URTeC: 1928491

Dynamic Characterization of Unconventional Gas Reservoirs. Field Cases Arévalo-Villagrán J.A., SPE, PEMEX E&P, Castellanos-Páez F., SPE, Martínez-Romero N., SPE, CNH, and Pumar-Martínez F., SPE, CBM. Copyright 2014, Unconventional Resources Technology Conference (URTeC)

This paper was prepared for presentation at the Unconventional Resources Technology Conference held in Denver, Colorado, USA, 25-27 August 2014.

The URTeC Technical Program Committee accepted this presentation on the basis of information contained in an abstract submitted by the author(s). The contents of this paper

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Abstract

Due to the complexity of unconventional gas reservoirs (shale, tight, and coalbed methane), various analytical

models have been developed to estimate formation parameters. This paper presents the results of the modification

and application of some analytical production data models. These analytical models were developed by Arévalo and

Bello. Additionally, dynamic characterization results are presented using a new model that considers the effects of

gas adsorbed in the formation and ultimately improve well production analysis and the evaluation of some formation

parameters. For this study, production data were taken from wells located in the Eagle Ford formation in

northeastern Mexico. The data were softened in order to utilize the Arévalo models for homogenous-isotropic and

low permeability heterogeneous-anisotropic formations. To identify the flow models and formation parameters, the

flow diagnostic plot [m(pi) – m(pwf)] /qg vs t and the specialized plot [(m(pi) – m(pwf)] /qg vs √t were used.

Afterwards, the Bello model was applied to analyze double porosity formations and horizontal wells with multiple

fracturing stages. Using the flow diagnostic and specialized plots, four flow regimes were identified (early linear in

the fracture system, bilinear in the matrix-fracture system, linear in the matrix, and boundary-dominated) and the

reservoir parameters were calculated for constant flowing bottomhole pressure. Finally, from these models, an

analytical model was developed that takes into account the effects of gas adsorbed in the formation. This analytical

model derives from the Bump model to consider changes in gas compressibility (cg) and the King model to consider

changes in the gas compressibility factor (z). It also considers Langmuir isotherms, which characterize gas

desorption in the reservoir due to pressure drops. With these models, the dynamic characterization of the

unconventional gas reservoirs was improved, allowing for a more accurate estimation of the reservoir parameters

and subsequently the optimal development and exploitation of these fields.

Introduction

The United States isn’t the only place in the world with large resources in shale oil and gas. In June 2013, the U.S.

Energy Information Administration (EIA) estimated technically recoverable international shale gas across 137

formations in 41 countries (excluding the U.S.) at 6,634 tcf, or nearly ten times the 665 tcf it estimated for the

United States. This is 10% higher than the EIA’s earlier estimates of 5,760 tcf in 2011. The international total

amount excludes some countries in the Middle East, which still have significant conventional natural gas reserves

still in place1.

Shale test wells have already been fracture-stimulated in Argentina, Australia, the United Kingdom, Poland, China,

and Mexico. Some important shale hydrocarbon basins have been identified in Mexico such as La Pimienta-La

Casita and Eagle Ford formations in which we estimate combined Technically Recoverable Resources (TRR) of 545

trillion cubic feet (tcf) of gas, which represents 27% of North American shale gas reserves and 7.5% of shale gas

reserves worldwide. Mexico began exploring its shale basins in 2011. These basins, similar to others, have

geological formations ranging from low permeability (less than 0.1 md) to extremely low permeability (nano-

URTeC 1928491 2

darcies). For this reason, to exploit them, it is necessary to drill horizontal wells with multiple fracking stages in

order to improve the fluid transmissibility from the formations to the producer wells.

