^{1}Department of Geology, Federal University, Birnin Kebbi, Nigeria

^{2}Department of Geology, Modibbo Adama University of Technology, Yola, Nigeria

- *Corresponding Author:
- Balumi WB

Department of Geology

Federal University

Birnin Kebbi, Nigeria

**Tel:**+234 700 000 000

**E-mail:**[email protected]

**Received date:** July 19, 2017; **Accepted date:** August 08, 2017; **Published date:** August 14, 2017

**Citation: **Balumi WB, Aminu MD (2017) Effect of Initial Pressure, Surface, Outlet Velocity, and Density of Adsorbed Gas on Transport and Production in Shale Gas Reservoirs. Oil Gas Res 3: 144. doi: 10.4172/2472-0518.1000144

**Copyright:** © 2017 Balumi WB, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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The recent technological advancements in horizontal drilling and hydraulic fracturing have led to a boom in gas production from unconventional shale gas reservoirs. However, knowledge and technologies required to successfully develop unconventional reservoirs are far beyond what is available in the industry at present. Shale gas reservoirs are extremely heterogeneous with ultra-low permeability and nano-pores. The flow of gas in this reservoir is non-linear, multi-faceted including adsorption/desorption, flow at high and low rates, solid-fluid interactions, etc., which makes it a significant challenge to quantify such flow. A pore-scale flow model was developed using a combination of CFD and COMSOL multi-physics 4.2 based on Darcy and Navier-Stokes equations to describe transport of adsorbed gas and free gas in pore spaces respectively. Parameters such as surface pressure, adsorbed gas density and initial reservoir pressure were used to study shale gas transport. The presence of adsorbed gas within the shale gas reservoir will decrease porosity while increasing total production and gas storage capacity due to the high affinity of surfaces of organic matter to methane found within the shale gas reservoirs and hence high gas-in-place estimates. Moreover, because production from the adsorbed gas phase is dependent on pressure, four different values of initial reservoir pressures were used to analyze the effect of reservoir pressure on flow velocity. It was observed that the higher the initial reservoir pressure, the greater the velocity of the flow and consequently higher production rates.

Unconventional; Shale gas; Reservoirs; Permeability; Nanopores; Darcy equation; Navier-stokes equation

CAD: Computer Aided Design; CFD: Computational Fluid Dynamics; DXF: Drawing Exchange Format; FIB: Focused ion beam; GIP: Gas Initially in Place; SEM: Scanning Electron Microscope

To achieve commercial production in unconventional reservoirs, stimulation of the reservoir is required. These types of reservoirs require fracture stimulation to release the vast amount of oil and gas stored in very low-permeability formations such as shale. Different processes combine to control gas storage and flow in shale gas reservoirs. Gas storage is achieved as compressed gas in pore spaces, as gas adsorbed on to the walls of pores, organic matter, clays etc., and in solid organic materials such as kerogen and clays where it exists as a soluble gas. The flow of gas in shale is achieved through a network of pores with different diameters ranging from nano-meters to micrometers [1,2]. These nanopores are important because of the following reasons:

(1) The surface area exposed is larger in nanopores than in micropores for the same pore volume. Since surface area is proportional to 4/d, where d=pore diameter, this large surface exposed leads to large volume of gas desorption from surfaces of the kerogen.

Gas flow in shale is through nanopores, which does not follow Darcy flow.

(2) This study was carried out to account for the contribution of adsorption phenomena in shale to production and gas initially in place (GIP) at pore scale. It is also aimed at improving our understanding of pore scale transport in shale gas reservoirs in general.

**Sample preparation**

The methodology for this study is divided into two: firstly, the creation of a pore-scale finite element mesh of shale rock from scanning electron microscope (SEM) images and secondly, the numerical simulation of shale gas transport at pore-scale level based on published reservoir data obtained from production and well completion of shale formation.

**SEM imaging and grey scale imaging**

Scanning electron microscope is an electronic microscope that
uses a concentration of electrons to generate different signals at the
surface of the shale sample under investigation. These signals reveal
salient information about the sample such as texture, structure and
grain orientation within the sample. Greyscale imaging which consists
of variety of grey shades devoid of any obvious color are the most
preferred format for image processing. Sometimes white shades may
appear as the brightest shades possible indicating the total presence of
light within the visible wavelengths. Black shades where present similarly
indicate the darkest shade possible and the complete absence of light in
such images. Brightness levels of equal measures including three major
colors such as green, blue and red are used to represent intermediate shades
of grey for reflected lights. Measurement techniques such as focused ion beam
(FIB) and scanning electron microscopy are capable of direct measurement
and visualization of shale matrix at nano-scale on high quality flat surfaces,
thereby providing a new insight on the nano-scale structures available in
shale matrix. **Figures 1** and **2** show SEM images, revealing area of interest
and etched geometric patterns in silicon wafer.

