of-frontal-advance equation may be rearranged to give
the time in days for a given displacing-phase satu- ration to reach
the outlet face of the linear sand body as a function of the slope
of the fractional flow vs. satura- tion curve (Fig. 40.7, Curve B)
as follows:
5.615NB, t= q,(df,,dS;) . . (30)
Note: Sk as used in this section is gas saturation as a fraction
of the hydrocarbon-filled pore space. When N is in cubic meters, q1
is in cubic meters per day.
The calculation procedure is first to calculate the
fractional-flow curve (Fig. 40.7, Curve A). The average gas
saturation in the swept area at breakthrough, which is equivalent
to the fraction of oil in place recovered, may then be obtained
from the fractional-flow curve by con- structing a straight line
tangent to the curve through the origin and reading Sk at fR = 1
.O. The time of break- through at the outlet face may be computed
from the slope of the curve at the point of tangency. The
subsequent per- formance history after breakthrough may then be
calcu- lated by constructing tangents at successively higher values
of Sk and obtaining Sh in a similar manner.
Example Problem 4. Welge2s presents a typical calcu- lation of
gas-cap drive performance for the Mile Six Pool in Peru.
Given:
Reservoir volume= 1,902 X lo6 cu ft, distance from original GOC
to average
withdrawal point = 1,540 ft,
average cross-sectional area = 1,902x IO6
1,540 =1.235x106 sq ft,
k, = 300 md, 8 = 17.50,
ps = 0.0134 cp, Po = 1.32 cp, q, = 64,000 res cu ft!D [I8 125
res m/d],
B,, = 1.25, B, = 0.0141
N = 44~ lo6 STB [6.996x106 m], R,, = 400 cu ft/bbl [71.245 m/mJ,
PO = 0.78 g/cm, and Ph = 0.08 g/cm 3
Solution. The performance history calculations are given in
Table 40.5 in a slightly simplified form.
Oil Reservoirs Under Gravity Drainage 29-37 Occurrence of
Gravity Drainage Gravity drainage is the self-propulsion of oil
downward in the reservoir rock. Under favorable conditions it has
been found to effect recoveries of 60% of the oil in place, which
is comparable with or exceeding the recoveries nor- mally obtained
by water drive. Gravity is an ever-present force in oil fields that
will drain oil from reservoir rock from higher to lower levels
wherever it is not overcome by encroaching edge water or expanding
gas.
Gravity drainage will be most effective if a reservoir is
produced under conditions that allow flow of oil only or
counterflow of oil and gas. This may be attained un- der pressure
maintenance by crestal-gas injection, which keeps the gas in
solution, or it may be attained by a gradual reduction in pressure,
so that the oil and gas can segregate continuously by counterflow.
It also may be obtained by
-
ESTIMATION OF OIL AND GAS RESERVES 40-15
first producing the reservoir under a depletion-type mech- anism
until the gas has been practically exhausted, then by gravity
drainage. A thorough discussion of the many aspects of gravity
drainage will be found in the classic paper by Lewis.32
y(, =22.5API, N,, for Jan. 1, 1957=44.6 million bbl of oil;
estimated ultimate 47 million bbl or I, 124 bbliacre- ft,
corresponding to 63% of the initial oil in place.
Several investigators 33m36 have attempted to formulate gravity
drainage analytically, but the relationships are quite complicated
and not readily adaptable to practical field problems. Most studies
agree, however, that the oc- currence of gravity drainage of oil
will be promoted by low viscosities, p,, , high relative
permeability to oil, k,, high formation dips or lack of
stratification, and high den- sity gradients (p, -p,). Thick
sections of unconsolidat- ed sand with minimal surface area, large
pore sizes, low interstitial water saturation, and consequently
high k, ap- pear to be especially favorable.
During the first 20 years the oil level in the field receded
almost exactly in proportion to the amount of oil produced, just as
in a tank.
2. Okluhoma City Wilcox Reservoir, OK. 29~32 The dis- covery
well, Mary Sudik No. I, blew out in March 1930, and flowed wild for
11 days.
The segregation of gas and development of gravity drainage began
to be important in 1934, when the aver- age pressure became less
than 750 psig, and was virtual- ly complete by 1936, when the
average pressure had dropped to 50 psig.
