Spe000426 the Behavior of Naturally Fractured Reservoirs

8/6/2019 Spe000426 the Behavior of Naturally Fractured Reservoirs

http://slidepdf.com/reader/full/spe000426-the-behavior-of-naturally-fractured-reservoirs 1/11

, q

The Behavior of Naturally Fractured ReservoirsJ OI% WARREN

P, J , RQOTMEMBERS AIME

ABSTRACT

An idealized model has been developed for thepurpose of studying the characteristic behaviorojapermeable medium which contains regions whichcontribute sigizificantly to tbe pore volume O! thesystem but contribute negligibly to the flow capacity;e.g., a naturally fractured or vugular reservoir, Vn-steady-state flow in this model reservoir has beeninvestigated analytically. The pressure buiid-upperformance has been examined insomedetait; and,

a technique foranalyzing tbebuild.up data to evaluatethe desired parameters has been suggested. The useof this ap$roacb in the interpretation of field datahas been discussed.

As a result of this study, the following generalconclusions can be drawn:

1. Two parameters are sufficient to characterizethe deviation of the behavior of a medium with “doubleporosity ”from that of a homogeneously porous medium.

2. These Parameters can be evaluated by theproper analy~is of pressure buildup data ob~ainedfrom adequately designed tests.

3. Since the build-up curve associated with thistype of porous system is similar to that obtainedfrom a stratified reservoir, an unambiguous interpre-tation is not possible without additional information.

4, Dif@rencing methods which utilize pressuredata from the /inal stages of a buik-kp test shouldbe used with extreme caution.

INTRODUCTION

In order to plan a sound exploitation program or asuccessful secondary-recovery pro ject, sufficientreliable information concerning the nature of thereservoir-fluid system must be available. Sincef itis evident chat an adequate description of the reser-voir rock is necessary if this condition is to be ful-filled, the present investigation was undertaken forthe purpose of improving the fluid-flow characteriza-

tion, based on normally available data, ofs particularporous medium.

DISCUSSION OF THE PROBLEM

For m an y ye a rs it was widely assumed that, forthe purpose of making engineering studies, two psram-. . -..

O ri gl m l m a n u s c ri pt r e c e iv e d fn e o ci at y of P e t r o le um E r t a t n e e r eoffiae AUS . 1 7 , 1 9 6 2 .Re v ie e d ma n u s c r i p t re c e i ve d .Ma r c h 2 1 , 1 9 6 3 .P e pe r p r+$ e en t e d a t t h e Fe t l Mee t in g of t h e %c iot Y o f. Pe t r e l eumE n gi n se r a I n Lo = Ar@Ies on Oc t . 7 -1 0 , 1 96 2 . ‘ .

GULF RE SE ARCH d DE VE LOP ME NT CO.P ITTSBURGH, PA,

eters were sufficient to desckibe the single-phaseflow properties of a prodttcing formation, i.e., theabsolute permeability and the effective porosity. It :later became evident that the concept of directionalpermeability was of more thin academic interest;consequently, the de$ee of permeability anisotropyand the orientation of the principal axes of permea-bility were accepted as basic parameters governingreservoir performance. 1,2 More recently, 3“6 it wasrecognized that at least one additional parameterwas required to depict the behavior of a porous systemcontaining region,s which contributed significantly tothe pore volume but contributed negligibly to theflow capacity. Microscopically, these regions couldbe “dead-end” or “storage” pores or, microscopi-cally, they could be discrete volumes of low-permeability inatrix rock combined with natural fis-sures in a reservoir. It is obvious thst some provisionfor the ;.ncIusion of all the indicated parameters,as weIl as their spatial vstiations$ must be made ifa truly useful, conceptual model of a reaetvoir is tobe developed.

A dichotomy Qfthe internaI voids of reservoir rockshas been suggested, r~s These two classes of porositycan be described as follows:

a. Primary porosity is intergranular and controlledby deposition and Iithification. It ie highly inter-coririected arid “usual ly can be correlated with perme-ability since it is largely dependent on the geometry,size distribution and spatial distribution of thegrains. The void systems of sands, sandstones andoolitic limestones are typical of this type.

b. Secondary porosity is foramenular and is con-trolled by fracturing, jointing and/or solution incirculating water although it may be modified by in-filling as a result of precipitation. It is not highlyinterconnected and usually cannot be correlated withpermeability. Solution channels or vugular voidsdeveloped during weathering or buriaI in sedimentarybasins are indigenous to carbonate rocks such aslimestones or dolomites. Joints or fissures whichoccur in massive, extensive formations composed ofshale, siltstone, schist, limestone or dolomite aregenerally vertical, and they are ascribed to tensionalfailure, during mechanical deformation (the permea-bility associated with this type of void system isoften anisotropic). Shrinkage cracks are the result

1 ~e f&en ce . a iven atend o f p@e r . ‘-



. ,

of a chemical process (dolomitization) and do notappear to have any preferred orientation.

In the most general case, both classes of porosityare present and the internal-void volume of the rockis intermediate in nature, i.e., rm independent systemof secondary porosity is superimposed on the primaryor intergranular system. The obvious idealization ofan intermediate porous medium is a compfex of dis-crete volumetric elements with primary porosity whichare aniaotrapicalIy coupled by secondary voids as

shown in Fig. 1. This heterogeneous, double=porositymodel will be investigated in detail since a significantnumber of petroleum reservoirs can be classified asintermediate., Although most intermediate-porosityrocks are limestones or dolomites, other sedimentaryrocks such as cherry shale or siltstone exhibit thischaracteristic. Since the,double-porosity model yieldsboth of the single-porosity systems as limiting cases,a-n internal check on the results is provided. *

Sources of useful information, for the characteri-zation of a reservoir with intermediate porosity arelimited and the information itself tends to be morequalitative than quantitative. The analysis of coredata ‘-1 1is complicated by the presence of extraneousfractures induced during coring and retrieval, poorcore recovery from intensely fractured zones, dis-placement of fracture surfaces and the size of thesample itself. In spite of these diff iculties, acceptablevalues for the porosity and permeability of the inter-granular material can be determined from plug-coredata, and an estimate of the total porosity can beobtained from whole cores. The vertical extent, width,spacing, dip, relative strike, degree of infilling andthe type of fracture can sometimes be qualitativelyascertained from a visual inspection of whole cores.

