Pore-by-pore modeling, analysis, and prediction of two-phase flow in mixed-wet rocks

A pore-network model is an upscaled representation of the pore space and fluid displacement, which is used to simulate two-phase flow through porous media. We use the results of pore-scale imaging experiments to calibrate and validate our simulations, and specifically to find the pore-scale distribution of wettability. We employ energy balance to estimate an average, thermodynamic, contact angle in the model, which is used as the initial estimate of contact angle. We then adjust the contact angle of each pore to match the observed fluid configurations in the experiment as a nonlinear inverse problem. The proposed algorithm is implemented on two sets of steady state micro-computed-tomography experiments for water-wet and mixed-wet Bentheimer sandstone. As a result of the optimization, the pore-by-pore error between the model and experiment is decreased to less than that observed between repeat experiments on the same rock sample. After calibration and matching, the model predictions for capillary pressure and relative permeability are in good agreement with the experiments. The proposed algorithm leads to a distribution of contact angle around the thermodynamic contact angle. We show that the contact angle is spatially correlated over around 4 pore lengths, while larger pores tend to be more oil-wet. Using randomly assigned distributions of contact angle in the model results in poor predictions of relative permeability and capillary pressure, particularly for the mixed-wet case.

Figure 1

The mean absolute discrepancy, $M_{AD}$, in filling state for the water-wet case. The black line represents the state in which pores and throats are occupied in a completely random way in the model, Eq. (6). The red line represents the discrepancy in filling state between the model and experiment, Eq. (4). The ratio of the areas shown, $A_M$ and $A_R$, defined by Eqs. (5) and (7), respectively, is used to define the error measure $E$, Eq. (8).

Figure 2

(a) A schematic representation of how contact angle is adjusted to obtain correct the filling order. (b) Schematic of the proposed algorithm to remove the mismatch between model and experiment by optimizing the contact angle.

Figure 3

Calibration algorithm, which optimizes contact angle on a pore-by-pore basis to minimize the error in filling order. Initially the filling state of the elements in the model is taken from the mapped distribution from the experiment at zero fractional flow. Then we run the model to the next fractional flow ($f_w^{j+1}$) with a water saturation $S_w^{j+1}$. At this fractional flow, by comparing the model to the mapped experiment we calculate $M_{AD}^{j+1}$, Eq. (4). Then we compare $M_{AD}^{j+1}$ with our error criterion: if $M_{AD}^{j+1}$ is greater, we need to update the contact angles, and perform another iteration with the updated contact angles, until $M_{AD}^{j+1}$ becomes less than our required error. We then proceed to the next fractional flow until the final fractional flow of 1. TD is the thermodynamic contact angle.

Figure 4

(a) The image of dry scan of Bentheimer sandstone used in the water-wet experiment. (b) The structure of the extracted pore network.

Figure 5

$M_{AD}$, Eq. (4), as a function of occupancy comparing the discrepancy between experiment and the model for constant contact angle, average thermodynamic contact angle, TD, and the distribution of contact angle from the optimization algorithm.

Figure 6

Pore and throat sizes that are filled with water at the end of waterflooding for the optimized contact angles (water-wet case).

Figure 7

Optimized contact angle for (a) pores and (b) throats for the water-wet sample. The thermodynamic contact angle of $47.8^{\circ }$ was used as the initial guess. For both pores and throats, approximately half of the optimized contact angles are unchanged.

Figure 8

Capillary pressure based on our optimization algorithm with the three initial conditions. The error bars represent the uncertainties in the experimental measurements.

Figure 9

Relative permeability results based on optimized contact angle distribution. For comparison, other experimental data on water-wet Bentheimer sandstone rocks in the literature [47, 48, 49, 50] are shown.

Figure 10

(a) A two-dimensional cross section of the three-dimensional dry scan of Bentheimer sandstone used in the mixed-wet experiment. (b) The structure of the pore network that was extracted.

Figure 11

$M_{AD}$, Eq. (4), as a function of occupancy comparing the discrepancy between experiment and the model for constant contact angle, average thermodynamic contact angle, TD, and the distribution of contact angle from the optimization algorithm.

Figure 12

Pore and throat sizes that are filled with water at the end of waterflooding for the optimized contact angle distribution (mixed-wet case).

Figure 13

Optimized contact angle for (a) pores and (b) throats for the mixed-wet sample. The thermodynamic contact angle of ${98}^{\circ }$ was used as the initial guess.

Figure 14

Capillary pressure obtained from the optimized contact angle distribution for mixed-wet Bentheimer sandstone using the thermodynamic contact angle as the initial guess. The error bars indicate the uncertainties in the experimental measurements.

Figure 15

Relative permeability based on the optimized contact angle distribution. For comparison experimental data on the corresponding mixed-wet Bentheimer sandstone are presented.

Figure 16

(a) Capillary pressure and (b) relative permeability for water-wet sample obtained using optimised distribution of contact angle, and ten randomly redistributed realizations of optimized contact angle.

Figure 17

(a) Capillary pressure and (b) relative permeability for mixed-wet sample obtained using optimized distribution of contact angle, and ten randomly redistributed realizations of optimised contact angle.

Figure 18

Spatial correlation, $\xi \left(d\right)$ of (a) water-wet and (b) mixed-wet samples. Also, results obtained by AlRatrout et al. [56], for (a) WW carbonate, (b) OW and MW carbonates are presented. Dashed vertical lines are the characteristic lengths representing minimum, mean, and maximum pore diameter size for each sample.