The initial phase in modeling the transport of shale gas in inorganic
component of shale using computational fluid dynamics (CFD)
modeling is the construction of pore-scale finite element mesh using
FIB or SEM, reproduced using a computer-based drawing program.
The generation of finite volume mesh model from FIB/SEM is preceded
by importation of the images into COMSOL Multiphysics software
prior to converting them into solids from computer aided design
(CAD) format. The model is subsequently reworked and defined again
to bring about the finite volume meshing. To avoid unnecessary errors
during the modeling process, the geometric profile and the ultimate
meshing of the model must be precisely defined. This is followed by
the definition of flow fields and importation of files into COMSOL
Multiphysics software to assist in bringing solutions to complex
flow equations. Parameters and variables such as flow domain,
fluid properties, output pressure, etc., are inputted to solve the flow
relationships in the software. Using the fundamentals of fluid dynamics
and by delineating boundary conditions, COMSOL Multiphysics
software solves the numerical variables [3]. **Figure 3** shows kerogen
content (shown as dark areas) and inorganic constituents (represented
in grey color) within the Barnett shale, using SEM imaging (**Figure 4**).

**Definition of the model**

The model covers an area of 6.4 × 10^{-4} m × 3.3 × 10^{-4} m (**Figure 5**). The flow of gas within the model is usually from the right-hand side to
the left-hand side across the model geometry. Laminar flow is assumed
in the pores and the gas does not go into the grains in the matrix.
Fluid pressures at the inlet (**Figure 6** and **Table 1**) and outlet (**Figure
7**, **Tables 2** and **3**) are known and defined. Symmetric gas flow is also
assumed at the top and bottom boundaries. The main area of interest
is rectangular with its topmost left corner set at (0,0) μm and lower
most right coordinates set at (5.6, 2.3) × 10^{-4} Gravity has not being
accounted for in this model due to its small-scale dimensions. The
physical boundaries of the model can be mathematically represented
by the equations used. The inlet and outlet pressures are known, to
achieve a no-slip condition the velocities at grain boundaries are set as
zero. The flow around the lower and upper boundaries is symmetric. **Figure 8** depicts walls within the boundaries in the model, and **Table 3** is a presentation of conditions at boundaries. **Table 4** shows parameters
for the model. The mathematical description for the detailed fluid
mechanics in the interstitial pore space is based on the assumptions
that we have steady state flow in isothermal conditions and the fluid is
continuum, Newtonian, and incompressible (**Figures 5** and **6**).

Description | Value |
---|---|

Boundary condition | Pressure |

Pressure | p0 |

Suppress backflow | On |

Flow direction | Normal flow |

Use weak constraints | Off |

Apply reaction terms on | All physics (symmetric) |

Constraint method | Elemental |

**Table 1:** Inlet boundary settings used in the model.

Description | Value |
---|---|

Boundary condition | Pressure |

Pressure | 0 |

Normal flow | Off |

Suppress backflow | On |

Use weak constraints | Off |

Apply reaction terms on | All physics (symmetric) |

Constraint method | Elemental |

**Table 2:** Outlet boundary settings used in the model.

Type of Boundary | Condition of the Boundary | Value |
---|---|---|

Outlet Boundary | Pressure, P | p=0 |

Inlet Boundary | Pressure, P | p=P_{0} |

Sides of the Symmetry | Symmetric | Nil |

Walls of the grain | - | Zero slip |

**Table 3:** Conditions at boundaries of the model. Note: where P_{0}=change in pressure.

Name | Expression | Value |
---|---|---|

rho | 0.67 (kg/m^{3}) |
0.67000 kg/m³ |

mu | 10.99E-6 (Pa*s) | 1.0990E-5 Pa·s |

p0 | 2000 (kPa) | 2.0000E6 Pa |

**Table 4:** Model parameters.

**Initial and boundary conditions**

The pressure distribution of the shale gas pore system at a time t=0 defines the initial condition. This distribution of pressure is assumed to be the same all through the shale formation and is equivalent to the initial pressure of the reservoir. The concentration of the shale gas on the surface of pores and matrix within the system is assessed at the pore scale level. The pore system is specified using the boundary conditions along the pores within the system.

