Prediction of Reservoir Compaction and Surface Subsidence: Field Application of a New Model .I.A. de Weel, SPE, Koninklijke/Shell E&P Laboratorium R.M.M. Smits, SPE, Koninklijke/Shell E&P Laboratorium

Summary. A new loading-rate-dependent compaction model for unconsolidated clastic reservoirs is presented that considerably improves the accuracy of predicting reservoir rock compaction and surface subsidence resulting from pressure depletion in oil and gas fields. The model has been developed on the basis of extensive laboratory studies and can be derived from a theory relating compaction to time-dependent intergranular friction.

The procedure for calculating reservoir compaction from laboratory measurements with the new model is outlined. Both field and laboratory compaction behaviors appear to be described by one single normalized, nonlinear compaction curve. With the new model, the large discrepancies usually observed between predictions based on linear compaction models and actual (nonlinear) field behavior can be explained.

Introduction Withdrawal of fluids from hydrocarbon reservoirs can result in reservoir compaction and surface subsidence. Well-known exam­ples are Goose Creek, Wilmington, Inglewood, Nigata, Bolivar coast, and Groningen. 1-3 A considerable amount of scientific ef­fort has been spent on the subject, which has resulted in a frame­work of theoretical and laboratory procedures that are commonly applied. 4·12

It has become apparent over the last decade, however, that the application of these procedures to a number of well-documented field cases results in strong discrepancies between predicted and observed field behavior. Two basic types of anomalies have been recognized.

1. Apparent nonlinear field behavior that has not been confirmed in laboratory compaction experiments. Examples are the Bolivar coast heavy-oil fields,13,14 Wilmington, IS-17 and the San Joaquin Valley. 18

2. Differences between laboratory-measured and field-derived reservoir rock compressibilities. 3

As compaction and subsidence can have environmental, techni­cal, and financial consequences, it was decided to start an investi­gation into the causes of these discrepancies. The following possible causes were put forward: (l) pressure-lag effects in neighboring clay layers; (2) plastic deformation, e.g., caused by grain breakage; (3) effects of the in-situ stress state; (4) previous deeper burial of reservoirs during their geologic history; (5) core disturbance; and (6) loading-rate effects.

Point 1 has been dealt with in a separate study 19 that demon­strated that the pressure-lag effect cannot explain the observed dis­crepancies. Therefore, Points 2 through 6 have been considered in more detail.

Experimental Result. Uniaxial compaction experiments were carried out in oedometer and triaxial compaction cells5 to study Points 2 through 6. The ex­periments were carried out on more than 200 sandstone samples covering a large range of atmospheric porosities (5 to 38%) and various degrees of consolidation (see Table 1). In these experiments, the axial deformation of the core samples was monitored continu­ously as a function of the applied axial effective stress.

Results of typical laboratory experiments are shown in Figs. 1 and 2. More experimental details on a large subset of the experi­ments carried out are given in Ref. 20. The compaction behavior of all samples studied in the various experiments was very similar, as summarized in Fig. 3. The following observations were made.

Copyright 1988 Society 01 Petroleum Engineers

SPE Fonnation Evaluation, June 1988

1. Compaction curves at different but constant loading rates form a fan of lines (dashed curves, further referred to as "virgin" com­paction curves). The lower the loading rate, the more the sample will be compacted at a given stress level. The compressibility along the virgin compaction curves will be denoted by C m,o'

2. The shifts, .:1O'l' between compaction curves for different con­stant loading rates, ul ' defined in Fig. 4 are quite systematic and related to stress level, according to

where b is a'material constant that depends on rock type (see Pre­diction of Field Compaction Behavior). Nonuniaxial compaction measurements showed that there is hardly any effect of the horizontal-ta-venical stress ratio on the measured b values.

3. Every time the loading rate is suddenly increased within a load­ing cycle, a compaction curve like A'-B' or B-C will result (Fig. 3). A sudden decrease in loading rate will result in a compaction curve like A-B. In both cases, the resultant compaction curve will finally reach the virgin compaction curve corresponding to con­tinuous loading at the new loading rate.

4. Interruption ofloading results in creep (C-D and B'-C'). When loading is resumed, the sample shows a compaction behavior similar to that after an increase in loading rate (D-E and C'-D').

5. The compaction behavior during reloading after partial un­loading (E'-F') strongly resembles that observed after creep or after an increase in loading rate. (For large amounts of unloading, a different compaction behavior results that is not important for the scope of this paper.)

6. The behavior described above has a large influence on the uniaxial compressibility, which is directly related to the slope of the curve of!lh vs. O'z. For example, when the loading rate is in­creased, the uniaxial compressibility initially becomes much low­er. Conversely, when the loading rate is decreased, the uniaxial compressibility initially becomes much higher.

7. No major differences with respect to these findings were found between the compaction behavior of artificially sedimented sand­packs and that of moderately consolidated and unconsolidated (origi­nal and remolded) reservoir samples.

8. The compaction behavior after an arbitrary loading history is restored to first-cycle behavior by not too large a mechanical dis­turbance (e.g., as encountered during the coring process and sample­preparation procedures).

9. No major differences were found between the compaction be­havior at 20°C [68°F] and that in an experiment at 90°C [194 oF].

