Thousands of horizontal wells are drilled into the shale formations across the U.S. and hydrocarbon production is substantially increased during past years. This fact is accredited to advances obtained in hydraulic fracturing and pad drilling technologies. The contribution of shale rock surface desorption to production is widely accepted and confirmed by laboratory and field evidences. Nevertheless, the subsequent changes in porosity and permeability due to desorption combined with hydraulic fracture closures caused by increased net effective rock stress state, have not been captured in current shale modeling and simulation.
Sina Hosseini Boosaria, Umut Aybarb, Mohammad O. Eshkalakb
aPetroleum and Natural Gas Engineering, Department at West Virginia University, USA. bPetroleum and Geosystems Engineering, Department at the University of Texas at Austin, USA
Received 15 November 2015-Accepted 21 February 2016
Hence, it is essential to investigate the effects of induced permeability, porosity, and stress by desorption on ultimate hydrocarbon recovery.
We have developed a numerical model to study the effect of changes in porosity, permeability and compaction on four major U.S. shale formations considering their Langmuir isotherm desorption behavior. These resources include; Marcellus, New Albany, Barnett and Haynesville Shales. First, we introduced a model that is a physical transport of single-phase gas flow in shale porous rock.
Later, the governing equations are implemented into a one-dimensional numerical model and solved using a fully implicit solution method. It is found that the natural gas production is substantially affected by desorption-induced porosity/permeability changes and geomechancis. This paper provides valuable insights into accurate modeling of unconventional reservoirs that is more significant when an even small correction to the future production prediction can enormously contribute to the U.S. economy.
Recoverable reserves of shale gas in the U.S. are estimated to be 862 Tcf . Although challenges associated to exploration and management of shale assets are yet to be resolved, decreased evaluated risk promises a secure gas supply for next decades. The large accumulation of gas shale formations serve as both a hydrocarbon source and a productive reservoir. Most of the gas is stored in organic-rich rock while less portion of gas in place is in pore spaces . Extremely low matrix permeability as well as highly complex network of natural fractures are unique characteristics of shale formations. Permeability of shale rocks is estimated to be between 50 nD (nano-Darcy) and 150 nD . Recent advances and innovations in hydraulic fracturing are key success of shale gas economic production as a viable global energy supply.
Nevertheless, complexities associated with flow mechanisms and existence of many pressure dependent phenomena, such as combined hydraulic and natural fracture conductivity losses, Klinkenberg gas slippage effect, desorption/adsorption and Darcy/non-Darcy flow, are not completely understood and need more attentions to reach our industry needs. In this study, desorption-induced porosity and permeability changes of shale matrix as well as closure effect of hydraulic fractures are focused in detail to evaluate their impact on production form four very productive U.S. shale resources.
Hydraulic fracturing creates highly conductive channels and paths for the reservoir fluid to flow from the reservoir pay zone to the well bore. Moreover, stress-induced natural fractures open with the hydraulic fracturing operation; thus a secondary fracture network is created in addition to hydraulic fractures. This secondary fracture network placed in the stimulated reservoir volume (SRV) area is caused by stress alterations during hydraulic fracturing treatment . Researchers named this secondary fracture network either as natural fractures or secondary fracture network .
The main difference between primary fractures, which are hydraulic fractures, and secondary fractures is that the secondary fracture network is unpropped. Typical proppant volumes used in the hydraulic fracture operations are very low to keep fractures open in the propped secondary fracture network. Therefore, these secondary fractures remain unpropped (Since the natural fractures lack proppant, their conductivities are much more pressure dependent compared to hydraulic fractures.). Pressure-dependency of hydraulic fractures and their impact on production are discussed by the researchers , ,  and are combined with desorption-induced porosity/permeability change in this study.
Reservoir simulation and modeling of unconventional resources have been given much more attention over the past years. Many numerical and analytical models are developed and extensive reservoir studies have been conducted. Commercial reservoir simulators are also improved to handle and capture fluid flow behavior and natural gas production from unconventional assets, such as shale. However, the developed models have ignored some of complex physics of shale and integrating the entire phenomena in shale is still a challenging target for the petroleum industry. Further, among analytical and semi-analytical methods, works done by Refs. , , ,  have provided comprehensive progress in the modeling of shale gas reservoirs.