According to Clarkson2, various methods have been developed to characterize the production behavior of

unconventional gas reservoirs. These methods include type curves for fractured wells, straight line techniques (flow

regimes), numerical simulation models, and empirical methods, all with or without desorption of adsorbed gas. For

the characterization of unconventional shale gas reservoirs and coalbed methane, it is very important to take

adsorbed gas into account because this is one of the main differences between conventional gas reservoirs and

unconventional low permeability reservoirs (tight gas) since the original gas in place represents an additional

amount of free gas that can be produced. As adsorbed gas is often not taken into consideration, frequently, mistakes

are made in the calculation of this original gas in place and reserves.

Most of the completed vertical and horizontal wells in unconventional gas reservoirs containing one or several

fracking stages generally show long-term transient flows (linear and bilinear) as a consequence of the extremely low

permeability in the formations. Additionally, there are large pressure and gas rate drops in short production times.

As such, an accurate characterization is needed to optimize field development and production.

Conceptual model for unconventional gas reservoirs

Organic material content and adsorbed gas are the governing factors that have a major influence on the behavior of

unconventional gas reservoirs with low permeability. This behavior can be represented through conceptual models

taking the following concepts into consideration: 1) storage mechanisms in unconventional gas reservoirs, 2)

transport in unconventional gas reservoirs, and 3) physical gas adsorption and desorption effects in unconventional

reservoirs. In the analysis and dynamic characterization, these models allow us to better understand these kinds of

reservoirs.

1. Storage mechanisms in unconventional gas reservoirs. There are two main types of gas storage in

unconventional gas reservoirs: a) free gas in the matrix pores and adsorbed gas in the matrix surface¡Error! No se

encuentra el origen de la referencia.. A theoretical triple-porosity model that includes both free gas and adsorbed gas is used to

consider all of the gas that is stored in formations that contain organic material3. A triple porosity system is a

combination of a double-porosity system, matrix fractures, and adsorbed gas in which free gas is stored in the

double-porosity system. Matrix micro-pores are considered porosity 1, natural fractures are porosity 2, and gas

adsorbed is porosity 3, although in reality, the storage of the gas adsorbed is not in the pore-fracture space, it is in

the surface of the formation particles in the matrix3 (Fig. 1). Porosity 3 is a virtual porosity that consists of the

storage capacity of adhered gas in unconventional reservoirs that exists due to the lack of permeability in

unconventional reservoirs. The reason why this virtual porosity does not exist in conventional reservoirs is because

the permeability in conventional reservoirs allows for fluid flow. As such, the organic material content might not be

adsorbed into the matrix.

2. Transport in unconventional gas reservoirs. In the primary porosity in unconventional formations that are rich

in organic material, there are large surface areas for gas adsorption that allow for the storage of large amounts of

gas. However, the rock pores are extremely small, which causes the system permeability of this primary porosity to

be substantially small, resulting in practically no gas or water flow4.

King’s gas transport model established that a diffusion process is present in the primary porosity. This diffusion

process can be divided into three different mechanisms a) rock matrix diffusion where molecule-molecule

interactions dominate, b) Knudsen diffusion where molecule-surface interactions dominate, and c) surface diffusion

from the adsorbed gas layer.

Depending on the gas and rock properties, the above mechanisms can act individually or together during the

transport process. The mechanism that dominates the transport in the primary matrix porosity obeys the first law of

Fick (concentration gradient). Darcy Flow is not reached due to the fact that the permeability is very low (equation

1).

(1)

URTeC 1928491 3

Where D is the diffusion coefficient in ft2/day, and C is the molar concentration in lb-mol/ft

3. In the case of

secondary porosity in the natural fracture system, diffusion can be present in two stages: 1) there is diffusion from

the matrix to the fractures due to the liberation of the adsorbed gas from the surface of the primary porosity as a

consequence of the pressure drop, and 2) free gas is transported by Darcy Flow into the natural fractures toward the

producer well. The fracture system acts in two ways: a) as an injection system for the primary porosity network and

b) as a conduct for well production3 (Fig. 2).

When gas is desorbed due to a drop in pressure, the gas molecules are able to move and diffuse into the porous space

of the surface particles. The diffusion time is negligible because the formation pores are generally very small in a

micro-scale. Once the gas is liberated from both the surface of the organic material and the rock pores, this gas is

free and follows the same transport mechanisms into the matrix pores and the fracture systems, similar to the

original free gas in the formation3.

