Page 8 - Transmissibility Corrections and Grid Control for Shale Gas Numerical Simulation

VA - DF: Transmissibility Corrections and Grid Control for Shale Gas Numerical Simulation p 7/23
Figure 7 – Comparison of pressure and pressure derivative curves between analytical
(markers) and numerical (continuous lines) results for a non-linear gas PVT. k=1E-4 mD.
As a consequence of these observations, we decided to develop a new transmissibility
derivation algorithm, which should account for non-linear pressure effects and ultimately
reduce the discrepancy between analytical and numerical results under similar assumptions.
Furthermore, an automatic adjustment of the grid (through numerical upscaling) to the
formation permeability and desired time-scale resolution was implemented, following the
observations made in [3]. Its purpose is to solve the early-time deviation due to the resolution
of the model, and better constrain the handling of gas compressibility in the vicinity of the
fractures, when non-linear PVT is considered.
2. Transmissibility calculations
The problem of correctly modeling fluid flow in the vicinity of the well is a difficult task. This is
due to the fact that the transmissibilities are computed based on the gridding geometry,
assuming a predefined simple pressure behavior around each cell face. These derivations are
usually made assuming linear, geometric or logarithmic pressure evolution away from the well
and account for pressure variation on cell faces throughout simple geometrical correction.
However, they do not account for pressure variation along the faces themselves. This usually
leads to satisfactory results for standard well geometry. For complex 3D geometries, however,
neglecting the complex variation of the pressure profile along the faces may lead to incorrect
flow representation. In order to correctly model the flow in this region, local pressure
variations have to be taken into account. This effect becomes predominant for low permeability
problems, where pressure variations are locally very important.
Several authors have investigated this problem by means of simple analytical solutions. This is
possible as long as the pressure field is 2D and is limited by simple well geometry. For complex
3D problems, if the fluid is assumed uncompressible, a potential solution has to be found
numerically. Lee [4, 5] introduced a boundary integral representation for the pressure field
around the wellbore and later used the slender body theory in order to correct the production
index. A similar method was developed by Ding [6, 7] and extended to the near wellbore
region to correct well index and the grid transmissibilities in the near wellbore region. Their
integral representation was based on simple kernels. In this paper, we present a methodology
using an integral representation of the potential field based on elementary Green’s function
surfaces that are given analytically. We derive the corresponding analytical kernels for each
surface type. The producing fracture surfaces are then discretized and elementary surfaces
0.1 1 10 100 1000
Time [hr]
1E+5
1E+6
Integral of normalized pressure
Integral of normalized pressure Derivative