347

TABLE 1-DETAILS OF THE SANDSTONE SAMPLES INVESTIGATED

Sandstone Type ,

Artificially sedimented sand packs Unconsolidated sandstone (reservoir rock) Friable sandstone (reservoir rock) Consolidated sandstone (reservoir rock) Consolidated sandstone (outcrop) Tight sandstone (reservoir rock)

,..-- - -,---- 1 1 hilI I m 1m: l[ (~ml~~--"~~~~~----------=------~

1 .. ...,

.. 000

I

II

:m: m:

21000

0, = 2300. bar th

at z. 62. barth

0, z. 6 .2 barth

0, = .62 bof/h

100

I [Xp NO. ! SAMI'\.[ A 16 RI

.oo 2DO

a,(bor)~

Fig. 1-lnfluence of changes In loading rate and partial un· loading on the compaction behavior of an unconsolidated reservoir sand (oedometer test).

Rate-Type Compaction Model (RTCM) Rate-Type Material Behavior. The compaction properties de­scribed earlier are similar to those of rate-type materials21 ,22

(Fig. 5) . I. When a sample is loaded at a constant loading rate, uz.2' start­

ing from an initial Point B, the compaction behavior (dashed line) is completely determined by the material constants cm,o and b, and the position of B relative to the virgin compaction line correspond­ing to uz.2' Consequently, it does not matter how Point B is ar­rived at; for instance, as a result ofa different loading rate (circles), creep after interruption of loading (squares), or partial unloading (crosses).

2. The position of the virgin compaction line that is finally reached depends only onuz and is independent of the previous stress/strain history . It appears as if this previous history is wiped out.

General Compaction Curve. It was observed that the transition curves after creep, after an increase in loading rate, and after partial unloading have a similar shape and position relative to the constant­loading-rate compaction curve finally reached. This even holds true

348

Number of Samples

10 50 50

100 10 20

D.D

ZDD.D

400.0

( 1-· ~

_ .0

Cm.o(lab)

(10 -5 bar- 1)

20 to 40 10 to 40 5 to 10 1 to 5 1 to 2

0.4 to 1

b

0.020 to 0.022 0.017 to 0.025 0.015 to 0.020 0.008 to 0.015 0.008 to 0.015 0.005 to 0.010

3000 1- r, I .. r/Ot'

.000 .D L-____ -..L ______ -'-_____ """-______ ..I..-___ --'

D.D ZOD.O 400.0 100.0 lOO.D tOOO. O

AXIAL ST"ISI titer t

Fig. 2-Exampl. of the compaction behavior of a friable reser· voir sandstone as a result of changes In loading rete at varloua stre .. levels (triaxial test).

for experiments carried out at different stress levels and on differ­ent samples. This suggests that all transition curves can perhaps be described by one general compaction curve after the applica­tion of an appropriate normalization procedure, using the compac­tion curve finally reached as a reference.

Such a normalization procedure has indeed been found. Basically, the axial stress, uz' is divided by a normalization stress, uz,o' which is related to the stress level (see Fig . 6). The axial displacement, E~. arbitrarily taken as zero at uz,o' is divided by uz•o and by the uniaxial compressibility, C m,o' of the compaction curve finally reached. As a result, a dimensionless plot of E~/(Cm.ouz,o) vs. uz/uz.o is obtained in which the compaction curve finally reached is represented by a straight line with slope unity (by definition).

Fig. 7 shows actual data points resulting from loading (at a con­stant rate) after creep, after increases in loading rate, and after partial unloading at various stress levels . The data points were measured ona number of unconsolidated sandstones from different reservoirs. All the data points have been normalized according to the above procedure. The normalized data points all fit one general compac­tion curve, which is shown as a solid line in Fig. 7. Normalized

SPE Formation Evaluation . June 1988

6to

.. OECIft:AS( I'll LQIOING RAT( tr r • it r, 3

• INCft(AS( N I.OlOWG RIoT( tTl ' 0". , 4

INT['UtuPFlON OF l OlDWG ii, = 0

M:' STlUn Of" Lo,t.DtNG trl ' crr , l

A' INCLIt[AS( IN UlIOING RAT( crt . iTI , l

.' IHT[lltRUPTIOfriI Of lOADING VI ' 0

C' .[ · STAliIT 0,. LOAOING G- rs cTt, l

O' """T'Al UHL.OlOtNG

£' IULOAOU~G

a'Z,4 :> rr %,3 ,. "'Z,2 > ITZ,i

[

ci"Z· ci"z, 3

ci"z· ci"z,Z

CTZ

Fig. 3-Condensed picture of compaction behavior.

o 0

0 o o

o o

o LOADING RATE ~I .~ ...

o

o SLOW LOADING RATE COMPo&CTION (cT" I)

o COMPACTION FOLLOWED BY CREEP

X PARTIAL UNLOADING

COMPACTION RESULTING FROM CONTINUOUS

LOADING AT ~" 2

COMPACTION RESULTING FROM LOADING AT ~ 1,2

STARTING FROM 9

Fig. 5-Rate-type compaction behavior.

