SPE 03730 Geertsma

Land Subsidence Above CompactingOil and Gas ReservoirsJ. Geertsma, SPE-AIME,Koninklijke/ShellExploratieen Produktie Laboratorium

IntroductionDuring the last 20 years, the Royal Dutch/ShellGroup has conducted extensive investigations intothe phenomenon of reservoir compaction and sub-sidence. These have included research projects tostudy subsidence above Bolivar Coast oil reservoirsin Venezuela and to examine the huge Groningengas reservoir in The Netherlands.

The latter investigation was conducted by a teamof specialists from both the Koninkh]ke/Sheii Ex-ploratie en Produktie Laboratonum (KSEPL) andB\’ Ncderlandse Aardolie Maatschappij (NAM), thelatter being the producing company owned jointlyby Sheii and Esso. Detaiis of the Gmniiigeii irmsti-g~!jo~ are pubiished eisewheres’-zs but as it mayhave consequences for other operating companiesworking in lowland and other subsidence-prone areas,

~-:~-- ka.- tha e.tt. ac mf cllhcirlence ahn~eW~Silaii CO II MUGL JJLIL L,IL, +Uu. vo “1 .Uwu . . . . . ..-w --- -

hydrocarbon-producing reservoirs in a more generalway, and review the state of the art of its prediction.A simple method will be presented for estimatingthe order of magnitude of both compaction andthe accompanying subsidence. Application of thismethod. which can be used to explore the need foran investigation in depth, requires hardly any spe-cialist knowledge. The objective is twofold: to dem-onstrate that land subsidence due to hydrocarbonproduction seldom leads to serious subsidence, andto pinpoint the few potential problem areas.

Earlier Field ObservationsThe literature on subsidence deals mainly with a few

notable examples, such as the Goose Creek oil andgas field in Harris County, Tex., where dramaticsubsidence occurred between 1918 and 1925,”2 andthe Wilmington field below Long Beach, Calif.,3-swhere almost 10 m of subsidence was experienced in1960. Further subsidence could be avoided in thislatter case after unitization and pressure maintenanceas a result of water injection. More recently, a searchfor adciitionai, documented surface depresshs overoil and gas fields in the U. S. was reported by Yerkesand Castle.s This search revealed only a few othersignificant cases, mainly fields close to Wilmington,-..~h . +kfi.~ .* R,,-ma victa Hllntinotnn Reach ~nd>W & tIIU& UL -w.- . w..., . . - . . . ...@----- —--- ,

Inglewood. From this concentration of subsidencebowls, it may be inferred that such events are some-how related to a similarity in reservoir conditions..

~h~ii has been confronted only once with a majorland-subsidence problem. It is related to the produc-tion of oil and gas in Venezuela, where subsidenceabove a number of important oil reservoirs borderingLake Maracaibo is a constant phenomenon. and hugedykes have been built to protect the coastal area fromflooding. Its cause is discussed by Van der Knaapand Van der Vlis.’ Subsidence data for oil and gasfields outside the Americas are very scarce indeed.OkumaraS and Hirono’ describe a case from theNiigata district of Japan that results from the produc-tion of methane dissolved in water. In Italy, AGIPhas been accused of contributing to subsidence inthe Po Delta by producing from a number of gasfields. However, this area is also plagued by a number

Notable subsidence above producing oil and gas fields is the exception rather than therule. A simple procedure is outlined to single out the exceptional but real problemareas. This exercise in potential-problem analysis shows that the huge Groningen gasfield in The Netherlands is a candidate jor causing subsidence troubles in a lowland area.

734 U& ’7’3 JOURNAL OF PETROLEUM TECHNOLOGY

of other subsidence-generating conditions.From all the field evidence collected so far, after

eliminating obvious hunches and speculations, wemust conclude that some or all of the following con-ditions are fulfilled when considerable subsidence is-1.”,....,..a ..L-..- -.,-.A..:.m-mhx,,r..r..-.nwl.n-.*. *-,,+...UU3G1 VGU auuv G pl UuuviIlg llyulubaLuull LGaL1 Vulla.

1. A significant reduction in reservoir pressuretakes place during the production period.

2. Production is effected from a large verticalinterval.

3. Oil or gas, or both, are contained in loose orweakly cemented rock.

4. The reservoirs have a rather small depth ofburial.

The major productive zones of the Wilmington oil,. ..,.nem, lor instance, cover seven stacked intervais dis-tributed over a vertical section from 500 to 2,000 mbelow surface. Oil and gas are produced from sandsof varying thicknesses and degrees of consolidation,interbedded with layers of shale or siltstone. Theadjacent Inglewood oil reservoir produces from Mid-dle to Upper Pliocene sands over a depth range of300 to 1,000 m. Commercial production from theGoose Creek field originated at depths between 350and 1,400 m from unconsolidated sands and claysconstituting a productive interval more than 300 mthick. The Lake Maracaibo reservoir rocks are post-Eocene loose sands interbedded with clay. The aver-age depth of burial of these reservoirs is 1,000 m.

“v’an tier Kriaap ana van aer v ns3 “-- ‘-”- ““-7 iiaVe cunse--- —--

quentiy already concluded that subsidence is the re-sult of reservoir compaction. Furthermore, only looseor weakly consolidated rocks seem to be candidatesfor considerable compaction,

However, we must be careful first to unravel allthe factors contributing to reservoir compaction.

