Chapter 13

Well TestingTerminologyin MultilayeredReservoir Systems

13.1 Introduction

This chapter discusses various types and testing of gas layered reservoirsystems including multilayered responses in fractured reservoirs. It also de-scribes crossflow identification and the nature and degree of communicationbetween layers. Performance equations for cases of constant flowing pres-sure and constant producing rate are presented and discussed. The chapteralso reviews "layer effect" on pressure and/or production behavior includingeconomic aspects of interlayer crossflow.

13.2 Classification of Layered Reservoir Systems

Layered reservoirs can be classified into four groups.

Crossflow Reservoirs

Figure 13-1 shows a crossflow reservoir, which consists of four continu-ous layers that are communicating at the contact planes. These layers are notentirely separated by impervious layers, therefore interlayer crossflow couldoccur during the test. The crossflow would be directed from the layer of lowpermeability to the layer of higher permeability, as shown in the figure. If k\is greater than k^ , then the pressure transients would travel faster in the layerof permeability k\ than in the layer of permeability k^. The duration of thecrossflow periods depends on the storage of each layer. If the storage of thelayer of permeability k\ is negligible when compared to the storage of the layerof permeability fe, then the crossflow will continue throughout the life of the

Figure 13-2. Three-layer without-crossflow reservoir.

well. If the opposite occurs, then the duration of the crossflow period will beshort.

Without-Crossflow Reservoirs

This type of layered reservoir consists of two or more separate layers andis carried to the surface through a common wellbore. Each layer has differentproperties as shown in Figure 13-2. Production is commingled at the well, solayers communicate only through the wellbore.

Commingled Reservoirs

This type of reservoir is also known as a composite reservoir. Layers com-municate only through the wellbore as shown in Figure 13-3. Investigators1"3

have conducted studies on wells with commingled fluid production from twoor more noncommunicating zones. In those cases, fluid is produced into thewellbore from two or more separate layers and is carried to the surface througha common wellbore.

impervious layer

impervious layer

impervious layer

Shale Barrier

Shale Barrier

Shale Barrier

Sand

Sand

Sand

Well

Figure 13-1. Four-layer crossflow reservoir.

Base rock

Communication layers

Cap rock

Layer 1

Layer 2

Layer 3

Layer 4

Well

Figure 13-4. Schematic view of a portion of a two-layer reservoir with inter-layer crossflow.

Interlayer Crossflow Reservoirs

Figure 13^4 shows that crossflows between the layers can occur; the pres-sure and production behavior of a gas well can be interpreted by the use ofhomogeneous reservoir theory. A gas well in a layered reservoir with crossflow

Baserock

Drainageboundary

Higherpermeability

layer

Lowerpermeability

layer

Direction of flowWell

Cap rock

Figure 13-3. Reservoir consisting of commingled zones and crossflow layers.

crossflow

crossflow

Well

Zone 1

Zone 2

Zone 3

Com

min

gled

Zon

es

behaves like a well in a homogeneous, single-layer reservoir that possesses thesame dimensions and pore volume as the crossflow system and a permeability-thickness product (kh) equal to the total kh of the crossflow system. The oc-currence of crossflow can be confirmed by the homogeneous-like appearanceof the pressure and/or production behavior.

133 Pressure Analysis Methods in LayeredGas Reservoirs

Figure 13-5 shows pressure buildup for a layered gas reservoir with cross-flow. Such a layered reservoir behaves like a homogeneous system. A semilogplot for any pressure transient test can be analyzed just as it can for homoge-neous systems and should yield an estimate of (kh)t. The following equationsare applicable to analyze these tests:

n

(kh), = J2 Wj (13-Dj=i

n

(4>Cth)t = J2 McMj (13~3)7 = 1

Shut-in time At days

Figure 13-5. Pressure buildup for crossflow gas reservoir.

V(PW

S)J

mm

psi

a2 /

cP

If {kh )t is known from a well test, individual layer permeabilities may beapproximated from

kj = (—[(kht)/hj]\, ./ = l , 2 , . . . . , n (13-4)

Pressure Buildup for Two-Layer Reservoirwithout Crossflow

Figure 13-6 shows a graph of dimensionless pressure Po versus dimension-less time to for a two-layer reservoir with permeability ratios k\ / ki of 1,2,10,and 100. All four curves are for re/rw = 2000. The dimensionless terms are

0.000264**tD = — — (13-5)

0.000264 kttDA = — -r- (13-6)

(frfigcAkh ( Pt - PJ)

p» = 140^7 <13-7)

tDA - 0.000264 kt / ^ c A

Figure 13-6. Muskat straight-line intercepts for two-layer reservoirs withoutcrossflow (after Cobb, Ramey, and Miller).4

Approximate end ofTransient state

where pg = 0.00504(zTPsc)/(pTsc ), bbl/scf, and

All +/?2

h = hi+ h2 (13-9)

hi +h2

Figure 13-6 indicates that during the early transient period, the slope of thestraight-line is 1.151 (2.303/2). The approximate semilogarithmic period endsat to — 5 x 105 and behavior beyond the end of the semilogarithmic period isstrongly influenced by permeability ratio.

