1. Introduction
Natural gas is a mixture mainly consisting of methane (~95% volume), ethane (~4% volume), and smaller proportions of other hydrocarbons, nitrogen, and carbon dioxide. Therefore, we will refer simply to methane or hydrocarbons as synonyms of natural gas. Unconventional reservoirs of natural gas have become a very important alternative to obtaining fossil energy in the United States. Other countries such as China, Canada, and Australia are currently investigating the potential of this promising resource. Unconventional reservoirs include several types of deposits: shale gas, tight gas sandstone, coalbed methane, and methane hydrates. This work aimed to study shale gas, which represents a large and growing share of the U.S.’s recoverable resources.
Unlike conventional gas reservoirs where methane is accumulated in underground voids and caverns, the distribution of methane in shale gas deposits is complex. The methane distributes in three sites: (1) free gas stored in the network of cracks and intergranular porosities in the shale, which is a very fine-grained sedimentary rock; (2) dissolved methane, which is distributed in the organic phase, called kerogen or bitumen, and is in turn distributed within the shale; and (3) adsorbed methane, located on the surface of micropores with a pore diameter (dp) < 2 nm and mesopores (2 ≤ dp ≤ 50 nm) formed in the organic matter. Etminan et al. [
1] carried out a laboratory study of the gas storage distribution in shale and determined the diffusion coefficient of methane in kerogen. The authors developed the batch pressure decay (BPD) technique to measure the gas volume stored in each one of the above-mentioned three sites in a single experiment. They used thin disks (~7 g) from a shale-core plug sample. The authors identified that at an early stage, ~4 min, free gas came out initially. Then, gas desorption from the organic pores followed at around ~2:30 h, and finally, gas diffusion from the organic material was detected until the end of the test, ~100 h. A mass balance was used to determine the gas proportion released from each site. Free gas and adsorbed gas contributed almost equally, 38% each, to the total gas-in-place, while dissolved gas was about 22% of the total gas-in-place. The rate of release of dissolved methane was controlled by a low diffusion coefficient, which the authors reported to be on the order of 10
−20 m
2/s. Chalmers et al. [
2] characterized the shale-gas pores by measuring the total porosity, pore-size distribution, surface area, organic geochemistry, mineralogy, and image analysis by field emission scanning electron microscopy/transmission electron microscopy, using samples prepared with focused ion beam milling (FIB). The authors analyzed samples form Barnett, Marcellus, Woodford, and Haynesville gas shales in the U.S. and the Doig Formation in northeastern British Columbia, Canada. They showed the nature and variability in pore structures within gas shales. They found that aggregates of kerogen, clay and/or carbonate contain mostly macropores (dp > 50 nm) and mesopores in the studied samples and that those samples with greater pore interconnectedness would show improved access to the surface area of the mesopores and micropores within the organic matter, enhancing the matrix permeability of the reservoir. Curtis [
3] presented a review of five principal shale-gas exploited systems in the U.S.: Antrim, Ohio, New Albany, Barnett, and Lewis shale basins. He showed some well data including the adsorbed gas content for these basins, which is between 20% (Barnett Basin) and 85% (Lewis Basin). Furthermore, all these gas reservoirs exhibit wide ranges of geological and geochemical parameters that affect the proportion of adsorbed gas. Therefore, gas flow rate production will depend not only on well stimulation treatments (fracking) but also on the thermal history of organic matter and the response of the shale matrix and organic matter to local and global stresses. For the previous reasons, it is important to develop models for predicting the rate of gas release from unconventional gas deposits.
Fathi and Akkutlu [
4] developed a gas transport model to investigate the effect of local spatial heterogeneities in porosity and the material content of the matrix on the rate of gas production from a laboratory-size sample. The free gas flow, diffusive transport, and adsorption/desorption rates were introduced in their model to carry out a perturbation analysis. The authors adopted weak-noise and mean-field approximations and used a statistical approach in spectral domain to find that the matrix heterogeneities generate non-trivial, macro-transport, and macro-kinetics effects on the system that retard gas release from the matrix and adversely influence the gas recovery. No measured data were used to validate the model results.
