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When analyzing the flow production data in a shale play notice that these measures are higher than those predicted by Darcy’s Law with which they were calculated.

Based on that we might ask ourselves why is the production of gas higher than expected? Is it that new factors have appeared that we must take into account? Are these still fulfilling classical laws of fluid mechanics used in conventional oilfields?

Answering these questions will help us to improve the understanding of fluid mechanics in nanoporous medium, with which we will achieve a better reservoir simulation, a more accurate production forecast, and ultimately predict with higher probability the profitability of investment.

The first thing to analyze in a shale play is its matrix, which is constituted of inorganic matter and fragments of organic matter dispersed within it, known as kerogen. These two components have petrophysical properties very different from each other.

The inorganic matter is formed by fine-grained sedimentary rocks. The presence of pores in it is scarce--micropores and natural microfractures--it is hydrophilic (wet water), presents high capillaries pressures, and multiphase flow.

On the other hand, the organic matter has greater porosity in a shale plays (up to 5 times greater than the porosity of the matrix, Javadpour 2007), with a well-formed poral network. The sizes of its pores are comprised between 5 to 1000 nanometers (Wang & Reed, 2009 (2)) and permeabilities in the order of the nanoDarcies, are oil wet and have mainly monophasic flow. This pore network generates a large surface area available for gas adsorption, that along with the free gas and gas dissolved within the kerogeno (organic matter), constitute the main source of storage of hydrocarbon in a shale field.

There is great heterogeneity present: hydraulic macrofractures, natural microfractures, micropores and nanopores generate zones with permeabilities several orders of magnitude greater than others.As we move in the direction of decreasing poral size the interacting forces between the pore walls and gas molecules are increasingly producing alterations in flow behavior. As a result new transport mechanisms appear such as slip flow, transitional flow and Knudsen diffusion.

In hydraulic fractures, natural microfractures and micropores uphold Darcy's Law, which was derived from the Navier Stokes equation based on continuum theory. This theory considers the fluid mass distributed uniformly in the element of volume, a compliance with the law of energy conservation between intermolecular collisions and assumes non-slip flow in the pore walls. This equation together with the Hagen Poiseuille equation correctly model the flow behavior at macrometrical and micrometrical scale where the main forces interacting are viscous.

At nanometric scale and/or at low pressures this theory begins to break because the main free path of gas molecules begins to be of the same magnitude as the pore size that contains them. The collisions of these with the walls are increasingly frequent forcing the gas molecules to slip by the walls.

The concept of slip flow first appeared in the oil and gas industry in 1941 when Klinkenberg observed in a laboratory setting that the gas flow obtained in a porous medium to low pressure was greater than that calculated by Darcy’s equation due to the slip. This meant that the speed of the molecules in the pore wall was not negligible and should be considered.

Klinkenberg made an empirical correction of the permeability of Darcy’s equation as a function of the pressure and introduced “the gas slip factor b” obtaining the following relationship:

Where ka is the apparent permeability, K is the liquid permeability, b is the slip factor and p is the average pressure.

This equation tells us that at infinite pressure the apparent permeability is equal to that of the liquid (absolute permeability), which is an intrinsic property of the porous medium.

The correction made to Darcy’s equation by the Klinkenberg effect has been and is enough to model gas flow in conventional oil fields whose porosities are on the micrometers order.

With the emergence of tight gas sands new corrections to the slip factor proposed by Klinkenberg appeared since this phenomenon is increased by decreasing pore size.

Jones-Owen in 1979 (3) through the study of more than 100 samples of tight gas sand in EEUU proposed the following equation for bk:

In 1981 Sampath and Keighin (4), with the analysis of 10 samples of tight gas sands, achieved the following relationship for the slip factor in function of the absolute permeability and effective porosity.

The dawn of shale plays fields necessitated a more solid theoretical support model to explain at the molecular level the present phenomena and the increase in flow measurements with respect to those predicted by these equations.

When the average mean free path of the gas molecules begins to be comparable to or greater than the pore size containing it, the result is a break in the continuum theory. The degree of deviation from this theory is measured with the Knudsen number:

Where d is the pore diameter, λ is the mean free path (the average distance that a molecule travels to collide with another) and can be calculated by the kinetic theory as:

Where kB is the Boltzmann constant, T the absolute temperature, d the pore diameter, and P the pressure.

From the Knudsen number we can divide the flow as: continuous flow, slip flow, transition flow and free molecular flow or Knudsen diffusion.

1-Viscous flow regime (Kn<0,001) the main free path of gas molecules is negligible compared with the characteristics of the flow geometry dimensions, the assumptions of thermodynamic equilibrium and the continuum are valid and the gas flow can be described by Darcy’s equation (Navier Stokes) with non-slip boundary conditions. This is the flow present in conventional reservoirs.

2-Slip flow regime (0,001<Kn<0,1) The effects of non-equilibrium dominate near the pore walls to increase the number of collisions between gas molecules and the wall pore. The condition of non-slip flow fails when comparing the values calculated with those measured in the laboratory, the continuum theory can be applied to describe the fluid core and can use the Navier Stokes equation with slip boundary conditions. (Darcy with Klinkenberg effect). This flow is present in tight gas sands.