Muskat Plot Characteristics

Figure 13-7 shows a Muskat plot for a well in the center of a closed two-layer reservoir with a permeability contrast of 2. Figure 13-8 illustrates Muskatstraight-line intercepts for two-layer reservoirs as a function of producing timefor selected permeability ratios. These plots can be used to estimate average khand reservoir pore volume for a two-layer system produced to pseudo-steady-

AtDA = 0.000264kt / ^ cA

Figure 13-7. Muskat plot for-layer reservoir with a permeability contract of2 (after Cobb, Ramey, and Miller).4

Producing time, tDApproximate start ofMuskat Straight-line

Figure 13-8. Muskat straight-line intercepts for two-layer reservoirs (afterCobb, Ramey, and Miller).4

state before shut-in by

u = HlJ2w.fi, (0-87) ( 1 3 _ n )

(PR ~ Pws)At=o

The effective skin factor for a commingled reservoir system will be given by

The reservoir volume may be obtained from

$Ac = 0.000264 — (slope, log-cycle/hr)"1 (13-13)

where m is the slope of Muskat plot straight line.

MDH Method2

Figure 13-9 shows a MDH plot, which also provides a straight line forthe early buildup, has an approximate slope for the early buildup, and has anapproximate slope of 1.15 for a producing time of any length. Thus the methodcan be used to estimate average kh. This method can also be used to estimatestatic pressure, if theoretical buildup curves exist.

Figure 13-9. MDH buildup curves (after Raghavan et a/.).7

13.4 Multilayered Responses in FracturedGas Reservoirs

Reference 7 has introduced the concept of reservoir layer conductivity CRDJ,given by

Cm = J=-^Nj (13-14)

where r\j = diffusivity of layer j , rj = k/((f)ctiiig), CRD = Y^=I CRDJ, andan equivalent fracture length and equivalent fracture conductivity are defined,respectively, by

n

Xf = ] P CRDjxjj (13-15)

and

kfw = '— (13-16)

Start of Muskat Straight-lineEnd of the straight-line

Producing time, tD

The dimensionless fracture conductivity is then defined by

CfD = & * (13-17)KXf

Camacho, Raghavan, and Reynolds6 have studied the correlations of mul-tiplayer responses with the single-layer solutions for a number of cases. Theyassume that the fractures are not in communication. If fractures are in com-munication, the values of CfD are somewhat higher and the ration hf/xjj isan important factor in the performance of the fractured well. When layers arestimulated by fractures, then maximum productivity will be achieved if thefracture tip in each layer begins to affect the well response at approximatelythe same time. For a two-layer gas reservoir system, the criterion for maximumproductivity is given by

CfQ2H2 _ CRD2Xf2

——— — — ^ o--i o;LfD\n\ tRDiXfi

where

r n - kfjWj

The vertically dashed line in Figure 13-10 represents Eq. 13-18.

Figure 13-10. Criteria for maximum productivity (after Camacho, Raghavan,and Reynolds).6

Rat

io =

[

CfD

2 h 2

/ C

fD1

h, ]

These results are based on the assumption that boundary effects are neg-ligible. The reservoir layer conductivity concept does not apply if boundaryeffects dominate the pressure response. During pseudo-steady-state flow thewell response is given by a well in a circular reservoir.

13.5 Pressure-Production PerformanceResponse Equations

During the period 1960 to 1962, seven papers1"4 were published on thetheoretical behavior of reservoir systems composed of intercommunicatinglayers. Russell and Prats3 summarized the practical aspects of the finding ofthese papers in a later paper. Performance relationships are given here for twocases.