**Navier-stokes equation**

By assuming a constant temperature and fluid density in the pores of the rock, Navier-Stokes and continuity equation can be used as follows [4,5].

*ρ (∇u ⋅∇u) = ∇. − pIη∇u (∇u)T) = ∇⋅ u = 0* (1)

Where,

η: Dynamic viscosity (kg/(m·s)),

μ: Velocity (m/s),

ρ: Fluid density (kg/m^{3}),

p: Pressure (Pa),

I: Identity matrix.

**Darcy law with adsorption**

The flow of gas and its transport processes in porous formations can
be described mathematically from the principle of mass conservation of
fluid/solute and Newton’s 2nd law of motion [6]. The general equation
for conservation of mass or volume in porous media is as follows:(rate
of mass inflow)-(rate of mass out flow)+(rate of mass production/
production)=(rate of mass accumulation) (**Table 5**). Desorption of
shale gas from the surface of pores is followed by diffusion through the
shale matrix until a fracture is met, after which the gas flow is based on
Darcy law. This statement of mass conservation can be combined with
a mathematical expression of relevant process to obtain a differential
equation that will describe flow or transport in shale gas reservoirs. Per
Darcy’s law for flow in a porous medium, the absolute permeability of the present model can be numerically calculated. The seepage or
superficial velocity in horizontal direction can be obtained from the
following expression [7].

(2)

From Equation (11), K (absolute permeability) can be easily estimated. Darcy’s Law is valid for the slow flow (inertial effects can be neglected) of a Newtonian fluid through a porous medium with rigid solid matrix.

(3)

Where,

V: Velocity of the fluid in the formation

K: Permeability of the rock

ρ: Fluid density

μ: Fluid viscosity

∇P: Pressure gradient

: Rate of change in mass in control volume.

Property | Value |
---|---|

Minimum element quality | 3.08E-04 |

Average element quality | 0.7342 |

Triangular elements | 13352 |

Quadrilateral elements | 4634 |

Edge elements | 2456 |

Vertex elements | 2232 |

Calibrate for | Fluid dynamics |

Maximum element size | 1.6 |

Minimum element size | 0.048 |

Curvature factor | 0.25 |

Maximum element growth rate | 1.08 |

Predefined size | Extra fine |

Custom element size | Custom |

**Table 5: **The table presents statistics of mesh used in the model, while other parameters used in the model which were selected based on published literature review on shale gas simulation [5,8] is tabulated in Table 6.

Conceptual flow models in shale can be single or two phase flow model based on the fluid content in the formation under study. Moreover, because we are mostly concerned with the transport of the gas in the reservoir, we can assume the system is immobile and has residual water saturation. This will simplify the formulation of the model into a single-phase flow. This single-phase flow can be further categorized into two based on the presence or absence of adsorption/ desorption as follows [8].

(4)

(5)

Where,

Ma: Amount of mass present in adsorbed state

Mp: Amount of gas present in the pores in the formation

The mass of adsorbed gas in the formation can be expressed as follows:

(6)

Where,

*ρ _{b}*: Bulk density of the rock

*ρ _{a}*: Gas density at standard condition

*V _{a}*: Adsorbed gas volume based on Langmuir adsorption isotherm.

By simply replacing *V _{a}* with

(7)

(8)

By taking away the constants

(9)

(10)

Hence

(11)

Mass of gas in the reservoir pore volume can be expressed as a function of the porosity of the reservoir and the density of the fluid in the reservoir:

(12)

Where,

Reservoir porosity

Flow of free and adsorbed gas in a shale matrix can be described using the following expression:

(13)

Where,

*v*: Darcy velocity

*M _{T}*: Total mass of gas in the shale formation:

*M _{T}: M_{ads}+M_{free}* (14)

*M _{ads}*: Mass of adsorbed gas in the shale formation

Mass of free gas in the shale formation

*M _{free}*: The masses of adsorbed gas and free gas are given by the
following equations:

(15)

(16)

The total flow in the organic-inorganic matrix system in shale can be represented by:

(17)

By rearranging and including the velocity term, we obtain the following:

(18)

**Assumptions of the model**

The model developed for this study is based on the following key assumptions:

I. A single-phase flow is present in the model,

II. Temperature in the reservoir is constant throughout the reservoir always,

III. Permeability is assumed to be the same throughout the formation,

IV. The gas obeys ideal gas laws,

V. The formation is uniform and rectangular,

VI. The effect of heterogeneity and gravity on gas flow are neglected,

VII. Gas adsorption-desorption is based on Langmuir curve.

**Implementation of the model**

COMSOL Multi physics software has a fluid flow physics interface
(based on Darcy’s law interface) which when selected along with the
time dependent study type in a model can be used to solve the above
flow equations used to describe the flow of gas in shale formations.
Material properties such as density, permeability, porosity, etc., are
inferred using the imported image in this model. The mesh setting was used to separate the model into several tiny triangular and rectangular
shapes as shown below in **Figure 9**.