These factors usually are combined in a rate-of-flow equation.
which states that such flow must be proportional to (k,,lp,)(p,,
-p,) sin 8, in which 8 represents the an- gle of dip of the
stratum. Smithj7 compared the values of this term for a dozen
reservoirs, some of which had strong gravity-drainage
characteristics and some of which lacked such characteristics.
Water influx played an effective role until 1936, when it came
to a halt after invading the bottom 40% of the reservoir. Gravity
has been the dominant mechanism since. The Wilcox sand consists of
typical round frosted sand grains, clean and poorly cemented.
When expressing k,,, in millidarcies, p,, in centipoises, and
p,, and pI: in g/cm, it was found that for reservoirs exhibiting
strong gravity-drainage characteristics the value of the term
(k,,ip,)(p, -P,~) sin 0 ranged from 10 to 203 and that in
reservoirs where gravity-drainage effects were not apparent, this
function showed values between 0.15 and 3.4.
The average depth is 6,500 ft; the formation dip is 5 to 15; 884
wells have been drilled on a total area of 7,080 acres. The net pay
thickness is 220 ft. The 890,000 net acre-ft of Wilcox pay
contained originally 1,083 million bbl of stock-tank oil, as
confirmed by material balance.
Reservoir data for this reservoir are pi =ph = 2,670 psi at
minus 5,260 ft, TR= 132F, $=0.22, k ranges from 200 to 3,000 md,
S;,.=O.O3 (oil wet), Rt,, =735 cu ft/bbl, B,;=l.361, y,i=40APl,
yoci=38 tO 39API.
Case Histories of Gravity Drainage After Pressure Depletion
The most spectacular cases of gravity drainage have been of this
kind. Following are the two best known.
According to Katz, z9 oil saturations found in the gas zone were
between 1 and 26%, while saturations between 53 and 93% were found
in the oil-saturated zone below the GOC. The oil saturation below
the WOC has been estimated at 43%, showing gravity to be more
effective than water displacement in this reservoir.
1. Lukeview Pool in Kern County, CA. 3~32 The dis- covery well
in the Lakewood gusher area blew out in March 1910, flowed wild for
544 days, and ultimately produced 8% million bbl of oil, depleting
the reservoir pressure. Gravity drainage thereafter controlled this
reser- voir. There was no appreciable water influx. The sand is
relatively clean and poorly cemented. The average depth is 2,875
ft. The formation dip is IS to 45. There are I26 producing wells on
588 acres. The net sand thick- ness averages 7 1 ft, the height of
the oil column is 1,285 ft. and there are 41,798 net acre-ft of
pay.
Cumulative production, N,, for Jan. 1, 1958, is esti- mated at
525 million bbl and the ultimate recovery at 550 million bbl. After
an estimated 189 million bbl displaced by the water influx is
deducted, the upper 60% of the Wil- cox reservoir will yield under
gravity drainage ultimate- ly 361 million bbl or 696 bbliacre-ft,
corresponding to 57% of the oil in place.
Oil Reservoirs With Water Drive- Volumetric Method9 General
Discussion
Reservoir data for this reservoir are pi =P/, = 1,285 psi&
PR on Jan. I, l957=35 psig, r,= 115F. 4=0.33, k ranges up to 4,800
md and averages 3.600 md (70% of samples above 100 md, 37% above
1,000 md), S,,, =0.235, R,,,=200 cu ftibbl, Boi= 1.106,
Natural-water influx into oil reservoirs is usually from the
edge inward parallel to the bedding planes (edgewater drive) or
upward from below (bottomwater drive). Bot- tomwater drive occurs
only when the reservoir thickness exceeds the thickness of the oil
column, so that the oil/water interface underlies the entire oil
reservoir. It is
TABLE 40.5~PERFORMANCE-HISTORY CALCULATION
s: = Flowing GOR = S near
Outget Face Recover; Fraction If,41 -01(&/Q
ro krok,, k f, df,lds; k of Oil in Place x5. l+R, I?? 0.30 0.197
0.715 0.496 - - - a 35 0.140 0.364 0.642 -
0.395 0.102 0.210 0.739 1 .a7 7.1 0.534 1.808 0.40 0.097 0.200
0.752 1.81 7.3 0.535 1.908 0.45 0.067 0.118 0.829 1 .25 10.6 0.586
2.811 0.50 0.045 0.0715 0.885 0.94 14.1 0.622 4.227
-
40-16 PETROLEUM ENGINEERING HANDBOOK
TABLE 40X-CONDITIONS FOR UNIT-RECOVERY EQUATION, WATER-DRIVE
RESERVOIR
ration as found by ordinary core analysis after multiply- ing
with the oil FVF at abandonment, B,)O, as the residual oil
saturation in the reservoir to be expected from flood- ing with
water. This is based on the assumption that water from the drilling
mud invades the pay section just ahead of the core bit in a manner
similar to the water displace- ment process in the reservoir
itself.