Constant Producing Rate

Pressure performance for transient flow period is given by

2 2 57.920 x lO%cTTZgzPsc^ (kh)tt ]Pwf = Pi TJTV^ l o 3 Z ^ 2 ~ 3 ' 2 3 (13-19)

J (kh)tTsc L (<t>h)tVgcrl Jif ^ if ^ 57.920 x l06qscTpsc f, (kh)tt ]

t(Pwf) = Ir(pi) — — log — - 3.23 (13-20)(kh)tTsc L ((ph)tfigcrl J

For larger times (semisteady state), the pressure behavior is described by

2 2 141.2^c7Zg/^[0.000528(£/z),r , r f ]Pw/= Pi TTT^— ~7IT\ 2~+/«( r g / r w ) -0 .75 (13-21)

J (kh)t L (4>h)tligcrl J

f(pw) = nPi) T7^-I 2 +In(re/rw) ~ 0.75 (13-22)(kh)t L (</>h)tV>gCr* J

where (kh)t = k\h\ +k2h2; ((ph)t = (0iAi) + (^2^2); ht = h\ + h2', t =time in hours; and figt = 0.00504z/r/?5C/p/ Tsc in rb/mmscfd.

The time at which the semisteady state starts is given by

/S1136.4(W^hr (13-23)(kh)t

For semisteady-state flow, the slope of the plot of flowing bottom-hole pressureversus time is given by

Slope = 0 . 0 7 4 5 5 - ^ - r psi/hr (13-24)((/)h)tcrj

Time / , hours

Figure 13-11. Idealized constant-rate pressure performance in two-layerreservoir with crossflow (after Russell and Prats).3

Figure 13-11 shows an idealized constant-rate flowing bottom-hole pressureperformance curve, and reservoirs of the type should possess the propertiesshown on this plot.

Constant Producing Pressure

For the case of constant bottom-hole producing pressure, the followingformula for producing rate was developed. Reference 5 has also provided themethod to calculate cumulative production from multilayered reservoirs:

(kh)t = k\h\ H- k2h2, and (<f)h)t = $\h\ + (J)2Ii2

_ (M), (P1 - Pwf)q&~ 141.2&/*g(/n£- 0.75)* ( I 3 " 2 5 )

where

AB = \ 0-0127 (fcft), 1lit h)tc^8 r? (In^-0.75)\

13.6 Flow Identification and Performance Analysis

Figure 13-12 presents the methods that can be used to identify the degreeof communication between layers and type of crossflow.

Eq. 13-24 gives slopeTime to reach semisteady state is

given by Eq. 13-23

Semi-steady state performance(Eq. 13-21 or Eq. 13-22

Transient performance ( Eq. 13-19 or Eq. 13-20)

\\j(P

wf).

mm

scfd

Figure 13-12. Systemic diagrams to determine degree of communication andtype of crossflow.

13.7 Pressure Buildup Behavior in LayeredReservoir Systems

Figure 13-13 shows pressure buildup behavior in layered reservoir systemsin a single-well, multiple-layer reservoir. Reference 3 has stated that, after theinitial semilog straight line, the buildup curve flattens, then steepens and finallyflattens toward the average reservoir pressure as indicated in the figure. Thisis not always correct. The C-D portion in Figure 13-13 can be insignificantfor some systems. That is particularly true for large contrasts in porosity orthickness, for more than two layers, or for nonsymmetrical systems. Classi-fications and pressure response characteristics including detailed analysis ofmultiple-layered reservoir systems are described in the previous sections.

Flow tests oncores

Petrophysicalanalyses

Formationtesting

Analyses ofproduction and

pressureperformance

Detecting degree ofcommunication

CrossflowWithout

crossflow

Gas flow willdecline

exponentiallyafter a short

period of time(see Figure 13-1)

Productioncharacteristics is ofexponential decline

type, after the one ofthe reservoir layersceases to produce,

the decline rate willgradually decrease(see Figure 13-2)

Degree of communication can bedetermined before the well is completed

Detecting type of flow

Determining degree of communication andtype of crossflow

Figure 13-13. Schematic pressure buildup curve for a layered reservoirsystem.5

Buildup Behavior Curve in Two-Layer Gas Reservoir

The pressure builds up first in the more permeable layer, giving a straight-line section as shown in Figure 13-14. Then the less permeable layer, which isat a higher average pressure, begins to feed fluid into the more permeable layer.This causes the rise above the straight line. Finally, equalization will occur,and the curve will flatten as indicated by the dotted line. Figure 13-14 showspressure buildup behavior in two-layer gas reservoir including the effects ofwellbore damage and storage. Total flow capacity can be obtained from theshape of a field data plot by using the following equations:

Pressure squared case (Eq. 6-5):

= 57.920 x IQ6 gscTPscJIgzmTsc

Pseudopressure case (Eq. 6-10):

n^ _ 51.920 x \06qscTpscyKn)t —

mTsc

Wellbore storage (WBS) durations for testdesigns are: For build-up test:twbs= 170,000 (u g c /kh)e 0 1 4 s

The initial semilog straight-line segment may beused to estimate the total permeability-thickness

product. The second steep slope is not analyzableby known methods.