In this work, we investigated the effect of parameters such as initial reservoir pressure, surface velocity, outlet velocity and the contribution of free and adsorbed gas on gas transport and production in shale gas reservoirs. These parameters are added to the model created to investigate their effect on fluid flow and production in shale formation.

**Single phase flow without adsorption**

Initial reservoir pressure is the sole driving force in gas reservoirs
and as such it is very essential to transport and production in gas
reservoirs. Here the effect of initial reservoir pressure on gas transport
is considered. Four different values of initial reservoir pressure - 500
KPa, 1000 KPa, 1500 KPa, and 20 KPa were considered as shown
in **Figure 10** at pore scale level. While varying the pressure during
the investigations the other parameters shown in **Table 6** are kept
constant. **Figure 10** shows the solution estimated using Navier-Stoke
analysis for fluid velocity field in the pore spaces of a micro-porous
shale formation. Higher velocity magnitudes which are indicated by
the red and cyan shades of color are concentrated on the narrowest
pores within the slide as the gas is moving from the inlet to the outlet
as shown by the arrows, this decreases as the fluid moves towards
the outlet due to the increase in cross sectional area available for
flow. Around the midsection of the slides, a high velocity magnitude
is recorded within a wider area which reveals that at relatively lower
pressures when multiple channels commingle at a central point, high
velocity magnitudes are obtained. The highest flow velocity is several
times greater than the lowest flow velocity within the shale sample.
This high velocity streamlines are believed to represent the favored
path for gas flow in the model. When the initial reservoir pressure
was increased 100%, the surface velocity was also doubled. This will
consequently mean an increase in production of shale gas. **Figure 11** gives the pressure contours across the sample slide at 2000 kPa. The
figure clearly shows the reduction in pressure across the slide from the
inlet to the outlet. The highest pressure occurs at the inlet while the
lowest pressure occurs at the outlet in the sample shown below. Since gases always travel from high velocity area to low velocity areas, a high
flow rate and production rate is expected from this model. **Figure 12** shows a domain plot that explains the x-velocity at the four different
pressures within the outlet. Because the flow is moving in the negative
side of the x-axis, the velocities are assigned negative values. This
buttresses the points raised above. Surface pressure plot of the model
is shown in **Figure 13**. Maximum pressure occurs at the inlet as the gas
travels because of pressure gradient and the pressure at the edge of the
inlet is several folds greater than the pressure at the outlet within the
boundaries of domain. This in addition to the results discussed above
further shows the importance of initial reservoir pressure as the driving
force within shale gas reservoirs.