Estimating Reservoir CompactionReservoir compaction or a reduction in reservoirvolume is primarily the result of a reduction in reser-voir height. Provided their lateral dimensions arelarge compared with their height, reservoirs deformpredominantly in the vertical plane.’” Formation com-p~ction can therefore be conveniently characterizedby the vertical strain in the reservoir. ., = d:/:, dur-ing production, which expresses the change in height(relative to the initial height) caused by an increasein effective stress due to a reduction in reservoir orpore pressure, p, under constant overburden. A uni-axial compaction coeflkient, c.,, can then be defined

.L r. ——-.!-—-- —---- :_- --- ..-:4 -1 -------:- ---,.ZN WE IOHIKiUOn LXJHIJJW1l UI1 p~l U1ll L (Xldllgc 111 pulc-

pressure reduction:

~ /i-

cm=-F- ;; ’or c:=c~,dp. . . . (ij

(The relationship between cmand other, better knowndeformation properties, such as rock bulk compres-sibility and Poisson’s ratio, will be discussed later.)The total reduction in reservoir height can then beexpressed as:

H PfAH= f fcti, (p, z)dpdz. . . . . (2)

o p,

Owing to the very nature of the structure of reser-

voir rocks, cm is usually not a constant but a functionof effective stress, and thus also of A p = PJ — pi,

the difference between future and initial reservoirpressure. However, in many instances it is quite pos-sible to assign a fixed value to the compaction co--w-;--+ f-. th- n-,m..t,ra Ymmm,m mrr.wail;nm Allr;nuGI1lLIL1l L AU1 ~~fl~ lJ~~-~1~ 1 at~e yl~ v -~R~~% --~,.,6

production. Under these circumstances, Eq. 2 simpi-fies to

AH= FGdZ)Ap(Z)d Z. . . . . . (3)6

This formulation enables us to recognize three indi-vidual influences on reservoir-compaction behavio~(1) the reduction in reservoir pressure, (2) the verti-cal extent of the zone in which pore-pressure reduc-tion takes piace, and (3) the order of rrragrritude ofthe relevant deformation property of the reservoirrock. Three of the contributions presumed character-istic for subsidence in the previous section are there-fore also apparent in this general formulation offormation-compaction behavior.

These elementary considerations show immediatelythat the combination of a large productive interval,or stack of smaller productive intervals, and a largedrop in reservoir-fluid pressure in unconsolidatedformations may lead to large compaction. On theother hand, a sizable degree of compaction can beexpected even in hard rock for the particular con-ditions of large pore-pressure reductions and a suffi-—,.—.,..,———-———J..-:-- :-. -— .-1clerruy large pruuuc]ng mivrva.

It is well known that the reduction in reservoirpressure as a function of place and time depends onmany factors, such as the mobility, volubility, density,and compressibility of the various pore fluids, as wellas on the reservoir boundary conditions (faults, edgeor bottom water, etc.). Gas reservoirs show a simplerbehavior than most oil reservoirs. In many cases thedrop in reservoir pressure from the start of the pro-duction period until abandonment is very small. Inother instances. particularly in gas and oil reservoirsthat produce mainly under the influence of a solution-gas drive. the pore pressure reduction may be con-side rable. ” In a gas reservoir, the rate and degreeof pore-pressure reduction depend on the permea-bility distribution, the location of the productionwells, and the production rate in relation to the rateof encroaching edge or bottom water. Reservoir simu-lators are of great help nowadays for predicting pres-sure distributions as a result of alternative produc-tion policies.

T.. +ha P..,w.;nmmn m.. A,31A arlm= ,.,. +-. l-wm”f=il-=t.=c111 L1l G VLU1llll~bll ~a> lZbl U, b-~ti WCL%bl rJti LBbLfi Utti O

into the reservoir, mainly from the north, far tooslowly to maintain the original reservoir pressure. Byabout the year 2000, iarge parts of the resemmir ‘W-WIhave experienced a pore-pressure reduction of some300 kg/cm’. The producing interval is large, varyingfrom 90 m in the south, where the production wellsare presently concentrated in clusters, to more than200 m in the north. These figures were sufficientlylarge to warrant an investigation of the possibility ofcompaction, the crucial parameter being the com-paction coefficient.

The compaction coefficient depends on a numberof factors, such as rock type, degree of cementation,

JUNE, 1973 735

porosity, and depth of burial. From a mechanicalpoint of view, it is the number of contact areas be-tween the individual rock particles and in particularthe size and shape of these contacts, that controls rockdeformation in sandstones. Similarly in limestones,the shape and strength of the rock skeleton determinerock deformability. Porosity is one of the factors thatis frequently influenced by these mechanical contactconditions, but it is certainiy not the only one. Effec-tive stress also influences the deformability of therock skeleton, which accounts for the nonlinear re-

lationship between stress and strain.The lowest compressibility for sandstone forma-

tions is 0.16 x 10-5 cm’/kg, the compaction coeffi-cient of pure quartz. The lower limit for limestonescorresponds to the value for calcite; i.e., 0.08 X 10-5cm:/kg. For sandstones, a rough classification canbe made in terms of the degree of cementation: hard,well consolidated, friable or semiconsolidated, and

----- - 1... A.m--- f.f -mm m+ ti*n rlef.ve~~~~,iousc said. As hlc UC51GG “L ~ti.l.e,..a..v.. _-v. wthere is a gradual transition in the deformation be-havior from elastic to cataclastic. Elastic behavior is

1—+++——+

IiIllporosity smles

-

Tit o Ond well -Consolkkzted

e roc& m 71 I

fig. ]—Uniaxial compaction coefficient, c., (vertical axes) for sandstone reservoirs. Effective verticalstress range U. = 100 to 200 kg/cm’, corresponding to depth of burial of 1,000 m

for normally pressured reservoirs.

736 JOURNAL OF PETROLEUM TECHNOLOGY

basically a reversible process, which implies a com-plete “rebound” during unloading. Catalastic defor-mation, on the other hand, involves crushing of theparticles or other constituents of the framework andis therefore an irreversible process; damage occurs ata number of critically loaded locations, and a perma-nently denser packing is thus obtained. In practice,—-... . .it is difficult to determine the elastic: cataclastic defor-mation ratio uniquely; if one could, it would be at-tractive to examine a correlation of this ratio withthe uniaxial compaction coefficient.