3. Physical gas adsorption and desorption in unconventional reservoirs. In unconventional gas reservoirs that

present organic material content, a storage mechanism that is different from conventional gas reservoirs (where the

gas is compressed in the formation pores and fractures) is the additional phenomenon of adsorption of the gas

molecules to the organic rock walls. This is known as physical adsorption or physisorption, a process in which the

electronic structure of the atom or molecule is barely perturbed upon adsorption. In other words, the adsorbed

molecules conserve their chemical nature.

The Langmuir model has been the most commonly used model in the petroleum industry to describe the gas

adsorption phenomenon in solids, which considers that a gas molecule is adsorbed in a single place and doesn’t

affect neighboring molecules. Additionally, these molecules are desorbed randomly. The Langmuir model6 can be

represented by equation 2:

(2)

where Va is the total volume of the adsorbed gas per volume unit that is in equilibrium in the reservoir at pressure p,

VL is the Langmuir volume or maximum adsorbed volume per volume unit in the reservoir at infinite pressure, and

pL is the Langmuir pressure that represents the pressure in which the adsorbed volume Va is equal to half of the

Langmuir volume, VL. A typical Langmuir isotherm is shown in Fig. 3.

The Langmuir isotherms describe the maximum gas amount in an unconventional reservoir that can be stored under

certain conditions with organic material content, pressure and temperature. There are different factors that can

decrease the adsorption capacity of gas in a reservoir, being less than the maximum capacity represented by the

isotherm. In these cases, saturated parameters are used in which the desorption or saturation pressure is equal to the

undersaturated initial reservoir pressure, which can be graphically represented by an initial gas content that is below

the isotherm (Fig. 4).

Other main parameters to consider during the exploitation of unconventional gas reservoirs include the Langmuir

parameter values since they determine the type of isotherm and in consequence the desorption pressure, the gas

storage volume, and the desorbed gas that can be produced during exploitation (Fig. 5).

Model modification to consider desorbed gas

The main topic of this work consists of the modification of the equations developed by Arévalo7 and Bello

8 for

conventional and unconventional reservoirs in order to take into account the desorption process as a modified

function of pseudotime. The development of these models is presented below.

Bumb et al.9 showed that the desorption phenomenon can be taken into account in the solutions for the dry gas

diffusion equation, using gas pseudopressure and modified total system compressibility ( ), represented by

equations 3 and 4, respectively.

URTeC 1928491 4

( ) ∫

(3)

and

( ) (4)

where:

( )

( )

(5)

In order to eliminate the nonlinearity of the diffusion equation, Geramy et al.10

and Clarkson11

used the Frame and

Wattenbarger pseudotime function¡Error! No se encuentra el origen de la referencia.

, which is expressed in terms of average

formation pressure, ( ), for drawdown pressure tests or production data. These authors studied flow regimes from a

production data analysis point of view through the derivative function for normalized rate. They defined the term

“time match function” taking into account the definition of modified apparent pseudotime (equation 6) that

considers gas desorption effects using ( ).

( )

∫

( )

(6)

These variables consider instantaneous desorption, which is a reasonable assumption for long-term gas production in

some low-permeability, shale, and coalbed methane reservoirs. The modified pseudotime function can be included

in the multi-fractured horizontal well models, in accordance with Table 1. The desorbed gas effect can be considered

in the pseudotime function to resolve the modified diffusion equation for adsorbed gas9, 11

.

Using this modification to the Arévalo7 and Bello

8 models that dynamically characterizes the reservoir using the

normalized rate derivative in the diagnostic log-log plots, flow regimes can be identified and reservoir properties can

be calculated in both conventional and unconventional gas reservoirs. These new models take into account gas

desorption in the pseudotime function and are presented in Tables 2 and 3, for vertical and horizontal wells,

respectively.

