Chapter 1

GAS MATERIAL BALANCE

1.1 Introduction Material balance is the application of the law of conservation of mass to oil and gas reservoirs and aquifers. It is based on the premise that reservoir space voided by production is immediately and completely filled by the expansion of remaining fluids and rock. As demonstrated later in this chapter, material balance is a useful engineering method for understanding a reservoir's past performance and predicting its future potential. To understand and analyze gas reservoirs, the following conditions will be applied.

1. Reservoir hydrocarbon fluids are in phase equilibrium at all times, and equilibrium is achieved instantaneously after any pressure change;

2. The reservoir can be represented by a single, weighted pressure average at any time (Pressure gradients in the reservoir cannot be considered by the method.)

3. Fluid saturations are uniform throughout the reservoir at any time (Saturation gradients cannot be handled.)

4. Conventional PVT relationships for normal gas are applicable and are sufficient to describe fluid phase behavior in the reservoir.

Material balance calculations can be used to:

1. Determine original oil and gas in place in the reservoir; 2. Determine original water in place in the aquifer; 3. Estimate expected oil and gas recoveries as a function of pressure decline in a

closed reservoir producing by depletion drive, or as a function of water influx in a water-drive reservoir;

4. Predict future behavior of a reservoir (production rates, pressure decline, and water influx);

5. Verify volumetric estimates of original fluids in place; 6. Verify future production rates and recoveries predicted by decline-curve analysis; 7. Determine which primary producing drive mechanisms are responsible for a

reservoir's observed behavior, and quantify the relative importance of each mechanism;

8. Evaluate the effectiveness of a water drive; 9. Study the interference of fields sharing a common aquifer.

Data requirements to accurately apply the material balance method consists of: (1) cumulative fluid production at several times (cumulative oil, gas, and water); (2) average reservoir pressures at the same times, averaged accurately over the entire reservoir; (3) fluid PVT data at each reservoir pressure as well as formation compressibility.

1.1

1.2 Basic concepts The general material-balance equation for a depletion-drive gas reservoir, neglecting water and formation compressibilities is expressed by:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−=

GpG

izip

zp 1 (1.1)

It is one of the most often used relationships in gas reservoir engineering. It is usually valid enough to provide excellent estimates of original gas-in-place based on observed production, pressure, and PVT data. During the life of a gas reservoir, cumulative production is recorded, and average reservoir pressures are periodically measured. At each measured reservoir pressure the gas z-factor is determined to calculate P/z, and the result plotted as shown in Figure 1.1 below. Notice that Equation 1.1 results in a linear relationship between P/z and Gp. That is, as gas is produced from the reservoir, the ratio P/z should decline linearly for a volumetric reservoir. Note that for an ideal gas, pressure alone would decline linearly.

0

500

1000

1500

2000

2500

3000

3500

4000

0 1 2 3 4 5 6 7 8 9 10 11 12

Cumulative gas produced,Bscf

p/z,

psi

a

(p/z)i

(p/z)a

Gpa=8.5 Bscf G=10 Bscf

Last measureddata point

extrapolate

Figure 1.1 Example of linear relationship between p/z and gas produced for a volumetric reservoir.

1.2

Example 1.1 Determine the original gas-in-place and ultimate recovery at an abandonment pressure of 500 psia for the following reservoir.

Gas specific gravity = 0.70 Reservoir temperature = 150°F Original reservoir pressure = 3000 psia Abandonment reservoir pressure = 500 psia Production and pressure history as shown in the following table.

eps: Determine z-factor and calculate P/z.

Fig. 1.1)

Gp must equal G, the OGIP.

5) m an abandonment pressure of 500 psia

In the exam ly-recognizable straight 1ine, but in practice

this example the recovery factor was estimated simply by the ratio of ultimate recovery

Gp, mmscf P, psia z, p/z, psia0 3000 0.822 3650

580 2800 0.816 34311390 2550 0.810 31483040 2070 0.815 2540

St

1)2) Plot P/z versus GP. (Example shown in3) Draw a straight line through the data points. 4) Extrapolate the straight line to p/z = 0, where

From Figure 1.1 G = 10.0 Bscf. Ultimate recovery is estimated froand z = 0.945, thus (p/z)a = 529 psia. From Fiqure 1.1, the ultimate recovery is estimated to be 8.5 Bscf, or 85% of the OGIP.

ple, the data points formed a easithis may not always be the case. Theoretically, there should be a linear relationship between p/z and G. However, in practice there are several factors that may cause the relationship to be nonlinear. Formation compressibility may be significant, as in an unconsolidated sand reservoir. If this extra stored energy is not accounted for, the measured data points will be extrapolated to an optimistically high value of OGIP. In the example, if the reservoir actually had an OGIP of 10.0 Bscf, but the formation was unconsolidated and had significant compressibility, the three measured reservoir pressures after production had begun would have been greater, and extrapolation to OGIP would have exceeded 10.0 Bscf. Secondly, average reservoir pressure may not have been accurately determined. This is a common problem, and the plot of p/z versus Gp frequently shows more fluctuations from a straight line than in Figure 1.1. Finally, if a water drive were acting on the gas reservoir, pressure support would occur and the plot of p/z versus Gp would give an overly optimistic OGIP. Into the original gas-in-place. Alternatively, it is possible to estimate ultimate recovery efficiency (RE) in a depletion-drive gas reservoir with negligible water and formation compressibilities by:

1.3

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−=

ipiz

azap

gaBgiB

volRF 11 (1.2)

For the example, ultimate recovery efficiency can be estimated from Equation (1.2),

OGIP of%8536505291 =⎟

⎠⎞

⎜⎝⎛ −=volRF

The decision on selection of an appropriate abandonment pressure is dependent on operational considerations, reservoir variables and economics. In general, the higher the reservoir permeability the lower the abandonment pressure. However, in tight gas sands, where permeability is in the milli- to microdarcy range, abandonment pressures are being reduced by a variety of methods targeting bottomhole flowing pressure. These methods can be on the surface, such as adding compression, or in the wellbore, such as adding plunger lift or capillary tubes. The added benefit can be quite substantial. For instance, in the previous example if the abandonment pressure could be reduced in half, the incremental gain in recovery would be approximately 1 Bscf.

