Fig 1.
A simple network used to describe the flow of water (white) from inlet to outlet
Fig 2.
A simple network showing water injection from the left (white), displacing oil (red), which leaves the system via the right-hand pore
Fig 3.
Schematic illustrating the multiple filling algorithm in a case where water (white) invades oil filled pores.
(a) represents the pores occupancy before the invasions and (b) shows the fluids distributions after the invasions. Note how the Pore element 1 is completely filled and all the other menisci advance according to Eq (21).
Fig 4.
An example of the networks used by Lenormand et al [18].
The non-wetting fluid is injected at the left of the picture [30]. Fig 4. is a reproduction of Fig 7 in Lenormand et al [30].
Fig 5.
Comparison between (a) simulations and (b) experiments of Mercury (Black) displacing air (white) (viscosity ratio M = 0.0111). NCa is the Capillary Number and measures the ratio of viscous to capillary forces (see Eq (22)). Fig 5 is composed of images from Fig 10c in Lenormand et al [18]. (Cambridge University Press, reproduced with permission).
Fig 6.
Comparison between (a) simulations and (b) experiments of Mercury (black) displacing hexane (white) (viscosity ratio M = 0.2). NCa is the Capillary Number (see Eq (22)). Fig 6 is composed of images from Fig 10B in Lenormad et al [18] (Cambridge University Press, reproduced with permission).
Fig 7.
Comparison between (a) simulations and (b) experiments of Glucose solution (white) displacing oil (black) (viscosity ratio M = 0.0098). NCa is the Capillary Number (see Eq (22)). Fig 7 is composed of images from Fig10d in Lenormand et al [18] (Cambridge University Press, reproduced with permission).
Fig 8.
Comparison between (a) simulations and (b) experiments of Glucose solution (white) displacing oil (black) (viscosity ratio M = 0.001). NCa is the Capillary Number (see Eq (22)). Fig 8 is composed of images from Fig 10E in Lenormand et al [18] (Cambridge University Press, reproduced with permission).
Fig 9.
Comparison between (a) simulations and (b) experiments of Air (white) displacing heavy oil (viscosity ratio M = 55555). NCa is the Capillary Number (see Eq (22)). Fig 9 is composed of images from Fig 10A in Lenormand et al [18] (Cambridge University Press, reproduced with permission).
Fig 10.
Comparison between (a) simulations and (b) experiments of Mercury (black) displacing oil (white) (viscosity ratio M = 645). NCa is the Capillary Number (see Eq (22)) Fig 10 is composed of images from Fig 11 in Lenormand et al [18] (Cambridge University Press, reproduced with permission).
Fig 11.
Comparison between (a) simulations and (b) experiments of Mercury (black) displacing oil (white) (viscosity ratio M = 64.5). NCa is the Capillary Number (see Eq (22)). Fig 11 is composed of images from Fig 11 in Lenormand et al [18] (Cambridge University Press, reproduced with permission).
Fig 12.
Comparison between (a) simulations and (b) experiments of Mercury (black) displacing oil (white) (viscosity ratio M = 3.6). NCa is the Capillary Number (see Eq (22)) Fig 12 is composed of images from Fig 11 in Lenormand et al [18] (Cambridge University Press, reproduced with permission).
Fig 13.
Different stages of displacement of a heavy oil by air after approximately 0.01, 0.03, 0.08 and 0.15 pore volumes throughput: (a) simulation and (b) experiment (viscosity ratio M = 55555). Capillary Number NCa = 5E-07. Fig 13 is composed of images from Fig 12 in Lenormand et al [18] (Cambridge University Press, reproduced with permission).
Fig 14.
X-Ray images of saturation maps of E7000 experiment for different injected pore volumes [2].
The water (white) is injected from the bottom to displace the oil (black). Note that a pore volume (PV) is a unit used to report the cumulative volume of water injected since the start of a displacement: e.g. 1PV means that the cumulative volume of water injected so far is equal to the total volume of the pore space.
