Extensions of the Black-Oil Table

Reasons for Extending the Black-Oil Table

The main reason for extending the black-oil tables above the initial saturation pressure was sumarized by Singh et al. and is repeated below

1. Saturated conditions may be encountered at pressures higher than the highest saturation pressure in a table – e.g. during pressure buildup where two-phase gas-oil flow exists prior to shut-in; gas percolation; or gas injection.
2. Saturated conditions may exist initially at pressures higher than the highest saturation pressure in a table – e.g. in a reservoir with compositional grading; or a saturated gas-oil system where only a down-structure low-bubble-point sample is available and was used to generate the black-oil PVT table.
3. Some simulators require that any gas-oil contact be “saturated”, even if the actual GOC is undersaturated with a critical mixture at the contact.
4. Some simulators require saturated PVT properties to extrapolate undersaturated PVT properties – e.g. gas FVF in an undersaturated gas condensate reservoir. Although this method can lead to non-physical extrapolations, it is still used and one needs to provide extrapolated saturated properties for the model to run at all.
5. Some simulators extrapolate saturated PVT properties using sub-optimal methods. These extrapolated values may never be used in a simulation, but they are still made during initialization and internal testing for “negative compressibilities”. With warnings about negative compressibilities during initialization, many engineers become unsure whether their simulation results have been affected by this “problem” – unknowing that the simulator never experienced negative compressibilities during the run.

Traditional Extension

The traditional black-oil extrapolation by Singh et al. was introduced in 2007 in the SEP paper "Consistent Black-Oil PVT Table Modification" ¹ together with a consistent method for controlling that the extrapolation was physically consistent. The procedure from the paper by Singh et al is given below.

whitson Generic Extension

Note

The main article on the methodology of the whitsonᵖᵛᵗ approach to black-oil extrapolation can be found in the user manual under methodologies: Black-Oil Table Generation

The approach of the whitsonᵖᵛᵗ software utilizes an equation of state (EOS) model, more specifically a cubic EOS, to find a critical mixture and depending on whether or not a critical mixture is found, the extension of the black-oil table is extrapolated to the critical point or some upper bound pressure.

There are three possible cases for the extrapolation scheme defined by the whitsonᵖᵛᵗ software. In the first case a critical pressure is found and is found to be on the correct side of the swell test curve (see whitsonᵖᵛᵗ manual for more details). The second case finds a critical mixture, but finds that the critical point is on the wrong side of the swell test curve. The third and final case is the case where no critical point is found because the swell test blows-up (i.e. the curve diverges to infinity and comes back as the opposite phase).

The procedure to find a critical mixture is as follows:

1. Flash the initial in-situ reservoir composition (\(z_{\mathrm{bo}i}\)) at its original saturation pressure. The obtained incipient phase compositions are called (\(x_{\mathrm{bo}i}\)) and (\(y_{\mathrm{bo}i}\)).
2. Re-combine the incipient phases using a ratio \(F_V\): $$\begin{equation} z_{\mathrm{c}i} = F_V \cdot y_{\mathrm{bo}i} + (1-F_V) \cdot x_{\mathrm{bo}i}. \end{equation}$$

3. Calculate the saturation pressure of the resulting composition \(z_{\mathrm{c}i}\) and its K-values \(K_i\).

4. Calculate a RMS (\(\delta_\mathrm{RMS}\)) quantifying the deviation from 1 of the K-values: $$\begin{equation} \delta_\mathrm{RMS} = \sum_{i \in \mathcal{C}} (1 - K_i)^2, \end{equation}$$ where \(\mathcal{C}\) is the set of components (e.g. \(\mathcal{C} = \left[ \mathrm{N_2}, \mathrm{CO_2}, \mathrm{C_1}, ..., \mathrm{C_{36+}}\right]\))

Minimize \(\delta_\mathrm{RMS}\) by changing \(F_V\).

Two cases can occur:

• The final mixture \(z_{\mathrm{c}i}\) verifies the critical criterion (for all \(i \in \mathcal{C}, K_i = 1\)). In that case, \(z_{\mathrm{c}i}\) is a critical mixture.

• The final mixture \(z_{\mathrm{c}i}\) does not verify the critical criterion. This usually indicates that the saturation calculation blows up at some point, see this for more details.

Consistency Check for Black-Oil Modeling

The set of three consistency checks defined by Sigh et al. are summarized below:

1. Basic Physical Consistency

   • \(\rho_g<\rho_o\)
   • \(R_s<1/r_s\)
   • \(B_o<B_{gd}/r_s\)
   • \(\mu_g<\mu_o\)

2. Critical-Point Consistency (Gas PVT = Oil PVT)

   • \(\rho_g=\rho_o\)
   • \(R_s=1/r_s\)
   • \(B_o=B_{gd}/r_s\)
   • \(\mu_g=\mu_o\)

3. Compressibilities at Saturated Conditions

   • \(\frac{(B_{gd}-r_s\cdot B_o)}{(1-r_s\cdot R_s)}\cdot\frac{R_s}{dp}>\frac{dB_o}{dp}\)
   • \(\frac{(B_{0}-R_s\cdot B_{gd})}{(1-r_s\cdot R_s)}\cdot\frac{r_s}{dp}>\frac{dB_{gd}}{dp}\)

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1. K. Singh, Oi. Fevang, and C. H. Whitson. Consistent black-oil pvt table modification. In SPE Annual Technical Conference and Exhibition, paper SPE–109596–MS. Society of Petroleum Engineers, 2007. doi:https://doi.org/10.2118/109596-MS. ↩