1.3 Advanced Topics The previous section provides the basic concepts in material balance for simple, volumetric gas reservoirs. However, nonlinearity can occur in the p/z vs Gp relationship as a result of water influx, or changes in rock and water compressibilities in geopressured reservoirs, or the inability to achieve average reservoir pressure such as in low permeability reservoirs. A comprehensive form of the gas material balance equation is given by:

[ ]( ) ( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−++−−⎟

⎠⎞

⎜⎝⎛

=−−

eWwBinjWwBpWgB

1swRpWinjGpG

Giz/p

izp

)pip()p(ec1zp

(1.3)

where Ginj and Winj are gas and water injection, respectively; Rsw is solution gas in the water phase, and ce(p) is an average effective compressibility term. From this general material balance equation we will investigate the affects of water influx, rock/water compressibilities and low-permeability systems. 1.3.1 Water-drive Gas Reservoir The impact of water influx is to provide pressure support, resulting in slower pressure decline. Subsequently, gas reservoirs associated with aquifers show a flattening of the p/z curve. Figure 1.2 shows p/z curves for gas reservoirs with varying strengths of aquifer support. In all cases, linear extrapolation of the water-drive cases to determine OGIP would lead to optimistically high values.

1.4

Figure 1.2 Water Drive Gas Reservoir p/z Curve

he rate of the gas withdrawal is directly proportional to the ability of water to encroach.

hen water invades a gas reservoir, the net volume of water influx reduces the gas

Assuming no injection has occurred, that rock and water compressibility changes are

p/z

Gp

(p/z)i

(p/z)a

Depletion drive

water drive

strength

(p/z)a

p/z

Gp

(p/z)i

(p/z)a

Depletion drive

water drive

strength

(p/z)a

TFor example, a high withdrawal rate coupled with a strong aquifer could lead to early coning and/or pockets of trapped gas. Agarwal, et. al, in 1965 attributed the low gas recovery in water drive reservoirs to the trapped residual gas saturation and a volumetric displacement efficiency less than unity. They showed that the size and properties of the aquifer and the withdrawal rates, along with residual gas saturation and volumetric sweep efficiency impact the ultimate gas recovery and thus are major factors in designing field development strategy. Wvolume. The material balance equation must reflect this addition, subsequently, we can write,

⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧

rcf influx,net water

rcf GIP, of volumeremaining

rcf GIP, of volumeoriginal

small and the solubility of gas in the water is negligible, then the general material balance equation reduces to:

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−+−⎟

⎠⎞

⎜⎝⎛= wBpWeW

gGB1

GpG

1iz

pzp (1.4)

here,

ulative water influx into the gas reservoir, rcf rnative expression is WpBw

earranging Eq. (1.4) to solve for gas-in-place results in the following expression,

wWe = cumWp = cumulative water production, rcf [Note: an altewhere Bw is the water formation volume factor in rbbl/stb. Then Wp can be expressed in stb] R

1.5

( )giBgB

pWeWgBpGG

−

−−= (1.5)

Early in the producing life of a reservoir, the difference in the denominator is small and

o estimate the ultimate recovery efficiency in a water-drive gas reservoir requires an

therefore could lead to erroneous values of gas-in-place. Subsequently, to obtain accurate results, Eq. (1.5) should be used over longer periods of time. Testimate of residual or trapped gas saturation. In a water-drive gas reservoir, gas saturation at abandonment (called residual or trapped) does not equal original gas saturation: Sgr = Sgt ≠ Sgi; therefore,

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

giSgrS

ipiz

azap

1giSgrS

gaBgiB

1wdRF (1.6)

Implicit in the derivation of Equation (1.6) is the assumption that volumetric sweep efficiency for gas, Ev, is 100%. This assumption is optimistic as frequently the displacement of gas by water results in unswept, bypassed portions of the reservoir, increasing the trapped gas saturation. Subsequently, a modified form of Eq. (1.6) is:

⎟⎟⎠

⎞⎜⎛ − vE1grSgiB⎜⎝

+−=vEgiSgaBvE1wdRF (1.7)

ome published values of residual gas saturation were given by Geffen (1952) and are

Porous Material Formation Sgr, %

Sshown in Table 1.1 below.

Unco nsolidated sand 16 Slightly consolidated sand

nsolidated sand Selas Porcelain 17

onsolidated sandstones 30-38

Bly

34-37

imestone eef

(synthetic) Synthetic co

21

Norton Alundum 24 C Wilcox 25 Frio Nellie 30-36 Frontier 31-24 Springer 33 Torpedo Tensleep 40-50 L Canyon R 50

Tabl al gas saturation af as measured on core plugs

e 1.1 Residu ter waterflood (Geffen,et. al, 1952)

1.6

Example 1.2 The reservoir is the same as described in the previous example except that pressure is fully maintained at its original value by a strong water drive. Assume that the entire gas reservoir is swept by water. Given: Sgi = 75% and Sgt = 35%, respectively. What if Ev = 60%? Since the pressure is constant for the life of the reservoir, a simplified form of Eq. (1.6) becomes,

