## Abstract

Hydraulic fracturing combined with horizontal drilling has been the technology that makes it possible to economically produce natural gas from unconventional shale gas or tight gas reservoirs. Hydraulic fracturing operations, in particular, multistage fracturing treatments along with horizontal wells in unconventional formations create complex fracture geometries or networks, which are difficult to characterize.

#### Authors:

##### Cong Wang, Yushu Wu

*Colorado School of Mines, United States*

21 April 2015

Copyright © 2015, Southwest Petroleum University.

The traditional analysis using a single vertical or horizontal fracture concept may be no longer applicable. Knowledge of these created fracture properties, such as their spatial distribution, extension and fracture areas, is essential information to evaluate stimulation results. However, there are currently few effective approaches available for quantifying hydraulic fractures in unconventional reservoirs.

This work presents an unconventional gas reservoir simulator and its application to quantify hydraulic fractures in shale gas reservoirs using transient pressure data. The numerical model incorporates most known physical processes for gas production from unconventional reservoirs, including two-phase flow of liquid and gas, Klinkenberg effect, non-Darcy flow, and nonlinear adsorption. In addition, the model is able to handle various types and scales of fractures or heterogeneity using continuum, discrete or hybrid modeling approaches under different well production conditions of varying rate or pressure. Our modeling studies indicate that the most sensitive parameter of hydraulic fractures to early transient gas flow through extremely low permeability rock is actually the fracture-matrix contacting area, generated by fracturing stimulation. Based on this observation, it is possible to use transient pressure testing data to estimate the area of fractures generated from fracturing operations. We will conduct a series of modeling studies and present a methodology using typical transient pressure responses, simulated by the numerical model, to estimate fracture areas created or to quantity hydraulic fractures with traditional well testing technology. The type curves of pressure transients from this study can be used to quantify hydraulic fractures in field application.

## 1. Introduction

For the unconventional gas reservoirs, hydraulic fractures characterization is important in assuring the maximum stimulation efﬁciency [1-3]. A lot of researches have been carried out on the ﬂow behavior analysis of vertical wells with ﬁnite- conductivity or inﬁnite-conductivity hydraulic fractures. Cinco- Ley and Samaniego summarized ﬂuids ﬂow in a hydraulic fractured well could be divided into four periods: fracture linear ﬂow, bilinear ﬂow, formation linear ﬂow and pseudo- radial ﬂow [4]. Nobakht and Clarkson pointed out that the dominant ﬂow regime observed in most fractured tight/shale gas wells is the third one, formation linear ﬂow, which may continue for several years [5]. Transient pressure analysis of this linear ﬂow behavior is able to provide plenty of useful information, especially, the total contact area between hydraulic fractures and tight matrix.

Therefore, quantifying ﬂow in unconventional gas reservoirs has been a significant challenge and traditional REV-based Darcy law, for example, may not be generally applicable. For gas ﬂow in these unconventional reservoirs, our previous work indicates that gas-slippage effect and adsorption/desorption play an important role to describe the subsurface ﬂow mechanisms, which cannot be neglected [7]. To the authors' knowledge, these two factors were not considered in the previous pseudo-pressure derivation. In this paper, a new derivation of pseudo-pressure is provided.

This paper presents our continual efforts in developing numerical models and tools for quantitative studies of unconventional gas reservoirs [8,9]. Speciﬁcally, we explore the possibility of performing well testing analysis using the developed simulator. The numerical model is able to simulate realistic processes of single-phase or two phase ﬂow in unconventional reservoirs, which considers the Klinkenberg effects and gas adsorption/desorption. We use the numerical model to verify our new derived pseudo-pressure formulation. We also apply it to generate type-curves of transient gas ﬂow in unconventional reservoirs with horizontal well and multistage hydraulic fractures. The type curves of pressure transients from this study can be utilized to quantify hydraulic fractures in field application.

## 2. Derivation of new pseudo pressure

In 1965, Al-Hussainy and Ramer derived the pseudo pressure which has been successfully used to analyze the ﬂow of real gas in the gas reservoirs.

where P_{0} is the reference pressure; P is gas pressure; µ is the gas viscosity and Z is gas pressure Z factor.

The pressure-dependent permeability for gas is expressed by Klinkenberg as:

where k_{∞} is constant, absolute gas-phase permeability in high pressure (where the Klinkenberg effect is minimized); and b is the Klinkenberg b-factor, accounting for gas-slippage effect.