SPE Formation Evaluation. June 1988

1\\ 1 \

I \ f7, • 0-: .•

I f 1

I I I I

Fig. 4-Deflnltlon of AU., U.,1' and U.,2'

"VIRGIN"COMPACTION CURVE - ("CTz LINE")

0.9"o'z LINE"

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 : I I I I I I I I

o al •O 0.9al Oi

"'.

Fig. 6-Determlnatlon of U Zoo on laboratory-measured tran­sition curve.

349

Fig. 7-Comparlson between the general compaction curve and laboratory data.

transition-zone data points from all other experiments in this study (not shown) were found to fit the same normalized compaction curve.

The part of the general compaction curve for which oz/oz.o is larger than about 1.0 was obtained from experiments carried out at loading rates between 0.062 and 400 MPa/h [0.62 and 4,000 bar/hr] or from the reloading behavior after creep. To determine the part of the curve for which oz/(1z.o is smaller than 1.0, loading­rate changes greater than 104 have to be applied. It was impossi­ble to obtain such changes in the laboratory variable-loading-rate experiments. Therefore, this part of the curve was obtained from unloading/reloading experiments, as illustrated in Fig. 8. An ex­ample of a real unloading/reloading curve can be found in Fig. 1.

For a specific experiment, the position of the normalized start­ing point on the general compaction curve is a function of the ap­plied change in loading rate (see Eq. I), or of the amount of unloading or creep. Conversely, the general compaction curve, in combination with Eq. I, can be used to predict the compaction be­havior upon arbitrary increases in loading rate, as discussed in more detail later.

PhysicolMathematical Background. The rate-type compaction be­havior may be caused by the rate effects that playa role in the fric­tional sliding of mineral (grain) surfaces.23•24 The contact between two mineral grains takes place at a limited number of spots called asperities. The average area of an individual asperity depends on its lifetime, which in tum depends on the loading rate. Therefore, changes in loading rate result in a different real area of contact be­tween the grains and a different friction coefficient.

Incorporation of the rate-dependent friction coefficient in a simple compaction model yields a rate-type compaction equation (see Ap­pendix A). Under uniaxial strain conditions, this equation becomes similar to an equation proposed previously by Kolymbas22 to describe the behavior of rate-type materials. Equations describing the general compaction curve, the distance between virgin com­paction curves, and creep can be derived from this equation. as shown in Appendix B. Together. these equations form the mathe­matical framework of the RTCM. A more detailed description of the physico/mathematical background is given in Ref. 20.

Application to Field Condition. Simulation of Field Compaction Behavior In the Laboratory. When a reservoir is taken into production, the rate of change of the effective stress exerted on the reservoir rock is suddenly in­creased from that imposed by the burial process over a geologic time span. to that induced by the depletion-pressure history. The ratio between the depletional and geological loading rates is of the order of 107 • This change in loading rate will have a large influ­ence on the in-situ compaction behavior if the rate effects observed at laboratory loading rates also occur at geological and depletional loading rates.

350

CHANGE IN LOADING RATE

SIMULATION

O'z,l

Fig. 8-Slmulatlon of compaction after an Increaae In loading rate by reloading after partial unloading.

The present laboratory experiments have been carried out at load­ing rates between 0.062 and 400 MPa/h [0.62 and 4,000 bar/hr). Whether the observed rate effects are also present to the same ex­tent under field conditions can therefore only be verified by com­paring predictions based on the RTCM and laboratory measurements with actual field data. The procedure to arrive at such predictions will be considered first. It will be shown later that this procedure appears to yield correct predictions of observed nonlinear field be­havior for a number of well-documented unconsolidated clastic reservoirs.

Prediction of Field Compaction Behavior. For a prediction of field compaction, we must know the transition zone that will arise upon a change in loading rate from the geological value, Uz(geol)' to the depletional value. c7z(dep)' The geological loading rate IS

roughly equal to the ratio of the initial effective stress over the age of the reservoir. The depletionalloading rate is determined by the production plan. Because of the almost logarithmic relation between ~oz and Uz.2/Uz.1 (see Eq. 4), fluctuations in the depletion rate are not very important.

The normalized field compaction behavior, as predicted by the RTCM, is given by the general compaction curve (see Fig. 9). Hence, the reservoir compaction at a certain value of the reservoir pressure drop. IIp. is given by

where on = (1z/oz.o. 0ni =(1z./oz.o' Oz =oz.i +~p. and H(u,n) is the general compaction curve given by Fig. 9. An algebraiC fit for H(on) is given in Appendix C. The determination of b, em•o• a~d 0:. 0 is discussed below. By substituting 0:.0 em.o ' and H(on) 1D

SPE Fonnation Evaluation. June 1988

'~ I! t ... " i1I,o l

A, , , , : , r l : ' ~ r : I~ I : : , , , , , , , , , , , , , , , , , , , , , , , , I , , , I ,

Fig. 9-Determlnatlon of 4U zlu Z,o and u z"lu z.o on the gener· al compaction curve for an a88umed Initial pOint.

Eq. 2, compaction can be calculated for various values of the reser­voir pressure drop, Ap.

The total amount of compaction expected at the end of the pro­duction period is obtained by taking (I" = (lz.//(lz.o' where (lz./ cor­responds to the fmal in-situ vertical effective stress (after depletion).