Nobakht et al. [
5] developed a simplified method of production forecasting for tight/shale-gas reservoirs showing an initial period of linear gas flow. The method is applicable to hydraulically fractured vertical wells and multi-fractured horizontal wells leading to isobaric gas flow. This model combines the theory of linear flow with hyperbolic decline during boundary-dominated flow. The authors claimed that the model does not require superposition time functions nor pseudo time. The only required major well parameter is the drainage area, which unfortunately is the parameter with the greatest uncertainty, particularly in the first year of gas production. Model validation was carried out using Marcellus and Barnett shale gas production data. Javadpour et al. [
6] presented a review on the gas flow models of shale. The authors pointed out that conventional flow models based on Darcy’s law cannot model shale gas production accurately, even after including molecular dynamics or lattice Boltzmann methods. This failure to match field data and laboratory-scale results has been attributed to the complexity of multi-scale pore size in shales. On the one hand, nanopores are present in the organic matter. The size of these pores is from a few to hundreds of nanometers. On the other hand, micropores are present in the inorganic matter and are generally on the order of microns, much larger than the organic nanopores. The authors found that pore network models have been used extensively by many researchers to upscale nanoscale gas flow from a single nanopore to a network of interconnected nanopores. However, most efforts have focused on modeling the microscale behavior of gas, while few models have been aimed on the scale-up calculation. The authors concluded that more research is needed to apply shale permeability models to field scale. Arevalo et al. [
7] performed a dynamic characterization of unconventional gas reservoirs where gas production could be estimated from two unconventional reservoir wells: Eagle Ford formation in Texas, and northeastern wells in Mexico. They concluded that Langmuir’s isotherm model is appropriate to predict the behavior of adsorbed and desorbed gas in unconventional formations that contain organic material. The authors pointed out that this result is important since, once gas desorption pressure is reached, there is an additional production mechanism of gas. Song et al. [
8] used the fractal theory to calculate the gas slippage factor in porous media considering both gas adsorption and no gas adsorption. The slippage factor is important for determining the apparent gas permeability of the porous medium. Two important model parameters that are required are the pore fractal dimension and tortuosity fractal dimension. The model predictions are in good agreement with molecular simulation and experimental data from other researchers. The authors conclude that the gas slippage factor depends on the structural parameters of porous medium, adsorption capacity, reservoir properties, and gas properties. Li et al. [
9] developed a simulation model to predict the production rate of a refracturing shale gas well. The model is based on the coupled solution of the equations for fluid flow and geomechanics in triple continuum media including kerogen, gas, and an inorganic matrix. A fracture network is proposed considering the multiscale flow characteristics of shale gas, the induced stress of fracture opening, and the pore elastic effect. A sensitivity analysis is performed to study the effects of the refracturing pattern, fracture conductivity, width of the stimulated reservoir volume (SRV), SRV length of new and initial fractures, and refracturing time on the production increase percentage (PIP). Younk [
10] applied the hyperbolic decline curve analysis method to data from the Bakken and Barnett formations. The coefficients of the non-linear regression equation were obtained for the most recent 5 years of data. The author presents some analysis guidelines to obtain satisfactory long-term predictions. Another recent effort to predict shale gas production and to optimize horizontal well parameters was reported by Wang et al. [
11], who developed a model based on machine learning to optimize multistage fractured horizontal wells. They combined Gaussian Process Regression (GPR) and Convolution Neural Network (CNN) to complement a numerical simulation model and achieve rapid optimization. Their sensitivity analysis identified porosity, permeability, fracture half-length, and horizontal well length as influencing factors. Furthermore, the n-factorial experimental design was applied to design the initial experiment, and the dataset was constructed by combining the simulation results with the case parameters. The authors reported that well parameter optimization using GPR was carried out in 1/720 of that period for numerical simulation. However, no details were described regarding the simulation model itself.
Laboratory experiments consistently confirm that natural gas is distributed as free gas in pores and cracks, adsorbed gas on the surface of nanopores, and dissolved gas in the kerogen, and these experiments also identify the role of production pressure on gas extraction. Sang et al. [
12] presented laboratory results for tests using four shale gas core samples from a shale gas reservoir in China and one sample from a tight sandstone reservoir in Canada. The authors found that the contributions of free gas, adsorbed gas, and dissolved gas to the cumulative gas production were affected by the production pressure. They reported a critical production pressure at which shale gas can be efficiently desorbed and diffused from kerogen. At high production pressures, the adsorbed gas extraction is mainly affected by the shale desorption properties, while at low pressures, the adsorbed gas extraction is affected by the stress sensitivity of permeability. These results show the importance of considering the production pressure for proper gas extraction modeling. Well pressure is a process variable that has been remarkably missed in most empirical models for gas production prediction.