Porosity, permeability and gas desorption of shale are considered the key parameters that affect shale ultimate gas recovery. However, least amount of simulation studies is conducted to account for porosity and permeability change due to desorption and rock compaction. In this paper, we first derived the porosity changes due to compaction and desorption, second, we plot the porosity and permeability versus the pressure for Marcellus, Barnett, New Albany and Haynesville shale. Afterward, we introduce a physical model of a horizontal well and the appropriate nonlinear partial differential equations created from governing equations are solved numerically through fully implicit method. The gas production from a single pair of hydraulic fractures is then scaled up to the entire horizontal well for each specific reservoir.
Shale desorption isotherms
Large portion of shale rock consists of organic matter, kerogenic media. Natural gas methane molecules are adsorbed on the organic rich strata (also they are stored in pore spaces and natural fractures). Thus, significant amount of natural gas can be produced from the surface of kerogen, which is also known as total organic carbon, TOC . By its very nature, in order to release methane stored within the shale, it is necessary to enhance fluid pathways (create fractures) and deplete the surrounding pressure. As the pressure decreases due to production, more and more adsorbed gas is released from the surface of matrix; this contributes to the total amount of gas produced. Therefore, an adsorption model is required to predict the gas desorbed from shale matrix that will also be served to determine the first objective of this study, calculating the desorption-induced porosity/permeability.
Langmuir adsorption model  is the most common empirical mathematical model used to quantify the amount of desorbed gas as a function of pore pressure at constant temperature. This analogy comes from the developments made in modeling coal bed methane (pre-shale technology), but it must be noted that sorptive characteristics of shale might not necessarily serve the same way as it does in shale .
Langmuir model simply represents a nonlinear relationship between the potential amount of releasable-gas and the pore pressure given by Eq. (1). This equation represents that the potential amount of releasable-gas is only a function of reservoir pressure.
where G is the potential releasable-gas content in scf/t, P is reservoir pressure (assumed to be the average reservoir pressure) in psi, and VL (Langmuir volume) in scf/t and PL (Langmuir pressure) in psi are Langmuir constants. Laboratory tests are necessary to determine VL and PL from core samples. Langmuir pressure is defined as the pressure at which 50% of gas is desorbed. By this definition, it is clear that the higher the Langmuir pressure reaches, more released-gas from the organic shale matter. Langmuir volume is the gas volume at infinite pressure representing the maximum storage capacity of gas, which is a function of TOC of the particular shale sample.
Fig. 1 shows the capability of four U.S. shale formations in releasing gas that is characterized through Langmuir model. These assets are, Marcellus, New Albany, Barnett and Haynesville shale.
Fig. 1. Desorption isotherms for four U.S. shale formations.
Table 1 provides the common values of properties used in this study for the aforementioned assets. All of them are gathered from the numerical modeling literature except the critical pressure that is calculated using Eq. (2), that is also explained in detail in the subsequent section.
Table 1. General properties of four U.S. shale formations.
Porosity and permeability change in shale
The changes in porosity and permeability of shale matrix occur when production starts. This variation in porosity and consequently permeability is because of two reasons; gas desorption from shale surface (unlike conventional reservoirs) and increasing effective stresses by pressure depletion. When gas molecules leave the surface of the shale rock and move toward the pore spaces, in fact the pore volume is increased as the matrix volume is decreased. This will result in an increase in porosity of the shale matrix.
Unlike desorption effect, the porosity of shale tend to decrease with production as a result of increased net stress effect. The change in porosity and subsequent permeability of the shale has not been studied in the reservoir simulation while focusing on its effect on a long-term production outlook. Therefore, the mathematical background of the porosity and permeability changes in shale matrix is presented herein.
Generally, in early stages of production from shale formations as the reservoir pressure is considerably high, there is no significant desorption from shale surface to contribute to production. This means that up to some stages of production, reservoir is encountered with porosity reduction due to increased net stress and compaction. Once a critical life time of a shale reservoir is reached, the porosity change due to desorption must be considered in which porosity will enhance due to increase of pore volume of shale and consequent reduction in rock volume.
This critical life time of a shale reservoir depends on its isothermal desorption behavior that should be measured experimentally. After the critical period of shale production is passed, three different scenarios are possible. First, the compaction effect on porosity dominates the porosity change against desorption and the total effect tend to reduce shale porosity. Second, the two effects may not overcome each other that means the porosity reduction due to compaction is balanced by its enhancement due to desorption. Last, porosity increases several orders of magnitude more than its reduction due to compaction. These three scenarios are all observed among results of this study.