Field case analysis

Following is a description of the application of the new models modified by Castellanos and Arévalo to field cases

(presented in this paper) in order to characterize dynamically the reservoir and estimate reservoir parameters and gas

volume. Actual well data were used to characterize and simulate the behavior of a producing well in a shale

formation both with and without adsorbed gas.

Analysis of Shale A well. Table 4 presents data from well located in the Eagle Ford shale formation in southern

Texas12

, and the map shows the location of areas according to the fluids produced (Fig. 6). Well Shale A is a

horizontal dry gas production well, which was completed with a stimulation treatment consisting of ten stages of

lateral fracturing at 4,000 ft., resulting in only 20 effective transverse fractures. A Stimulated Reservoir Volume

(SRV) of 169 MMft3 was estimated. The thickness of the production zone is 283 ft. Table 5 presents the general

well data12

.

Fig. 7 shows the gas rate and the cumulative gas production history, and Fig. 8 illustrates the bottomhole flowing

pressure log. In both cases, the data correspond to 250 days of production. To detect the flow geometries from the

production data, the normalized gas rate is plotted against time and modified apparent pseudotime ( ) as shown in

Fig. 9 and considering the case of a well with and without gas desorption, respectively. Two slopes can be observed.

The first slope corresponds (m2 =1/4) of a probable bilinear flow for a period of 5-50 days and the second slope (m3

URTeC 1928491 5

= 1/2) corresponds to a later linear flow for a period of 60-200 days. Finally, starting at t = 225 days, it can be seen

how the flow is dominated by the boundary effects.

Subsequently, specialized graphs of the normalized rate vs. a specified function of time were created for bilinear and

linear flows (Fig. 10 and 11, respectively). To estimate formation gas desorption parameters and the modified

apparent pseudotime values from Tables 6 and 7 were used15

.

Once the present flow regimes are, the formation parameters and the original gas in place can be estimated. Table 8

shows the matrix and fracture permeabilities without and with gas desorption that were estimated using the

equations to characterize horizontal wells from Bello8 (Table 3) using the slopes values of m3 and m4 obtained from

the specialized plots (Fig. 10 and 11), respectively.

To estimate the Original Gas in Place (OGIP), equation 22 can be used, presented by Wattenbarger et al.16

, which is

based on the assumption that boundary dominated flow begins when the pressure in the center of the matrix blocks

begins to decline. The Langmuir equation that quantify the adsorbed gas volume in the producer formation at initial

condition is given as

( )

√

(22)

Where tlr is the time in which the straight line stops in the specialized graph of the square root of time. These

allowed estimating OGIP of 3.15 Bscf and 4.06 Bscf without and with gas adsorption, respectively. There is

variation between the values estimated from the permeabilities and the original gas in place since, depending on the

well isotherm (Fig. 9) and the bottomhole pressure, the adsorbed gas is being released to the free gas phase, which

causes the gas compressibility in the formation to be modified along with pseudotime function. Likewise, the

original gas in place considering gas adsorption is modified by an additional 25% due to the gas adsorbed in the

formation. With these results the modified models can be used to characterize the formation and calculate OGIP in

unconventional gas reservoirs considering the gas desorption phenomena.

Analysis of Shale B well. In February 2011, Shale well B was drilled and completed with a horizontal geometry in

Eagle Ford’s upper Cretaceous formation, with a vertical depth of 8,300 ft and a horizontal path of 13,356 ft. During

its completion, 17 fractures were made, 2 more than what had originally been planned in the well design. On

average, the fractures are 856 ft in length, 459 ft in height, and an average width of 0.8 inches. During completion, a

fish was left in hole at 11,030 ft, behind which, according to production logs, there is currently no production. As

such, of the 3,970 ft horizontal stimulated, only 1,837 ft are producing via 8 hydraulic fractures, which is less than

what had been anticipated. Fig. 12 illustrates the mechanical state of Shale B well.