The most accepted and widely used fundamental approach to predict the rate of gas production is based on Darcy’s law, which is given by the following expression:
where W is the fluid mass flow rate (kg/s), ρ is the average gas density (kg/m
3), k is the permeability (m
2), A is the cross-sectional area (m
2), μ is the fluid viscosity (Pa s), and Δp is the pressure difference (Pa) between two points separated by a distance L (m) that is measured along the flow direction. The permeability is a property of porous media that represents the ease with which a fluid flows through these media when a pressure gradient is applied. Singh and Cai [
13] presented a critical discussion of papers on the permeability of fractured shale and also presented a summary of their previous work on brine-gas relative permeability in fractures. The authors included experimental and modeling results aimed to understand gas transport processes in shale that determine the permeability at different times during the well’s lifespan. They explained that experimental work has been focused on characterizing permeability using steady-state or unsteady-state laboratory methods. From these measurements, the permeability of laboratory samples can be studied as a function of other variables, such as the pore diameter, pore porosity, gas pressure, residual brine saturation, etc. On the other hand, shale permeability models have been developed considering the complex characteristics of shale-gas systems, namely, (1) gas rarefaction in pores and cracks during gas extraction, which leads to correction formulae specified as a function of the Knudsen dimensionless number (Kn =
λ/L,
λ = mean free path of gas molecules, L = representative physical length scale); (2) the compressibility of the gas, shale matrix and shale bulk; (3) the kinetics of gas adsorption–desorption on pore and crack surfaces; (4) stress, strain, and damage of cracks; (5) local heterogeneity; (6) the imbibition of slick frack water into the matrix by osmosis (counter-imbibition); and (7) gas dissolved in organic matter. The parameters of these permeability models can be determined from laboratory tests of core samples. On the other hand, the determination of field-scale permeability requires a further method; for example, for unconventional reservoirs, history-matching has been used successfully. It consists of estimating the parameters included in the permeability models by minimizing the difference between the computed and the measured gas production rates. Therefore, using historical production data is possible to determine a more accurate forecast for the field-scale permeability. Another method that has been used traditionally for conventional reservoirs is the well logs (measurements of properties, e.g., density and hydrogen index). In this case, the permeability is determined using mathematical relationships between permeability and the well logs. These relationships can be obtained from either empirical data from core samples or well-known porous media models, for example, the Blake–Kozeny model. Application of well logs to unconventional reservoirs is questionable for two reasons: (i) the shale properties do not represent a statistically significant correlation with permeability, and (ii) kerogen exhibits log responses that are indistinguishable from pore fluid, causing the logging tools to read excessively high or low values. Finally, the authors present a summary of their formulation of the two-phase relative permeability in shale fractures. Their work also models the upscaled permeability by accounting for distinct ground characteristics using a finite difference grid that contains fractures and matrix.
Some selected examples of permeability models developed for shale matrix are the following. Darabi et al. [
14] studied the gas flow process in ultra-tight porous media that included nanometer- to micrometer-size pores. They formulated a pressure-dependent permeability function assuming Knudsen diffusion and slip flow (the Klinkenberg effect). Slip flow occurs when 0.001 < Kn < 0.1, which represents moderate gas rarefaction, but Knudsen diffusion corresponds to ultra-rarefaction levels, Kn > 10. The model predicts that in nanometer-size pores, the gas permeability values are as much as 10 times greater than the results obtained by continuum hydrodynamics predictions (Kn < 0.001). However, with increasing pore size (i.e., on the order of micrometers), the gas permeability converges to continuum hydrodynamics values. The authors used the pulse-decay laboratory method (an unsteady-state technique) to estimate the main parameter of the model, which is the Darcy permeability.
Civan [
15] developed a correlation of apparent gas permeability in tight porous media. His model is based in a unified Hagen–Poiseuille-type equation representing the porous media as a bundle of tangled tubes and includes parameters such as the intrinsic permeability, porosity, tortuosity of the apparent gas permeability, rarefaction coefficient, and Klinkenberg gas slippage factor. His formulation was used to obtain an improved correlation of available laboratory data with respect to previous models.