3-Transition flow regime (0,1<Kn<10) rarefaction effects are the dominant ones, the assumptions of the continuum theory and the thermodynamic equilibrium are no longer valid. This is the area most difficult to model due to the non-linear variation of the stress tensor by the growth of the Knudsen layer and must be modeled at a microscopic level by the Boltzmann equation or by using the Karniadalis and Beskokequation. This kind of flow is present in the majority of shale plays.

4-Free molecular flow or Knudsen diffusion (Kn>10) intermolecular collisions are negligible compared to the collisions between the molecules and the pore walls, there is a full development of the Knudsen layer.

Now that we know the classification of the different types of flow we can say that the Klinkerberg correction is a model of first-order boundary conditions which is no longer valid for Knudsen numbers greater than 0.13 (Sreekanth). For being unable to incorporate the effect produced by the growth of the Knudsen layer and the free molecular flow.

To be able to model the flow in nanopores where a part is produced under transition flow and Knudsen diffusion, we must turn to models of second-order boundary conditions. In this way, we can cover the flow regimes present in shale fields.

Beskok and Karniadakis (1999) developed a unified model that predicts the mass flow and volumetric gas in channels and pipes covering all the flow regimens (0<Kn<∞). Their results were validated by means of molecular dynamics simulation.

The rarefaction coefficient

is used to model the effect of the reduction in the molecular interactions to increase the Knudsen number and varies between zero in the slip flow to a constant value (asintotic) that is obtained when it reaches the free molecular flow.

Based on this model, Florence et al 2009(5) proposed the following equation for apparent permeability:

where the rarefication coefficient is given by

Replacing it in the above equation gives us the following equation for the apparent permeability:

Also Civan 2010 (6) based on the same model of Kardiniakys and Beskok obtained an equation of apparent permeability by introducing the following equation for the coefficient of rarefaction:

where the values of the coefficients were set in α0=0,1358, A=0,178, B=0,4348 and b=-1 to analyze tight gas sand, with which the apparent permeability equation gives us:

Javadpour 2009 (7) on the other hand obtained a new equation for apparent permeability combining the flow produced by the Knudsen diffusion:

and the slip flow:

to obtain the following equation for the apparent permeability:

Where J is the Knudson diffusion, M is the molar mass, D is the Knudson diffusion coefficient, R is the gas constant, T is the temperature, p is the pressure, Ja is the advective flow due to pressure forces, ρ is the average density, r is the pore radius, µ is the viscosity, L is the pore length, P is the pressure and F is a correction coefficient by slip flow and is given by:

Now let's look at the results obtained by applying the equations presented under operating conditions similar to that of a shale.

First we will analyze the variation of apparent permeability with respect to the Darcy permeability for different models (where kD or k was determined by the equation of Hagen Poiseuille by r2/8). For this, we consider a pore radius of 8 nm (which is a representative of the pore size throats found in the shale), and we'll see how it varies the apparent permeability when we diminish the pressure from of 10000psi to a final pressure of 150psi for a temperature of 200F.

We can see from the graph that the apparent permeability increases significantly below the 1000psi. Where the increase with respect to the Darcy permeability in this point is 11% for the Klinkenberg model, 6% for the Jones & Owens model, 18% for the Sampath & Keighn model, 28% for Florence model, 23% for the Civan model and finally 27% for the Javadpour model. When we decrease pressure until the 150 Psia, permeabilities of up to 3 times greater than the theoretical permeability appear to the Florence model.

In the next graph we will analyze how the ratio varies between apparent permeability and Darcy permeability according to the Knudsen number.

We can see from the graph the present flow mechanisms in a shale field; as starting from a slip flow regime (0.001 <Kn<0,1) the apparent permeability is incremented until you reach 45% greater than the rate of permeability in the theoretical models, when it reaches the transition zone. From the beginning of this regime (0,1<Kn<10) the increases of relative permeability become more abrupt as a result of the growth of the Knudsen layer where the slip flow regime and the start of the Knudsen diffusion reside.

With a Knudsen number equal to 1, apparent permeabilities are reached of 6.7 times greater than the Darcy permeability and 2.2 times greater than that predicted by the Klinkenberg effect. The free molecular flow zone (Kn>10, total development of the Knudsen layer ), corresponds to very low pressures for this pore diameter that are not found in the reservoir development. We can conclude then that the flow within the shale matrix occurs mostly between the regimes of slip flow and transition flow.

Finally we will analyze the variation of the gas flow through a cylinder of 8nm of radius and a length of 60 µm for all the models proposed.

We see that the increase in the flow rate becomes more significant to move us in the direction of decreasing pressure, under1000psi, in the same way that happened with the apparent permeability (something which is expected given that the permeability and flow rate are directly proportional).

This graphic is of interest to analyze what happens along the life of a shale play while it is depleted. As the reservoir pressure decreases we pass from the slip flow zone to the transition flow zone which is accompanied by a strong increase of the flow rate due to the contribution of new mechanisms present. Flow volumes up to 3 times higher than the theoretical are possible to find in a shale field. But not only must we wait until the reservoir is depleted, if not, it will also produce strong increases in areas near the hydraulic fractures and the walls of the borehole where the fluence pressure is lower than the center of the reservoir.