%5375.035.01

giSgrS

1wdRF =−=⎟⎟⎠

⎞⎜⎜⎝

⎛−=

Similiarly, if the volumetric sweep efficiency is accounted for, then Eq. (1.7) results in RFwd = 32%. Both are much less than typical recovery efficiencies in depletion-drive gas reservoirs. Note that a partial water drive does not maintain pressure completely, and thus would allow some gas production by pressure depletion, and recovery efficiency would improve. In general, recovery efficiency in a gas reservoir is much better under depletion-drive than under water-drive. A modified material balance for water drive gas reservoirs was proposed by Hower and Jones (1991) and Schafer, et al (1993) to account for pressure gradients that develop across the invaded region. Previous theory assumed the invaded zone pressure is equivalent to the reservoir pressure and is constant. The modified approach accounts for pressure gradients in the invaded zone due to capillary pressure. The method predicts a higher pressure at the original reservoir boundary and a much lower pressure in the uninvaded region of the reservoir. Both water influx calculations and reservoir performance predictions are influenced by the pressure gradient term. The pressure drop in the invaded zone is given by the steady state radial flow equation.

khrwktrorwwq

tpopinvp)/ln(2.141 μ

=−=Δ (1.8)

where po = pressure at the original reservoir boundary pt = pressure at the current reservoir boundary ro = radius at the original reservoir boundary rt = radius at the current reservoir boundary The relative permeability to water is evaluated at the endpoint; i.e., at residual gas saturation; therefore it is not required to obtain the entire relative permeability curve. The residual gas saturation is assumed constant throughout the entire invaded region. The water flow rate can be estimated using the water influx term, qw = dWe/dt. The resulting modified material balance equation becomes:

pWwBeWgBgtBtGgiBgBGgBpG −+−+−= )()( (1.9)

1.7

where Gt is the volume of trapped gas in the invaded region of the reservoir and is a function of Sgr and average pressure in the invaded region. Results from the proposed modified material balance method agreed with a numerical simulation model and demonstrated the influence of relative permeability on the reservoir performance. Figure 1.3 from Hower and Jones (1991) illustrates the excellent match with the simulator if a krw = 0.06 is assumed. Also, notice the difference in reservoir performance between the conventional and modified material balance techniques.

Figure 1.3 Comparison of reservoir performance for conventional and modified material balance methods and numerical simulation. (Hower and Jones, 1991) Example 1.3 GWINFLUX is a software application for the modified gas material balance method provided by GRI. Use the DEMO.DAT file and determine the OGIP. When applying the material balance equations for oil and gas reservoirs the typical solution is to rearrange to solve for N and then used to compute OOIP or OGIP at different times during a reservoir's life. Naturally the results of each calculation vary somewhat from one time to another. Thus, there are often major questions about the correct value for OOIP or OGIP, especially if a gas cap or water influx is present.

1.8

A powerful method of removing much of the doubt concerning the accuracy of computed results was presented by Havlena and Odeh in two papers in 1963 and 1964. The first paper presents the theory; the second presents field case studies. Havlena and Odeh's method rearranges the material balance equations into an algebraic form that results in an equation of a straight line. The procedure requires plotting one variable group versus another. The shape of the plot and sequence of plotted points provide important insight into the validity of the assumed reservoir drive mechanism. The linearized material balance equation for gas reservoirs is:

tEwBeW

GtE

F+= (1.10)

where G represents the original gas in place at standard conditions, and

F = total net reservoir voidage

wBpWgBpGF += (1.11)

Et = Eg + Ecf = total expansion

Eg = expansion of gas in reservoir

giBgBgE −= (1.12)

Ecf = connate water and formation expansion * Bgi

(⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

= pipwiS

fcwcwiSgiBcfE

1* ) (1.13)

Since G and Gp are usually expressed in SCF, the units of Bg and Bgi are in RCF/SCF. A plot of F/Et vs WeBBw/Et should result in a straight line with intercept of G, the original gas in place, and slope related to water influx (see Figure 1.4).

F/E t

,stb

Intercept=G

tEwBeW

We correct

We too small

We too large

F/E t

,stb

Intercept=G

tEwBeW

We correct

We too small

We too large

Figure 1.4 Material balance linear plot for gas reservoirs with aquifer support

1.9

To appropriately interpret Figure 1.4, the material balance equation must be coupled with a water influx model. For example, for the Fetkovich aquifer model, the slope of the straight line should be equal to one. If not, different aquifer properties must be used. Others are the unsteady state Van Everdingen and Hurst and steady state Schilthuis water influx models. In the former, a straight-line slope provides an estimate of the water influx constant, B. If the data does not plot as a straight line then different aquifer properties must be estimated. In the latter, the slope is equal to the water influx constant, k. Further discussion on water influx properties is beyond the scope of this chapter. For details the reader is directed to the references at the end of this chapter. The drive indices for a gas reservoir are defined as follows:

Gas drive index: gBpG

gGEGDI = (1.14)

Water drive index: gBpG

wBpWwBeWWDI

−= (1.15)

Compressibility drive index: gBpG

cfGECDI = (1.16)

Where the sum of the drive indices is equal to one.

1.10

Example 1.4 The performance history for a gas reservoir with water influx is given in Table 1.2 below. Solve for the correct gas-in-place using the linearized method.

time Gp Bg Wpdays mscf rbbl/mscf stb

1 816 0.837 031 25299 0.843 061 52191 0.849 092 83814 0.855 0

123 140817 0.868 16563153 210174 0.885 37934184 260921 0.894 54497214 313742 0.905 75868245 392389 0.925 108994276 429366 0.930 120036304 504400 0.952 144969335 618738 0.993 183615365 733260 1.035 221015396 825832 1.067 259662426 909022 1.098 297061457 949839 1.104 335708488 991534 1.117 385396518 1031923 1.132 396082549 1074335 1.152 478896579 1109657 1.167 516296610 1145626 1.185 554942641 1196897 1.221 571505670 1275411 1.289 612823701 1315925 1.316 651469731 1353185 1.340 688869762 1385369 1.360 721995792 1427744 1.407 754052823 1463357 1.442 853428854 1501825 1.488 958326

Table 1.2 Performance history data for Example 1.4 The example is setup to demonstrate the influence of water influx on identifying the correct straight line. A cumulative water influx of 831 mbbl, results in a straight line and estimate of 2.107 Bscf of gas-in-place (See Figure 1.5). A decrease in cumulative water influx of 484 mbbl. resulted in a concave upwards trend and an increase in cumulative water influx of 1427 mbbl resulted in a concave downwards trend. The GDI is approximately constant at 60% and the WDI at 40% for the +2 years of production history.

