Capillary pressure and relative permeability investigation in counter-current spontaneous imbibition. Numerical resolution from Schmid's theory

Spontaneous imbibition is a displacement process of a wetting fluid in a solid matrix impregnated with non-wetting fluid. This process occurs frequently in nature. For example, the water rainfall soaking into the soil is a diffusive process called spontaneous imbibition, in which the water enters displacing the air. In human life, spontaneous imbibition influences many important phenomena. For example, humidity in the air can penetrate in the wall house allowing mold to grow. Another example, when a baby's nappy is filled, spontaneous imbibition is manifested. The analysis of the spontaneous imbibition is also important in the oil industry. The majority of reservoirs are composed of fractured rocks with a significant difference of permeability between the fractures and the unfractured rock, the matrix. The water injected first fills the regions with high permeability. To enter in the low permeability rock, a huge amount of energy is required to push the water inside, using just the viscous pressure force. The invasion of water is called waterflooding. This technique in fractured rocks is uneconomic is the water simply flows along the high permeability fractures and does not displace oil from the matrix. However, spontaneous imbibition permits the invasion of water into low permeability regions. This process is entirely controlled by capillary forces, and the analysis of spontaneous imbibition could predict some important parameters, for example the productivity using the water injection in a fractured rocks, to aid the design of water injection. Schmidt's theory is adopted to analyze spontaneous imbibition. The quantities used in the mass balance and the constitutive equation, Darcy's law extended to multiphase flow, are defined and discussed. From mass balance and the constitutive equation, the multiphase flow equation is obtained. After this, Schmidt's theory is applied to obtain the spontaneous imbibition equation which is solved semi-analytically. I wrote a program to compute solutions. The input of this numerical program, written in Matlab, are the multiphase properties and the output is the water saturation profile and the oil recovery curve. Spontaneous imbibition can also be studied experimentally where the oil recovery is measured as a function of time. We use our analytical solutions to find the multiphase flow properties which are consistent with the measurements. The unknowns are three, water and oil relative permeabilities and capillary pressure. This implies the problem is undetermined. A deterministic approach cannot obtain a unique combination of inputs that match the oil recovery curve, the output. The data considered are given by Zhou et al. [2000]. The waterflooding experiments was also performed by Zhou, so Buckley-Leverett analysis, which considers displacement driven by viscous forces only, is used to match the oil recovery curve during waterflooding. Matching both the waterflood and imbibition recoveries reduces the number of possible relative permeabilities and capillary pressures. The input multiphase flow functions obtained are displayed graphically. Considering the work of Valvatne et Blunt [2004], the relative permeabilities curves are shown in waterflooding case, and the difference from that obtained from my program is discussed. The results show that during spontaneous imbibition, the water relative permeability is low in the experiments, which limits the recovery rate. It is this relative permeability which controls the process. On the other hand, a wide range of capillary pressure can match the results.