35. Calculating permeability (25 marks total)
(i) A cylinder of rock is held vertically. The rock sample has a length of 40 cm and a diameter of 5 cm. Water is pumped through the sample at a rate of 0.06 ml/s upwards. The pressure at the inlet is 130 kPa and the outlet pressure is 101 kPa. Calculate the permeability ot the rock. What properties have you had to estimate or look up?
(10 marks)
(ii) A rock of the same permeability is used to store carbon dioxide in the subsurface. Under the influence of gravity, in which direction will carbon dioxide move in an aquifer initially saturated with water?
Estimate the Darcy velocity for flow under gravity if the viscosity of carbon dioxide is 4* 105 Pa.s, its density is 600 kg/m3 and the density of the water (brine) is 1050 kg/m3. (8 marks)
(iii) The rock has a porosity of 0.2. The storage aquifer has a height of 100 m. Estimate the time taken for the carbon dioxide to rise 100 m. Comment on your answer - what will prevent escape of the carbon dioxide? (? marks)
36. Pore scale displacement and Leverett J function
(25 marks fatal)
(i) Draw diagrams to illustrate how piston-like advance and cooperative pore filling supresses trapping, while snap-off leads to trapping in imbibition.
Provide brief notes to explain the diagrams. (15 marks)
(ii) The capillary pressure for a laboratory displacement in a rock of permeability 150 mD and porosity 0.18 is 3.5 kPa. The interfacial tension is 50 mN/m.
Use Leverett J-function scaling to estimate the capillaiy pressure in the field for a permeability 200 mD and porosity 0.15 with an interfacial tension of 21 mN/m. What approximations have you made? (10 marks)
37. Relative permeability
(25 marks total)
(i) What approximations are made in the multiphase Darcy law and the definition of relative permeability?
When are these assumptions valid? (5 marks)
(ii) Draw example relative permeabilities for waterflooding for a water-wet and an oilwet rock. Explain briefly in your own words the features of the relative permeability functions and how they are linked to pore-scale displacement processes. Also state what type of displacement process - invasion percolation or normal percolation -occurs in the two cases and how this relates to overall oil recovery. (13 marks)
(iii) An oil field has a cross-sectional area of 100,000 m2. There is a pressure drop between two wells of 2 MPa - the wells are placed 400 m apart. Estimate the total flow rate of oil in the horizontal direction if the permeability is 150 mD, the oil viscosity is 3 mPa.s and the oil relative permeability is 0.12. (7 marks)
38. Gas diffusion layers
(25 marks total)
Gas diffusion layers are used in fuel cells. In a fuel cell hydrogen and oxygen react to produce water, while also generating an electrical current. The gas diffusion layer is made of fibres - usually a mix of hydrophilic (water-wet) and hydrophobic (oil and gas-wet) fibres. The pore space must allow gas to flow to the electrodes to react and allow water that is formed to escape.
Imaging experiments are performed and the curvature of the water/gas meniscus in two orthogonal directions (the first and second curvatures) were measured as shown in the table below. The interfacial tension is 70 mN/m.
Fraction of hydrophobic fibres First curvature (mnv1) Second curvature (mm’1)
0 31.2 28.7
0.25 12.5 6.8
0.5 10.1 -9.8
0.75 2.5 -15.2
1 -25.7 -27.9
(i) The fibres are made hydrophobic by coating them in a plastic. Explain carefully why this should result in hydrophobic conditions. In this context, what does this imply about the contact angle between water and gas? (5 marks)
(ii) Calculate the capillary pressure, defined as the gas pressure minus the water pressure, for the five experiments in the table. A positive curvature indicates a positive capillary pressure. (13 marks)
(iii) What fraction of hydrophobic fibres is likely to give the best performance for the fuel cell? Explain your answer. (7marks)