Name | Expression | Value |
---|---|---|

V_std | 0.0224 (m^{3}/mol) |
0.022400 m³/mol |

M | 0.016 (kg/mol) | 0.016000 kg/mol |

rho_s | 2560 (kg/m^{3}) |
2560.0 kg/m³ |

PL | 3.05e6 (1/Pa) | 3.0500E6 1/Pa |

VL | 9.80E-4 (m^{3}/kg) |
9.8000E-4 m³/kg |

k | 1.0E-19 (m^{2}) |
1.0000E-19 m² |

epsilon | 0.5 | 0.5 |

rho_g | 0.66 (kg/m^{3}) |
0.66000 kg/m³ |

mu | 0.0184 (cP) | 1.8400E-5 Pa·s |

**Table 6:** More parameters used in the model.

**Single phase flow with adsorbed gas**

The Langmuir isotherm describes the effect of pressure on
adsorption in shale gas reservoirs based on a Langmuir volume and Langmuir pressure of 9.8000e-4 m³/kg and 3.0500e^{6} Pa respectively.
Fluid viscosity at initial reservoir condition is 0.01804 cp. The effect
of adsorption on pore size and transport in shale gas reservoirs was
also considered. It is important to note that in this work we compared
a situation where adsorption is considered and where it is ignored.
Where adsorption was ignored, only free gas in the rock matrix
was considered. Adsorption in shale gas reservoir could lead to the
modification of the porosity and permeability of the reservoir rock as
the reservoir is being depleted and its pressure gradient is reduced [9].
The pressure across the surface of the sample is less than 0.50E7 Pa
as shown in **Figure 14**. The effect of adsorption on shale gas transport
depends on pressure within the matrix of the rock. This low pressure
means low adsorbed gas density and volume. Total gas production in
a shale gas reservoir is the sum of adsorbed gas within surfaces in the
rock and free gas found in the pores, and natural fractures. Moreover,
any analysis of the producible gas within a shale gas reservoir must
include desorption, and failure to do so will result in erroneous
estimate. The density of the gas is usually affected by the interaction of
the gas molecules with its environment and this depends on the nature
of the surface of the matrix and its bulk material properties. The density of adsorbed gas is found to be much lesser than that of the free gas, but
by not considering it during analysis and estimations of GIP, a very
significant volume of gas production will be neglected. For example,
in comparing the free gas and adsorbed gas density across the surface
of the sample, free gas had a density of 3.3e-1 Kg/m^{3} while adsorbed
gas had a density of -2.97e-7 Kg/m^{3}, as such, based on this comparison
we can see that adsorbed gas has a considerably lower density than the
free gas but it is still significant for consideration in any analysis of gas
production in shale gas reservoirs.

This study presents the result of numerical simulation of gas transport and the effect of adsorption on production in shale gas reservoirs. Initially, SEM image of the shale sample was converted to a grey image of the pore-scale finite element mesh. The CAD image formed was imported into COMSOL Multiphysics 4.2 and the organic components of the shale removed to form a pore scale model. The flow fields were defined to obtain accurate solutions to complex flow equations by delineating boundary conditions and using fundamentals of fluid dynamics in COMSOL Multiphysics software to solve for the numerical variables. Pore-scale modelling was carried out by describing the morphology of the porous medium and subsequently importing the SEM image as drawing exchange format (DXF) file into COMSOL Multiphysics software. The outlined software was then used to simulate the effect of initial reservoir pressures, surface pressures and adsorption on gas transport using the fluid flow module in the software. The transport of single phase gas in the pore space was described based on Navier-Stokes equation while fluid flow incorporating free and adsorbed gas was described based on Darcy law. The direction of gas flow in the model was set from the right-hand side to the left-hand side based on several assumptions that were meant to further validate and simplify the difficulties to be encountered. Parameters such as fluid pressure, inlet and outlet pressures were known and defined. At time t=0, the pressure distribution in the shale gas pore system was defined as the initial condition. This was assumed to be the same throughout the formation and it’s the same as the initial pressure of the reservoir. The effect of initial reservoir pressure, surface velocity, outlet velocity and density of adsorbed gas on gas transport and production was studied. The analysis of the results was divided into two based on single phase flow transport where the effect of adsorption was considered and where the effect of adsorption was ignored (in which only free gas in the rock matrix was considered). Furthermore, in analyzing single phase flow without adsorption four different values of initial reservoir pressures were used to gauge the effect of the parameter on shale gas transport and production. The results showed that a direct relationship exists between the initial reservoir pressure and velocity of the shale gas within the pores in the shale reservoir and hence the greater the initial reservoir pressure, the higher the velocity of the shale gas and consequently the greater the production. The effect of adsorption on single phase flow was considered based on surface pressure and density of the adsorbed gas and a comparison was made between the density of adsorbed gas and free gas. Hence, considering the effect of desorption in any analysis of shale gas reservoirs will lead to more accurate estimates of total gas production. The following conclusions were drawn from this study:

i. High velocity magnitudes are concentrated on the narrowest pore channels and at junctions where there is a commingling of channels within the shale formation. This high velocity channels are believed to be the preferred channel for shale gas transport.

ii. Initial reservoir pressure has a hugely significant effect on gas production. By doubling the initial reservoir pressure, the surface velocity of the shale gas was significantly increased.

iii. There is a pressure gradation across the model from the inlet to the outlet. The highest pressure occurs at the inlet while the lowest pressure occurs at the outlet. This further confirms the fact that pressure is the main driving force in shale gas reservoirs.

iv. Adsorption is very important to shale gas production. By ignoring the impact of adsorbed gas on shale gas production and cumulative production during estimation of the resource, we significantly reduce the gas production estimate.

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