A readily measurable rock property is porosity.High porosity values, up to 40 or 45 percent, applyfor loose sands. Hard sandstone formations usuallyhave a low porosity. Even within the above-mentionedcategories of degree of cementation, porosity plays acertain role in delineating the compaction coet%cient.

Figs. 1 and 2 are an attempt to summarize ourlarge amount of experimental data collected so farfor sandstones; they apply to two different preload-ing”z conditions of interest (corresponding, respec-tively, to burial depths of approximately 1,000 and

1(7

cdfk

165

cm?fkg

2

1

0 10 20 30

= md till-cansalidated.=.. - ----- .---. .--.. ——-——● rock

I I I I

4H=#

EEEE15 35

%mi.rnncnlidnt~d

&

1

~1

105

:n?/kg

40 ‘

30-

20 ‘

..::::7.:.:... .. .,., .

Fig. 2—Uniaxial compaction coefficient, cm, (vertical axes) for sandstone reservoirs. Effective vertical-1---- ----- 9nm .- erifi l.- /--? -- ..-.=.-.” .4;-”stress rarlge u. = cwu LU cwu ng/r.111 , bUIIGapUIIUIIIg

hli.; .t -f 2 nnn Mtc depth d Uul l-1 WI *,WWW

for normally pressurized reservoirs.

JUNE, 1973 737

3,000 m). Figs. 3 and 4 provide a similar summaryof data obtained for limestone reservoirs. Additionalfigures can be derived from a series of pore-volumecompressibility data for both sandstones and lime-stones published by Newman.*g A few additional datafor unconsolidated sands have been reported bySawabini et al.” (For the appropriate conversionformulas, see the section on Procedure for a DetailedIrivestigatiorl.)

The use of such data can be illustrated by takingthe Rotliegend sandstone deposit in which the Gron-ingen gas was found as an example. The material inthe Rotliegend reservoir may be described as semi-consolidated. Its depth of burial is about 3,000 m,

and the reservoir rock is subjected in situ to aneffective prestress of approximately 300 kg/ cmz.From Fig. 2 it may be concluded that the compac-tion coefficient must be rather low; somewhere be-tween 1 and 3 X 10-‘ cm’/kg. This is approximatelythe compressibility of a good concrete. A small reser-voir in this type of rock could hardly show any com-paction. However, a combination of the three factors~on~rib[~ting to Co.mPactiQn.— i,e, j nressure reduc-‘.. –tion, height of productive interval, and compactioncoefficient — produce a compaction figure rangingfrom 50 to 150 cm; a figure that cannot be over-looked. It is clear that in this case fear of sizablecompaction, and thus subsidence, could be experi-

0 10 20 30 tl 20 z 30 3s%

porosity smlas

W1l- consoltiuf 1-

105&~/~g

1

0 10 m 30152025303S%

porosityScolss

Well - consohdotcd Im===lFig. 3—Uniaxial compaction coefficient, c~, (vertical

axes) fOr carbonate rock. Effective vertical stressrange C, = 100 to 200 kg/cm’, corresponding

to depth of burial of 1,000 m fornormally pressured reservoirs,

Fig. 4--U niaxial compaction coefficient, c., (vetilcalaxes) for carbonate rock. Effective vertical stress

range O, = 300 to 600 kg/cm2, correspondingto depth of burial of 3,000 m for

normally pressured reservoirs.

738 JOURNAL OF PETROLEUM TECHNOLOGY

enced only after the full reservoir dimensions wereknown in 1964.

If this simple analysis, which takes little time oncethe appropriate reservoir data are available, predictscompaction figures lower than, say, 10 cm, there islittle reason to pursue the matter further. If, on theother hand, larger values are obtained, the compac-tion/subsidence relationship must also be clarified toestablish any consequences in terms of subsidence.

The Subsidence/Compaction RelationshipA first attempt to arrive at a mathematical analysisof subsidence caused by oil-reservoir depletion is dueto McCann and Wilts.” The objective of their studywas a better understanding of the subsidence behaviorabove the Wilmington field. They investigated theconsequences of two different models, both basedon elastic continuum mechanics, labeled the “tensioncenter” and the “vertical pincer” model (see Figs.5A and 5B). Although they concluded that the “ten-sion center” model showed features closely resem-bling field behavior, whereas the “vertical pincer”model did not, these authors did not subsequentlyproceed in the proper direction. Consequently, theydid not arrive at an explanation for the cause of thissimilarity to natural behavior, but instead made pre-dictions for future subsidence on the basis of un-realistic combinations of “tension centers”.

The problem must be treated mechanically as oneof an isolated volume of reduced pore pressure ina porous or nonporous, but elastically deforming,half-space with a traction-free surface. The displace-ment field in this continuum. and thus also that ofits free surface. results from the shrinkage or compac-tion of the inclusion; if the pore pressure decreases,the effective stress on the rock skeleton increasescorresponciingiy throughout ihe ii2CiiiS~Ofi. We ‘wish to

determine the interaction between the shrinking in-clusion and its surroundings to which it is connected.This interaction can be calculated with the heip ofthe theory of poroelasticity, sometimes inappropri-ately called the “theory of consolidation”. This theory

is mathematically similar to that of thermoelasticity.A number of authors’’-” have discussed and appliedthis similarity in the past. The present poroelasticproblem can be analyzed most conveniently with thehelp of the so-called nucleus-of-strain concept in thehalf-space, as introduced by Mindlin and Cheng’6and, independently, by Sen’7 in the theory of thermo-elasticity. From this concept and its results, it followsthat the subsidence — i.e., the displacement perpen-dicular to the free surface due to a nucleus of strainof small but finite volume, V — under the influenceof a pore-pressure reduction, 3P, amounts to

Z&(r,o)= — -!--C,n(l —v)D

T (r, + D,),,, JP~.

