We investigate the relationship between apparent electrical resistivity and water saturation during unstable multiphase flow. We conducted experiments in a thin, twodimensional tank packed with glass beads, where Nigrosine dyed water was injected uniformly along one edge to displace mineral oil. The resulting patterns of fluid saturation in the tank were captured on video using the light transmission method, while the apparent resistivity of the tank was continuously measured. Different experiments were performed by varying the water application rate and orientation of the tank to control the generalized Bond number, which describes the balance between viscous, capillary, and gravity forces that affect flow instability. We observed the resistivity index to gradually decrease as water saturation increases in the tank, but sharp drops occurred as individual fingers bridged the tank. The magnitude of this effect decreased as the displacement became increasingly unstable until a smooth transition occurred for highly unstable flows. By analyzing the dynamic data using Archie’s law, we found that the apparent saturation exponent increases linearly between approximately 1 and 2 as a function of generalized Bond number, after which it remained constant for unstable flows with a generalized Bond number less than −0.106.
Multiphase fluid flow in porous media is an important problem for applications including petroleum production [
Electrical resistivity measurements are commonly used to investigate fluid saturations in multiphase flow systems [
The saturation exponent is usually determined experimentally from measurements of
It is well known that variations in the magnitude and connectivity of permeability could lead to flow channeling in reservoirs and consequently to a reduction in oil production [
The Bond number (
In these expressions,
The value of the generalized Bond number plays a critical role for determining the occurrence of viscous instabilities. For
Comparison of (a) stable flow conditions (
The main goal of this work is to determine the relationship between the saturation exponent in Archie’s law and the degree of flow instability in a porous medium as quantified by the Bond and Capillary numbers. To achieve this objective, resistivity index curves were measured during the displacement of mineral oil by water in a 2D tank packed with glass beads. Fourwire resistance measurements were collected throughout the experiment while the light transmission method was used to simultaneously monitor changes in saturation. The effect of gravity on flow instability is controlled by changing the orientation of the tank to achieve different Bond numbers. Experiments at different flow rates were conducted to control the relative importance of viscous forces by varying the capillary number.
The fluids used in these experiments are water and mineral oil (EMD Chemicals, NJ, USA). The properties of each fluid are given in Table
Properties of the fluids used in the experiments.
Wetting phase, water (with 0.05 g/L nigrosine)  

Density, 
1000 kg/m^{3} 
Dynamic viscosity, 

 
Nonwetting phase, mineral oil  
 
Density, 
880 kg/m^{3} 
Dynamic viscosity, 
0.068 N.s/m^{2} 
Interfacial tension [ 
0.049 N/m 
 
Viscosity Ratio, 
0.015 
The experiments were conducted in the specially designed 2D acrylic tank shown in Figure
Physical properties of the flow system.
Length  40 cm 
width  45 cm 
Thickness  1.25 cm 
Porosity, 
0.30 
Formation factor, 
3.04 
Permeability, 
57 Darcy 
Grain Size, 
2 mm 
Side view sketch of the experimental setup for the resistivity cell with light transmission imaging and resistivity measurement systems.
The tank could tilt to arbitrary angles so as to vary the effect of gravity on flow and control the Bond number. The component of gravity acting on the flow system is determined by
Summary of the tank inclination angle (
Experiment index  1  2  3  4  5  6  7 