The mass of adsorbed gas in formation volume, V, is described by Refs. [10,11,7]:

where m_{g}(V) is absorbed gas mass in a volume V,_{ρK} is rock bulk density; ρ_{g} is gas density at standard condition; f(P) is the adsorption isotherm function. If the adsorbed gas terms can be represented by the Langmuir isotherm (Langmuir, 1916), the dependency of adsorbed gas volume on pressure at constant temperature is given below,

where V_{L} is the gas content or Langmuir's volume in scf/ton (or standard volume adsorbed per unit rock mass); P is reservoir gas pressure; and P_{L} is Langmuir's pressure, the pressure at which 50% of the gas is desorbed.

For real gas,

Substitute Equations (3)e(6) into Equation (2),

From the deﬁnition of the isothermal compressibility of gas:

We also deﬁne the “compressibility” from the adsorption:

Let the total compressibility:

Equation (7) will be:

Assume the viscosity and gas law deviation factors change slowly with pressure changes; the second part of Equation (11) becomes negligible. Equation (11) becomes:

We deﬁne the new pseudo-pressure m(P) as follows:

Equation (2) can be rewritten in terms of variable m(P) using the deﬁnition of ct(P) given by Equations (8)-(10) as

Based on Equation (14), we could apply this form of the ﬂow equation, quasi-linear ﬂow equation, to the analysis of real gas ﬂow behavior in unconventional reservoirs.

## 3. Linear gas ﬂow

The early ﬂow regime observed in fractured tight/shale gas wells is linear ﬂow from formation, which may continue for several years. Sometimes decline curve may indicate outer boundary effects, but no pseudo-radial ﬂow. Wattenbarger et al. gave the “short-term” approximations for this linear ﬂow with constant rate production conditions [12];

In Equation (15), m_{D} is the pseudo pressure; q is the gas production rate; B is the gas FVF; Φ is the formation porosity; c_{t} is the total compressibility, k is the formation permeability; A is the hydraulic fracture area; t is the time; and subscript i refers to initial condition and subscript w_{f} refers to the wellbore condition.

Equation (15) indicates that for the constant-ﬂowing-rate production condition, linear ﬂow appears as a straight line on the plot of normalized pressure vs. the square root of time. The slope of this square-root-of-time plot can be used to estimate the total contact area between hydraulic fracture and the tight matrix. The estimation accuracy is inﬂuenced by initial pressure, formation average permeability and total compressibility [5].

This model is limited by the assumptions of only one inﬁnite-conductivity hydraulic fracture and the neglect of gas adsorption/desorption effect. For the gas ﬂow analysis in unconventional reservoirs, these assumptions are generally unacceptable. Multi-stage hydraulic fracturing is the key technology in developing the unconventional shale gas or tight gas reservoirs. In addition, our previous work indicates that adsorbed gas could contribute more than 30% to the total production in some unconventional shale gas reservoirs [7]. For the above situations, will be more accurate and it will avoid the overestimation of total hydraulic fracture area.

One efﬁcient approach to include the adsorption is by adding an “adsorption compressibility” term into the total compressibility. Based on the derived “adsorption compressibility” formulation in Equation (9), the value of adsorption compressibility is calculated below. Table 1 lists the data used in this calculation and Fig. 1 shows the gas corresponding Z factor value with pressure.

Fig. 2 is the calculated adsorption compressibility value and this value is in the same magnitude with the rock and ﬂuids total compressibility. These calculation results indicate that when the pressure increases from 2000 psi to 5000 psi, this compressibility drops signiﬁcantly from 2.71x10^{4}/psi to 4.29x10^{5}/psi.

## 4. Numerical model

The unconventional oil/gas reservoir simulator developed coupled multiphase ﬂuid ﬂow with the effects of rock deformation, gas slippage, non-Darcy ﬂow and chemical reaction of adsorption and desorption processes in unconventional reservoirs.

In numerical formulation, the integral ﬁnite difference method [13] is used for space discretization of multidimensional ﬂuid and heat ﬂow in porous and fractured reservoirs using an unstructured/structured grid. Time is discretized fully implicitly as a ﬁrst-order backward ﬁnite difference. Time and space dis- cretization of mass balance equations results in a set of coupled non-linear equations, solved fully implicitly using Newton iteration [7].

