The global updating results in a suite of representative models calibrated with respect to global and regional production and pressure data. Next, we use the streamline-based generalized travel time inversion to match the production response for each well. The details of the generalized travel time inversion can be found elsewhere (Cheng et al.
2005; He et al. 2002). Fig. 2.26 illustrates a flowchart of local update.
Fig. 2.26 Illustration of local model calibration
Step 8: Finite Difference Simulation and Streamline Tracing
The production response and fluid flow in the reservoir might be obtained using either a streamline simulator or a finite-difference simulator. Streamline models are not only computational efficient in run-time , but also offer some unique advantages for dynamic production data integration to field-scale geologic models. In streamline simulations, 3-D fluid flow calculations are approximated by a sum of 1-3-D calculations along streamlines. The streamline approach is extremely effective for modeling convection-dominated flows in the reservoir especially when heterogeneity is the predominant factor controlling oil recovery, for example in water-flooding (Datta-Gupta and King 2007).
This is often the case when we have global model update that calibrates large scale reservoir fluids and movements such that the major model uncertainty lies on permeability heterogeneity.
Another important advantage of streamline formulations is that the sensitivity of travel time misfit with respect to logarithm of grid permeability exhibits quasi-linear relationship with the tight-of-flight of streamlines passing the grid block of interest (Cheng et al. 2005), which greatly facilitates inversion process. In this step, we’ll utilize finite difference simulation for its versatility and robustness in modeling complex flow physics, such as three-phase, unfavorably displacing etc, and geologic features, such as faults, non-neighbor connections etc, while streamlines are constructed based on numerical velocity fields generated by the finite difference simulator, sensitivity of travel time misfit to logarithm permeability can be formulated from streamlines (Cheng et al. 2006; He et al. 2002).
Step 9: Generalized Travel Time Inversion Using Streamline Analytical Sensitivity The production data integration via local updating involves the minimization of a penalized misfit function as given below
R
In the above equation, the first term includes data misfit, which is quantified by taking individual well response (either water-cut or GOR) and systematically shift observed to simulated response to get max correlation between the two, this correlation defined by
We call the optimal time-shift when r maximized in Eq. 2.13 generalized travel time (GTT) misfit (He et al. 2002), as denoted by ∆t~ in Eq. 2.12. Fig. 2.27 shows the optimal time-shift according to the correlation function.
Fig. 2.27 Generalized travel time misfit and correlation function (Cheng et al., 2004)
We are try to minimize this misfit using streamline analytical sensitivities (Cheng et al. 2006), S denotes the sensitivity of the GTT at each well with respect to grid log permeability while δR corresponds to the change to make to minimized GTT misfit.
The second term is the ‘norm’ penalty and the third term is the roughness penalty, and L is a second-spatial-difference operator. The first term ensures that the difference
between the observed and model simulated production responses is minimized. The second term, ‘norm’ penalty minimizes the changes to the globally updated model. This ensures that the global misfit is not adversely impacted by the local updates, and is an important characteristic of our approach.
The third term is a roughness penalty that ensures that the changes to the model are smooth and consistent with the large-scale and low resolution of the production data.
The weights β1 and β2 determine the relative strengths of the prior model and the roughness term. The selection of these weights can be somewhat subjective although there are guidelines in the literature (Parker 1994). In general, the inversion results will be sensitive to the choice of these weights. The minimization of Eq. 2.12 leads to an augmented least-squares system of equations is given as follows.
An iterative least squares solution approach via the LSQR algorithm (Paige and Saunders 1982) is used to solve Eq. 2.14.
For sensitivity calculation, consider two-phase incompressible flow of oil-water in a nondeformable, permeable medium, the transport equation can be written in the streamline time-of-flight (TOF) coordinates as follows (Cheng et al. 2005; Datta-Gupta and King 2007)
where τ represents time of flight which is the travel time along a streamline,ψ, and s(x) is the “slowness” defined as the reciprocal of the total interstitial velocity
∫
With the assumption that streamline paths do not shift significantly because of small changes in reservoir properties we can relate the change in travel time to the change in reservoir properties and thus slowness by
∫
In this research effective permeability are primary parameters to calibrate, therefore
∫
Which can be further related to arrival time of a particular concentration (e.g., water front) by
And this generalized travel time (GTT) sensitivity can be calculated by averaged sensitivities for all time steps for the well of interest. It is:
m t dm
t
d j j
∂
−∂
∆ =
= ~
S ...(2.21)
where m is the reservoir parameter, j is the well number, and S is the sensitivity, and this sensitivity will be used for dynamic data integration.
Step 10: Model Reexamination
Before finishing inversion, models from local update need to be checked for consistency with geology, as well as global model calibration, since local model calibration starts with multiple dissimilar models, some of which might be unrealistic in terms of matching dynamic production data. This is especially true because β1 in Eq. 2.12 serves to limit the globally-updated models from changing dramatically and may result in models that fail to provide good data matches as quantified by the local objective function terms. However, from the global updates we are provided a large population of models with diversity and, subsequently, those (typically few models according to our experience) that are potentially viable for global model calibration but unviable for local model calibration are discarded during the last workflow step, i.e. model re-examination while the majority of models give reasonable improvements in the local objectives with limited permeability updates. Therefore, looping back to global calibration is typically unnecessary because the local calibration induces small, grid-cell-scale changes to the permeability that do not deteriorate the global update. Additionally, from the initial ensemble there are typically several alternative models that are consistent with the observation data after both global and local updates.