Fig 15.
X-Ray images of saturation maps of E2000 experiment for different injected pore volumes [2].
The water (white) is injected from the bottom to displace the oil (black).
Fig 16.
Comparison between (a) Oil Recovery, (b) Water cut and (c) Differential Pressure in E2000 and E7000 experiments as functions of Pore Volumes injected.
Fig 17.
comparison at water breakthrough between 2D and 3D simulations for M = 7000 and matched capillary numbers: (a) Nca = 3.86E-11, (b) Nca = 3.86E-08, (c) Nca = 3.86e-07 and (d) Nca = 1.54E-06.
Fig 18.
Water saturation at breakthrough versus capillary number.
In this figure two sets of simulations are presented: a 2D case (blue) and a 3D case using 3 seeds (red, green and purple). The viscosity ratio was equal to M = 7000 for both sets of simulations.
Table 1.
Table of parameters used in the simulations.
Fig 19.
Different stages of viscous fingering initiation and growth in E7000 simulation for different pore volumes injected.
Fig 20.
The simulated differential pressure in (a) E7000 and (b) E2000 experiments as functions of Pore Volumes injected.
Fig 21.
Comparison between the recovery in E7000 and E2000 simulations as functions of Pore Volumes injected.
Fig 22.
Saturation maps from E7000 simulation for different injected pore volumes.
White pores correspond to injected water and red pores contain oil.
Fig 23.
Saturation maps from E2000 simulation for different injected pore volumes.
White pores correspond to injected water and red pores contain oil.
Fig 24.
Location of the water invasions that occurred after injected phase breakthrough.
These are depicted in red at different stages of the simulation of experiment E7000.
Fig 25.
Pressure of the resident fluid in each pore (a) and the saturation path (b) in E2000 simulation at breakthrough of the injected phase. Note that some pores contain water and some pores contain oil–the pressures shown here correspond to the pressure within the resident fluid.
Fig 26.
The saturation path at breakthrough (BT) of the injected fluid (left-hand column), the saturation path after 5 Pore Volumes (5PV) of injection (middle column), and the pores displaced between these two times (right-hand column), for different network lengths: 30cm (a), 15 cm (b),7.5 cm (c) 3.25 cm (d). In this set of simulations: Q = 0.005 cc/hr, Nca = 6.9 E-09, theta = 100° and μoil = 7000 cP.
Fig 27.
The differential pressure from simulations for different network lengths and mean Pc in the network (calculated using Young-Laplace’s law and the average pore radius in the network).
In this set of simulations: Q = 0.005 cc/hr, Nca = 6.9 E-09, theta = 100° and μoil = 7000 cP.
Fig 28.
Comparison between the differential pressures in two simulations with different contact angles: θ = 100° (blue) and θ = 170° (red).
The mean Pc in the network (calculated using Young-Laplace’s law and the mean pore radius in the network). In this set of simulations: Q = 0.005 cc/hr, Nca = 6.9 E-09 and L = 30 cm.
Fig 29.
The saturation path at breakthrough (BT), the saturation path after 2PV injection, and the water invasions that occurred between these two times for simulations with different contact angles: θ = 170° (a) and θ = 100° (b). In this set of simulations: Q = 0.005cc/hr, Nca = 6.9 E-09, L = 30 cm and μoil = 7000 cP.
Fig 30.
Comparison between the differential pressures in two simulations with different injection rates: Q = 0.005 cc/hr (a) and Q = 0.05 cc/hr.
The mean Pc was calculated using the average pore radius in the network. In these simulations θ = 170° and L = 30 cm.
Fig 31.
The saturation path at breakthrough (BT), the saturation path after 1PV injection, and the water invasions that occurred between these two times for simulations with different injection rates: Q = 0.005 cc/hr (a) and Q = 0.05 cc/hr.
In these simulations θ = 170°, L = 30 cm and μoil = 7000 cP.