Determination of the Material Constllnt b. As can be derived from Eq. 1, the value of b required for the calculation of cm.o and (lz,o can be determined from the compaction behavior in labora­tory experiments after an increase or decrease in loading rate ac­cording to

b In(uz.2/O'l.l)

In(ol,2/ul.l) .... . .. . ...... . ...... ..... ..... ... (3)

An alternative procedure for determining b is to plot the value of AO'z"n(uz.2/Uz.I) vs. the values of (lz.1 at which the loading rate was changed. This yields straight lines with slope b, because Eq. 1 can be approximated by

A(llo:buz.lln(uz.2/UZ,I) . . .. .. .... ....... .. . ... ..... . (4)

for not too large values of Uz,2/Uz,1 (approximately up to 103). Eq. 4 shows that b characterizes the relative width AUz/Uz,! of

the transition zone. Fig. 10 depicts b as a function of l/cm,o(1ab)

for some 150 sandstones. This plot shows that loading-rate effects in the compaction of clastics become less pronounced with increasing rock stiffness, As a result, accurate b values become increasingly difficult to measure at higher values of l/Cm.o(l.b),'

Determination 01 the Virgin FieUl Compressibility. The virgin field compressibility, cm,o(fieJd), can be calculated by correcting the Cm,o(llb) determined from uniaxial laboratory compaction experi· ments for the difference in loading rate between reservoir deple­tion and laboratory measurement (see Eq. 1 and Fig. 3) according to

where the factor (l-Rc) is brought into the equation for reasons discussed by Geertsma. 25

Comparison of field compaction behavior and predictions based on the new model (see Comparison With Actual Field Data) has shown that for cm,o(lab)' one should use the first-cycle (constant­loading-rate) uniaxial compressibility, as measured in the labora­tory under simulated in-situ stress and stress-ratio conditions. This is in agreement with Point 8 discussed earlier.

Detennination of the Nonnalization Stress, The normalization stress, (lZ,OI is determined in the following way. The normalized

SPE Formation Evaluation , June 1988

0 .050

0 .04'

0.040

0.035

o

0.005

00

0 8 0

o o

o

o 0

O.OOO+----r-----,----,-----r------. 0.0 50.0 1000 150.0 200.0 2500

1Ic • . , (lib) (1Qw)

Fig. 10-b v •• lIcm,o(\8b) for a large .. t of sandatone sample ••

a'ii IUz,o o.a,--------------------::r--,

0 .•

1.0

II

I.

"

0.1 o.t 0.' 0 .•

[(::~::!))' .,]

Fig. 11-0'1,,10'1,0 as I function of 11.0' rIO' 1,' determined graphl. cilly from the generll completion curve.

:--~----~~~~~ I I I I I I I .. Aa I,oe

C7z.i

Fig. 12-Effect of overconsolidation.

-_0",

stress az.i/az.o corresponding to the initial in-situ state must satisfy Eq. 1; hence,

(4az!az•o) I [ uz(dep) Jb } (az./a

z.o

) ={, Uz(geol) -1. . . . ... ...... . . . .... .. . (6)

Fig. 9 shows the general compaction curve. For a given Point A on the curve, the denominator and the numerator of the left side ofEq. 6 are indicated. Hence, the right side ofEq. 6 can be calcu­lated. This calculation has been made for every point of the general compaction curve. The result is shown in Fig. 11, and can be ap­proximated by the following algebraic fit:

az.; az 0= , . .•.... . •...... . .. ..... . . ... (7)

, 1.4-0.804yo.314

with

y=[Uz(dep)/Uz(geol))b -1. ...... .. . . . .. ............... (8)

Effects of overconsolidation (as a result of previous deeper burial) can, in a frrst approximation, be taken into account by replacing az.; in Eq. 7 by az.; +4az.oc (see Fig. 12).

Relation Between Reservoir Compaction and Surface Subsi­dence. It is realized that the observed nonlinear compaction be­havior might have some influence on the relation between reservoir compaction and surface subsidence. To evaluate this fully, a three­dimensional rate-type compaction equation would be required, but such an equation is not available currently. The magnitude of this effect is expected to be limited, however. This suggests that elastic models, based, for example, on the nucleus-of-strain approach dis­cussed by Geertsma in Ref. 5, can still be used to calculate subs i-

352

o '. ' . .....

Start of depletion ,

Abandonment condition

•

". . ............. .

..................... Unear extrapolation .... ......

o

..•..... .•....

............ ".

". .... ' . .... ....

........ .....

'.

........ of early field

measurements

Real field behavior (KSEPL's model)

" . .•.•....

Prediction according to ....... . classical linear model using

" . LBb-measured compressibility ..•..

'. RESERVOIR PRESSURE DROP -

Fig. 13-Consequences of nonlinearity from loadlng·rate ef· fects for the prediction of field compaction.

dence from compaction. This assumption is supported by the field evidence discussed later.

Comparison Between the RTCM and the Linear Compaction Model. The new compaction model is more complex than the linear compaction model. The linear model is still applicable for a worst­case prediction, because in practice the compaction according to the linear model is more than that predicted with the new model as a result of the relatively small amount of compaction in the tran­sition zone. A schematic comparison of an old and a new predic­tion is given in Fig. 13, where the extrapolation of early compaction or subsidence measurements is also included. It is clear that such an extrapolation will lead to an underestimation of ultimate com­paction and subsidence. Use of the linear model, on the other hand, will lead to too pessimistic an estimate. For an accurate predic­tion, therefore, application of the new model, although more elaborate, is recommended.