Cao et al. [
16] developed a fully multiscale shale deformation–gas transport model. Their model consisted of the solution of a set of partial differential equations to calculate the following processes: (1) the mechanical equilibrium equation that defines the shale body deformation; (2) gas flow in the kerogen system of matrix; (3) gas flow in the inorganic system of matrix; and (4) gas flow in the hydraulic fracture system. For each gas flow system, a permeability or diffusivity model was derived to define its evolution. The verified model was applied to investigate the impact of the adsorption parameters, flow regimes (defined by the Knudsen number), the initial permeability of the inorganic matrix, and the effective stress variations on the gas production rate. Model predictions were compared with the respective production rates in the Marcellus and Barnett shales, in the U.S.
Yuan et al. [
17] presented an analytical model of apparent gas permeability for tight porous media. Their approach also considers a porous medium as equivalent to a bundle of tangled tubes. However, the distribution of capillary diameter and tortuosity were computed using fractal theory. The authors studied the effect of pore size distribution, fractal dimensions for pore size and tortuosity, porosity, surface diffusivity, and Langmuir parameters on flow rate processes in tight rocks.
Sheng et al. [
18] also used fractal theory to consider the effect of multi-scale pore spaces on shale-gas permeability. The authors considered various flow regimes that occur in these multi-scale pores. They confirmed that gas permeability varies with the pore-size distribution of the samples, even when they have the same intrinsic permeability value.
Yang et al. [
19] developed a model of Dynamic Adsorption–Diffusion (DAD) to represent gas transport and storage in crushed shale at three different particle sizes. The mathematical model for this phenomenon was validated on dynamic adsorption experiments with a constant pressure condition. It was found that the absolute permeability decreases with crushing to smaller particles. Furthermore, particle size plays a more important role on diffusion than on permeability.
All the above mathematical models have been focused on predicting the evolution of flow of gas released mainly from a laboratory sample or in some cases from an unconventional reservoir. To the best of our knowledge, very few reports include pressure evolution in the reservoir. It should be clear that both the gas flow rate and pressure evolve during well production. Moreover, pressure cannot be dependent only on the amount of recovered gas but also on the amount of gas that is being released from the kerogen. Recently, Gu et al. [
20] reported a study of the gas pressure field and production rate when using proppants with two different particle sizes. They solved the mass conservation equation for gas and water phases and the corresponding Darcy’s equations for them. They found that using proppants with a larger particle size will lead to a lower permeability and therefore to larger gas production rates.
In this paper, we propose a completely new model that introduces a semi-empirical pressure source equation representing the gasification kinetics from the kerogen. The model is based on the transient solution of the conservation of mechanical energy equation (Bernoulli’s equation) and the conservation of gas mass (continuity equation) to predict the evolution of gas flow and pressure in an unconventional reservoir. This approach is more computationally efficient than those based on the numerical solution of the conservation of mass and momentum differential equations. Moreover, it is consistent with previous models based on Darcy’s equation and the Langmuir desorption model. Actual production data from unconventional natural gas wells were used for model validation.
4. Summary and Conclusions
This work presents and validates a new mathematical model to calculate the evolution of gas pressure and flow rate during the production period of unconventional reservoirs. This model is based on the principle of conservation of mechanical energy, expressed by Bernoulli’s equation. The transient nature of the gas extraction process is approached by steady-state stepwise periods. The pressure is computed from the free gas amount available in the cracks, pores, and main duct in the well. A pressure source term accounts for the kinetics of desorption and diffusion of methane. The five model parameters were obtained from gas production data of wells in the Eagle Ford [K’ = 8.61, B = −62.2, C = 62.2, a = 5, and b = 3] and Burgos basins [K’ = 408, B = −7.0, C = 7.0, a = 5, and b = 3].
The predicted evolution of gas pressure and flow rate fully agree with the corresponding smoothed data points curves, finding Pearson correlation coefficients above 0.95 for flow rate and pressure drop data and for both studied wells. This means that constant model coefficients allow for the calculation of the evolution of both the gas pressure and flow rate for each well.
A comparison of the curves obtained from regression on smoothed production data using the PSM, Arps’ model, PLE model, SED model, and Duong model showed a remarkable improvement when using the PSM. A reason for this may be that, differently from the available decay models, the pressure source model considers both the gas flow rate and gas pressure evolution and is based on mechanical energy and mass conservation principles. The decay models only consider gas flow rate data.
The postulated power-law function representing the pressure source term was able to mimic the desorption and diffusion phenomena. These processes provided free gas to the cracks and macropores after decay of the initial gas pressure.
The measured gas pressure–flow rate curves follow a parabolic function in agreement with the Ergun equation for porous media. The present model results also agree with this parabolic function.