. . . . . . . . . . . . (4)

For the elastic deformation constants that can beintroduced into this relationship, we again selectedthe uniaxial compaction coeffic-ient, c,,,, in additionto Poisson’s ratio, ,. The depth of burial of the nu-cleus of strain is indicated b! D, and r is tl:e radialdistance from the vertical axis through the nucleus.

Similarly, the horizontal surface movement, whichmay be of interest if serious surface deformations areto be expected, amounts to

u,(r,O) = + -+- c,,,(1 –,) (r, + ~,),,, Apv.

. . . . . . . . . . . . (5)

Apart from the proportionality constant, which inthe present formulation acquires a clear physicalmeaning, these expressions are similar to those foundby McCann and Wilts for their “tension center”model. It also follows that the ratio of the horizontaland verticai surface ciispiacenierits above s“ucha tiu-c{+=llcof <train amnunts IQ — r/D..,Iw””v. “.. -.. . -.. .-—...

The results of the nucleus-of-strain concept can beapplied to real reservoir conditions in a number of.... . NA..al. :,.4,..-”+;ways. lVIUL1l 111 Lvlllla L1or, car, ~~ ~~i~~~~~ f~~rn. an

analysis of the deformation pattern around a disc-shaped reservoir of thickness H and radius R at depth

%7Ur(r,o]

z r /% 2,0)

D urUz

L

Uz(r,d = + F%iviUrk,o)=++

~= ti 03_; ti = TENSION PER IMT AREA WER SURFACE

2OF TENSION SPHERE;

o =RAOIUS OF TENSION SPHERE

Fig. 5A—Spherical-tension model of McCann and Wilts.”

JUNE. 1973

8;G,,2+%2i--wUz(z,o)=- . .w (r,o)=a r

8- (~2+D2)~ [+ [j2-(1-2v,v- q3;2v) IJM xPAh p= =REmH OF VERTICAL FORCES

Ah= EFFECTWE SEWRATION OF FORCES

Fig. 5B—Vertical-pincer model of McCann and Wilts.”

739

D for a uniform reservoir pressure reduction Ap

throughout the reservoir. Such an idealized reservoirshape has been sug~ested before’S and has beenstudied in more detad by Evangelist and Poggi.*gThe latter authors did not include an explanation ofhow solutions could be obtained in closed form, butinstead provided computer data. To perform rathersimple calculations, one need only assume that thesolution is not too adversely influenced by treatingboth the reservoir and its surroundings as homogene-~~~ with ~eqe~f to their defo~ation properties: bothcm and v must be assumed to be constant throughoutthe entire half-space. With the help of this assump-tion, the subsidence above a disc-shaped reservoircan be found by integrating the nucleus solution overthe entire reservoir volume. z’ After the necessarymathematical manipulations, one obtaiiis

uz(r, O).= –2c., (1 —v)AP~~

7e-Du.1, (a Z?) JO (a r) da, . . (6)6

and

t4, (r,0j= +Zc)), (i —vj ApHR

Ye-D”J, (a R)J, (ar)da , . . (7)b

in \vhich 1,, and .lI are Bessel functions of zero andfirst order. respectively. Numerical values of these“Hankel-Lipschitz integrals” have been reported byEason c’t al.”) After introducing the dimensionless

m

ratios p = r/R and q = D/R and the shorthandnotation

u,(r, o)= —2cn, (l —v)Ap HA(p,7),

u, (r, @ = + 2C,,, (1 .– I)AP~~ (p, q) ,

we can use Table 1 and 2, which list A and B fora selected number of values of p and ~.

It has been shown in Ref. 8 that the first row inTable 1 can also be determined, where A = A (O, q),from the more simple formulation

(u,(o, o)=–2cm(l–v)Ap H 1– q\~ )

. . . . . . . . . . . (6a)

Also, since reservoir compaction amounts to cmAPH,we may write

SubsidenceReservoir compaction =

– 2(1 – lJ)A,

which means + — 1.5 A . . . . (8)

Horizontal surface displacementU*Q.afiIAircnmnartinn

= 2(1 – v)B,.,WOWL.“., w“.... . . ... ..

orz 1.5 B.

. . . . . . . . . . ?: (9)

Thus the ratio between maximum subsidence andreservoir compaction is in essence determined by theratio ~,between depth of burial and the lateral extentof the reservoir. Small, deeply buried reservoirs are

TABLE l—VALUES OF A = R (- J, (a R) J, (a r) e ‘“R da FOR RANGES OF VALUES OF p = r/R AND TI = D/Rb

P

m0.20.40.60.81.01.21.41.61.82.03.0

0.0

1.00001.00001.00001.00001.00000.50000.00000.00000.00000.00000.00000.0000

0.2

0.80390.79830.77890.73490.63010,38280.15440.07170.04000.02490.01680.0042

0.4

0.62860.62010.59240.53770.44330.31050.18710.11010.06820.04490.03120.0082

,,0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 3.0—. —— — ——

0.4855 0.3753 0.2929 52318 0.1863 0.1520 0.1258 0.1056 0.05130.4771 0.3683 0.2876 0.2279 0.1835 0.1500 0.1244 0.1045 0.05100.4508 0.3473 0.2720 0.2167 0.1754 0.1442 0.1202 1.1014 0.05020,4043 0.3124 0.2470 0.1989 0.1628 0.1351 0.1135 0.0965 0.04880.3368 0.2658 0.2147 0.1762 0.1465 0.1234 0.1049 0.0901 0,04700.2559 0.2130 0.1787 0.1510 0.1286 0.1102 0.0951 0.0827 0.04490.1795 0.1621 0.1433 0.1257 0.1103 0.0965 0.0848 0.0748 0.04240.1216 0.1197 0.1120 0.1024 0.0925 0.0831 0.0744 0.0667 0.03980.0829 0.0876 0.0865 0.0824 0.0768 0.0707 0.0646 0.0589 0.03700.0580 0.0647 0.0668 0.0659 0.0633 0.0597 0.0557 0.0516 0.03430.0418 0.0485 0.0519 0.0528 0.0520 0.0502 0.0477 0.0450 0.03150.0118 0.0149 0.0174 0.0193 0.0207 0.0216 0.0221 0.0222 0.0198