189  251  67  119  157  27  67 

90  90  90  90  90  90  90 

98  93  90.2  93.7  81.4  126  94.7 
























 
Experiment index  8  9  10  11  12  13  14 
 

119  27  189  251  358  67  99 

90  90  90  90  90  90  90 

82  78.7  79.5  76.7  78.4  80.1  83.1 
























 
Experiment index  15  16  17  18  19  20  21 
 

89  52  146  119  67  27  67 

90  90  90  90  90  90  90 

74.8  77.6  74.2  79  77.6  78.3  87 
























 
Experiment index  22  23  24  25  26  27  28 
 

189  251  67  67  67  67  67 

90  90  30  30  30  0  45 

71.8  72.8  78  91.5  90  81.1  80.5 
























 
Experiment index  29  30  31  32  33  34  
 

67  67  67  67  358  358  

15  60  60  60  90  90  

82.8  84  79.2  84.7  89.1  88.5  























The bulk DC resistance of the tank was determined using the fourelectrode method [
The system developed for the light transmission measurements [
The background reference image obtained before water is injected is subtracted from each subsequent image to overcome problems related to variations in light intensity due to the specific arrangement of light bulbs in the array. The intensity (
This equation is obtained from calibration experiments using a small chamber with the same material, thickness, and packing of glass beads to obtain a porosity of about 0.30, consistent with the flow cell.
With the experimental setup described above, 2 main series of 34 experiments are conducted: one series is at constant Bond number of 0.0165 and the other is at constant capillary number of 0.0494. Table
The range of the Bond numbers that can be achieved by rotating the tank, that is, 0 to 0.0165, is smaller than the range of capillary numbers that can be achieved by changing the flow rate, that is, 0 to 0.264. Therefore, we can obtain the largest range of generalized Bond numbers by changing flow rate. The maximum generalized Bond number used in the experiments is −0.0035 because the digital multimeter was not able to read the high resistivity of the mineral oil in completely stable situations where water uniformly displaced the oil. The lowest (i.e., most negative) generalized Bond number investigated is −0.248 as the medium tended to compact under high internal pressures if higher flow rates were applied in the closed cell.
The average water saturation and resistivity index of the tank over time are shown in Figure
Changes in (a) average tank saturation and (b) resistivity index through time for varying values of
At small negative values of
Dependence of resistivity index on saturation for different values of
Based on experimental data, Méheust et al. [
Figures
Direct comparison of resistivity index changes with saturation for different generalized Bond numbers.
It is notable that the resistivity index versus saturation curves for
Archie’s law (equation (
Illustration of how the saturation exponent was determined for experiments with discrete resistivity drops (data shown for experiment index 7).
Table
Saturation exponent determined for each experiment.

Data  Mean  Standard deviation  



 
−0.0035  Experiment Index  6^{1}  9^{1}  20  0.70  0.11  

0.59  0.80  0.71  
−0.022  Experiment Index  16  0.91  —  

0.91  
−0.033  Experiment Index  3  13^{1}  7^{1}  19  0.96  0.08 

0.96  0.90  0.90  1.08  
−0.049  Experiment Index  15  1.30  —  

1.30  
−0.057  Experiment Index  14  1.51  —  

1.51  
−0.071  Experiment Index  4  8  18^{1}  1.73  0.27  

1.73  1.99  1.46  
−0.091  Experiment Index  17  1.43  —  

1.43  
−0.099  Experiment Index  5  1.71  —  

1.71  
−0.12  Experiment Index  1^{1}  10  22  1.96  0.10  

2.06  1.97  1.86  
−0.17  Experiment Index  2  11  23  1.93  0.02  

1.92  1.96  1.92  
−0.25  Experiment Index  12  33  34  1.94  0.02  

1.93  1.96  1.94  
 


 
−0.035  Experiment Index  30  31  32  1.38  0.18  

1.41  1.17  1.54  
−0.038  Experiment Index  28  1.22  —  

1.22  
−0.041  Experiment Index  24  25  26  1.28  0.19  

1.10  1.47  1.27  
−0.045  Experiment Index  29  1.40  —  

1.40  
−0.049  Experiment Index  27  1.59  —  

1.59 
^{ 1} Significant resistivity drops observed in the data.
We find that the apparent saturation exponent increases when flow becomes increasingly unstable, that is,
Saturation exponent as a function of generalized bond number
The observed dependence of the saturation exponent on generalized Bond number demonstrates that relationships used to estimate fluid saturation from resistivity measurements, for example, Archie’s law, must be dynamic and take into account the way in which a reservoir is managed and produced. The saturation exponent is fundamentally related to the geometry of the conductive and nonconductive phases within a porous medium, specifically the connectivity of the conductive phase. Given that instability has an overwhelming influence on fluid distributions during multiphase flow, this process will also strongly influence the saturation index. We identify the increasing connectivity of the water phase between the current electrodes as the primary cause for dependence of the saturation exponent on the generalized Bond number for
The influence of flow instability on electrical resistivity measurements was investigated by displacing a light, but viscous mineral oil by water in a homogeneous porous medium. The experimental setup allowed the effects of gravity and flow rate to be controlled, thereby permitting the generalized Bond number to be changed between different experiments. Video showing the distribution of the fluid phases in the tank allowed us to determine the overall tank saturation throughout the experiment, while continuous measurements of average tank resistivity were also collected throughout each experiment.
In the resulting data we observed a transition between stable and unstable displacement. By analyzing the saturation and resistivity data using Archie’s law, we found that the resulting apparent saturation exponent depends linearly on the generalized Bond number until it reaches a maximum of 1.94 for highly unstable flow systems, that is, when
Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for support (or partial support) of this research.