1.11

Figure 1.5 Influence of water influx on gas material balance for Example 1.4 Note if no connate water and formation expansion occurs, then Eq. (1.10) reduces to Eq. (1.5), the standard expression for gas material balance with water drive. Also, if the reservoir drive mechanism is purely by gas expansion (depletion drive), then Eq. (1.10) reduces to:

tGEF = (1.17) A plot of F vs Et should be a straight line through the origin with slope of G. 1.3.2 Abnormally pressured gas reservoirs In high-pressure, depletion drive type reservoirs formation and fluid compressibility effects result in a nonlinear p/z vs cumulative plot. Figure 1.6 is a schematic illustrating this behavior. The rate of decrease of pressure during the early time is reduced due to support of these compressibility components. Extrapolation of this initial slope will result in an overestimation of gas-in-place and reserves. As pressure reduces to a normal gradient, the formation compaction influence on the reservoir becomes negligible and thus the remaining energy comes from the expansion of the gas in the reservoir. This accounts for the second slope in Figure 1.6.

1.12

y = 1.0602x + 2.1074R2 = 0.9843

3.0

3.2

3.4

3.6

3.8

4.0

4.2

4.4

4.6

4.8

5.0

1.0 1.5 2.0 2.5 3.0

We/Et

F/E t

p/z

Gp

(p/z)i

Gas expansion

Gas expansion+

Formation compaction+

Water expansion

Overestimate of G

p/z

Gp

(p/z)i

Gas expansion

Gas expansion+

Formation compaction+

Water expansion

Overestimate of G

Figure 1.6 nonlinear p/z vs Gp plot due to formation and water compressibility effects. Assuming no water influx or production and no injection, then the general material balance equation (1.3) reduces to:

[ ])pip()p(ec1GpG

1izip

zp

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

= (1.18)

where ce(p) is a pressure-dependent effective compressibility term. An early definition of ce(p) by Ramagost and Farshad (1981) where in terms of constant pore and water compressibilities.

)wiS1(fcwiSwc

ec−

+= (1.19)

Average values were assumed thus removing the complication of pressure dependency. To determine gas-in-place, the p/z term (y-axis) is linearized by plotting,

( )pGvs

)wiS1()pip(fcwiSwc

1zp

⎥⎦

⎤⎢⎣

⎡−

−+−

Example 1.5 Estimate the original gas-in-place for the data given by Duggan (1972) for the Anderson L sand. Apply both the conventional and geopressured material balance equations. Given:

pi = 9,507 psia cw = 3.2 x 10-6 psi-1

Swi = 0.24 cf = 19.5 x 10-6 psi-1

Original pressure gradient = 0.843 psi/ft

1.13

p,psia z Gp,Bcf9507 1.440 09292 1.418 0.39258970 1.387 1.64258595 1.344 3.22588332 1.316 4.26038009 1.282 5.50357603 1.239 7.53817406 1.218 8.74927002 1.176 10.50936721 1.147 11.75896535 1.127 12.78925764 1.048 17.26254766 0.977 22.89084295 0.928 28.14463750 0.891 32.56673247 0.854 36.8199

Figure 1.7 displays the results of the conventional material balance. Gas-in-place is estimated to be 89.3 Bcf.

Volumetric (normal pressured)

y = -74.679x + 6667.5R2 = 0.9929

0

1000

2000

3000

4000

5000

6000

7000

8000

0 20 40 60 80 1

Gp, Bcf

p/z,

psi

00

a

G=89.3

Figure 1.7 Conventional material balance solution for Example 1.5

Figure 1.8 shows the results when using the geopressured approach. The resulting gas-in-place is 70.7 Bscf. Thus if the conventional approach is taken, the gas-in-place will be overestimated by more than 25%.

1.14

Volumetric (geopressured)

y = -92.336x + 6532.2R2 = 0.9979

0

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 10

Gp, Bcf

mod

ified

p/z

, psi

0

a

G=70.7

Figure 1.8 High-pressure material balance solution for Example 1.5

The above example assumed that formation compressibility was both known and constant. However, frequently formation compressibility varies during pressure depletion, and is difficult to obtain in the laboratory. Roach (1981) developed a material balance technique for simultaneously estimating formation compressibility and gas-in-place and was later applied by Poston and Chen (1987) to the Anderson L example. The revised material balance equation is:

( ) ( ) wiS1fcwcwiS

ipzzip

pippG

G11

ipzzip

pip1

−+

−⎥⎥⎦

⎤

⎢⎢⎣

⎡

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−

− (1.20)

If the formation compressibility is constant then a straight line will develop with a slope = 1/G and an intercept = -(Swicw + cf)/(1-Swi). Example 1.6 Repeat example 1.5 and estimate both gas-in-place and formation compressibility. Figure 1.9 shows the results from this analysis. The original gas-in-place is estimated from the slope,

BscfG 8.75199.13

1000==

and the formation compressibility from the intercept, 16105.12)1(610 −−=−−−−= psixwcwiSwiSbxfc

The deviation from the straight line at early time is due to the pore fluids supporting the overburden pressure. However, as fluids are withdrawn the formation compacts, thus transferring more of the support on the rock matrix.