The volume and nature of the solution voi !,canbe estimated from the examination of thin s{ ~ionsand/or polished surfaces. While it is upually notpossible to correlate, well logs on a well-to-wellbasis, an indication of the presence and verticalextent of fractued zones in an individual well maybe obtained from sonic logs; similar qualitative in-

*An a lo go u s p h y si ca l p r ob le m s a t ia e fr o m t h e c on s id e ra t io n o fh ea t or m as s t r a n s f e r ina h e t er og en e ou s m e d iu m o r e le ct r ic alpower t r a nn m i ss lo n . Ma t h em a t ic ally a n alo go u s p r ob le m s o cc u rwh en t wo flow p ro ce na es a re c ou ple d by lin ear fu nc tio ns In -.v ol vi n a b o t ’m de p e n d en t v a ri ab le s ; es., c h e r n l cu r ea ct io n , r r d-e o r p t i on o r r r h a a e e q r r i li br i um .

\\VINS MATRIX FRACTU)E MATkX FRAC>URES

ACTUAL RESERVOIR - : MODEL REsERvOIR

FIG. 1 — IDEALIZATION OF THE HETE ROGENEOUS,POROUS MEDIUM.

,. ,

*~~-- - :. . - , . . . -,. . . . -. , -.-,

formation may be obtained from spinner surveys,observed loss of circulation and down-hole photo-graphs, [t is clear that data obtained from thesesources, which reflect conditions in the immediatevicinity of the well, give only a p>sitive indication .of the presence of secondary porosity and a descrip-tion of the primary-porosity mater ial. Some additionalinformation can be acquired from non-routine labora-tory tests on cores 3-57 l% 13 if the scale of variation‘in the secondary porosity is sufficiently small.

Multi-well interference tests 14 or tracer tests canbe utilized to establish the components of the aniso-tropic permeability and the orientation of the principalaxes. The analysis of pressure build-up or fall-offdata Is, 16 permits the evaluation of the apparentpermeability, (the geometric mean of the directionalpermeability components) the completion damageand, possibly, the primary and secondary porevolme .17-19 This well-test information indicatesthe gross properties of the intermediate formation onan interwell scale.

To develop a plausible model for an intermediaterestirvoir, it is essential that all of the availablemeasurement and observations,, cited in the precedingparagraphs, be utilized; furthermore, the model must

be consistent with the physical inferences obtainedfrom the performance of actual reservoirs of thisparticular type. Applicable studies involving pro-ducing reservoirs located in many geographic areas”are to be found in the literature. Reflecting theseperformance studies ate a number of mechanisticmodels which have been proposed to describe theresponse of an intermediate reservoir to variousnatural and/or art if ic i al drives. 20-29 Since our ob-jective is to suggest “a model which will simulate. the behavior of a formation with intermediate porositvduting single-phase flow, the more esoteric ~eature~of the proposed models can be discarded; the resultis a ph ysi’kal ide alizat ion which incorporates acommon set. of characteristics which seem to besignificant,

The model to be employed in this ~nvestigationis based on the following general aa sumptions:

a. The’ material containing the primary porosityis homogeneous and isotropic, a“nd is containedwjthin a systematic array of identical, rectangularparallelepipeds.

b. All of the secondary porosity is containedwithin an orthogonal system of continuous, uniformfractures which are oriented so that each fracture isparallel to one of the principal axes of permeability;the fractures normal to each of the principal axesate uniformly spaced ~nd are of constant width, adiff&ent fracture spacing or a different width mayexist along each of the axes to simuIate the properdegree of anisotropy.

C. The complex ofprimary and secondary porositiedis homogeneous albeit anisotropic; flow ,can occurbetween the primary and secondary porosities, butflow through the primary-porosity elemem-s can notoccur,

Additional assumptions of more particular naturewill be made at appropriate points in the mathematical.,

s OCKET-Y OF PETROLEUM ENC’f.NEE RS J .OGRNAr : .’ .-’. _, ,



.q

.

treatment.While the assumed model certainly implies hetero-

geneity on.a macroscopic scal~, it maybe consideredto be”homogeneous if the dimensions of the homoge-neous blocks ate small in comparison with the dimen-sions of the reservoir, see the discussion by Warren,”et aL 30 The same type of argument is often used tojustify the measurement of potosity or permeabilityon a small core. The integration of the solution andfissure porosities is obviously necessary if the modelis to be practically useful; ftuthermore, it is reason-able on the basis of similarity of flow performance.A certain amount of freedom has been allowed in thedescription of the fracture system to permit the useof all of the qualitative information that might beavailable.Because of the discrete nature of the primary.

porosity elements and evidence which indicates thatnormal and log-normal distributions of porosity andpermeability y are typical, 31 the arithmetic-meanporosity and the geometric-mean permeability shouldbe used to obtain a “most probable” modeI. 32 Thecomponents of directional permeability and theirorientation should be determined by means of inter-ference tests. While this completes the description

of the reservoir via current teclmi ques, there are twoparameters which are not yet determinate — thesecondary porosity and a shape facror which describesthe communication between the primary and secondaryregions. An attempt will be made to develop methodsof determining approximate values for these parametersfrom the known information and pressure data fromwd tests. ‘

THEORY

Since it has been aasunied that the reservoir canbe treated as though it were homogeneous, let usdefine two press~es at each, point. (x, y,z, t) in thefoilowing rnannec ‘ ,--