In our model, a hybrid-fracture modeling approach, deﬁned as a combination of explicit-fracture (discrete fracture model), MINC (Multiple Interacting Continua) approach [13] on the stimulated zones, and single-porosity modeling approaches on unstimulated areas (Figs. 3 and 4), is used for modeling a shale gas reservoir with both hydraulic fractures and natural fractures [14-16]. This is because hydraulic fractures, which have to be dealt with for shale gas production, are better handled by the explicit fracture method. On the other hand, natural fractured reservoirs are better modeled by a dual-continuum approach, such as MINC for extremely low-permeability matrix in shale gas formations, which cannot be modeled by an explicit fracture model.

Fig. 4 illustrates the conceptual model including a horizontal well, multistage hydraulic fractures and their corresponding stimulated reservoir volume (SRV). The stimulated reservoir volume (SRV) is deﬁned as the 3D volume of microseismic-event cloud nearby the hydraulic fractures, inside which complex natural fracture networks are stimulated.

We apply the method of MINC to model this the porous and fractured medium. A single-porosity model is applied in the region outside the SRV. In this method, hydraulic fractures are represented by grids matching the real fracture geometric data. Then the properties of the fractures, such as permeability and permeability, are assigned to the corresponding fracture grids.

## 5. Model applications

In this section, we apply the numerical model to analyze transient pressure behaviors vs. fracture areas. Here we present a set of simulation cases to study the gas ﬂow characteristics in fractured wells.

In the ﬁrst simulation case, the formation is homogeneous with no natural fractures. A reﬁning grid system is built for the simulation. The simulation input parameters are summarized in Table 2. The ﬂuids include gas and water, but water is at residual or immobile, so it is a single-phase gas ﬂow problem.

The hydraulic fractures are represented by a discrete fracture with inﬁnite conductivity. With the input parameters as shown in Table 2, the dimensionless fracture conductivity is calculated by its deﬁnition, C_{fd}=k_{f} w_{f} =k_{m} w_{m} . Its value is large enough thus the hydraulic fracture could be treated as inﬁnite conductivity.

Three different fracture models with the same total fracture- matrix contact area are built, as demonstrated in Figs. 5-7. Fig. 8 illustrates the simulated wellbore bore pseudo-pressure vs. time for these three cases. Their pressure change with time are almost same, which indicates that fracture number and fracture geometry, as long as the total fracture area keep the same, have little inﬂuence on the wellbore pressure change for gas production from extremely low permeability rock.

Next we compare the transient pressure behavior in early time with different fracture-matrix total contact area. Two cases with different area are run and Fig. 9 is the simulation result. A larger surface area will lead to a slower increase of the dimen- sionless pseudo-pressure. Slope of this line is inversely propor- tional to the fracture area.

Gas adsorption inﬂuence is also analyzed. Two cases with different initial pressures are run, one is 2350 psi and the other is 3800 psi, respectively. The results are illustrated in Figs. 10 and 11. Two conclusions could be drawn from these simulation works:

- Gas ﬂow with adsorption will also behave straightly in the normalized pressure v the square root of time plot for the linear ﬂow period. This is identical with our previous analysis that adsorption could be treated by a compressibility factor if the pressure changes a little.
- Adsorption will have different inﬂuences on the linear ﬂow behavior at different initial pressure, as shown in Fig. 2 that “adsorption compressibility” drops with pressure increases.

## 6. Conclusions

In this paper, we derive a new formulation of gas pseudo-pressure for the transient pressure analysis in unconventional gas reservoirs. We prove its efﬁciency and accuracy by running a set of relevant simulation examples. We use the developed numerical model to simulate shale gas production with a continuum, discrete or hybrid modeling approach.

Our modeling studies indicate that the most sensitive parameter of hydraulic fractures to early transient gas ﬂow responses through extremely low permeability rock is actually the fracture-matrix contacting area, generated by fracturing stimulation. We observed that gas ﬂow with adsorption will also behave straightly in the normalized pressure vs. the square root of time plot for the linear ﬂow period, if an adsorption term is included in the total gas compressibility.

Based on this observation, we demonstrate that it is possible to use transient pressure testing data to estimate the area of fractures generated from fracturing operations. A methodology using typical transient pressure responses, simulated by the numerical model, to estimate fracture areas created or to quantity hydraulic fractures with traditional well testing technology is presented. The methodology as well as type curves of pressure transients from this study can be used for quantify hydraulic fractures in ﬁeld application.

#### Acknowledgments

This work was supported in part by EMG Research Center and UNGI of Petroleum Engineering Department at Colorado School of Mines, and by Foundation CMG.

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Corresponding author.

E-mail addresses: cowang@mines.edu, congwang268@gmail.com (C. Wang), ywu@mines.edu (Y. Wu).

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