Reservoir pressure Interstitial water,
bbllacre-ft Reservoir oil,
bbllacre-ft Stock-tank oil,
bbllacre-ft
Ultimate Initial Conditions Conditions
Pi Pa
7,75848,, 7,75&S,,
7.756@(1 -S,,) 7,758@,,
7,7584(1 - S,,)IB,, 7,75&S~B,,
further possible only when vertical permeabilities are high and
there is little or no horizontal stratification with im- pervious
shale laminations.
In either case, water as the displacing medium moves into the
oil-bearing section and replaces part of the oil originally
present. The key to a volumetric estimate of recovery by water
drive is in the amount of oil that is not removed by the displacing
medium. This residual oil satu- ration (ROS) after water drive,
S,,, plays a role similar to the final (residual) gas saturation,
S,, , in the depletion- type reservoirs.
To determine the unit-recovery factor, which is the the-
oretically possible ultimate recovery in stock-tank barrels from a
homogeneous unit volume of 1 acre-ft of pay pro- duced by complete
waterflooding, the amount of intersti- tial water and oil with
dissolved gas initially present will be compared with the condition
at abandonment time, when the same interstitial water is still
present but only the residual or nonfloodable oil is left. The
remainder of the original oil has at that time been removed by
water displacement.
Unit-Recovery Equation The unit recovery for a water-drive
reservoir is equal to the stock-tank oil originally in place in
barrels per acre- foot minus the residual stock-tank oil at
abandonment time (Table 40.6).
By difference, the unit recovery by water drive, in stock-tank
barrels per acre-foot, is
.(31)
where N,,. is the unit recovery by water drive, in stock- tank
barrels, and S,, is the residual oil saturation, frac- tion. The
ROS at abandonment time may be found by ac- tually submitting cores
in the laboratory under simulated reservoir conditions to flooding
by water (flood-pot tests). Another method commonly used is to
consider the oil satu-
Recovery-Efficiency Factor
The unit recovery should be multiplied by a permeability-
distribution factor and a lateral-sweep factor before it may be
applied to the computation of the ultimate recovery for an entire
water-drive reservoir.
These two factors usually are combined in a recovery- efficiency
factor. Baucum and Steinle3 have determined this
recovery-efficiency factor for five water-drive reser- voirs in
Illinois. Table 40.7 lists the recovery efficien- cies for these
reservoirs, together with some other pertinent data.
Average Recovery Factor From Correlation of Statistical Data In
1945, Craze and Buckley,39,40 in connection with a special API
study on well spacing, collected a large amount of statistical data
on the performance of 103 oil reservoirs in the U.S. Some 70 of
these reservoirs pro- duced wholly or partially under water-drive
conditions. Fig, 40.8 shows the correlation between the calculated
ROS under reservoir conditions and the reservoir oil vis- cosities
for these water-drive reservoirs. The deviation of the ROS from the
average trend in Fig. 40.8, vs. per- meability, is given by the
average trend in Fig. 40.9. The deviation of the ROS from the
average trend in Fig. 40.8, vs. reservoir pressure decline, is
given by the average trend in Fig. 40.10.
Example Problem 5. In a case where the porosity, 4=0.20, the
average permeability, k=400 md, the in- terstitial water content,
Si,=O.25, the initial oil FVF, B,, = 1.30, the oil FVF under
abandonment conditions, B, = 1.25, the initial reservoir oil
viscosity, pLo = 1 .O cp, and the abandonment pressure, pu =90% of
the initial pressure, pi, determine the average ROS.
