Combined mechanistic and machine learning method for construction of oil reservoir permeability map consistent with well test measurements

Abstract

We propose a new method for construction of the absolute permeability map consistent with the interpreted results of well logging and well test measurements in oil reservoirs. Nadaraya-Watson kernel regression is used to approximate two-dimensional spatial distribution of the rock permeability. Parameters of the kernel regression are tuned by solving the optimization problem in which, for each well placed in an oil reservoir, we minimize the difference between the actual and predicted values of (i) absolute permeability at the well location (from well logging); (ii) absolute integral permeability of the domain around the well and (iii) skin factor (from well tests). Inverse problem is solved via multiple solutions to forward problems, in which we estimate the integral permeability of reservoir surrounding a well and the skin factor by the surrogate model. The last one is developed using an artificial neural network trained on the physics-based synthetic dataset generated using the procedure comprising the numerical simulation of bottomhole pressure decline curve in reservoir simulator followed by its interpretation using a semi-analytical reservoir model. The developed method for reservoir permeability map construction is applied to the available reservoir model (Egg Model) with highly heterogeneous permeability distribution due to the presence of highly-permeable channels. We showed that the constructed permeability map is hydrodynamically similar to the original one. Numerical simulations of production in the reservoir with constructed and original permeability maps are quantitatively similar in terms of the pore pressure and fluid saturations distribution at the end of the simulation period. Moreover, we obtained an good match between the obtained results of numerical simulations in terms of the flow rates and total volumes of produced oil, water and injected water.

Figures (12)

• Figure 1: A synthetic reservoir model; $N$ vertical wells (blue cylinders) cross the formation by perforating it along the entire thickness (red colored zones).
• Figure 2: Kernel functions $\mathcal{K}^{\mathrm{far}}(r)$ and $\mathcal{K}^{\mathrm{near}}(r)$ (see Eq. \ref{['eq:kernel_regression_2']}) at $r_d = 150 ~ \mathrm{m}$, $r_g = 30 ~ \mathrm{m}$, $\alpha = 1, ~ \beta = 1, ~ \gamma = 0.5, ~ \delta = 0.5$.
• Figure 3: An example of the synthetic reservoir model generated according to the algorithm described in the main text. Here, the minimal distance between any two wells $\Delta d$ equals 500 m. Black crosses denote the positions of the vertical wells, while the squares with dotted blue boundary mark the domains $\mathcal{B}^{\mathcal{D}}$ used for the creation of the samples for the synthetic dataset.
• Figure 4: Plot (a) shows an example of the synthetic permeability distribution $k(x, y)$ in the square domain $\mathcal{B}^{\mathcal{D}}$; contributions from well $k^{\mathrm{well}}$ located in the center of the domain and neighbor wells $k^{\mathrm{neigh}}$ are shown in plots (b) and (c); distribution $k^{\mathrm{neigh}}$ is fitted by the quadratic polynomial \ref{['eq:quadratic_polynomial']}, and plot (d) shows the result of approximation.
• Figure 5: Plot (a) shows comparison of bottomhole pressure dynamics obtained in in-house semi-analytical reservoir model (dashed black curve), numerical model in MUFITS simulator (solid gray curve) and semi-analytical model implemented into Kappa Saphir (dashed red curve); in plot (b) we show the results of solution to minimization problem \ref{['eq:minimization_2']} (welltest interpretation using semi-analytical model) in a reservoir with the non-uniform absolute permeability distribution shown in Fig. \ref{['fig:database_sample']}a.