Comparison With Actual Field Data To test the applicability of the RTCM to field conditions, a com­parison has been made between the observed field behavior and that predicted by the RTCM for two unconsolidated field cases, namely the Bolivar coast reservoirs and the Wilmington field.

Bolivar Coast Reservoirs. The nonlinear compaction and subsi­dence behavior of the Bolivar coast reservoirs has been the subject of several studies. 13,14 In Ref. 13, the concept of "threshold com­paction behavior" was introduced to describe the nonlinear behavior observed. The physical cause of the increase in width t:.az of the transition zone with increasing depth (see Fig. 14), however, could not be explained at that time. In Ref. 14, the nonlinear compaction behavior was attributed, without detailed consideration, to pressure­lag effects.

An explanation of the observed compaction behavior is obtained by using the RTCM. For example, when the Bachaquero compaction

SPE Formation Evaluation, June 1988

5

4

t.h 3 (%) h 2

5

4

t.h (%) 3

h 2

0

5

4

t.h %) 3

h 2

o

4

t.h (%) 3 h

o

JJ - 6

DEPTH (FT)

2400'

600 1200 6P (PS;)

MM-6

3010'

600 1200 A PIps;)

PP-6

3860'

600 1200 A PIp,,)

RR-6

4360'

600 1200 AP(ps;)

Fig. 14-Bachaquero, field data: compaction behavior at different depths (after Ref. 13). 4h = compaction; h = Initial gross thickness.

behavior of Block PP6 is normalized and plotted according to the procedure described earlier, good agreement with the laboratory­derived general compaction curve is obtained (Fig. 15). The in­crease in width of the transition zone with depth is described by Eq. 1. Similar results were obtained for other Bolivar coast fields. Finally, the observation made in Ref. 14 that after the transition zone the field-observed compressibility becomes comparable with the laboratory-measured first-cycle uniaxial compressibility is in good agreement with the RTCM (see Fig. 13).

Wilmington. The field subsidence behavior of Wilmington is strongly nonlinear. 15 Application of a linear compaction model re-

-0.1 E~/lt",'OO'\D)

'\,. '\,. ... '\,. . '\,.

'\,. '\

:-. 0.'

~

- IENVlAL. CO .. "crtON CU"Y[

• ."CHAQUEftO f'1£LD DATA 0.2 IlOac pp-a (IN-SITU CO,. ... CTIONI DEPTH 3.'0'

-1'·1 .• 0.'

0.' 0.' I .• 1.1 1.2 I.' I.'

Fig. 15-Comparlson between general compaction curve and Bachaquero field data.

-0.1 "'I/{C",,oO"r •• ,

'\,. '\,. ... '\,.

'\,.

\. '\

0.1 ~ ~

'£ME"AI.. COM~ACTIOH CURVE

0.2 • WilMINGTON SUeSID£NC( OF TYPICAL I[NCHWAltK

·a/-I .• ... 0.' 0.' 1.1 '.2 ...

Fig. 16-Comparl80n between general compaction curve and Wilmington field data.

suIted in too large a subsidence prediction. 16 Furthermore, the ob­served subsidence bowl is much more restricted to the central region of the field than predicted. A nonlinear model of the cap type had to be introduced to describe the observed subsidence behavior 17 properly.

The width of the transition zone taken in the cap model is roughly equal to the value predicted by the RTCM. Moreover, the RTCM predicts that the outer region of the field, which is less depleted than the center, will be compacted much less because of the non­linear form of the general compaction curve. As in the Bolivar coast fields, the observed reservoir compressibility after the transition zone becomes more or less equal to the laboratory-measured first­cycle uniaxial compressibility,I7 as predicted by the RTCM.

As illustrated in Fig. 16, good agreement exists between the nor­malized subsidence behavior of a typical benchmark (derived from Ref. 15) and the general compaction curve of the RTCM.

Discussion. The above field data suggest that rate effects are also present under field conditions, and that the RTCM can certainly be applied in the field for unconsolidated reservoirs. Additional work is currently being carried out to investigate its applicability to more consolidated reservoirs.

Conclusions 1. Loading-rate effects in the compaction behavior of reservoir

rock are much more important than has previously been recognized. 2. Changes in loading rate within a loading cycle, creep, and par­

tial unloading strongly influence the compaction behavior. 3. By applying an appropriate normalization procedure, the com­

paction behavior after an increase in loading rate, creep, or partial unloading for various unconsolidated rock types and at various stress

levels can be described with one general normalized compaction curve. Hence, unconsolidated reservoir rock behaves as a rate-type material.

4. The observed effects are described by a constitutive equation of the rate type. This equation can be derived from a model in which rate-dependent intergranular friction is taken into account. For uniaxial conditions, this equation has a form similar to an empirical constitutive equation of the rate type postulated earlier in soil­mechanics literature.

5. Good agreement was found between predictions of compac­tion based on the RTCM and observed field behavior for two differ­ent well-documented field cases. For these unconsolidated reservoirs, the RTCM explains the apparent discrepancies observed in the past.