For the formulation of our first objective, the procedure and mathematical background is provided in the following. The porosity is expressed as a function of pore volume (PV) and the bulk volume (BV) as given in Eq. (3), and then expanded over pressure by Eq. (4).
Bulk volume can be represented as a function of pore and rock volumes and by manipulation we get Eq. (5).
Also, the equality of (dVR/dp)= (dVrockads/dp) holds, since the volume of grains are constant.
Further, since the adsorbed gas volume measurement is difficult, we need to relate the volume of the released gas to the adsorbed gas volume through the mass balance of the desorbed gas given by Eq. (6).
where ρgads and Vgads. are the density of adsorbed gas and the released gas volume, respectively. Represents ρgfree the density of free gas that is further defined Eq. (7).
where M, Z, R, T are the gas molecular weight, compressibility factor, gas constant, and temperature, respectively. Released gas volume at any pressure defined as:
when plug these back into Eq. (6), we get Eq. (10).
Incorporating Eqs. (8), (9), (10) with Eq. (4), Eq. (11) is obtained:
In order to use Eq. (11) to determine and plot the porosity changes, first we need to calculate VP because it varies slightly during each time step due to desorption and rock compaction. With a simple integration of Eqs. (6), (10)), we get Eq. (12).
and for the permeability, we used simple empirical formula introduced by Ref.  that is expressed in Eq. (15).
Now, proper equations are derived that suffice determination of desorption-induced porosity and permeability changes for shale. By numerical integration of Eq. (11) and incorporating Eqs. (12), (13), (14), (15), the porosity change versus reservoir pressure is calculated and plotted versus pressure in the subsequent sections.
For the permeability, from a petrophysical point of view, there is no true relationship between porosity and permeability, specifically for shale rock. However, permeability is proportional to the squared of a characteristic length of the system. Hence, considering the same trend and using an empirical expression for variation of permeability dependency on porosity will satisfy our quest for calculating such data, and will not add very much error if the permeability ranges in an acceptable bound. Another assumption is homogenous and isotropic through the entire field. This might not seem realistic but is a common practice in reservoir engineering studies, as the results will provide invaluable insights.
For the Marcellus shale, the plot of porosity and permeability changes versus the pressure is given in Fig. 2. At the beginning of production, reservoir pressure is 7500 psi and by pressure depletion of the reservoir as a result of production, the porosity is decreased and consequently the permeability does until reservoir pressure reached its critical pressure around 3000 psi. A sharp increase in porosity due to desorption after the critical pressure is observed that is in agreement with the high capability of Marcellus shale in releasing gas from its nano-pores that is shown in Fig. 1. The permeability has the same trend, but sharper decline.
Fig. 2. Marcellus shale porosity/permeability change.
In the case for Barnett and New Albany shales, Fig. 3, Fig. 4, similar trend is observed but with smoother decrease and increase for porosity and ultimately the permeability. This result demonstrates less capability of these formations in releasing the gas and subsequently providing more pore volume in the bulk of porous rock.
Fig. 3. Barnett shale porosity/permeability change.
Fig. 4. New Albany shale porosity/permeability change.
The slope of porosity change versus pressure is observed to be positive along the pressure depletion of the Haynesville shale (Fig. 5). This is mainly associated to the small amount of Haynesville rock desorption of methane. This is in an agreement with the experimentally measured isothermal desorption curves given in Fig. 1. The change in permeability is sharper, but has followed the same trend.
Fig. 5. Haynesville shale porosity/permeability change.
Closures of hydraulic fractures
The permeability (or conductivity) of both hydraulic and natural fractures are easily influenced by changes of stress and strain during gas production from shale. Therefore, incorporating pressure dependency of fractures permeability in reservoir modeling and simulation process is a significant step towards more realistic assessments of production behavior of shale reservoirs.
The geomechanical properties of Marcellus shale is investigated by Ref. . They generated rock mechanical properties, geomechanical well logs, and studied various characteristics such as minimum horizontal stress, Young, bulk shear modulus, as well as Poisson’s ratio that play an important role in defining the stress profiles of an unconventional reservoir. Moreover, having an access to rock’s geo-mechanical properties enhances the understanding of parameters, such as conductivity and pressure dependency of permeability .