The well turned out to be a producer of dry gas. Fig. 13 illustrates the pressure-production history of the well during

the first 250 days, and Table 9 and Fig. 14 present the general well data and drainage volume of the well. From the

well completion data, we obtain a ye = 246 ft and the following:

(23)

As such, the cross-sectional area to flow is:

Acw = 2xeh = 2 x 1837 x 492 = 1,807,411 ft2 (24)

Therefore, the matrix–natural fracture area between the blocks formed by the hydraulic fractures is:

Acm = 2 x 2yehL = 2 x 2 x 246 x 1837 x 492 x 8 = 3,873,024 ft2 (25)

For the analysis of the flow regimes and geometries, the normalized rate was plotted against apparent pseudotime

(Fig. 15). Subsequently, due to the fact that linear flow with a slope of m = 1/2 was observed, the specialized graph

for normalized rate vs. √ was made (Fig. 16).

URTeC 1928491 6

Using the m3 slope of Fig. 16 and equation 20 from Table 3, matrix permeability (km) was estimated, assuming that

the fracture porosity is negligible compared with matrix porosity13

,

( ) ( ) (26)

The reservoir is saturated in its desorption pressure; as such, it is considered that the desorption began with reservoir

production. Additionally, rock compressibility (cf) is negligible in comparison with gas compressibility; therefore,

we have,

= cg + cd (27)

Applying the adsorbed gas values from Tables 6 and 10, pertaining to Eagle Ford in the U.S. and published by

Mullen14

, the gas volume that is desorbed from the formation was estimated. Following are the diagnostic and

specialized graphs (Figs. 15 and 16), as well as the results obtained in the estimation of parameters using the

modified Bello model in order to consider desorbed gas. To estimate the parameters, instantaneous desorption was

taken into account as if all of the released gas would be produced.

The parameters obtained from the characterization are matrix permeability (km) of 3.85 x 10-6

md, and OGIP of 1.70

Bscf. Fig. 17 shows the results of the gas rate match vs. √ and the variation of the normalized pseudopressure vs.

√ with actual data.

In Fig. 17 it can be observed that the match of the pressure-production history data with the modified model by

Castellanos–Arévalo is good. With this model, desorption can be considered in any stage of well production since

the desorbed gas produced can vary depending on the type of isotherm and the initial reservoir pressure and the gas

desorption pressure.

Conclusions

This is the first sentence of the third sample section. Pages 2-4 will need to include the paper number in the upper

left-hand corner with the page number in the upper right-hand corner. In unconventional gas reservoirs with high

organic material content, it is important to consider the gas that is adsorbed in the formation since this can

significantly alter the original gas in place and the estimated parameters such as primary and secondary

permeabilities. The following conclusion can be drawn from this study.

1. It was proven that, by applying Langmuir’s isotherm model, it is possible to take into consideration and

predict the behavior of adsorbed and desorbed gas in unconventional formations that contain organic

material. This is significant since, once gas desorption pressure is reached, there is an additional production

mechanism in the reservoir.

2. It was proven that the pseudotime developed for the characterization of conventional gas reservoirs can be

effectively applied to unconventional formations, taking into account instantaneous gas desorption in the

total compressibility of the system and depending on the average reservoir pressure.

3. Through the analysis of the well data, it was possible to confirm the applicability of the modified models to

analyze production and characterization data from unconventional gas reservoirs, taking into consideration

the phenomenon of adsorption using Langmuir’s isotherm and modified pseudotime at any moment during

well production.

4. For the characterization models used in this work, the assumption was made that the desorbed gas is

instantaneous, obtaining good results. However, it is important to bear in mind that desorption is not

instantaneous in all reservoirs. As such, it is recommended to adjust the models taking into account real gas

desorption time.

5. Another important point to keep in mind regarding the adsorption and desorption models is that Langmuir

isotherms only consider the monocomponent fluid, methane gas. However, it has been observed that in

some reservoirs there are multicomponent blends. As such, it is recommendable to utilize the

multicomponent Langmuir isotherm or to study how to adjust a cubic state equation, allowing us to better

characterize the desorption phenomenon.