JI (q)’,z, t ) = * P(x,xz, t )g l (v)dv/

/ 1~g,V)dt f . . . ‘ (1)*

J 72(v)dv. . . . . .“(2)

whore q M={

i, in piitimry’ porosity -O,Outsideof primary pwosity

g2(v)= {

1, in sscondary porosityO,outsidsof secondaryporosity

If useful solutions tire to be obtained utilizing the“smoothed; t pressures in the primary and seconday

Wh e s@sc ? ip t s 1 a r id 2 a lway s r e fe r t o t h e p r im ar y a n ds e co n d av y p o re v olu m e s r e sp e ct iv ely , fo r d e fin it ion s of o t h e rs YI@o ls , s ee No me rd at we .

f

regions, pi and &, (shown in Fig. 2) it is obviousthat the volume considered must be small in compar-ison with the volume of the reservoir and must belarge in comparison with the size of the matrixelements; i.e., the following conditions must besatisfied:

J q(wv=f$ . . . . . . . . (3)

If the connate-water saturation in the secondaryporosity is negligible, the average measured inter-granular porosity #m is simply related to +1 and 4.

For a uniformly thick reservoir that is horizontal,homogeneous and anisotropic, the single-phase fIowof a slightly compressible liquid is partially describedby applying Green’s theorem to the volume V to obtainthe applicable form of the continuity equation.

dP ;.........(6)g~c ~ + “ “

where the x-axis and the y-axis “coincide with theprincipal axes of permeability. In this expressionC1 and C2 are total compressibilities; however, if itis assumed27 that the external forces ate constant,that there is no interaction between the two regions(4 I is independent of ~ and +2 is independent ofp I) and that the variation of 42 with respect to p2is negligible, the following approximations obtain:

Cp+,$w&Cwc,=co+

I-swc “ ‘ “ “. “. (7)

c~= co. . 0 . . . . . . . . . (8)For many of the pertinent cases, the second term, inEq. 7 will ,be at least as large as the compressibilityof the flowing liquid,

In addition to Eq. 6, continuity on a local basis isnecessary. If it is’ assumed that a quasi-steady stateexists in the primary-porosity elements at all times,the following equation must be satisfied in the volumdV surrounding each point in the reservoir:

LATRiXC

FRACTURE

FIG. 2 — SCHEMATIC “REPRESENTATION OF THEPRWSURE DXSTRBUTION IN THE MODEL.,, , .!,.

r SEPTEMBER. 1 9 6 S i t 4 7



~’*(P2-Pl) . . . . . ‘(9)@lCI at

The parameter a ‘has the dimensions of reciprocalarea; it is a shape factor which reflects the geometryof the matrix elements and it controls -the flow betweenthe two porous regions. * The assumption of a quasi-steady state will introduce some error into the solu-tions when times are small; but, since the results

ace ultimately to be used in conjunction with well-test data, the approximation should be adequate.Eqs. 6 and 9 can be cast in dimensionless form

by introducing appropriate tran sformaticns. For aninfinite reservoir with a uniform initial pressurewhich is to be produced at a constant rate, the trans-formed equations, initial condition and boundaryconditions can be written as follows:

4’!2 . . . . . . . . . .. (10)= ‘T

“J’ISA(*2-*,) . : . . .(1-u) —dr

. (11)

*,=+2’O; for r =0, all g and 8

*2= O;fcw TZO, $—0,011 6

J27r

I a+2K ~de= ‘I;T>O,R’

a

/J KCOS26 + ~sh2f? “ . . (12)=

/

qF oI ”a m o r e d e t ai le d d is cu s si on , s e e Ap p en d ix A.

J -Zw”=:. . - -, ---- . - . . .. . . . . . . .

,.

.

@ = $2C2 (@, c, +J&22)

It is obvious that the well, which becqmes an ellipseafter transformation, causes the pressure distributionto lack axial symmetry. To avoid the necessity ofinciuding K as a parameter in the solution, it is as-aumed that the well-bore pressure obtained by solving(10-12) differs by a constant amount from the well-bore pressure obtained bf solving the symmetricalproblem (K = 1), After a very short time, the differencein these pressures due to the distortion of the pzes-sure distribution in the region near the well will beconstan~ 2 therefore, since it has already been as-sumed that errors which are restricted to small rimesare permissible, the following expression completesthe statement of the problem:

()+ Iwhere S* = In —2%/7?

$2* (l,r) = solution of symmetrical problem, de-scribed by (10-12) with K = I, evaluated at # = 1.

Using the Laplace transformation and solving theresulting equat ion subject to the boundary conditions,the desired solution can be written as follows:

[

-1

LK.(m)

*2*(1, d= 1 14)S- K I (~~)

The asymptotic solution which is valid for r > 100**is obtained by making the usual substitutions, Ka(v)‘=(-y +lnv - in 2) and KI (u) ~ I/v, and inverting.

“{2* (I,r) ~ I(2 ln~+.80908+E,[-hr/4i*)]

, ~ - E, [-At/(1-a)]} . . . q q (15) :

soexp (-u) dawhere -Ei (-v)= “

v

The solution for the case of a finite reservoir canbe obtained in a similar manner if the. condition atinfinity in (12) is replaced by

*=O; for r>Ot Cat

=R, all 9.”. (16)

If R2 ‘>> K, Eq. 13 is vrilid for this case; and,

*rnT~B~~&~~ is s tific ie nt fo r a ll va hle e Of h an d @h oweve r , If ~ << 1 , t h e con d it io n la ? > MWG or, if @ << 1,r> (IOOA- DIA

. S OCIE TX O F P ET ROLE Ur n .. ENC I NE A~R s J O fJ R NAL

1



.

. . . . . . . ...*. •**~** , . (17).