6. In the laboratory, loading-rate effects have also been meas­ured on consolidated rock samples. It is still being investigated whether these effects are also present in situ-i.e., whether the RTCM is fully applicable to consolidated reservoirs.

Nomencl.tur. a ... constant used in extended rate-type compaction

equation (Eq. B-6) A = constant used in Dieterich's friction equation

(Eq. A-2) b ... material constant describing influence of loading

rate on compaction (see Eq. 1) cb.o = bulk-volume compressibility along virgin

compaction curve, Pa -I [bar - I ] C m = uniaxial compressibility, Pa -I [bar - I ]

cm•o = uniaxial compressibility along virgin compaction curve, Pa -I [bar-I]

cm.o(field) = uniaxial compressibility finally reached in field, Pa- I [bar-I]

cm.oClab) = uniaxial compressibility on virgin compaction curve in (first-cycle) constant-loading-rate laboratory compaction experiments, Pa - I [bar-I]

C r = rock matrix compressibility, Pa -I [bar -11 C = constant used in Dieterich's friction equation

(Eq. A-2), seconds- I

D 1,D2,D) = compressibilities used in Appendix A, Pa- I [bar-I]

e = total compaction (see Appendix A) e = time derivative of e, seconds-I

e' = time derivative of e, seconds-2

en = matrix compaction component (see Appendix A) e s = compaction component as a result of

intergranular sliding (see Appendix A) f = auxiliary function (see Appendix C) F = function of e (Eq. A-4) h = sample height (laboratory) or reservoir thickness

(field), m eft] !MJ = change in sample height (laboratory) or change

in reservoir thickness (field), m eft] H = general compaction curve of Fig. 7 (see

Appendix C) p = pore pressure, Pa [bar] p = time derivative of p, Pals [bar/sec]

IIp = change in pore pressure, Pa [bar] Rc = ratio of rock-matrix compressibility to rock-bulk

compressibility t = time, seconds u = integration variable used in Eq. B-2 U = auxiliary variable (see Appendix C) y ... auxiliary variable given by Eq. 8 Y = Ez' seconds-I ex = parameter in Eq. B-2

fZ = vertical strain as a result of compaction=llh/h

E; = translation of Ez• such that E; =0 at Clz =Cll •O

Ez = time derivative of Ez• seconds- I

E; = time derivative of Ez• seconds- I

Ez.creep = vertical strain as a result of creep IlEz = change in Ez

I' = intergranular friction coefficient 1'0 = rate-independent part of the intergranular

friction coefficient CI ii = trace of stress tensor. Pa [bar] Cln = normalized stress CI/Uz•o un! = value of Cln at end of depletion Un; = value of Cln at start of depletion CIt = vertical effective stress, Pa [bar]

ClZ.! = vertical effective stress at end of depletion, Pa [bar]

Clz.; = vertical effective stress at start of depletion, Pa [bar)

Clz•o = normalization stress, Pa [bar) Clz.1 = vertical effective stress at which loading rate

increases, Pa [bar) Clz.2 = Clz.! +IlClz' Pa [bar) IlClz = distance between constant-loading-rate

compaction curves, given by Eq. 1, Pa [bar) lIz = loading rate (time derivative of Clz ), Pals

[bar/sec) "z.! = value of lIz before increase in loading rate, Pals

[bar/sec) Uz.2 = value of Uz after increase in loading rate, Pals

[bar/sec) "z(dep) = depletionalloading rate, Pals [bar/sec] uz(geol) = geological loading rate, Pals [bar/sec]

c7z(1ab) = laboratory loading rate, Pa/s [bar/sec] II; = time derivative of "z' Pa/s2 [bar/sec2]

Acknowledgment. We thank R. Fernandez-Luque, J. Geertsma, G. Gudehus, A.C. van der Viis, A.J.T. Grimberg, and K. Mess for stimulating dis­cussions; I.D. de Graaf and W. Moor for carrying out the meas­urements; I.F.C. van Kooten for critically reading the manuscript; and the management of Shell Research B. V. for permission to pub­lish this paper.

Relerence. 1. Proc .• Intl. Symposium on Land Subsidence. lASH and UNESCO

(1969) Publication 89. 2. Proc. , Forum on Subsidence Due to Fluid Withdrawals, Checotah, OK

(1982). 3. Schoonbeek, J.B.: "Land Subsidence as a Result of Natural Gas Ex­

traction in the Province of Groningen. " paper SPE 5751 presented at the 1976 SPE European Spring Meeting, Amsterdam. April 7-9.

4. Chilingarian. G.V. and Wolf, K.H.: Compaction o/Coarse-Grained Sediments. Amsterdam (1975-76) I, 2.

5. Van Kesteren, J. etat.: "The Analysis of Surface Subsidence Resulting from Gas Production in the Groningen Area, The Netherlands. " Trans .• Royal Dutch Soc. of Geologists and Mining Engineers. Nederlandse Aardolie Mij. B.V. (ed.) (1973) 28.