URTeC 1928491 7

Acknowledgments

We would thanks to PEMEX E&P for permission to present the data and results of some tight and shale wells. We

are also thankful to Cynthia Sperry who effective work and participation contribute to do this paper.

Nomenclature

A = area, L2, ft

2

Acm = matrix-natural fracture area between the blocks formed by the hydraulic fractures, L2, ft

2

Acw = cross –sectional area to flow, L2, ft

2

bSSPDP = intercept for boundary-dominated flow of ( ) vs. t plot, psia2-D/Mscf-cp

C = molar concentration, m/L3, lb-mol/ft

3

cd = Desorbed gas compressibility, Lt2/m,1/psia

cf = Formation compressibility, Lt2/m,1/psia

CA = shape factor

cg = gas compressibility, Lt2/m,1/psia

ct = Modified total system compressibility, Lt2/m,1/psia

t* = Modified total system compressibility, Lt

2/m,1/psia

cw = water compressibility, Lt2/m,1/psia

D = diffusion coefficient, L2/t, ft

2/day

EIA = Energy Information Administration

k = formation permeability, md

kf =fracture permeability, L2, md

km = matrix permeability, L2, md

L = Effective producing length in the horizontal section of the well, L, ft

m = straight line slope

m(p) = real gas pseudo-pressure at average reservoir pressure, m/Lt3, psia

2/cp

m(pwf) = pseudo-pressure at bottomhole flowing pressure, m/Lt3, psia

2/cp

mLPC =slope for constant pwf for bilinear or linear flows from specialized plot, psia2-D/Mscf-cp

mCPRPC = slope for constant pwf for radial flow from specialized plot, psia2-D/Mscf-cp

mCRSDP = slope for constant qg for spherical flow from specialized plot, psia2-D/Mscf-cp

p = pressure, m/Lt2, psia

= average reservoir pressure, m/Lt2, psia

pL = The Langmuir pressure that represents the pressure in which the adsorbed volume Va is equal to half

of the Langmuir volume, VL, m/Lt2, psia

PSS = pseudostationary state flow

gg = gas production rate, L3/t, Mscf/D

R = gases universal constant, equal to 10.732 (psia -ft3)

/ (lbm-mol-°R)

rw = well radius, L, inches

S = skin factor, adim

SRV = Stimulated Reservoir Volumen, L3, MMft

3

Sw = water saturation, %

t = time, days

T = reservoir temperature, °R

= modified apparent pseudotime, t, days

tlr = time in which the straight line stops in the specialized graph of the square root of time, t, days.

TRR = Technically Recoverable Resources, L3, tcf

TVD = true vertical depth, L, ft

URTeC 1928491 8

Va = total volume of the adsorbed gas per volume unit that is in equilibrium in the reservoir at pressure p,

L3, ft

3.

VL = The Langmuir volume or maximum adsorbed volume per volume unit in the reservoir at infinite

pressure, L3, ft

3.

w = width of the fracture, L, inches

x = distance, ft

ye = distance from the well to the formation boundary, L, ft

z = real gas compressibility factor

ρ = density, m/L3, lb/ft3

µ = viscosity, m/Lt, cp

φ = porosity, %

ω = acentric factor

tcf = trillion cubic feet

Subscripts

= apparent

cw = cross sectional area to flow

cm = matrix-natural fracture area between the blocks formed by the hydraulic fractures

d = desorbed

f = fracture

m = matrix

mf = matrix-fracture system

sc = standard conditions

g = gas

i = initial

r = rock

t = total

w = water

References

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Energy: International Shale – Hope Shifts but Inherent Challenges Continue to Stymie Development,

Bernestein Research.

2. Clarkson, C. R., Jensen, J. L., & Blasingame, T. (2011, January 1). Reservoir Engineering for

Unconventional Reservoirs: What Do We Have to Consider? Society of Petroleum Engineers.

doi:10.2118/145080-MS.