TABLE 2—VALUES OF B = R ~ J, (a R) J, (CYr) e ‘“~ d. FOR RANGES OF VALUES OF p = r/R AND ~ = D/Ro

&

0.00.20.40.60.81.01.21.41.61.82.03.0

740

.,0.0 0.2 0.4 0.8 1.0

0.00000.10150.21340.35300.5721

x

0.52350.32930.23380.17670.13900.0580

0.00000.09540.19790.31630.45730.54560.42780.30260.22280.17110.13580.0576

0.00000.08040.16220.24430.31510.34220.30720.24820.19620.15660.12720.0562

0.6

0.00000.06280.12380.17890.21970.23550.22370.19580.16500.13770.11520.0541

0.00000.04720.09170.12980.15700.16930.16660.15350.13580.11800.10180.0514

0.00000.03500.06750.09490.11470.12520.12650.12080.11100.09970.08850.0483

1.2 1.4 1.6 1.8 2.0 3.0—— —— —0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.02590.05000.07030.08540.09450.09760.09580.09070.08380.07620.0449

0.01940.03750.05290.06480.07260.07640.07660.07430.07030.06530.0414

0.01470.02850.04050.05000.05670.06050.06190.06110.05900.05590.0380

0.01130.02200.03140.03910.04480.04850.05040.05060.04960.04780.0346

0.00890.01730.02480.03110.03590.03930.04140.04220.04200.04100.0314

0.00320.0062

0.01170.01390.01580.01740.01850.01940.01990.0190

JOURNAL OF PETROLEUM TECHNOLOGY

.

therefore hardly capable of producing significant sub-sidences, even if their rese-moir compaction cannotbe neglected. In contrast, extermely large reservoirsat large depths may be potential candidates forsubsidence,

In order to illustrate the deformation pattern ofcompacting reservoirs and their surroundings in moredetail, Fig. 6 exaggerates the vertical-displacementdistribution at the surface, as well as at the top andbottom of the reservoir. Two practical ratios q = D/Rare used for this illustration: 1.0 and 0.2. For q = 1.0,subsidence is about 0.45 times reservoir compaction.For q = 0.2, subsidence is much larger for the samedegree of compaction; i.e., 1.20 times reservoir com-paction. It is thus seen that subsidence may exceedcompaction for homogeneous rock conditions, themaximum ratio being 2(1 – v). Figs. 7 and 8 pro-vide details of these deformation patterns.

To date no field cases have been reported of sub-sidence above reservoirs at depths exceeding 2,000 m.However, the theory certainly does not exclude thispossibility. In practice, the chances of severe sub-sidence above deeply buried oil or gas reservoirs are~m.~11for the f~]low@ reasons:

1. The value of uniaxial -compaction coefficientdecreases with increasing effective stress (see Figs. 1through 4). Because reservoirs frequently experiencea hydrostatic pore pressure before production, theoriginal effective stress will increase with depth ofburial of the reservoir, and the degree of compactionwill therefore be reduced.

