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OH – ET – VA - LL: Analysis of Dynamic Data in Shale Gas Reservoirs – Part 1 – Version 2 (December 2010)
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Improved stress dependence models
As described in section 11, we account for stress-dependence through our “unconsolidation”
model, i.e. we can input tables describing the dependence of permeability and porosity to
pressure. It is worth noticing that this dependency is numerically handled using a fully implicit
scheme, in other words the model is unconditionally stable and does not suffer from time-step
restrictions when using this option.
The main restriction in the current version is that this pressure dependence only impacts the
matrix, while the main influence may be due to the stress-dependence of the secondary
(unpropped) fracture systems. As a consequence, an extension of the use of unconsolidation to
fractures is required.
A threshold model for the unpropped fracture system may also be developed, where the
network becomes conductive only when the local pressure gradient becomes high enough. In
this case, we would use a new permeability (conductivity) function ruled by the pressure
gradient.
Ultimately, we may have to couple our reservoir model with geomechanics, in order to
simulate advanced phenomena related to stress changes, such as solid stress-strain relations.
Complex processes
The quantity of gas locally adsorbed (or desorbed) during a time step is classically simulated
using an adsorption isotherm, which is a function of pressure only. So far, we have limited
ourselves to the Langmuir isotherm. Although this is by far the most widely used model in our
industry, many other isotherms are available and could be implemented. Thermal effects could
also be investigated, which would require the extension of the current description of adsorbed
gas quantities to a more general function of pressure and temperature.
The transport of gas molecules from the microporous organic matter to the production well is
an extremely complex process, involving many scales of porosity. At each scale, a different
physical process occurs: molecular diffusion in the organic matter, desorption from micropore
walls, Knudsen diffusion in the micropores, Darcy flow in macropores, Forchheimer flow in the
hydraulic fractures…
Our model currently uses Fick’s law to simulate gas diffusion inside the matrix, and Darcy’s law
to simulate free gas flow in the macropores. In micropores, however, the mean path of gas
molecules is not negligible compared to the average throat size. Free gas hence flows under
slip regime (gas slippage), usually simulated as an increase of the effective permeability at low
pressure. This can be simulated using the Klinkenberg model (which reduces to a pressure-
dependent permeability model) or any generalization to model Knudsen diffusion. It is
interesting to notice that this approach nicely lumps advection and Knudsen diffusion in a
single equation with dependence on pressure gradient only (i.e. similar to Darcy’s law in
essence), while Fick’s law induces dependence on the concentration gradient. Note also that
this additional permeability dependence to pressure is an effective way to account for
molecular effects, and is not related to the stress-dependence previously described. Both
effects can, however, be simultaneously simulated.