1.15

Geopressured

y = 13.199x - 17.511R2 = 0.993

0

50

100

150

0 2 4 6 8 10 12 14

x, mmscf/psi

y, p

si- 1

Figure 1.9 Simultaneous solution of gas-in-place and formation compressibility for a volumetric geo-pressured gas reservoir, Example 1.6 Fetkovich, et al. in 1991 further expanded the effective compressibility term by including both gas solubility and total water associated with the gas reservoir volume. The resulting expression accounts for pressure dependency.

)1(

])()([)()()(

wiSpfcptwcMpfcwiSptwc

pec−

+++= (1.21)

The cumulative total water compressibility, ctw, is composed of water expansion due to pressure depletion and the release of solution gas in the water and its expansion. The associated water-volume ratio, M accounts for the total pore and water volumes in pressure communication with the gas reservoir. This includes non-net pay water and pore volumes such as in interbedded shales and shaly sands, and external water volume found in limited aquifers. The authors defined both terms as:

( )( )

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛ −=

+=

12

/

/1

rraqr

rhaqh

ghnhghnh

r

nnp

aqMNNPMM

φ

φ

φ

φ (1.22)

where nnp - non-net pay property r - reservoir (net pay) property aq - aquifer property hn/hg - net to gross ratio The proposed method of obtaining gas-in-place requires a trial and error solution. Historical pressure and production data is coupled with an assumed gas-in-place value to

1.16

back-calculate values of effective compressibility from a rearranged material balance equation.

( ) ( )( ) ( )pip

1GpG

1z/p

iz/p1atedbackcalculec

−⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−= (1.23)

The effective compressibility from Eq. (1.23) can be plotted as a function of pressure, and compared to values determined from rock and fluid properties in Eq. (1.21). A reasonable fit between the two methods provides an estimate of gas-in-place and a measure of physical significance to the results. Example 1.7 Fetkovich, et al tested their method on the Anderson L sand data from Duggan (1971). Additionally, they calculated total water compressibility as a function of pressure. Using their values of ctw, a G = 72 Bscf, cf = 3.2 x 10 -6 psi -1, and M = 2.25, the following results were obtained.

05

101520253035404550

0 2000 4000 6000 8000 10000

pressure, psia

ce, p

si- 1

ce(back) assuming OGIP

ce(p) generated from rock & water properties

Figure 1.10 Comparison of back-calculated effective compressibility assuming OGIP with the rock and fluid property derived values. Figure 1.10 shows best fit results after varying M, cf, and G, respectively. (The authors also varied Swi and decided on Swi = 0.35. In Fig 1.10 the original Swi = 0.24 was maintained.) Figure 1.11 shows the performance match and prediction using the variables listed above. The estimated gas-in-place of 72 Bcf is within the range of the previous methods. Notice the first data points at high pressure in Figure 1.10 do not fit the correlation drawn. These points correspond to the same data points which deviate from the straight line in Figure 1.9, and thus the same explanation is believed valid for this method as well. The disadvantages of the Fetkovich, et al method are the requirement of rock and water properties to build the effective compressibility correlation, the assumption that cf is constant, and the non-uniqueness of the solution since multiple combinations of the

1.17

variables can yield the same outcome. The advantage is the addition of the pressure dependency of the water compressibility and the development of a physical basis for the analysis.

0

1000

2000

3000

4000

5000

6000

7000

0 20 40 60 80 10

Gp, Bscf

p/z,

psi

0

ahistorical performance datamodel

Figure 1.11 Performance match and prediction for the Anderson L reservoir.

1.3.3 Low permeability gas reservoirs Gas material balance in conventional, volumetric reservoirs is described by a linear relationship between pressure/z-factor (p/z) and cumulative production. Unfortunately, tight gas reservoirs do not exhibit this type of behavior, but instead develop a nonlinear trend (see Figure 1.12), which is not amenable to conventional analysis.

Gp

p/z

Conventional response

G

Tight gas response

(p/z)i

(p/z)int m1

m2 = m

1

?

Gp

p/z

Conventional response

G

Tight gas response

(p/z)i

(p/z)int m1

m2 = m

1

?m2 = m

1

?

Figure 1.12 P/z response for conventional gas reservoir and a tight gas reservoir.

1.18

The nonlinear trend is a function both of the pressure measurement technique and the reservoir characteristics. Typical shut in periods are not of sufficient duration to achieve a representative average reservoir pressure. This concept can be reinforced by examining the criteria for reaching pseudosteady state flow.

DApsstk

Aticipsst

φμ3790= (1.24)

Assuming a well located in the center of the drainage area and substituting typical reservoir and gas properties for a tight gas formation (φ = 11%, k = 0.1 md, μgi = 0.012 cp., cti = 0.001 psi-1), results in a time to reach pseudosteady state of 2 years for an 80-acre drainage area and 16 years for a 640-acre drainage area. Subsequently, a single buildup pressure measurement after seven days of shut in will not achieve such a boundary condition. To analyze low-permeability reservoirs the following constraints are applied: (1) no water influx, (2) constant reservoir temperature, (3) no rock compressibility effects, and (4) only single phase dry gas; i.e., no phase changes occur in the reservoir. Furthermore, to simplify the analysis the bottomhole flowing pressure will be assumed to be constant over the life of the well. A reasonable assumption for dry gas wells controlled by surface line pressure. Referring to Figure 1.12, three trends are exhibited on the p/z plots for low permeability reservoirs. During the early time period a rapid decrease in pressure occurs. If this trend is extrapolated to p/z = 0, the gas-in-place (G) will be seriously underestimated. The behavior has been previously explained as the response to transient flow (Slider, 1983); however, additional analysis did not confirm this hypothesis. An alternative solution is the rapid depletion of a stimulated well in a reservoir consisting of a natural fracture network; in simple terms, the flush production associated with such a condition. Coupled with this behavior is the inability of the pressure measurement technique to capture reservoir pressure within the testing time. Subsequently, as the drainage radius is expanding the testing pressure deviates more and more from the average reservoir pressure. The intermediate period exhibits uniform slope over an extended period of time, even though, the magnitude of the pressure measurement observed is significantly below the average reservoir pressure. During this period, the test time is too short to capture the average pressure response; however, consistency of the data suggests that a similar region is being repeatedly investigated by the pressure test. For example, notice in Figure 1.13 the difference in pws and pr is approximately constant for an extended period of time. Several researchers (Stewart,1970) (Brons and Miller, 1961) have presented methods to correct measured data to average reservoir pressure by pressure buildup techniques.