DISCUSSION OF RESULTS

The solutions obtained emphasize the need fortwo parameters with which to describe a reservoir ofthe intermediate type; this agrees with the conclusionsof Stewart, et al, 6 w“hich were based on experimentalwork. The next step is to examkte the effects of theparameters on reservoir behavior;, then an attempt toevaluate them must be made. \

If the derived results are to tie-accepted, it isnecess~y that the limiting behavior described byEqs. 15 and 17 be physically reasonable. As CO+O,both equations approach the asymptotic solutionobtained .by Van Everdingen and Hurst 33for the sameboundary conditions. * This behavior is proper sincethe primary porosity, or its effective compressibility,

must vanish if co is to approach unity; then, thereservoir contains only the homogeneous secondaryporosity. Similarly, the equations indicate homoge-neous behavior as A + 00. This is alao correct sincethere is no impedance to interporosity flow when Aapproaches infinity, i.e., either k 1 or a must becomeinfinite. Furthermore, the necesssr y condition thatthe infinite reservoir responds in a homogeneousmanner for very large values of time is properlydescribed.

Some >esults for the special case in which @ = O(negligible storage capacity in the secondary porosity)in an infinite reservoir are shown in Fig. 3; thisparticular form of the model has been discussedquite extensively. ls~271t is apparent that the dimen-sionless pressure drop +2*( I,r) increases discontin-uously. when production begins, and~it asymptoticallyapproaches the homogeneoias solution when Arbecomeslarge. The pressure discontinuity is the result of alack of fluid ca acitance ,in the secondary porosity;

rhe magnitude o the discontinuity is %(-y – lnA) forthe cases of interest. Fig. 4 indicates the deviationfrom the asymptote for each case. The intercept atr = 1 is equaI to the initial pressure discontinuity,and all of the linear segments have the same slope,i.e., -1. 15/cycle. The apparent linearity of the curvespersists until A? >0.05.

In Fig. 5; behavior patterns that are associatedwith finite values of OJ are shown; once aga”in the

reservoir is infinite. The most notable feat~e i a thesecond linear segment which is parallel to the asymp-tote bu~ vertically displaced by an amount that isequal to In( l/@). The transitional curve which con-nects the two linear portions represents the inter-action bet ween A~and ~,, Deviations from the asymp-

o~d d 10’ m ’ 10” 106 10’ 10’ 10’ 10’

OIMENS1ONLESS TIME , r

INFINITE RESERVOIR

W. O FOR ALL CASES

<

d K? 10’ 104 ma !0’ 10’ .- to’ 10’

*Frs v >> I, + (-v ) ~ exp (v)/ V; and, for V << 1 , % (+)~ - ~ + In u ) wh e r e y = 0.577!2. . .

DIMENSIONLESS TIME , f



tote are presented in Fig. 6. The initial constantlevel is equal to the displacement of the secondparallel segment from the homogeneous asymptote.The linear poreion of thedifference cwvehas a slopeof -I. ~5/cycle and an intercept of %[-y-ln~+ln( 1~)1at r = L Although the linear part of the deviationti~ve will not always be clearly defined, a line of theproper slope paasing through the point of inflectionwill generally yield an intercept that will permit A tobe approximated quite adequately.

No results are presented for finite reservoirs sincethe behavior described by Eq. 17 is so familiar. Theform of the equation $4 that which results in theclassic time-lag curve encountered commonly inthe quasi-steady state. The aaymptote for the dimen-sionless draw-down #*(1,7) is a linear function of rwith a slope of %(R‘~1) and an intercept approximatelyequal to [In R - % + 2(1 - d2/AR 21 .TheIowithmof the deviation from the asymptote is also a linearfunction of c the dope k [-h/2.3w( ~ - d l wl-and the intercept at r = O is log [2(1 - W)2/A(R~-0].In principle, it is possible to evaluate R, A and ofrom the asymptote and the difference curve.

For practical use, equations, expressed. in fieldunits, for the draw-down and build-up histories andfor the deviation of the actual build-up curve from itsa sym p t o t e a re Iiated in Appendix B. Theoreticalbuild-up curves for various combinations of A and coare shown in Figs. 7 and 8 to illustrate the type ofbehavior that can be expected in an infinite reservoir.The asaumed parameters and operating conditionsused to generate the data are the following:

It is obvious that the behavior when there is only“@imsry porosity u = O is so discontinuous that thebuild-up occurs “almost instantaneously when A isvery small. The other cases for the smaller value ofA, 5 x 10-9, give results that cause the reservoir toappea CO have a closed boundary; furthermore, ifonly the early portion of the curve were recorded,

the value of P, determined by extrapolation to(At/t~ + At) = 1, would be erroneously high by the, quantity ti’ log ( I/@) and the value of the skin resist-

ance Sd would be high by 1.15 log (l/m). * Although

~h e a pp si? e n tv alu e of t h e s kin r es ie tsn ce in a n a nie ot rop icre s e r vo ir ie a ct u ally e qu al t o S d - S *; t h er efo re , t h e c or re ct io ne hot dd be m a de if we ll c om ple tion s a re t o be e va lu at ed o n t h fsba s is . If a n ege tlve sk in r e s is t an ce is m ea su red in a we t t t ha th ss n ot be en s t im u la te d, it m ay be u se d t o obt ain a lower l imi to n t h e a n is ot r op y p a ra m e t er K .

TABLE 14

X2 =“40mdh=2Qftreew

= 0.316 ft#lCl+#2$ = 2.64 x 10-6 psi’1

p=2cp/3 = 1.23s~ = S*P = 4000 psia‘ g = 115 STB/Dts = 21 days

2s0 . :.-–-”... .----’=... .

these particular results may be rather artificial, theydo suggest, the posaibilit y of misinterpretation e.g.,the latter part of the transitional curve between theinitial linear portion and the asymptote can be

11I /

tieK~#6 -

:

~

3

INFINITE RESERVOIR

— (0.0,001------ Lo.o.ol—-— (J).0,1

011 (? 10’ 10* !d 10” !0’ Id. [0’ 10* 10’

D IMENSIONLESS T IME , 7

ana-

FIG. 5 — SOLUTIONS OBTAINED BY APPROXIMATEINVERSION.