3. Lane, H. S., Lancaster, D. E., & Watson, A. T. (1990, January 1). Estimating Gas Desorption Parameters

From Devonian Shale Well Test Data. Society of Petroleum Engineers. doi:10.2118/21272-MS.

4. Bo Song (2010). Pressure transient analysis and production analysis for New Albany shale gas wells,

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Reservoirs. Society of Petroleum Engineers. doi:10.2118/20730-MS.

6. Langmuir, I. (1918), The Adsorption of Gases on Plane Surfaces of Glass, Mica and Platinum, Am. Chem.

Soc. 40, 1361.

7. Arevalo Villagran J. (2001). Analysis of long-term behavior in tight gas reservoirs: Case histories. PhD

Dissertation, Texas A &M U, College Station, Texas, E.U.

8. Bello, R.O. (2009, May). Rate Transient Analysis in Shale Gas Reservoirs with Transient Linear

Behaviour. PhD Dissertation, Texas A &M U, College Station, Texas, E.U.

9. Bumb, A.C., McKee, C.R. (1988). Gas-Well Testing in the Presence of Desorption for Coalbed Methane

and Devonian Shale. SPEFE 3 (1): 179 – 185. SPE-15227-PA. doi: 10.2118/15227-PA.

URTeC 1928491 9

10. Gerami, S., Pooladi-Darvish, M., Morad, K., & Mattar, L. (2008, July 1). Type Curves for Dry CBM

Reservoirs With Equilibrium Desorption. Petroleum Society of Canada. doi:10.2118/08-07-48.

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URTeC 1928491 10

Table 1. Flow regimes identification through the derivative function for normalized rate and pseudotime function2, 3

.

Flow regime Log-Log diagnostic Derivative

function

slope

Type of plot

Bilinear flow

Radial derivative m = 1/4 | ( ) ( )|

( )

(7)

Bilinear derivative m = 0 | ( ) ( )|

√

(8)

Linear flow

Radial derivative m = 1/2 | ( ) ( )|

( )

(9)

Linear derivative m = 0 | ( ) ( )|

√

(10)

Radial flow

Radial derivative m = 0 | ( ) ( )|

( )

(11)

PSS flow.

Radial derivative m = 1

| ( ) ( )|

( )

(12)

Table 2. New models of the derivative function for normalized gas rate for vertical wells7

(modified by Castellanos

and Arévalo, 2014) to take into account gas desorption in the pseudotime function.

Flow regime Specialized plot Interpretation equation

Lineara ∑( )

√

( )

| ( ) ( )|

√

√ ( ) ( ) (

) (13)

Bilinearb ∑( ( ))

√

| ( ) ( )|

⁄ √

√

√ [( ) ( ) ]

(

) (14)

Radialc ∑( ( ))

(

( ) )

| ( ) ( )|

( )

(15)

Sphericalb ∑( ( ))

√ ( )

| ( ) ( )|

[√ [( ) ( ) ] (

)]

(16)

Boundary

dominated

effectsb

∑( ( ))

( ( )

)

| ( ) ( )|

( )

[ (

) ]

(17)

Solution:

a. and constant ,

b. For constant

c. For and constant

URTeC 1928491 11

Table 3. New models of the derivative function for normalized gas rate for horizontal wells8

(modified by

Castellanos and Arévalo, 2014) to take into account gas desorption in the pseudotime function.

Flow regime

Derivative

function

plot

Type of plot Interpretation equation

Early linear flow m = 1/2 | ( ) ( )|

√

√

√ ( )

(18)

Bilinear flow m = 1/4 | ( ) ( )|

√

√

[ ( ) ]

(19)

Linear flow

m = 1/2

| ( ) ( )|

√

√

√( )

(20)

Transitory linear

flow in the

matrix

m = 1/2 | ( ) ( )|

√

√

√( )

(21)

Table 4. Data from a well located in the Eagle Ford shale formation in southern Texas12

.