6. Biot, M.A.: "General Theory of Three-Dimensional Consolidation." J. Appl. Phys. (1941) 12, 155-64.

7. Tenaghi. K.: Theoretical Soil Mechanics. John Wiley Publishing Co., New York City (1943) 510.

8. Van der Knaap, W.: "Nonlinear Behavior of Elastic Porous Media," Trans., AIME (1958) 216, 179-87.

9. Van der Knaap. W. and Van der Viis. A.C. : "On the Cause of Subsi­dence in Oil Producing Areas, ,. Proc .• Seventh World Pet. Cong., Mex­ico City (1967) 13, 85-95.

10. Holzer, T.L.: "Preconsolidation Stress of Aquifer Systems in Areas of Induced Land Subsidence, " Water Resources Research (1981) 17, 693-704.

11. Martin. J.C. and Serdengecti. S.: "Subsidence Over Oil and Gas Fields." GSA, Reviews in Engineering Geology (1984) 6, 23-35.

12. Helm, D.C.: "Field-Based Computational Techniques for Predicting Subsidence Due to Fluid Withdrawal," GSA. Reviews in Engineering Geology (1984) 6, 1-22.

<:Pt:' P ..... --... ,.., ,, : ......... t:' . ... , •• ~ .. : _ _ T ..... _ '('10 ('1

13. Merle, H.A. et al.: "The 8achaqucro Study-A Composite Analysis of the Behavior of a Compaction Drive/Solution Gas Drive Reservoir, ., paper SPE 5529 presented at the 1975 SPE Annual Technical Confer­ence and Exhibition. Dallas, Sept. 28-Oct. 1.

14. Schenk. L. and Puig, F.: "Aspects of Compaction/Subsidence in the Bolivar Coast Hcavy Oil Fields. Highlighted by Performance Data of the M-6 Project Areas." hoc., 1982 Forum on Subsidence Due to Fluid Withdrawals. Checotah. OK (1983) 1()9-20.

15. Lee, K.L.: "Subsidence Earthquake ata California Oil Field," Proc., 1978 Inti. Conference on Evaluation and Prediction of Subsidence, Pen­sacola Beach, FL, 549-64.

16. Kosloff, D. and Scott, R.F.: "Finite Element Simulation of Wilming­ton Oil Field Subsidence: 1. Linear Modelling." Tectonophysics (1980) 65.339-68.

17. Kosloff. D. and Scott, R.F.: "Finite Element Simulation ofWilming­ton Oil Field Subsidence: n, Nonlinear Modelling." Tectonophysics (1980) 70, 159-83.

18. Prokpovich, N.P.: "Tectonic Framework and Detection of Aquifers Susceptible to Subsidence," Proc., 1982 Forum on Subsidence Due to Fluid Withdrawals, Checotah, OK (1983) 33-44.

19. Smits, R.M.M. and de WaaI, J.A.: "A Comparison Between the Pressure-Lag Model and the Rate-Type Model for the Prediction of Reservoir Compaction and Surface Subsidence," SPEFE (June 1988) 3.57-63.

20. de WaaI, I.A.: "On the Rate-Type Compaction Behaviour of Sand­stone Reservoir Rock, .. PhD dissertation, Delft U. ofTechnology, The Netherlands (May 1986).

21. Truesdell, C. and Noll, W.: "The Non-Linear Field Theories of Mechanics," Handbuch der Physik, Sprilller Verlag (196.5) me, par. 36.

22. Kolymbas, D.: "Ein nichtlineares viscoplastisches Stoffgesetz filr Boden," MS thesis,Inst. filr Bodenmechanik, U. of Karlsruhe (1978).

23. Dieterich, I.H.: "Time-Dependent Friction and the Mechanics of Stick­Slip," Pageoph (1978) 116, 790-806.

24. Teufel, L.W. and Logan, J.M.: "Effect of Displacement Rate on the Real Area of Contact and Temperatures Generated During Frictional Sliding of Tennessee Sandstone," Pageoph (1978) 116,

25. Geertsma, J.: "The Effect of Fluid Pressure Decline on Volumetric Changes of Porous Rocks," Trans., AIME (1957) 210, 331-38.

26. Teller, 0.: Taschenbuch tier Mathematik, Lindauer Verlag, Munich (1965).

Appendix A-Derivation 01 a Rate· Type Compaction _quatlon It is assumed that the influence of the complicated statistical grain deformation and grain-movement processes during compaction can be described by writing the total compaction as the sum of two com­ponents: a matrix compaction component, e,., taken to be propor­tional to the mean hydrostatic total stress, and a component e, from intergranular sliding, taken to be proportional to the average inter­granular maximum shear force (which is taken to be proportional to the mean hydrostatic effective stress). Hence,

According to Dieterich,23 the friction coefficient ~ in this equa­tion is coupled to the strain rate by

~=~o+A In(l+Cle) . ....... ..... .. .... . ...... . .. (A-2)

Substituting Eq. A-2 into Eq. A-I and differentiating with respect to time (while neglecting e2 with respect to Ce, which is valid for all practical cases), yields

D]A .. e=(D I +D3F)(Uij/3 -P)- --(uij/3 -p)~ +D1P, .. . (A-3)

iJ.o e where

D3 =D21lo .................................... (A-4a)

and

F=I+~ln(I+~) ............................ (A-4b) Jl.o e

Hence, F is a slowly varying function of e.