Solution. S,, may be estimated as 0.35+0.03-0.04= 0.34 and the
average water-drive recovery factor from Eq. 31 is
l-O.25 0.34 N,,.=(7,758)(0.20)
>
=473 STBlacre-ft
TABLE 40.7-RECOVERY-EFFICIENCY FOR WATER-DRIVE RESERVOIR
Unit-Recovery Actual Reservoir
Recovery Factor Recovery Efficiency
Number $I S,, B, S,, (bbl/acre-ft) (bbllacre-ft) (O/o) 1 0.179
0.400 1.036 0.20 526 429 82 2 0.170 0.340 1.017 0.20 592 430 73 3
0.153 0.265 1.176 0.20 504 428 85 4 0.192 0.370 1.176 0.20 500 400
80 5 0.196 0.360 1.017 0.20 653 482 74
From flood-pot tests
Average = 79
-
ESTIMATION OF OIL AND GAS RESERVES 40-17
0 0.2 0.4 06 I 2 4 6 IO 20 40 60 100 EC0
OIL VISCOSITY AT RESERVOIR CONDITIONS; CENTIPOISES
Fig. 40.8-Effect of oil viscosity on ROS water-drive sand
fields.
In another statistical study of the Craze and Buckley data and
other actual water-drive recovery data on a total of 70 sand and
sandstone reservoirs, the API Subcom- mittee on Recovery Efficiency
t6 developed Eq. 32 for unit recovery for water-drive reservoirs,
N,,. In stock- tank barrels per acre-foot,*
-0.2159
, . . . (32)
where symbols and units are as previously defined ex- cept
permeability, k, is in darcies, and pressure, p, is in psig.
Example Problem 6. For the same water-drive reser- voir used
previously and assuming pwi =O.S cp, the API statistical equation
yields the following unit recovery factor:
N,, =4,259 (0.20)(1-0.25) .0422
1.30 1
1.0 ( > -0.2159
x- 0.9
= 504 STB/acre-ft
Because data were arrived at by comparing indicated recoveries
from various reservoirs with the known pa- rameters from each
reservoir, the estimated residual oil and the average recovery
factor based on these correla- tions allows for a
recovery-efficiency factor (permeability- distribution factor times
lateral-sweep factor) that is not present in the unit-recovery
factor based on actual residual oil as found by flood-pot tests or
in the cores. because Eq 32 IS empirlcally darned, conversion to
metric units jmJ/ha.m) requires mulbpl~cark?m of Nup by 1.2899
Fig. 40.10--Relation between deviation of ROS from average trend
in Fig. 40.8 and pressure-decline water-drive sand flelds.
l o.30 . .
5, F :: *a20 Lsk 3a LiL
1 8 l O.O 02 20 ?I+ 0 OIL hi0
g 6 -o .,o & L 4
EE
2 -0.20
g
-0.30 20 40 100 200 400 lcco EOW 4oM) Io.ow
AVERAGE PERMEABILITY OF RESERVOIR; MILLIDARCIES
Fig. 40.9-Relation between deviation of ROS from average trend
in Fig. 40.8 and permeability water-drive sand fields.
Water-Drive Unit Recovery Computed by Frontal-Drive Method26-28
The advance of a linear flood front can be calculated by two
equations derived by Buckley and Leverettz6 and simplified by
Welge** and by Pirson. These are known as the fractional-flow
equation and the rate-of-frontal- advance equation. This method
assumes that (1) a flood bank exists, (2) no water moves ahead of
this front, (3) oil and water move behind the front, and (4) the
relative movement of oil and water behind the front is a function
of the relative permeability of the two phases.
If the throughput is constant and the capillary-pressure
gradient and gravity effects are neglected, the fractional- flow
equation can be written as follows:
1 fw =
1 +(k,lk,,,,)(pJp,) . . (33)
0 20 40 60 SO 100 RESERWR PRESSURE DECLINE: PER CENT
-
40-18 PETROLEUM ENGINEERING HANDBOOK
3 1.0
5 0.9
2 k-~ 0.8
d 5 0.7 I- z - 0.6 ii? : 0.5
1.0 5
Iv.. I .2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
WATER SATURATION, S,,
FRACTION OF PORE SPACE TIME YEARS +
Fig. 40.11-Fraction of water flowing in total stream f, and
slope off, curve df,/dS,, vs. water saturation S,, (example:
frontal-water-drive problem).