INFINITE RESERVOIR

— Limo.ool-------- CU. O .01—-— K1. o,l

30 -

j

%

2 L?.-g

:

c:

.

g

zK&l .5-

g

z

~m~,1 .0

z’6

a5 -

0,I& lot 10* !0+ id Id !0’ 10’ 10’ 10$

DIMENSIONLESS TIME , T

FIG. 6- DEVIATION FROM THE ASYMPTOTE.,,-.

----- s OCIE TY , OF. P.S T, SOLE Ur n E NGINE ER S J O URNAL

.

-,



..

lyzed by Pollard’s method 17’or its extension 19 justas if the reservoir actually had a finite ~ainage radius-

For A = 5 x 10-6, the curves, with the exceptionof o = O, exhibit a double-slope type of build-upperformance; appatentl , this behavior is typical offractured reservoirs. Ii, 17,35 Unfortunately, it ischaracteristic of stratified reservoirs as well, so theevaluation of such data maybe uncertain. If, however,a, plot of the deviation from the asymptotic solutionbehaves ‘in the prescribed manner, it is safe to con-clude that the reservoir can be considered to beintermediate in nature for predictive purposes. It maybe postulated that evidence of this pressure lagindicates the presence of macroscopic heterogeneity yor that A is actually a direct measure of the scale ofheterogeneit~ e.g., for A >> 1, the reservoir behaveshomogeneous y.

When both of the parallel segments are discernible,the build-up analysis presents no problem. The laterportion is interpreted in exactly the same manner as‘it would be if the reservoir were homogeneous. Inaddition to the possible errors which can result fromusing the wrong portion of the build-up curve, em-ploying an incorrect method of analysis or assuming

the wrong type of spatial permeability distribution,there is the possibility that a significant portion ofthe curve may be masked by afterflow effects. Thereis also a finite probability y that the use of the ‘clog(P - pw) vs time “ method lG,17)lgcould result in acomplete description of a reservoir soIel y on” thebasis of after production. This difficulty can be cir-cunivenred by utilizing a recently developed toolwhich permits the well to be closed”. in at theformation. 36

./

No results are presented to demonstrate the ex-pected behavior of a finite reservoir. This is donedeliberately to avoid any chance of misunderstanding

I4000 -

3 0 s 0 -

g ~*oo -

c ?,.wuz 3e30 -co;

g 360 0 -

m~

,

$ 3?6 o- INFINITE RESERVOIR

A. SKl@* FOR ALL CASES

q.11$ STB/D3700- 1,.21 OAYS

Fit% 7 — THEORETICAL BUILD-UP CURVES.

. s EPTEMBER, 1 96 9 . . . . , ,J ’

or misuse. Until the effect of the closed boundary isfelt, the reservoir will act as though it were infiriite,so the previously discussed methods can be applied.Beyond that time, the difficulties inherent in theprocurement of sufficient, accurate data and thefundamentrtl uncertainties in the available methodsof analysis force us to question the value of suchtests. The time required to achieve quasi-steadystate conditions in a heterogeneous reservoir is oneor two orders of magnitude greater than it is in a

homogeneous r~servoir. 35The acc~ate determinationof S1OWI varying pressures is quite clifficult unlessa pressure-sensing device that is operating in thecorrect range is being used (it might be neceasatyto change pressure bombs several times to accomplishthis). Finally, the analysis of these small changesin pressure may become largely a matter of personaljudgment; and, if higher eider pressure differencesare involved, the results may be more artistic thanscientific. To emphasize these points, the dimen-sionless pressure history of a single, undamagedwell producing at a constant rate from an infinitehomogeneous reservoir 33was used together with thereservoir data in Table 1, q = 200 STB/Dand t~ = 10 days, to obrain the synthetic build-upcurve shown in Fig. 9. The data are plotted in rheprescribed manner 17, Ig and thk graphical analysisis shown. The interpretation of the parameters sodetermined leads tc the following results:

pore volume of the coarse voids = 1,100,000 cu ft

4900

..3930 -

asoo -(0.0

.s: mm -

;

g

3w WOO -K

~

T.5g 3760 -

s700

t

INFINITE RE5ElwolR

k .5 xl Cf S FOR AU IMSES

t

.,3 6 s 0

) q .115 STWO

to. a OAYS

,

FIG. 8 — THEORETICAL BUILD-UP CURtiS.!,!.-.

SE]. ...’. . . . . .. .. . ., :..” .. . . . .



pore volume of the fine voids ==5,670,000 CUft

radius of the reservoir = 904ftradius of the damaged region = 130ft

It is obvious that this approach yields erroneousresults. The fault cannot be assigned entirely to themodel upon which the technique is based since themodel is physically reasonable during quasi-steadystate flow, Rather, the difficulty lies in the nature ofthe data used. It is probable that any decaying func-

tion can be approximated by a series of exponentialterms over an interval of gradual change; thus, am-biguity results.

As a consequence of the examination of the approx-imate mathematical treatment of the idealized physicalmodel, it has been suggested that the two additionalparameters which ate significant in the behavior ofan intermediate reservoir can be evaluated from pres-sure draw-down or build-up data. The data shouldbe obtained before the effect of the reservoir boundaryis felt ar the well. Extreme caution should be usedin the analysis of data s~cured during the later stagesof build-up in a finite reservoir.