For constant-loading-rate experiments, we assume that the e· term in Eq. A-3 is negligible. Hence, for those experiments

e=(D I +D3F)(uii/3-P)+D1P . ................... . (A-5)

Comparison with Ref. 5 yields

Dl +D3F=Cb.o and Dl =cr . ....•..... . ...•....... (A-6)

Substitution of Eq. A-6 in Eq. A-3 yields

where

A b=- . ...................................... (A-8)

~aF

The function b is a slowly varying function of e.

Appendix a-Mathematical Formulation 01 the RTCM General Solution of the Rate-Type Compaction Equation. For uniaxial laboratory compaction experiments at atmospheric pore pressure and with cr""O, Eq. A-7 reduces to the following rate­type compaction equation:

E: i; "z=--+buz-;- . ..... . .. . . . .. . .... . ... .. . . .... (B-1)

cm•o E:

Substituting Y(t) =iz' we obtain a particular case of the differ­ential equation of Bernoulli. From its solution,26 the following par­ticular cases have been derived.

Shape of the TransItion Zone. In the case of constant loading rate, one obtains

E~ J tlt/tlt •o( 1 ) -.--..:=.- = du, . .. . ............ (B-2) C 11I.0UZ.O 1 1 + au -lib

where

During a conventional constant-loading-rate experiment a=O, which yields the normal constant-loading-rate solution when linear elasticity applies. From the numerical solution of Eq. B-2, an ex­cellent fit to the general compaction curve is obtained with a=4.56 and b=O.056 (sec Fig. B-1). Hence, the general compaction curve is a constant-loading-rate solution of the rate-type compaction equa­tion (Eq. B-1).

Creep. For creep experiments, one obtains

where Ez(O) is the initial creep rate after the interruption of load­ing (where t:cO). Such logarithmic creep behavior is indeed con­firmed by experimental data.

Distance Between Constant-Loadlng-Rate Compaction Curves. For infinitesimal stress changes we obtain from multiplying Eq. B-1 by dt and substituting AEz "'0,

dUt =buzdOn Ez) .. . ... . ....................... . . . (B-4)

- 0.1 .. './(C ..... CT • •• )

1 "-"-

0.0 "-"-

"-"-~

0 .1 ~

- O[H(IItAL. C;OWACTI ON cu .. ..,[

0.' • __ 0.,.1S CAL.CUUT{O US ING [ooA,,,* 82

0.' 0 .1 0.' 1.0 1.2 I.' I.'

Fig. B-1-Comparlson of theoretical and experimental general compaction curves.

For the virgin compaction curves, we have approximately (neglect­ing the nonlinearity in the virgin compaction curves for the pres­ent purpose) E, =Cm •o c7z . Neglecting the small loading-rate dependence of cm.o• one then obtains

U,.2/U,.1 =(iI,.2/ilz.l)b . ........................ .. . . (B-5)

Substituting dUz =u,.2 - u,.I' we obtain Eq. 1. Hence, Eq. 1, which was obtained empirically by fitting the re­

sults of variable-loading-rate experiments. is obtained here from the theoretical model. This agreement between theory and experi­mentsfurther supports the hypotheses underlying the physical model.

Extensions of the Rate-Type Compaction Equation. The value b=O.02 given in Table 1 for unconsolidated sands agrees with the value of A reported by Dieterich (see Ref. 23 and Eq. A-8) . The b value obtained from fitting Eq. B-2 to the general compaction curve differs from that in Eq. B-5. and from the value of b required to fit the creep experiments. This indicates that Eq. B-1 cannot be integrated through changes in loading rate, which suggests that a term containing iI; is missing in Eq. B-1. At first sight, an ad hoc

356

solution is obtained by adding an additional term to Eq. B-1, which results in

. uz;'; Ez ulE; u, +a-- = --+ b-- . ..... .. . . . ..... .. .... (B-6)

ilz Cm•o El

Multiplying Eq. B-6 by dt and taking into account that during a sudden change in ill' duz =dEz =0, yields

a d(ln Ez)=-d(ln ill)' .... ... ... ......... .. .... . ... (B-7)

b Eq. B-6 properly describes the compaction behavior except for the case of creep, because under creep conditions Eq. B-6 yields Ez.creeP' =0, contrary to experimental evidence. This problem has to be investigated in more detail.

Appendix C-Algebraic Description of the Oeneral Compaction Curve The following algebraic fit describes the general compaction CUf'\~ of Fig. 9 with sufficient accuracy.

For U" <0.80,

For U" >0.80.

H(un) = -0.OlO+/(U)2.969U3.485 +[I-/(U)](U-O.30I.

with U=un-O.80 andf(U)=e - (Ulo.43)7.s8

51 Metric Conversion Fectors bar x 1.0* E+05 Pa

ft x 3.048* E-Ol m in. x 2.54* E+OO cm psi x 6.894757 E+OO kPa

'Converslon factor is exact. SPEf"E

Original SPE manuscript recetved for revi_ Sept. 22. 1985. Paper acx:eptea lOt DuDOot.'" May 22. 1987. Revised manuscript received Sept. 16. 1987. Paper (SPE 142"'''''' ~ ed at the 1985 SPE Annual Technical Conference and Exhibition held in Las VeQaO S-22- 25.

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