Fig. 40.12-Example of frontal-drive problem, unit-recovery fac-
tor, and WOR vs. time.
wheref,, is the fraction of water flowing in the reservoir at a
given point, k,. is the water relative permeability, fraction, and
pn, is the reservoir water viscosity, cp. Since k,,lk,. is a
function of water saturation, f,+, can be determined by Eq. 33 as a
function of water satura- tion for a given water/oil viscosity
ratio (see Fig. 40.11, Curve A).
The Buckley-Leverett rate-of-frontal-advance equation may be
rearranged to give the time in days for a given displacing phase
saturation to reach the outlet face of the linear sand body as a
function of the slope of the frac- tional flow vs. saturation curve
(Fig. 40.1 I, Curve B) as follows:
5.615 NB, t= qr(df,,,dSi,*,) ( . . . . (34)
where df,ldS,,. is the slope of thef, vs. Si, curve; the time,
t, is in days; and the total liquid flow rate, qr, is in reservoir
cubic feet per day.
The average water saturation behind the flood front at
breakthrough, and therefore the oil recovery, may be ob- tained
from the fractional-flow curve by constructing a straight line
tangent to the curve through S;, atf,=O, and reading S ;,, at f, =
1 .O. The time of breakthrough at the producing well may be
computed from the slope of the curve at the point of tangency. The
subsequent per- formance history after breakthrough may be
calculated by constructing tangents at successively higher values
of S;, and obtaining Si, in a similar manner.
Table 40.8 illustrates the calculation procedure for a water
drive at constant pressure in a homogeneous reser- voir and with a
water-influx rate equal to the production rate.
Fig. 40.12 is a plot of the results of the performance- history
calculation from Table 40.8. If the economic limit is taken to be a
WOR of 50, then it can be noted from Fig. 40.12 that the
unit-recovery factor will be 575 bbllacre-ft to be recovered in
20.7 years.
Effect of Permeability Distribution t41-44 In some reservoirs
there may be distinct layers of higher and lower permeabilities
separated by impervious strata. which appear to be more or less
continuous across the reservoir. In such a case, water and oil will
advance much more rapidly through the higher-permeability streaks
than through the tighter zones, and therefore the recovery at the
economic limit will be less than that indicated by the
unit-recovery factor.
Methods for computing waterflood recoveries that take into
account the permeability distribution were proposed by Dykstra and
Parsons,4 Muskat. and Stiles.43
In the Dykstra-Parsons paper4 it is assumed that in- dividual
zones of permeability are continuous from well to well, and a
computation procedure as well as charts are presented for the
coverage or fraction of the total volume of a linear system flooded
with water for given values of (1) the mobility ratio knvpolkropw,
(2) the pro- duced WOR, and (3) the permeability variance.
This permeability variance is a statistical parameter that
characterizes the type of permeability distribution. It is obtained
by plotting the percentage of samples larger than the sample being
plotted vs. the logarithm of per- meability for that sample on
log-probability graph paper and then dividing the difference
between the median or 50% permeability and the 84. I % permeability
by the me- dian permeability. Although the Dykstra-Parsons
method
-
ESTIMATION OF OIL AND GAS RESERVES 40-l 9
TABLE 40.8-WATER-DRIVE PERFORMANCE-HISTORY CALCULATION*
Time Residual Oil Unit-Recovery Saturation Factor WOR =
s S,, w df,JdS,w 1w ~ f (years) (1 -S,,) (bbl/acre-ft) f,/l -f,
0.545 0.619 0.800 2.70 3.94 0.381 441 4.0 0.581 0.655 0.875 1.69
6.29 0.345 484 7.0 0.605 0.675 0.910 1.29 8.24 0.325 507 10.1 0.634
0.697 0.940 0.95 11.19 0.303 534 15.7 0.673 0.720 0.970 0.64 16.61
0.280 561 32.3 0.718 0.748 0.990 0.33 32.21 0.252 594 99.0
N = 597,000 STB, ao, = 1 30, o=o 20. S,, =0 25, and qr = 200 E/D
x 5 615 cu ftlbbl = > ,222 ,esewow cu fl/D
does not allow for variations in porosity, interstitial water.
and floodable oil in the different permeability groups, it has
apparently been used extensively and successfully on close-spaced
waterfloods. mainly in California.