1.19

Pi

rw

.472re

re

Pwfri

Pi

rw

.472re

re

Pwfri

Figure 1.13 Schematic of a partial buildup response in a tight gas reservoir, indicating the difference in measured pws and average reservoir pressure, pr. The constant slope provides an opportunity to estimate the hydrocarbon-pore volume, Vhc. Defining the slope (m) as;

pG)z/p(m

ΔΔ

= (1.25)

and substituting into the gas material balance equation, results in an expression to determine Vhc.

m1*

scTscTP

hcV = (1.26)

From volumetrics,

)wS1(Ah43560hcV −φ= (1.27)

thus providing a method to determine the drainage area. Furthermore, from the observation of a constant slope, three scenarios can be developed to determine the gas-in-place as illustrated in Figure 1.14. The problem is defining the relationship between the determined slope and the actual slope if one could measure the actual reservoir pressure. Case A exhibits two parallel trends of constant slope; i.e., m1 = m2. Gas-in-place can readily be obtained from,

m1*

izip

G⎟⎟

⎠

⎞

⎜⎜

⎝

⎛= (1.28)

The difference in gas-in-place between the two lines is due to the initial reservoir pressure difference; and not the hydrocarbon pore volume, which is the same for both lines.

1.20

G1≠ G2Gp

Figure 1.14. Three possible relationships between the conventional response and the tight gas response. To have equal slopes suggests the radius of investigation of the pressure test is expanding at the same rate as the radius of drainage of the reservoir. That is, ri ≈ constant* re over an extended period of time. The magnitude of gas-in-place will be overestimated by this method and therefore provides an upper bound to the well. In case B the slopes are different, but the intersection point occurs at the same gas-in-place. Estimation of G is obtained by,

2

1*1

1*int miz

ip

mzpG

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=⎟

⎠⎞

⎜⎝⎛= (1.29)

where the (p/z)int is the intercept value from the identified pressure trend. To solve for the correct Vhc requires the substitution of m2 into Eq. (1.26). Subsequently, the hydrocarbon pore volume is corrected to reflect the difference in reservoir pressures. For this behavior to occur means the investigative volume seen during subsequent pressure tests is approaching the average drainage volume of the well. In other words, ri → 0.472re. This is as expected for depleted reservoirs where the pressure gradient is approximately uniform throughout the reservoir.

P/z m1 =m

2

Case A

Gp

zP/

G1= G2

Case B

m2

(p/z)int

G p

P/z m

1 ≠ m 2

G 1 ≠ G 2

Case C

G p

P/z m

1 ≠ m 2

G 1 ≠ G 2

Case C

G p

P/z m

1 ≠ m 2

G 1 ≠ G 2

Case C

G p

P/z m

1 ≠ m 2

G 1 ≠ G 2

Case C

G p

P/z m

1 ≠ m 2

G 1 ≠ G 2

Case C

G p

P/z m

1 ≠ m 2

G 1 ≠ G 2

Case C

m1 ≠m

2

m1 ≠m

2

G1≠ G2Gp

zP/ m1 =m

2

Case A

Gp

zP/

G1= G2

Case B

m2

(p/z)int

G p

P/z m

1 ≠ m 2

G 1 ≠ G 2

Case C

G p

P/z m

1 ≠ m 2

G 1 ≠ G 2

Case C

G p

P/z m

1 ≠ m 2

G 1 ≠ G 2

Case C

G p

P/z m

1 ≠ m 2

G 1 ≠ G 2

Case C

G p

P/z m

1 ≠ m 2

G 1 ≠ G 2

Case C

G p

P/z m

1 ≠ m 2

G 1 ≠ G 2

Case C

m1 ≠m

2

m1 ≠m

2

1.21

The third and final scenario (Case C) exhibits both a different slope and intercept between the measured pressure trend and the actual reservoir behavior. Unfortunately, the measured data does not reflect the actual reservoir behavior. The best is to estimate a range for gas-in-place using Case A as the upper bound and case B as the lower bound. A final stage of the life of the well occurs when depletion has been significant (see Fig. 1.12). At this time the measured pressure curve flattens and becomes constant; converging to the actual average reservoir pressure. In many cases the gas-in-place was estimated by extending a straight line from the initial p/z point through this late time point. Experience has shown this method typically underestimates gas-in-place, due to the late time measured pressure slightly underpredicting the actual reservoir pressure. Also, as Fetkovich, et.al. (1987) correctly point out, a rise in pressure can be a rebound effect due to a decrease in withdrawal from the reservoir.

Example 1.8 The example well produces from the Pictured Cliffs sandstone in the San Juan Basin of northwest New Mexico. Picture Cliffs is a low permeability, sandstone to shaly sandstone gas reservoir found at a depth of approximately 3200 feet and developed on 160 acre spacing (Dutton, et al, 1983). The example well (No.114) was initially completed in 1958 and included a hydraulic fracture treatment to be commercially productive. Other well and reservoir data are listed below. The long history of production and pressure data make this well an excellent candidate for investigation.