SUiiMAflY AND CONCLUSIONS

An investigation has been conducted for the pur-pose of improving the description of a formationwhich contains both primary (intergranular) porosityand secondary (fissure and/or vugukw) porosityBased on the assumption that the primary-porosityregion contributes significantly to the pore voIume

but contributes negligibly to the flow capacity, anidealized model has been developed to study thecharacteristic behavior of such a reservoir. Unsteady-state flow in this model reservoir has been describedmathematically, and asymptotic solutions have beenobtained. The pressure build-up performance has beenexamined in some detail; and, artechnique for analyzingthe buiid-up data to evaluate the desired parametershas been suggested. The use of this approach in theinterpretation of field data haa been discussed. ?

As a result of this study the following generalconclusions can be drawn:

L Two parameters are sufficient to characterizethe deviation of the behavior of a medium with’ ‘doubleporosiry” from that of a homogeneous, porous medium.One of the parameters, ~, is a meaaure of the fluidcapacitance of the secondary porosity and the other,A, is related to the scale of heterogeneity that ia

.

present in the system.2. These parameters can be evaluated by the

proper analysis of pressure build-up data obtainedfrom adequately designed testa.

3. Since the build-up curve associated with thistype of porous medium is similar to that obtainedfrom a stratified reservoir, an unambiguous interpre-tation is general] y not possible without additionalinformation.

4. ,Methods which utilize differences obtainedfrom ~resaute data recorded during the final stagesof a buiid-up test should be used with extreme caution.

NOMENCLATURE

C = total compressibility, LT 2/M

K = ~py, degree of anisotropy, dimension-

P = initial pressure, M/LT 2

R = re/rw, radius of closed boundary, dimension-less

S“ = In [(k+ 1) /2 @], effective skin resistancedue to ‘iinisotropy, dimensionless

S d = skin resistance due to completion, dimen-sionless

Swc = connate water saturation, dimensionlessco = compressibiIit y of flowing liquid, .LT 2 /M

‘P =effective pore compressibility y, .LT2/M

c w = compressibility of connate water, LT2/Mh = thickness of formation, f..k = absolure permeability, L2

~2 = = , effective permeability of anisotrop-ic m%difim, L 2

.4. = characteristic dimension of heterogeheo,~s ‘region, L

p = pressure, M/L-T2p ~ = well-bore pressure, M/LT2 ,,

~P 71J= deviation of actual pressure from the value

on its asymptote, M/L T* . ,,q = production rate, L3/T

s =’ Laplace operator, l/Tt =’:irpe, T

t s = time of shut-in,T .,=. ,’

.’h = t - t~, time elapsed since shur- in, Tx, y, z = rectangular coordinates; the axes coincide -.

with the principal axes of permeability, La = kzeometric Darameter for heterogeneous

INFINITE RWCRVOIRA.m ORU J.lq. KU srsm

t,. m O N i

“

,,

‘* IOmwmsO wmsowrnt. nvawnowo

-J SIWT.111IME . HOURS

FIG. 9 — WORK CURVE FOR BUILD-UP ANALYSIS(REFERENCES 19, 21).

26s ~~ ..----: --- “. - -i .“ . .

~egion, l/L2’@ = formation v?lume factor, dimensionlessy = 0.5772. . ., Euler-hfascheroni constant, di-

mensionless

A = C&lrw2/K2, the parameter governing inter-porosity flow, dimensionless

F = viscosity, M/LT

~=~—, radial coordinate, dimen-

W ,W ..aionless . .

,. ... sOCIS .T Y 9 F . PE9ROL.EUM EN”GJNEi!s,,J ou~ NAL -



I

I

.q

r = E2t/(q51I + q52C2)p7w21ime, ditn-i-

rs =Ar =

4=*=

$2’ =

lesstime of shut-in, dimensionlessr-rti time elapsed since shut-in, dimension-lessporosity, dimensionless2mK2h (P-p)/q@ pressure decline, dimen-sionlesspressure decline in secondary porosity at

r,,, dimensionless

‘W = f~o~fls:resSWe decline at well, dimen.

~ = #2C2/(t#1C1i+2C2), pa r am et er r e la t in g fl~d

capacitance of the secondary porosity to thatof the combined system, dimensionless

SUBSCRIPTS

1 = primary porosity2 = secondary porositym = matrix (refers to properties measured on smalI

core samples)

&y, z = vecror component

1 .,

2,

3,

4.

5.

6.

?,

8.

9.

10.

11.

12.

13.

14.

15.

16.

17,

18.

19.

REFERENCESBar fie ld , E . C., J or d a n , J . . K, a n d h foore , W. D.:J o u r . Pe t . T ec h . (April, 19S9) Vol. XI, No. 4, 15.Mwt sda , h f. a n d Nabo r , G. W.: Trans., AIME (1 9 6 1 )vol. 22 2 , 11 .

Klu t e, C. I-L: ]o t i . Po lym er Sc ien ce (1 95 9 ) Vol. 4 1 ,3 0 7 .

Fatt, I.: Traas., AIME (1 95 9) Vo l. 2 16 , 44 9.

Good kn igh t , “R. C., Klik off, W. A. ad Fa t t , L: ]OUT.Pb y ei ca 2 Cb er e. (1 96 0) Vo l. 6 4, 1 16 2..

St ewa r t , C, R,, LubinMc i, A. an d Blen k am , K , A.:Tr am s,, AIME (1 96 1) Vol. 22 2, 3 S3 ,

E3 uln ee , A. C, a nd F it t t ng, R. V,: Tru rr s . , AIME (1945)vol. 1 60 , 1 79 ,

Im bt , W, C. a n d Ellia on , S , P .: API Dvilt . t r d Prod .%7c , (1 94 6) 3 64 .