Johnson4 in 1956 published a simplification of this method and
presented a series of charts showing the frac- tional recovery of
oil in place at a given produced WOR for a given permeability
variance, mobility ratio, and water saturation. Reznik er al. 4s
published an extension to the Dykstra-Parsons method that provides
a discrete analytical solution to the permeability stratification
prob- lem on a real-time basis.
In the Stiles method4 it again is assumed that individu- al
zones of permeability are continuous from well to well and that the
distance of penetration of the flood front in a linear system is
proportional to the average permeabil- ity of each layer. Instead
of representing the entire per- meability distribution by one
statistical parameter, Stiles tabulates the available samples in
descending order of per- meability and plots the results in terms
of dimensionless permeability and cumulative capacity fraction as a
func- tion of cumulative thickness. From these data, Stiles com-
putes the produced water cut of the entire system as the watering
out progresses through the various layers, start- ing with those of
the highest permeability. Stiles then as- sumes that at a given
time each layer that has not had breakthrough will have been
flooded out in proportion to the ratio of its average permeability
to the permeability of the last zone that had just had
breakthrough, and then constructs a recovery vs. thickness
relationship. This then is combined with previous results to yield
a recovery vs. water-cut graph. The Stiles method is used
extensively and successfully, mainly in the midcontinent and Texas,
for close-spaced waterfloods. It does not make allowance for the
difference in mobility existing in the formation ahead of and
behind the flood front. which the Dykstra- Parsons method allows
for. It also does not provide for differences in porosity,
interstitial water, and floodable oil in the various permeable
layers.
Arps introduced in 1956 a variation of the Stiles meth- od,
called the permeability-block method. This method handles the
computations by means of a straightforward tabulation and does make
allowance for the differences in porosity, interstitial water, and
floodable oil existing in the various permeable layers. Since it is
designed primarily for the computation of recoveries from water-
drive fields above their bubblepoint. no free-gas satura-
tion is assumed. The method further assumes that (I) no oil
moves behind the front, (2) no water moves ahead of the front, (3)
watering out progresses in order from zones of higher to zones of
lower permeability. and (4) the ad- vance of the flood front in a
particular permeability streak is proportional to the average
permeability.
This method, applied to a hypothetical pay section 100 ft thick,
is illustrated in Table 40.9, which is based on data from a
Tensleep sand reservoir in Wyoming where good statistical averages
of more than 3,000 core analy- ses were available. Part of these
cores were taken with water-base mud that yielded the residual-oil
figures on Line 6. Another portion was taken with oil-base mud and
yielded the interstitial-water figures of Line 7. An oil/water
viscosity ratio of 12.5 was used in calculating the WOR of Line
13.
In Group I the recovery of 61.7 bbliacre-ft for WOR= 15.5 is the
product of the fraction of samples in the group and the
unit-recovery factor. In all other groups for WOR = 15.5 the full
recovery is reduced in the propor- tion of its average permeability
to 100 md. The total recovery at WOR= 15.5 is shown as 175.6
bbliacre-ft. The cumulative recoveries for WORs of 35.9, 76.5,
307.7, and infinity are calculated in a similar manner. Fig. 40.13
is a plot of WOR vs. recovery factor. From Fig. 40.13 it can be
seen that, if the economic limit is taken to be a WOR of 50, the
recovery factor would be 297 bbliacre-ft.
It should be stressed that the permeability-block method is
applicable only when the zones of different permeabil- ity are
continuous across the reservoir, or between the source of the water
and the producing wells. When the waterfront has to travel over
large distances, nonunifor- mity of permeability distribution in
lateral directions be- gins to dominate, and recoveries will
approach those obtainable if the formation were entirely uniform
(per- meability distribution factor= 1). In such a case, an esti-
mate based on the permeability-block method may be considered as
conservative, except for the fact that one of the basic assumptions
of this method is that the WOC, or front, moves in pistonlike
fashion through each per- meability streak, sweeping clean all
recoverable oil. In reality, part of this oil will be recovered
over an extend- ed period after the initial breakthrough, which may
tend to make the estimate optimistic. Those using the
permeability-block method hope that these two effects are more or
less compensating.