φ 11 μgi, cp 0.0134 h, ft 40 cti, psi-1 x 10-4 5.77 γg 0.67 Tr , deg F 106 Sw, % 44 rw, ft. 0.229 Pi, psi 1131

Table 1.3 Input well and reservoir properties

Figure 1.15 is the p/z vs cumulative production plot for this well. In the San Juan Basin, pressure data is recorded over a 7-day shut in period and reported annually until 1974 and every other year until 1990. The primary purpose of collecting this information was for deliverability testing and proration. Notice the typical tight gas well response of a rapid decrease in pressure within the first year. This behavior does not correspond to the end of the transient period, which occurs 8 to 10 years later according to decline curve analysis. The majority of time and hence cumulative production exhibits case B behavior; i.e., constant p/z decline. Applying Eq. (1.29) this trend results in an estimate of 660 mmscf of gas-in-place.

1.22

P/Z vs. Cumulative Production

No. 114

y = -0.8367x + 552.88R2 = 0.958

0

200

400

600

800

1000

1200

1400

1600

0 100 200 300 400 500 600 700

Cumulative Production, mmscf

P/Z,

psi

a

Figure 1.15. Field example of tight gas response (Case B) on p/z plot and estimation of gas-in-place. Also shown on Figure 1.15 is an extrapolation between the initial p/z and the anomalous increase in p/z found in the latest data points; resulting in 520 mmscf of gas-in-place. Frequently this extrapolation is applied to tight gas wells to estimate gas-in-place and recovery. The validity of the last points is pivotal to this method being successful or not. These pressure points were acquired during a time of extended cycles of shutin and production due to external constraints. The resulting bottomhole flowing pressure is increased which subsequently translates into an increase in recorded shutin bottomhole pressure. This is the same conclusion as drawn by Fetkovich, et al. in 1987. Unless this pressure data is obtained very late in the life of the well it is likely this method will underestimate gas-in-place and reserves. Cumulative production (through 2006) for this well is 526 mmscf; therefore 80% of the gas-in-place has been recovered. A rate – cumulative plot (Figure 1.16) also provides a linear trend, which when extrapolated results in gas-in-place of 700 mmscf or 75% recovery. Both methods are within reasonable agreement.

1.23

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 100 200 300 400 500 600 700

Cumulative production, mmscf

flow

rat

e, m

scf/m

onth

Figure 1.16. Extrapolation of Rate – cumulative trend for gas-in-place. A key to tight gas development is the drainage area of existing wells and the feasibility of infill drilling. Using Equation (1.29) to adjust the slope, the hydrocarbon pore volume is calculated to be 7.544 mmrcf. Substitution of the known gas and well properties results in a drainage area calculation of 70 acres. To further investigate the tight gas, pressure behavior, a single well, simulation model was developed for single-phase flow. As a simplification, the reservoir properties were assumed to be homogeneous and isotropic. The well was bottomhole pressure constrained, initially at 250 psi and then reduced to 150 psi ten years later. This change reflects the actual pressures measured during the annual deliverability tests. Figure 1.17 illustrates the excellent match between the results from the simulator with the measured data for both gas rate and shutin bottomhole pressure. The success of the model verifies the linear trends seen on the gas material balance plots and the slow pressure response of tight gas reservoirs. Furthermore, to obtain this match the areal extent of the simulation model was 86 acres, which is in agreement with the previous methods. The analysis suggests this well has drained 70 to 90 acres of the dedicated 160-acre proration unit and has recovered approximately 70% of the gas-in-place within that volume. The paradox is the boundary-dominated flow exhibited by the decline curve. The nearest well is approximately 1850 feet away from the subject well, farther than the estimated drainage area. Two explanations can be given. First, the drainage calculations are based on isotropic conditions and therefore a circular drainage pattern. However, if anisotropy exists, then the two wells are sufficiently close enough to provide interference. Investigation of production and geological trends show a dominant northwest/southeast direction, the exact direction of these two wells. Second, a thinning of the reservoir net pay thickness over the areal extent of this well would increase the drainage area. For example if thickness is reduced by half then the drainage area doubles to approximately 160 acres.

1.24

1

10

100

1000

0 5 10 15 20 25

time, years

prod

uctio

n ra

te, m

scf/m

o

0

200

400

600

800

1000

1200

SIBH

P, p

si

simulated

measured

Figure 1.17 Comparison of simulation results with measured data for Pictured Cliffs example.