At kin s on ,’ B. an d J oh ns t on , D.: Tr um s t , AIME (1 9 49 )vol. 17 9 , 1 2 8 . ,

Ke lt on , F . C.: Tr am . , AIME (19 50 ) Vol. 1 S9 , 2 25 . “$$Ge o io gi cs l R e po r t ,, S t e e lr n a n Ma in Mi da le BectaUnit”~su bm it t e d t o Oil a n d Gaa Con se rva t ion Boa rd ofSa sk at ch ewa n, Oc tobe r, 1 96 0.

St ewa r t , C. R., Cr a ig, F . F . an d Mors e , R. A.: Tr a n s . ,AIME (1 95 3) Vol. 1 98 , 9 3.

S t ewa r t , C. R., Hu n tr E . B., Sch ne id e r , F . N., Ge ff~p ,T. M. a n d Be r r y , V. J .: Tra n s. , AIME (1 9 54 ) VOL 20 1 ,2 9 4 .Elk ia e , L, F . an d Sk ov,; A. M.: Tr a n s . , AIME (19 60 )vol. 2 19 , 3 01 . ~ .,Dyee , A. B. an d J oh ns t on , O, C.: Tmi z ; , AtME (1 9 53 )vol. 1 9 8 , 13 5 .

Ma tt h ews , C-. S .: ]ow~ Pe t . T ec h (S ep t ,, 1 9 61 ) Vol.XIII, “No, 9 , 86 2.

Polla rd , P .: Tw z z s . , AIME (1 9 59 ) ” V0 1 . 2 1 6, 3 8 .’Sam ara , H.: ~~Estimation o f R es er ve s from PresaUTe”Ch an ge s in F r a c t ur ed Reee rvoiraDr , p re se n ted atSe con d Arab Pe t ro leum Con gre ss ; Be ir u t , Le ba n on, (O ct ., 1 9 60 ).

P ir son . R. S . an d Pir aon . S . j. : “An Ext en eic t n--of t hePolla rd An alys ie Ma th & of-We ll Pr e ssu re B~~d -Upa n d Drawdown Tea t s9 ~, p re se n ted a t t he 3 6 t h AnnualFal l Meet ing of Soc ie t y of Pe tro leu m Engineers,

,-_sEFTEM.BER; 1 9 6 3. r . . . . .

.r

t. : . -. ? : , --. , . . .– . ’ . ’-’- --- . .. . .

2 0 .

21 .

22 .

2 3 .

2 4 .

2 5 .

2 6 .

2 7 .

28 .

29 .

3 0 .

3 1 .

3 2 .

3 3 .

3 4 .

3 5 .

3 6 .

Dallaa, Texas (Oct. 8-11, 1961),Pirson, S, J.: Bufl., APG (1 95 3 ) Vol. 3 7 , 2 32 .u ak e r , W. J .: Pr oc ,, Fou fih Wor ld Pe t , Con gr e s s ,Se c t . X (1 95 5) 37 9.

Bir ks , J , : Proc. , Fou rt h Wor ld Pe t , Con gre s e , Sec t . II,(1 9 55 ) 4 2 5.

J on e s-Pa r t a , J . and Reytor, R. S.: .Tw vw .t ~M~(19 59 ) vol. 21 6, 3 95 .

F r e e dm an , H, A. an d Natanson, S. G.: f%oc., FifiWor ld Pe t . Con ~e s s , se c t . II (19 59 ) 29 7 .

Aron ofsk y , J . S. a n d Na t en eon , S . G.: l’r u t rs , , AIME

(19 SS) Vol. 2 13 , 17 ,Mek sim ovich , G. K.: Geologtya Nef:i f Gaza (Pe t ro-le um Ge ology) (1 95 S) 2 5S.

Ba r e n bls t t , G. I. , Zh e lt ov, I. P . s n d Koch in a , L N.:PMM (sovie t Ap p lie d Me t bem et ic s sn d Mech an ic s )(19 60 ) Vol. 2 4, S5 2.

Ma t t ex, ‘C. C. en d Kyt e , J , R.: Sot . PeL .%g. J o u r .(Ju ne , 19 62 ) Vol. 2 , No. 2 , 17 7 .

Te r r , C. M. an d Heu e r , L% J .: ‘# Fac t ors In flu e n c in gt h e Op tim um Tim e t o St ar t wa te r In je ct ion ”, p re se nt eda t t he Rocky Mou nt ala Re gion al Mee t in g of t he Societyof Pe t ro leu m En gin ee rs , Billin gs , Mon t . (May 24 -25 ,

Wa rr en , J . E . , Sk iba , F . F . s n d Pr ic e , H. S .: jou r .Pe t . T ec h (Au g., 19 61 ) Vol. XIfI, No. S , 7 39 .

Bu ln es , A, C.: Tr an s. , AIME (19 46 ) Vol. 1 65 ; 2 23 .

.,.

War ren , J . E , a n d Pr ic e , H, S .: Tr a n s . , AIMK (1 9 6 1 ]

Va n Eve rd in gen , A. F . an d Hwa t , W.: Tra n s . , AIME(19 49 ) Vol. 1S6 , 3 05 .

Ba r re r, R . M. : Diffusion In ar.rdThrough Solids, Uni-ve ra it y P re ss , Ca mbr id ge (1 95 1).

Le fk ovit s , H. C., Haze broek , P ., Allen , E . E , an dMa tt he wa , C. S .: T ra ns ., AIME (1 9 61 ) VOL 2 22 , 1 1.

Pit se r , S . C., Ric e , J . D. s n d Th om as , C. E.: Tr a n s . , ,AIME (1 9 59 ) vol. 2 16 , 4 16 .

APPENDIX A \

ADDITIONAL THEORETICAL CONSIDERATIONS

Pr ior t o the derivation of Eqs. 6 and 9, it wasassumed that the secondary porosity was containedwithin an array of homogeneous, geometrically iden-tical elements. Since it is ulcimateI y necessary to Iintroduce average values in order to obtain usefulresults, this simplification is justified. To show thatthere is no 10SS of generality, an alternate approachwill be .consideted.