-
40-20 PETROLEUM ENGINEERING HANDBOOK
TABLE 40.9-WATER DRIVE PERMEABILITY-BLOCK
Group
(1) Permeability range, mud (2) Percent of samples in group (3)
Average permeability, md (4) Capacity, darcy-ft (2) x (3) + 1,000
(5) Average porosity fraction $ (6) Average residual-oil fraction
Sgr (7) Average interstitial-water fractron S,, (8) Relative water
permeabrlity behind front k (9) Relative oil permeability ahead of
front k,,
(10) Unit-recovery factor (B,, = 1.07) (11) Cumulative wet
capacity, E(4) (12) Cumulative clean oil capacity, 3.241 - (11)
(13) Water-oil ratio WOR= (~00~c)(8/9)(1 l/12) (14) Cumulative
recovery at WOR = 15.5 bbllacre-ft
Min k wei =I00 md (15) Cumulative recovery at WOR = 35.9
bbllacre-ft
Min k,,, =50 md (16) Cumulative recovery at WOR = 76.5
bbl/acre-ft
Min k we, = 25 md (17) Cumulative recovery at WOR = 307.7
bbllacre-ft
Min k we, =lO md (18) Cumulative recovery at WOR =
mbbllacre-ft
Min k wer =0 md
Effect of Buoyancy and Imbibition In limestone pools producing
under a bottomwater drive, such as certain of the vugular D-3 reef
reservoirs in Al- berta, one finds an extreme range in the
permeabilities, often running from microdarcies on up into the
darcy range. Under those conditions the modified Stiles method
heretofore described yields results that are decidedly too
400, I I I I r f n
/
200. 1
0 G.-- ~100
I I I I I I I I
g 80- 1 I I
- ECONOMIC , .9 !
5 50 60kIMIT WOR=5Ojmi
-T---q---
5 40
20 RECOVERY FACTOR =297 BBL/ACRE-
, FT@ WOR =50
lOI 31 , , I 0 200 400 600
RECOVERY FACTOR, BBL/ACRE-FT
Fig. 40.13-Example of modified Stiles permeability-block method
WOR vs. recovery factor.
>lOO 8.5
181.3 1.541 0.159 0.173 0.185 0.65
0.475 726
1.541 1.700 15.5 61.7
2
50 to 100 10.9 69.0
0.752 0.150 0.195 0.154 0.63 0.53 693
2.293 0.948 35.9 52.1
CALCULATIONS
3
25 to 50 14.5 34.4
0.499 0.152 0.200 0.131 0.60 0.61 722
2.792 0.449 76.5 36.0
61.7 75.5 72.0
61.7 75.5 104.7
61.7 75.5 104.7
61.7 75.5 104.7
4 5
10 to 25 0 to 10 21.2 44.9 16.1 2.4
0.341 0.108 0.130 0.099 0.217 0.222 0.107 0.185 0.56 0.54 0.66
0.47 623 415
3.133 3.241 0.108 0 307.7 21.3 4op5
42.5 8.9
85.1 17.9
132.1 44.7
132.1 186.3
Total
100.0
3.241
175.6
260.6
344.9
418.7
560.3
low. The reason is that, in pools like the Redwater D-3, there
is a substantial density difference between the ris- ing salt water
and the oil. While the water rises and ad- vances through the
highly permeable vugular material, it may at first bypass the
low-permeability matrix mate- rial, leaving oil trapped therein.
However, as soon as such bypassing occurs, a buoyancy gradient is
set up across this tight material, which tends to drive the trapped
oil out vertically into the vugular material and fractures. In the
case of Redwater D-3, where the density difference between salt
water and oil is 0.26, while the vertical per- meabilities for
matrix material are only a fraction of the horizontal
permeabilities, a simple calculation based on Darcys law applied to
a vertical tube shows that during the anticipated lifetime of the
field very substantial addi- tional oil recovery may be obtained
because of this so- called buoyancy effect.
To calculate the recovery under a buoyancy mechanism it is
necessary first to determine by statistical analysis of a large
number of cores the average interval between high- permeability
zones or fractures. A separate computation is then made for each of
the permeability ranges to deter- mine what percentage of the
matrix oil contained in a the- oretical tube of such average length
may be driven out during the producing life of the reservoir under
the ef- fect of the buoyancy phenomenon.
Surprisingly improved recoveries are sometimes indi- cated by
this method over what one would ex