When we plot the ratio between the apparent permeability against the Knudsen number for the majority of the new models we can see that these exhibit a straight line (despite of their different equations) with different slope and the same origin ordinate (ka).

This suggests to us that the apparent permeability in a shale field follows a linear trend variation when is plotted against the Knudsen number in the next form:

Where k is the Darcy equation an m is the slope of the line.

Now if we make laboratory measurements with core samples of a particular shale play for different Knudsen numbers, plot the results, extrapolate through a straight line and obtain the slope of the same; we can obtain the variation rule of apparent permeability with respect to the Knudsen number for that field in a simple way.

1-We have found a theoretical explanation for the greater gas flow rates found in the shale play matrix that could not be explained by traditional fluid mechanics. We saw how this broke the continuous theory and validated the need to incorporate to the discipline of Petroleum Engineering new theoretical basis for a better understanding of the new mechanisms present: slip flow and transient flow.

2- I proposed an empirical model that enables us to obtain a variation rule of the apparent permeability against Knudsen number with measurements easy to obtain in laboratory.

3- In both models of second order derived from the Karniadakis and Beskok equation as in the Javadpour model we noticed that the permeability in nanopores is no longer just an intrinsic property of the rock but that also depends on the type of gas, the pressure and the temperature. As a result the permeability should be incorporated as a variable to analyze the shale fields.

4-We have found that the flow within the shale matrix to a pore radius of 8nm occurs mostly between the regimes of slip flow and transition flow.

5- The poor connectivity of these make that the heterogeneity present in the flow mechanism would be even greater since there is not a uniform depressurization of the field, originating different regimes of flow in areas close to areas of lower pressure coupled with a non-uniform distribution of the poral size which increases the variation in the flow mechanisms present.

6-We can say then that when we talk about shale plays we are talking about reservoirs of high complexity in which there are present and working together a large variety of storage and flow mechanisms, this being a real challenge for the simulators.

7- This paper did not address the effects of gaseous adsorption and the diffusion inside the body of the kerógeno which will be addressed in a future paper.

8-If you need the Excel file with I made this post you have to click here: "Excel Worksheet to calculate the permeability in Shale Gas Fields".

I would like to thank Mrs. Francisca Ellis for having me supported since the beginning of the paper (when I started to study the gas flow in nanopores) and for helping me with the wording of the same.

Unfortunately I have to say that this post is a paper that I made for the Journal of Petroleum Resources Economics and would be published this February 1st 2016. Hours ago (to be publicated) I received a mail from the president of SPRE J.C. Rovillain that the same would not be part of the Journal. I see unethical that He has not informed me at least one week in advance that my paper would not be part of the Journal so I could look for other oil magazines to publicate it.

1- Figure 1-Reed, R. M., Loucks, R.G., Jarvie, D. M., and Ruppel, S. C., 2007, Nanopores in the Mississippian Barnett Shale: Distribution, Morphology, and Possible genesis (abs.): Geological Society of America Abstracts with Programs, v. 39, no.6, p. 358.

2- Wang, F. P., and Reed, R. M.: ―Pore Networks and Fluid Flow in Gas Shales,‖ SPE124253, paper presented at the Annual Technical Conference and Exhibition, SPE, New Orleans, LA, October 4-7, 2009.

3- Jones, F.O. and Owens, W.W.: “A Laboratory Study of Low Permeability Gas Sands,” paper SPE 7551 presented at the 1979 SPE Symposium on Low-Permeability Gas Reservoirs, May 2022, 1979, Denver, Colorado.

4-Sampath, K. and Keighin, C.W.: “Factors Affecting Gas Slippage in Tight Sandstones,” paper SPE 9872 presented at the 1981 SPE/DOE Low Permeability Symposium, May 27-29, 1981, Denver, Colorado.

5-Beskok, A. and Karniadakis, G.E. 1999.“A model for flows in channels, pipes, and ducts at micro and nano scales.”Microscale Thermophy. Eng. 3(1), 43–77 .doi:10.1080/108939599199864

6-Florence, F. A., Rushing, J. A., Newsham, K. E. and Blasingame, T. A. 2007. Improved Permeability Prediction Relations for Low-Permeability Sands. Paper SPE 107954 presented at SPE Rocky Mountain Oil & Gas Technology Symposium held in Denver, Colorado, 16-18 Apr. doi: 10.2118/107954-MS

7-Civan, F. 2010. “Effective Correlation of Apparent Gas Permeability in Tight Porous Media.” Transport in Porous Media, 82(2), 375-384. doi:10.1007/s11242-009-9432-z

8-Javadpour, F. 2009.“Nanopores and Apparent Permeability of Gas Flow in Mudrocks (Shales and Siltstone).”Journal of Canadian Petroleum Technology, 48(8), 16-21. doi: 10.2118/09-0816-DA

FLOW MECHANICS IN SHALE GAS by Emanuel Omar Martin is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

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