2. To provide similar q-values, deep reservoirsmust have a larger lateral extent than shallow ones.

On the other hand, deep resewoirs can be sub-jected to a larger ultimate reduction in reservoirpressure compared with what is physically possiblein shallow reservoirs. This means, for instance, thatgiant gas reservoirs are in principle candidates forsubsidence, even if they are situated at great depth.We have come to the conclusion that the Groningengas field, for which q <0.20 and for which ultimatecompaction may reach a value of 1 m, is such a case.

~~~~~~.~~~ f~~ ~ ~c*zJilaA l“”@Q*~~~*i~~WLuna”” . .. v w“..&. . . . . .

A survey ofthe !k!ihm! d both cmnpaction andsubsidence in the way indicated above reveals thatthere are only a few candidates for further analysis,if we take 10 cm as an acceptable but already diffi-cult to analyze subsidence measure. According to

Fig. 6-Compaction and subsidence.

core analyses, NAM’s Schoonebeek oil reservoir mustshow a small subsidence of this order of magnitude.Careful Ievelling surveys indeed show small surfacemovements, but the displacement rate is too smallto aiiow a detailed comparison ‘M-w-eeti‘&eory dpractice.

It is not even justified to take a fixed order ofmagnitude as a universal tolerance limit; the iatterdepends both on the location of the reservoir in re-lation to residential or industrial areas, and on theacceptability of flooding or other surface calamities.The compaction of offshore reservoirs, or reservoirsbeneath the desert or a tropical forest, may have tobe analyzed for reservoir-engineering studies, butprobably not in the light of local subsidence, uniess,for example, a large dam is or will be located in theneighborhood, or an active fault plane may bemobilized.

Nevertheless, our technical survey confirms whathas been observed in practice; namely, that detailedand costly investigations are necessary only in a veryrestricted number of cases. Such an investigation in-volves a series of steps, which we shall outline in thefollowing section.

Laboratory Measurement of the Uniaxial-CompactionCoefficient on Representative Core SamplesThe technical difficulties are related to equipmentdesign, the selection of representative core material,and the interpretation of the measurements.

Equipment Design. The uniaxial compaction of loose,sands and clays can best be measured with the helpof an oedometer-type cell. A great deal of care mustbe taken when cutting the samples from rubber-sleevecores and mounting them in the cell with the leastpossible distortion. The most compressible parts ofthe core material are frequently the most vulnerableduring both recovery and laboratory handling. Wellconsolidated and friable rocks can be studied eitherin a triaxial cell with zero lateral deformation, or ina hydrostatic loading cell. As has been pointed out byTeeuw,” the first method is the more accurate and~1-- -l I_...” -- A.. -a-a” ,.4 Dn;..r,ri$c -If~fi. ~C%~\r~~,dlsu illluw> Iuccxidl Glllwt U1 1 UIS?.W1l s , a.!”,

it is also the more elaborate technique. The secondprocedure can be carried out rapidly in a rathersimple setup and is thus better suited for routinemeasurements. A formula relates uniaxial and hydro-static compaction data:

1c.=—

3 ()~ (l–~)cb, . . (10)

where

Ch= hydrostatically determined bulk compressi-bility.

v = Poisson’s ratio, for which an estimated valueflf 075 to 030 can he used for most res-., ..,--- .- ._.=__ .—----

ervoir rocks. v can also be measured for aselected number of cores in the triaxiai ceil

~ = ratio between rock matrix and rock bulkcompressibility = c,/cb. This ratio can bedetermined for sands and sandstones onthe basis of the c. value of quartz, and for

JUNE, 1973 741

.,

limestones on the basis of the c, value ofcalcite. The larger c~, the smaiier /3 andthus the smaller its influence on c.,.

As a less practical alternative, Eq. 10 can be writtenin the form

C,,, = ~ (1 - p) (11-_2,,”) , . ~ (11)

where G represents the shear modulus.Eq. 10 is also useful to obtain an impression of

the appropriate deformation constant for subsidenceanalysis from rock bulk -compressibilityy or pore-compressibility data published elsewhere, the latterbeing 1/+ times the former, if @represents porosity.Note that because as an average

13-( )

~ (1 – ~) = cn,/cb = 0.5/0.7,

cm is considerabley smaller than the bulk compressi-bility, ch.

Selection of Representative Core Material. The de-gree of homogeneity or heterogeneity of the reservoirmust be established, which means that logs must beexamined and information must be obtained fromreservoir geologists to gain a good impression of~e.ew~ir conditions. For a recently discovered field,many questions cannot be answered in detail untilsufficient wells have been cfriiied and analyzed.

I* -,.. * OICfi h- r.-ali7d that both the least com-11 IIIU>L Lb,a” “w .- ----- - -pressible and the most compressible rock samplesmust be tested in proportion to their relative abun-dance in the reservoir.

Interpretation of Measurements. The measurementsprovide a curve of displacement or strain as a func-tion of effective stress (above natural level). Theoriginal stress level cannot be measured in the field,but must be estimated on the basis of reasonableassumptions. The precise initial loading condition forIhc lahorato~ experiment cannot therefore be estab-

001 2 P“& 3

V*, t Id dlspl.a.,mt s4b~ Cuw&

-1

of Lmtfm/of r69rvar

// Ruarvor ~tm & r:R

Fig. 7—Compaction and subsidence for q = D/R = 0.2and I, = 0.25.

742

lished with certainty.A m~~e Serk?US pd!i?rn, which has yet to be over-

come, is unloading of the core material during corerecovery in the well; thus an unloading/reioadingcycle always precedes a laboratory experiment. Asexplained previously, part of the rock deformationprocess is irreversible, so that we are always dealingwith a somewhat distorted sample. As a consequence,the laboratory experiments probably provide com-pressibility figures for the first loading cycle that aretoo high.

Determination of the Compaction DistributionIn the Reservoir

The compaction distribution can be derived byy@-diciously combining the data obtained using the ap-proach described in the section of Equipment Designwith other relevant reservoir data, which include thefollowing:

