This paper summarises the results of a benchmark study that compares a number of mathematical and numerical models applied to specific problems in the context of carbon dioxide (CO 2 ) storage in geologic formations. The processes modelled comprise ad-H. Class (B) · A. Ebigbo · R. Helmig · M. Darcis · B. Flemisch vective multi-phase flow, compositional effects due to dissolution of CO 2 into the ambient brine and nonisothermal effects due to temperature gradients and the Joule-Thompson effect. The problems deal with leakage through a leaky well, methane recovery enhanced P. Audigane BRGM, French Geological Survey, 410 Comput Geosci (2009) 13:409-434 by CO 2 injection and a reservoir-scale injection scenario into a heterogeneous formation. We give a description of the benchmark problems then briefly introduce the participating codes and finally present and discuss the results of the benchmark study.

Transport properties of a tubular nanofilter with amphoteric properties have been investigated by means of the SEDE (steric, electric, and dielectric exclusion) homogeneous model. Within the scope of this 1D model, the separation of solutes results from transport effects (described by means of extended Nernst-Planck equations) and interfacial phenomena including steric hindrance, the Donnan effect, and dielectric exclusion (expressed in terms of (i) the Born dielectric effect, which is connected to the lowering of the dielectric constant of a solution inside nanodimensional pores, and (ii) the interaction between ions and the polarization charges induced at the dielectric boundary between the pore walls and the pore-filling solution). The effective volume charge density of the membrane has been determined from tangential streaming potential experiments coupled with conductance experiments in a potassium chloride solution at various pH values ranging from 2 to 11. The inferred values have been used in the SEDE model to compute the ion rejection rates with the dielectric constant of the solution inside the pores as a single adjustable parameter. The model provides a relatively good description of experimental data even at extreme pH values for which a ternary system has been considered (K+, Cl-, and H+ or OH- depending on the pH). The fit to experimental data at the membrane isoelectric point indicates that the confinement effect decreases the dielectric constant inside the pores only slightly (with respect to its bulk value). However, the (pH-dependent) ionization of surface sites has been found to lead to a substantial lowering of the dielectric constant inside the pores.

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Tangential streaming potential technique is an attractive way to characterize the electrokinetic properties of various kinds of materials such as films or flat membranes with a dense or a porous structure. However, the interpretation of data in terms of zeta potential is usually carried out by doing the implicit assumption of nonconducting substrates. In the present paper, we investigate the electrokinetic properties of a commercial ultrafiltration membrane, the porous structure of which affects the streaming potential because streaming and conduction currents involved in the streaming potential process do not flow through identical paths (the streaming current flows only through the slit channel formed by the two membrane samples facing each other whereas a non-negligible part of the conduction current is likely to flow through the membrane pores filled with electrolyte solution). The correct zeta potential value is determined from an extrapolation method for which a set of measurements with various channel heights is required. A very good agreement is obtained with zeta potential values deduced from consecutive streaming potential and total conductance measurements (referred in the text as the direct method). The ratio of the correct zeta potential to the apparent one (given by Helmholtz-Smoluchowski equation) is dependent on the pH which suggests a non-negligible contribution of surface conductance within the membrane pores.

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