Let us define pl (x, y,Z,t) within a representativevolume in the fo llowin g m an ne r:

m#

.. . . . . . . . ...0 . . . . . . . . (1- 1)

a

where @l = J v(zMJi’(. OcU

‘“[. Wc(A’lrn, ‘ (1) = @m(z) l-s

# ‘

I

2ss. . . ,---

. . . . . . .



.

v(A u!! = bulk vaiunm of matrix materi01

with chor,@tristic dimensionfrom “A to 4+dJ Wr unit bulk

A’ #

Jt

v(i) = A (U )dv = charcteristi c voiume

oassociated with the dimension2,

The interporosit y‘ f low per unit volume, q, is givenh v --f

[s ,1(2)J?J’x )441) w)(#&@7!!V* ~. . . . . . . . . . . . . . . . . . . . (l-2)

where p2 “(x,y,z, t) is assumed to be only a functionof time over the region of interest. Let

. . . . . . . . . . . . . ..*.. (1-3)S&s~i tu t ing (l-3) in (1-2),

(.

. . . . . . . . . . . . . . . . . . . . . ( I-4)

Utilizing previously defined quantities,‘\, < a~l‘,~= -jr (P~-p, )’. , . . . . . . (l-5),,

Similarly,’,

Therefore, quite generally,’

It is apparent that the characteristic dimension# must be examined more closely. IfI principIe, the

. .

*S4 ‘ ‘-.., - .

three dimensions of the identical matrix blocks andthe width of the fracture (or one dimension and thethree fracture widths) can be calculated if k.2 , k2 ,and k2 and +2 are known. Since this infor$atignwill no? generally be availabIe, it is necessary toresor t to approximations. The simplest approach isto assume uniformly spaced fractures and to allowvariations in the fracture width to satisfy the con-ditions of anisotropy. Then,

a=4fl(fl+2)/42 . . . . : . (1-8)*.where n = number of normal sets of fractures = 1,2,3.Let us define # in terms of measurable parameters.

z 1/2

( )’2=2[dn+2)]”2 * . . . (l-9)

The following min-max approximations obtain:

()1/2

k,rwzJ?= 4.78 — in= It213. (l.lO)

AZ2

where the maximum error is 38 per cent.

,( ),fwz “2.4= 4.30 - ~ n u 1,2. (I-Ii)**)ikz

where the maximum error is = 24 per cent.

J=6 .55 (tiY’2ia=2t3 .“(1-12)

where the maximum error is = 16 per, cent.If the blocks actually have the dimensions z = %y= banciz= c, rhe equivalent value ~f ,4 c an beestimated from the surface-volume ratio; x.e.,

A=3a&c

c (ob+bc+cu) i ‘=3” . . (1-13)

4=~;n =2.:,., . (1-14)

*= ’oi”n=l . . . . . . . . (1-15)

APPENDIX B J

BEHAVIOR EQUATIONS

In the equations that follow, these definitionsobtain:***

l-= (2.637 x lo-4) If2~@ Cl + fi5#2) ILG2

%n fic at ive o f t h e in se ns it ivit y o f Ct to t h e e xa ct ge om e tr y isthe e c t t h a t t h e r e su lt = fo r e n n -d ir n e n si on@l t i p fx er ea gr e e w it h(1 - S) if# is re pla ee d Z.

*Wh ik ‘ie - t h e m o st probable c a se s inc e i t 1$ unftkeIy thath o r i zo n t a l f ra c tw e s c o n t r i bu t e s i gr d fi c e n t ly to t he fl ow c apm it y.

• ** u~t ~ ue. psIa , S’1’B/D, res , b j i , rnd , f t , ps l%d h ou rs .

SOCIE TY o ? P ETh OLE UM .ENOINE Er is Iou RNAI.. . . . .. . . . . . .. .. . . . . . ....--.’

‘~,.. ”--”--’.- -.-.’:-”- ---—”- -“. .”-’ .. ”-”



q

Sd = skin resistance due to completion

Apw(t )= difference between the asymptoteand the actual curve

NFNITE ‘RESERVOIR

a) Draw-Downl r := 100aI if A <c I or‘L-> 100-+ if w<< 1

or r >100 (011 cases)

[

log r + .351+ .435E/[Wl-w~Pw(r)= P-m

1-.435E, [“k/( j-W)l -,87@t<87Sd

. . . . . . q ’.*O I (II- 1)

b) Build-UpJ ATS>3AT > IOWI if X << I orAr>100-fif w <<1

..

{

r$+Arpw (7S +AT) ~ P-m log ~ -.435E~LAr/

{[ 1pw (r* +Ar)%4Z15mEi - Mlrlw (1-u) -El*

[-hAr/(1-w)]} ;

all values of w and all values of Ar (11-3)

(constant portion of curve)

Apw(r~+ Ar )= m-@51-iog h+iogk)-bgA;i

all vaiues of w and intermediate values of

Ar . . . . . . ..cooc*”(I~-5)

(linear portion of curve)‘F~lTE RES~RVOIR

a) Draw-Down,rWOwR2if h cc I orT>10W2~

-,87m(ln l?-.75-S*+Sti) ; q I s o (~-d)

b) Build-Up, kr~> 5WAT > iOOw/?2if h cc IAr> 100 R2-~ if w<< I

. . . . .“. . . . . . . . . . . . . . . (11-;)

.w(l-w~ +,435 E/[-AAr/(1-a]} : q “ “ \!~2) _.435 [hAr/w(\-w)] .1 . ...’... (II-8)

***

Date post:	07-Apr-2018
Category:	Documents
Upload:	consultariska
View:	225 times
Download:	0 times

Spe000426 the Behavior of Naturally Fractured Reservoirs

Documents