1.25

References Agarwal, R.G., Al-Hussainy, R. and Ramey, Jr., H.J.: “The Importance of Water Influx in Gas Reservoirs”, JPT (Mar. 1965). Brons,F and Miller, W.C.:”A Simple Method for Correcting Spot Pressure Readings,” (1961) Trans., AIME 222, 803-805. Carter,R.D. and Tracy, G.W.: “An Improved Method for Calculating Water Influx”, JPT, (Dec. 1960) Duggan, J.O. “The Anderson L – An Abnormally Pressured Gas Reservoir in South Texas, “ JPT 24, No. 2, pp. 132-138, (Feb. 1972). Dutton,S.P., Clift,S.J., Hamilton,D.S., Hamlin,H.S., Hentz, T.F., Howard, W.E., Akhter,M.S., and Laubach,S.E.: “Major Low Permeability Sandstone Gas Reservoirs in the Continental United States”, GRI/BEG Report No. 211 (1993) Engler, T.W.: “A New Approach to Gas Material Balance in Tight Gas Reservoirs”, SPE # 62883, presented at the ATCE in Dallas, TX (2000). Fetkovich, M.J.: “A Simplified Approach to Water Influx Calculations- Finite Aquifer Systems”, JPT, (July 1971), p814. Fetkovich,M.J., Vienot,M.E., Bradley,M.D. and Kiesow, U.G. : ”Decline-Curve Analysis Using Type Curves-Case Histories,” SPEFE (Dec. 1987) 637-656. Fetkovich, M.J., Reese, D.E. and Whitson, C.H.: “Application of a General Material Balance for High-Pressure Gas Reservoirs”, SPE 22921, presented at the ATCE in Dallas, TX. (October 1991) Geffen, T.M., Parrish, D.R., Haynes, G.W., and Morse, R.A.: “Efficiency of Gas Displacement from Porous Media by Liquid Flooding”, Trans AIME 195, pp 29-38 (1952). Hammerlindl, D.J. “ Predicting Gas Reserves in Abnormally Pressure Reservoirs,” paper presented at the SPE ATCE in New Orleans, La., (Oct 1971). Havlena, D. and Odeh, A.S.: “The Material Balance as an Equation of a Straight Line,” Trans. AIME Part 1: 228 I-896 (1963), Part 2: 231 I-815 (1964). Hower, T.L. and Jones, R.E.: “Predicting Recovery of Gas Reservoirs Under Waterdrive Conditions”, SPE 22937, presented at the ATCE in Dallas, TX (Oct. 1991)

1.26

Ikoku, C.U.: Natural Gas Reservoir Engineering, Krieger Publishing Co., Malabar, FL (1992) Lee, J. and Wattenbarger, R.A.: Gas Reservoir Engineering, SPE Textbook Series, Vol 5, Richardson, TX 1996. Poston, S. W. and Chen, H-Y.: “Simultaneous Determination of Formation Compressibility and Gas-in-Place in Abnormally Pressured Reservoirs,” SPE 16227 presented at the 1987 Production Operations Symposium in OKC, OK (March 1987). Ramagost, B.P. and Farshad, F.F: “p/z Abnormal Pressured Gas Reservoirs,” SPE 10125, presented at the ATCE in San Antonio, TX (Oct. 1981). Roach, R.H. :”Analyzing Geopressured Reservoirs – A Material Balance Technique,” SPE paper 9968, Dallas, Tx, (Dec. 1981). Schafer, P.S., Hower, T.L., and Owens, R.W.: Managing Water-Drive Gas Reservoirs, published by GRI (1993) Slider,H.C.: Worldwide Practical Petroleum Reservoir Engineering Methods, Pennwell Publishing, Tulsa, OK (1983) Stewart,P.R.: Low-Permeability Gas Well Performance at Constant Pressure,” JPT, (Sept. 1970) 1149-1156. Van Everdingen, A.F. and Hurst, W.: “Application of the Laplace Transform to Flow Problems in Reservoirs”, Trans AIME 186, pp 305-324, (1949)

1.27

Problems 1. One well has been drilled in a volumetric (closed) gas reservoir, and from this well

the following information was obtained:

Initial reservoir temperature, Ti = 175ºF Initial reservoir pressure, pi = 3000 psia Specific gravity of gas, γg = 0. 60 (air = 1) Thickness of reservoir, h = 10 ft Porosity of the reservoir, = 10% Initial water saturation, Swi = 35% After producing 400 MMscf the reservoir pressure declined to 2000 psia. Estimate the areal extent of this reservoir.

2. Reservoir temperature is 180ºF. Reservoir pressure has declined from 3400 to 2400

psia while producing 550 MMscf. Standard conditions are 16 psia and 80ºF. Gas gravity is 0.66. Assuming a volumetric reservoir, calculate the initial gas-in-place and the remaining reserves to an abandonment pressure of 500 psia, all at the given standard conditions.

3. A gas field with an active water drive showed a pressure decline from 3000 to 2000

psia over a 10-month period. From the following production data, match the past history and calculate the original hydrocarbon gas in the reservoir. Assume z = 0.8 in the range of reservoir pressures and T = 600ºF.

time p Gp

months psia mmscf0.0 3000 0.02.5 2750 97.65.0 2500 218.97.5 2250 355.4

10.0 2000 500.0 4. The material balance plot below is for Well No .88, completed in the Picture Cliffs

Formation in the San Juan Basin as described in Example 1.4. Well and reservoir properties are given below.

φ, % 11 μgi, cp 0.0131 h, ft 67 cti, psi-1 x 10-4 6.22 γg 0.67 Tr , deg F 103 Sw, % 44 rw, ft. 0.229 Pi, psi 1045

1.28

Estimate the gas-in-place and drainage area for this well. If cumulative production was 752 mmscf, what has been the recovery factor?

0

200

400

600

800

1000

1200

1400

0 200 400 600 800 1000

cumulative production, mmscf

p/z,

psi

a

5. Ramagost and Farshad (1981) provided the following information for an offshore

Louisiana gas reservoir. pi = 11,444 psia cf = 19.5 x 10-6 psia-1

Swi = 0.22 cw = 3.2 x10-6 psia-1

p,psia z Gp,Bcf p/z11444 1.496 0 765010674 1.438 9.92 742310131 1.397 28.62 72529253 1.330 53.60 69578574 1.280 77.67 66987906 1.230 101.42 64287380 1.192 120.36 61916847 1.154 145.01 59336388 1.122 160.63 56935827 1.084 182.34 53755409 1.054 197.73 51325000 1.033 215.66 48404500 1.005 235.74 44784170 0.988 245.90 4221

Estimate the original gas-in-place a. assuming a normally pressured gas reservoir b. assuming a geopressured reservoir and known cf c. assuming a geopressured reservoir with an unknown, but constant cf.

1.29