WellboreStorage

Pres

sure

Initial semilogStraight line

Finalflatteningbehavior

Static or averageReservoir pressure, PR

finalpressure increase

Secondpressure

rise

Nonsymmetricalsystems showing largecontrast in porosity or

thickness

Changingwellborestorage

Log shut-in time

Slope oc (*/),where {kh\ = I (kh)-t

Figure 13-14. Pressure buildup behavior in a two-layer gas reservoir.

13.8 Determining Reservoir Characteristicsin Commingled Systems

References 2 and 4 have provided techniques to determine average reservoirpressure in commingled systems. It requires some knowledge of the layerproperties and correlations for specific systems. The following types of testsmay be used to estimate individual zone properties for a two-layer reservoirwith communication only at the wellbore:

• Single well test• Pulse tests• Flow meter surveys

Apparent kh/fig is always equal to or greater than the actual total (kh/fjig)t forthe reservoir. Apparent (j>cth is always equal to or less than the total ((pcth)tforthe reservoir. Deviation of apparent values from actual total values depends onthe pulse duration.

13.9 Factors Affecting Performance

The following factors can affect the performance of multilayered reservoirsystems:

• Relative permeability. If both layers have the same relative permeabilitycharacteristics, average gas saturation will be higher in the tighter layer

\\f (p

ws)

, m

mp

sia2 /

cP

Flowing pressure

Straight-line sectionSlope m

than in the more permeable layer, because of the average pressure isalways higher in the less permeable layer.

• Pore size. If pore size in the tight layer is smaller than that in the morepermeable layer, then it will tend to reduce crossflow. This effect can beestimated from capillary pressure curves.

• Reservoir geometry. The geometrical nature and extent of interlayer com-munication have some effect on observed field performance.

• Permeability anisotropy. In most petroleum reservoirs, vertical perme-ability is significantly less than horizontal permeability.

• Reservoir n-layer system. Analysis of performance can be handled to ac-ceptable accuracy merely by the previously presented formulas YTj kjhjand YTj <t>jhj f° r (kh)t and ((/)h)t respectively.

13.10 Economic Aspects of Interlayer Crossflow

The absence or presence of crossflow between interlayers can control theeconomic success of a gas production venture. Some of the advantages ofinterlayer crossflow are as follows:

• Shorter operating life• Higher ultimate gas production• Reduced perforating and completion costs• Generally less engineering time required for interpretation of routine tests

Note: A without-crossflow reservoir can be converted into a crossflow reser-voir by fracturing. Thus a vertical fracture can help to establish vertically ad-jacent gas-production strata, which were not in communication prior to thefracture job except at the wellbore.

References and Additional Reading

1. Lefkovits, H. C , Hazebrock, P., Allen, E., and Matthews, C S . , "A Study ofthe Behavior of Bounded Reservoirs Composed of Stratified Layers," Soc.Petroleum Eng. J. (March 1961) 43-58.

2. Cobb, W., "A Study of Transient Flow in Stratified Reservoirs with Com-mingled Fluid Production," Ph.D. dissertation, Stanford University, Stan-ford, CA, 1970.

3. Russell, D. G., and Prats, M., "The Practical Aspects of Inter Layer CrossFlow," /. Petroleum Technol. (June 1962) 589-594.

4. Cobb, W. M., Ramey, H. J., Jr., and Miller, F. G., "Well Test Analysis forWells Producing Commingled Zones," /. Petroleum Techol. (Jan. 1972)27-37; Trans. AIME 253.

5. Earlougher, R. C , Jr., Kersch, K. M., andKunzman, W. J., "SomeCharacter-istics of Pressure Buildup Behavior in Bounded Multiple Layer Reservoirs

without Crossflow," /. Petroleum Technol (Oct. 1974) 1178-1186; Trans.AIME 257.

6. Camacho, V., Raghavan, R., and Reynolds, A. C, "Response of WellsProducing Layered Reservoirs, Unequal Fracture Length," SPE FormationEvaluation (Feb. 1987) 9-28.

7. Raghavan, R., Topaloglu, H. N., Cobb, W. M., and Ramey, H. J., Jr., "WellTest A Analysis for Wells Producing from Two Commingled Zones ofUnequal Thickness," J. Petroleum Technol (Sept. 1974) 1035-1043; Trans.AIME 257.