Depth, ft 2,500 - 14,000

Net thickness, ft 50 - 300

Pressure gradient, psia/ft 0.4 - 0.8

TOC, % 2 - 9

Gas saturation, %: 83 – 85

Permeability, nd 1 - 800

Table 5. Data from Shale A well12

.

Well radius, ft 0.33

Lateral length, ft 4,000

Thickness, ft 283

Depth, TVD, ft 10875

HC* porosity (%) (φhc = φef (1-Sw)) 5.76

Reservoir pressure, psia 8,350

Temperature, °R 745

Gas compressibility, 10-5

psia-1

6

Gas viscosity, cp 0.03334

Number of effective fractures 20

Stimulated Reservoir Volume (SRV), MMft3 169

URTeC 1928491 12

Table 6. Gas adsorbed values in shale formation in the U.S.15

.

Tabla 7. Rock parameters to estimate gas desorption in Shale A well12,

.

VL = 720 scf/Ton ρr = 2.5 gr/cm3

PL = 550

SRV = 16900000 ft

T = 285 °F mr = 1197306.06 Ton

φ = 0.0576

Table 8. Matrix and fracture permeabilities for Shale A well for both without and with gas desorption.

Permeability Without gas desorption With gas desorption

matrix ( ):

fracture ( ):

Table 9. General data from Shale B well.

Well radius, ft 0.375

Lateral length, ft 1837

Thickness, ft 492

Depth, TVD, ft 2530

HC* porosidad (%) (φhc = φef (1-Sw)) 6.0

Reservoir pressure, psia 5,100

Temperature, °R 667

Gas compressibility, 10-4

psia-1

1.3

Gas viscosity, cp 0.0239

Number of effective fractures 8

Stimulated Reservoir Volume (SRV) (MMft3) 445

Table 10. Rock parameters to estimate gas desorption en Eagle Ford

15.

VL = 60 scf/ton ρr = 2.8 gr/cm3

PL = 250

SRV = 446 MMft3

T = 207 °F mr = 35280000 Ton

φ = 0.06

URTeC 1928491 13

Fig. 1. Triple porosity storage model for unconventional gas reservoirs3.

Fig. 2. Transport sketch in unconventional gas reservoirs with adsorption process4.

Fig. 3. Langmuir typical isotherm6.

URTeC 1928491 14

Fig. 4. Langmuir isotherms in a gas reservoir: a) saturated in adsorbed gas (left sketch), and b) undersaturated in

adsorbed gas (right sketch).

Fig. 5. Behavior of the adsorption isotherm upon changing the Langmuir pressure.

Fig. 6. Extension of the Eagle Ford formation in southern Texas12

.

URTeC 1928491 15

Fig. 7. Gas rate and cumulative production from well A12

.

Fig. 8. Bottomhole flowing pressure history for well A12

.

Fig. 9. Diagnostic plots of normalized pseudopressure: a) | ( ) ( )|

vs t (left sketch) and b)

| ( ) ( )|

vs

(right sketch) detecting both bilinear and linear flows.

URTeC 1928491 16

Fig. 10. Specialized plots to characterize bilinear flow:

a) | ( ) ( )|

vs (left sketch) and b)

| ( ) ( )|

vs

(right sketch).

Fig. 11. Specialized plots to characterize linear flow:

a) | ( ) ( )|

vs (left sketch) and b)

| ( ) ( )|

vs

(right sketch)

Fig. 12. Mechanical state of Shale B well.

URTeC 1928491 17

Fig. 13. Pressure-production history of Shale B well.

Fig. 14. Drainage volume of Shale B well.

Fig. 15. Diagnostic plot of | ( ) ( )|

vs

for Shale B well.

URTeC 1928491 18

Fig. 16. Specialized plot of | ( ) ( )|

vs √

for Shale B well.

Fig. 17. History matches: a) gas rate (qg) vs √ y b)

| ( ) ( )|

vs √

.