1. A map showing the vertical and lateral distribu-tions of the productive zone. The words “productivezone” must be considered here in a somewhat differ-ent context compared with their interpretation inreservoir engineering. The “productive zone” forcompaction analysis includes all zones where thepore pressure will be reduced, irrespective of whetherthey contain hydrocarbons or not.

Z!. A prediction of the reservoir-pressure distribu-tion as a function of piace and time, presiimiablyL-..~ -- .- .nmrnnriat.n nl]mefi~~l analysis (reservoirUC3>cu Ull al, CLyy. ”y .. ...-...-...- -

simulator).Both these items gain in accuracy the better the.,

reservoir nas Deen tmpmcd. I 1113 ,lluall. .Aiw..1. ~---l——.—-. .-1--,. l%:= .no wc that in nrac-

tice the predictions of compaction and subsidencemust be updated periodically when more pertinentinformation becomes available.

3. Attempts have been made to correlate labora-tory compaction data with petrophysical propertiesmeasured or derived from logs. Even a vast numberof core samples represent a very small fraction ofthe reservoir rock, Sonic and formation-density logshave been considered for this purpose, but such at-

mr

Fig. 8-Compaction and subsidence for ~ = D/R = 1.0and I, = 0.25.

JOURNAL OF PETROLEUM TECHNOLOGY

tempts have not been successful.28 The best correla-tion established so far is between compaction coeffi-cient and log-derived porosity values, although thereis still a wide scatter. The sonic log shows a goodcorrelation with compaction for hard rocks, but theseare the ieast interesting for a W@ of coinpactkfiand subsidence. The cataclastic deformation proper-ties of friable and loose rock material are not reflectedin the known petrophysical properties measured ina wellbore.

Determination of the Subsidence Profile

This can be derived from the compaction distribu-tion on the basis of the theory of poroelasticity. Be-cause complex geometries are usually involved, aswell as various degrees of heterogeneity both in andoutside the reservoir, numerical techniques such asfinite-element” or finite-dillerence procedures can beused to calculate the compaction/subsidence relation-ship. To apply these techniques in three-dimensionalspace to any advantage, however, one must be ableto specify the input data with sufficient detail tojustify such a sophisticated approach.

A certain degree of inaccuracy or uncertainty mustalways be accepted in specifying the spatial distribu-tion of compaction within the reservoir. The valueand distribution of the deformation properties of thereservoir surroundings are known with even less accu-racy, as there are fewer core and log data available.For more or less homogeneous conditions, we havedeveloped a method that treats reservoirs of arbitrarythree-dimensional shape and pressure distribution.2GThe analytical concept of elastic deformations causedby nuclei of strain according to Eqs. 4 and 5 can stillbe used in this application, but integration over theentire reservoir volume must be replaced by summa-tion of the effect of a finite number of nuciei of strain,which together represent the reservoir volume. Anindhidual pore-pressure reduction can be assigned toeach nucleus. The summation is performed by adigital computer.

Important aspects of the subsidence/compactionrelationship, such as the ratio between the volumeof the subsidence bowi and the reduction in reser-vrm..-— .-:_

volume, as well as the subsidence profile, are influ-enced by contrasts in deformability between reservoirand underlying formations. For the Groningen gasfield the deformability of the reservoir is rather smalland seems to be of the same order of magnitude asthat of the overlying and underlying formations. Thismeans that, according to the theory, the subsidencevolume may exceed the reduction in pore volume.Conversely, under Bolivar Coast conditions thesevolumes are approximately equai, because here thebasement rock is very stiff compared with the highlydeformable loose sands and clay layers that consti-tute the reservoirs.

The nuclei-of-strain concept can even be appliedto analyze the latter type of conditions, provided thesurface deformations u, (r, O) and u, (r, O) for a nucleusare tabulated from a finite-element analysis of theheterogeneous half-space. There is a restriction inthat the deformation contrasts must be due to hori-zontal layering. Provided this approximates the real

situation, the numerical procedure of summing theeffect of a finite number of nuclei of strain is still avalid one for obtaining the subsidence profile abovea reservoir of arbitrary lateral shape.

Lastly, in problem areas such research work mustk. .,-o -~.~;.~ h~, in-citn cnmnactirm r7_iC~SUrern@SUb aedhllp,.lwu “, 1.. “.. - --------------- ---

in observation wells and levelling surveys during theproduction period. The costs of this field work con-stitute the main expense and it should be carried outonly if the severity of the problem justifies it.

ConclusionsSubsidence is the result of reservoir compaction,which in turn depends on the product of reservoir-pressure reduction, height of productive interval, androck compressibility. The relevant compressibilityfactor is the uniaxial-compaction coefficient, whichvaries between 0.3 X 10-5 cm?/kg for tight rock and20 to 40 X 10-5 cm’/kg for loose sands. The lowerthe effective stress level the higher the maximum pos-sible compaction coefficient for a loose sand. Theamount of subsidence resulting from reservoir com-paction depends mainly on the ratio between thedepth of burial and the lateral extent of the reservoir.For subsidence to equal compaction, a reservoir ata depth of 1,000 m must have a surface area of notless than 50 km’.

In applying these findings to a judgment of thefrequency of occurrence of major subsidence aboveoil and gas fields, it must be remembered that in manyoil reservoirs the reduction in reservoir pressure issmall throughout the production period. Exceptionsare oil reservoirs that produce by means of adissolved-gas drive. Pressure maintenance may be ofadvantage from a recovery point of view, and thereservoir engineer will therefore automatically con-.: s--- : :--.:+- -: ,.,,.+a- --------- +-- -Aifit-”nl nr-ccl)rpslu~r IrlJCCUUll U1 WdLG1 U1 ga~ lu1 ~, u..du. ~.--....-

maintenance. Gas reservoirs can be produced onlyby expansion; water injection may have an adverseeffect on the recovery factor and is therefore usuallyunattractive.

On the basis of these crude generalizations, it maybe concluded that oil reservoirs of the depletion type

. ..J -w+. --I., 1ow-- naQ ract=rvnirc ini~l i~~S~ S~i@. cI1lU GA L1&lluy wl& &O .U-W .V .= . . .

either loose or friable rocks, are most sensitive tosubsidence. Other reservoirs cannot give rise to realconcern in this respect.

NomenclatureCtj= rock bulk compressibilitycm = uniaxial compaction coefficientc, = rock matrix compressibilityD = depth of burialG = shear moduiusH = height of productive interval

AP = pore (reservoir) pressure reductionr = radius

R = reservoir radiusu. = radial displacementus = vertical displacementV = volume of a nucleus of strainz = vertical coordinatep = c,/cb

ez = vertical strain = dz/z

JUNE, 1973 743

.,

~ = D/Rv = Poisson’s ratiop = r/R+ = porosity

AcknowledgmentsProf. U,S. Grant kindly provided Ref. 12, and Prof.G. L. Chierici, Ref. 19. I wish to thank in particularD. Teeuw for providing the details on laboratoxy-compaction measurements, G. van Opstal for stimu-l=tim~ &CllQCifinQm-sfhe subsidence,/compaction re-.u.s..~ . ...-.. ””.-. . . .lationship, and the management of Shell ResearchNV for permission to publish this paper.

References1. Pratt, W. E. and Johnson, D. W.: “Local Subsidence of

the Goose Creek Field,” J. oj Geology ( 1926) 34, 577.2. Snider, L. C.: “A Suggested Explanation for the Surface

Subsidence in the Goose Creek Oil and Gas Field, Tex-as,” Bu//., AAPG (1927) 11, 729.

3. Gilluly, J. and Grant, U. S.: “Subsidence in the LongBeach Harbor Area, California,” Bull., GSA ( 1949) 60,461.

4. Mayuga, M. N. mtd Allen, D. R.: “Subsidence in theWilmington Oil Field, Long Beach, California, U. S. A.,”

m L,.i.mfi Suh5idence, ruvuraiion No. %J, -11.9-U, .LUA lUC.1 TNECCQ

(no date of issue) I, 66.5. Allen, D. R.: “Physical Changes of Reservoir Properties

Caused by Subsidence and Repressurizing Operations,Wilmington Field, California” J. Per. Tech. (Jan., 1968)23-29.

6. Yerkes, R. F. and Castle, R. O.: “Surface DeformationAwocia[ed with Oil and Gas Field Operations in theUnited States:’ Lard Subsidence, Publication No. 88,AI HS-UNESCO (no date of issue) I, 55.

7. Van der Knaap, W. and Van der Vlis, A. C.: “On theCause of Subsidence in Oil-Producing Areas,” Proc.,7th World Pet. Corsg., Mexico City ( 1967) 3, 85.

8. Okumara, T.: “Analysis of Land Subsidence in Niigata,’”Land Sub.ndence, Publication No. 88, AIHS-UNESCO(no date of issue) 1, 130.

9. Hirono, T.: “Niigata Ground Subsidence and GroundWater Change,” Land Subsidence, Publication No. 88,AI HS-UNESCO (no date of issue) I, 144.

!O, c.e~rt~rna, J,: “The Eff@ of Fluid Pressure Decline onVolume Changes of Porous Rocks,” Twm~., AIME( 1957) 210, 331-339.

1I. De Haan, H. J. and van Lookeren, J.: “Early Resultsof the Large-Scale Steam Soak Project in the Tia JuanaField. Western Venezuela.” J. Pet. Tech. (Jan., 1969)101-110: Trans., AIME. 246.

12. McCann, G. D. and Wilts, C. H.: “A MathematicalAnalvsis of the Subsidence in the Long Beach-San PedroAV.=D”’.,.. $-. ~nierna! r.-nnrt. ~alifornia institute of Tech-.- r-. .,nology. Pasadena (Nov., 1951).

13. Lubinski, A.: “The Theory of Elasticity for PorousBodies Displaying a Strong Pore Structure,” Proc.,Znd U. S. Nati. Cong. of Applied Mechtics ( !954)247.

14. Biot, M. A.: “General Solutions of the Equations ofElasticity and Consolidation for a Porous Medium,”J. Applied A4ech. (19S6) 23, 91.

15. Geertsma, J.: “A Remark on the Analogy BetweenThermoelasticity and the Elasticity of >aturded PorousMedia,” J. Met/r. Phys. Solids (1957) 6, 13.

16. Mindlin, R. D. and Cheng, D. H.: “Tbermo-ElasticStress m the Semi-Infinite Solid,” J. Applied Phys.(1950) 21, 931.

17, Sen, B.: %clei of“Note on the Stress Produced byThermoplastic Strain in a Semi-Infinite Elastic Solid;Quarterly Applied Math. ( 1950) 8, 635.

18. Geertsma, J.: “Problems of Rock Mechanics in Petro-leum production Engineering,” Proc., 1st Cong. of theIntl. Sot. of Rock Mech., Lisbon (1966) I, 585.

19. Evangelist, G. and Poggi, B.: “Sopra i fenomeni d!deformazione dei terreni da variazione dells presslone dlstrato~’ Memorie Serie II, Atti dells accademia dellsscienze dells instituto di Bologna (1970) No. 6.

20. Eason, G. et al.: “On Certain Integrals of Lipscisiiz-Hankel Type Involving Products of Bessel Functions,”Phil. Trans., Royal Sot., London, A 247, 529.

21. Teeuw, D.: “Prediction of Formation Compaction fromLaboratory Compressibility Data; Sot. Pet. Eng. J.(S@., 1971 ) 263-271; Trans., AIME, 2S1.

22. Na~r, K.: “Analytical Methods for Predicting Subsi-dence,” Land Subsidence, Publication No. 69, AIHS-UNESCO (no date of issue) 2, 588.

23. Sandhu, R. S. and Wilson, E. L.: “Finite ElementA. . .. .. .. -f I m“~ Mrsid!me,

ruaaay an> “, tic... “ ~gnd Subsidence, Pub-~3ti&n No. 89, AIHS-UNESCO (no date of issue)

24. T~euw, D.: “Laboratory Measurements of CompactionProperties of Groningen Reservoir Rock~’ Verhande[in-gen Koninklijk Nederlandsch Geologisch Mijnbow -kundig Gersootschap ( 1973) 2S, 19.

25. Geertsma, J.: “A Theoretical Analysis of the Compac-tion-Subsidence Relationship: The Homogeneous Case,”Ibid., 43.

26. Geertsma, J. and van Opstal, G.: “A Numerical Tech-nique for the Prediction of Subsidence Above Com-pacting Reservoirs, Based on the Concept of Nuclei ofStrain,” [bid., 63.

27. Van Kesteren, J.: “Prediction of Possible Future Sub-sidence Resulting from Gas Production in the Gronin-gen Field,” /bid, 11.

28. Van Kesteren, J.: “Estimate of Compaction Data Repre-sentative of the Groningen Field,” Ibid., 33.

29. Newman, G. H.: “Pore Volume Compressibility ofReservoir Rocks Under Hydrostatic Loading,” j. f%f.Tech. (Feb., 1973) 129-134.

30. Sawabini, C. T., Chilingar, C. T. and Allen, D. R.:“Triaxial Compaction of Unconsolidated Oil Sands,”paper SPE 4058 presented at SPE-AIME 47th AnnualFall Meeting, San Antonio, Tex., Oct. 8-11, 1972.

J_PT

Paper (SPE 3730) was presented at SPE-AIME European SpringMeeting, held in Amsterdam, May 16-18, 1972. @ Copyright 1973American !r!stitute of Mining, Metallu~ical, and Petroleum En-gineers, Inc.

744 JOURNAL OF PETROLEUM TECHNOLOGY