Mathematical model of methane gas extraction from Lake Kivu

Transcription

1 Mathematical model of methane gas extraction from Lake Kivu Moderators: N Fowkes, G Hocking, A Hutchinson, D Mason Industry Representative: D Ndanguza, University of Rwanda 19 January 2018 Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19D January Ndanguza, 2018 University 1 / 41 o

2 Geography: Lakes Kivu, Monoun, Nyos Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19D January Ndanguza, 2018 University 2 / 41 o

3 Lake Kivu Surface area: 2370 km 2 Average depth: 240 m Maximum depth: 485 m Maximum length: 89 km Maximum width: 48 km Approximately two million people in the area Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19D January Ndanguza, 2018 University 3 / 41 o

4 Geological Structure Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19D January Ndanguza, 2018 University 4 / 41 o

5 Gas Accumulation 60m 200m 200m LAKE DENSITY LAYERING Biozone nutrients descending bubbles escaping dissolved gases CO 2 CH 4 in salt in thermocline moderate density picnocline high density Lake Kivu is stratified into layers. Due to stratification, gases accumulate in the bottom layer due to biological and geological processes. Lowest layer (depth of 250 m to 485 m) estimated to contain: 300 km 3 carbon dioxide (CO 2 ) km 3 methane (CH 4 ) Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19D January Ndanguza, 2018 University 5 / 41 o

6 Methane Concentrations Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19D January Ndanguza, 2018 University 6 / 41 o

7 Gas Accumulation Risks According to Schmid et. al. (2002), there are three major risks that could potentially trigger a gas eruption: 1 relatively small uplift of water by a strong internal wave ; 2 a volcanic event could produce sufficient thermal energy that would lift water with high gas concentrations to a level where it is oversaturated and bubbles could form. ; 3 a large amount of gas could be injected into the lake, e.g., by a gas release from the sediments triggered by intruding magma. Catastrophic eruption predicted within years. Such eruptions have occurred in Lakes Monoun and Nyos, Cameroon in 1984 and 1986 respectively. Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19D January Ndanguza, 2018 University 7 / 41 o

8 Lake Nyos: before and after Moderators: N Fowkes, G Hocking, A Mathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 8 / 41 o

9 Remedial action Extract the methane gas via venting However, methane is four times worse as a greenhouse gas than carbon dioxide. Alternatively, extract the methane gas and use it for power generation. Current project at Gisenyi and Kibuye. OUR TASK Modelling of various features to better understand the processes involved. Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19D January Ndanguza, 2018 University 9 / 41 o

10 Gas extraction, and water return Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 10 / 41 o

11 Outline 1 Gas accumulation and stability issues 2 Bubbles: creation and movement and the effect of eruptions 3 Geologically induced waves: eruptions 4 Gas removal: pipe flow 5 Conclusions Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 11 / 41 o

12 Gas accumulation and concentration dynamics Layer 1 ρ 1, P 1, C 1, Low nutrients 80 m M 1, S 1 Layer 2 ρ 2, P 2, C 2, M 2, S 2 High nutrient concentration after re-injection 120 m Layer 3 ρ 3, P 3, C 3, M 3, S 3 Confines layer 4 so gas can't escape 60 m 485 m Layer 4 ρ 4, P 4, C 4, M 4, S 4 High concentration of CH 4 and CO m Inflow Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 12 / 41 o

13 Governing Equations: Methane M i concentration of methane (%) dm 4 dt dm 3 dt dm 2 dt dm 1 dt = α 4 (M 4 M 3 ) + k 1 = α 3 (M 3 M 2 ) + α 4 (M 4 M 3 ) = α 2 (M 2 M 1 ) + α 3 (M 3 M 2 ) + I (t) = α 2 (M 2 M 1 ) + α 1 (M A M 1 ) M 4 > M 3 > M 2 > M 1 Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 13 / 41 o

14 Governing Equations: Carbon Dioxide C i concentration of carbon dioxide (%) dc 4 dt dc 3 dt dc 2 dt dc 1 dt = β 4 (C 4 C 3 ) + k 2 = β 3 (C 3 C 2 ) + β 4 (C 4 C 3 ) = β 2 (C 2 C 1 ) + β 3 (C 3 C 2 ) = β 2 (C 2 C 1 ) + β 1 (C A C 1 ) C 4 > C 3 > C 2 > C 1 Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 14 / 41 o

15 Governing Equations: Salts and Nutrients S i concentration of salts and nutrients (%) ds 4 dt ds 3 dt ds 2 dt ds 1 dt = γ 4 (S 4 S 3 ) = γ 3 (S 2 S 3 ) + γ 4 (S 4 S 3 ) = γ 2 (S 2 S 1 ) + k 3 (t) γ 3 (S 2 S 3 ) = γ 2 (S 2 S 1 ) S 4 > S 2 > S 3 > S 1 Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 15 / 41 o

16 Density and Temperature Variations The density in each layer is given by ρ(t, S, C, M) = ρ(t ) (1 + β s S + β c C + β m M), where and β s = kg.g 1, β c = kg.g 1, β m = kg.g 1, ρ(t ) = ρ w T + 1. Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 16 / 41 o

17 Concentration (%) 0.10 Fun 0.08 with concentration Concentration (%) 0.15 M4[t] M3[t] 0.10, M2[t] M1[t] 0.05 S4[t] S3[t] S2[t] S1[t] t (years) t (years) Concentration (%) (%) Concentration Density (%) C4[t] M4[t] C3[t] M3[t] 0.10,, C2[t] M2[t] C1[t] M1[t] ρ4[t] S4[t] ρ3[t] S3[t] ρ2[t] S2[t] ρ1[t] S1[t] t 300 t (years) t (years) 500 t (years) Concentration (%) Density Figure: Concentration of Methane (left). Density of each layer (right). 0.4 C4[t] C3[t], C2[t] C1[t] ρ4[t] ρ3[t] ρ2[t] ρ1[t] Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, t (years) model D Mason[2em] of methaneindustry gas50 extraction Representative: 100 from 150 Lake200 Kivu 19DJanuary 250 Ndanguza, t (years) University 17 / 41 o

18 M4[t] M3[t], M2[t] M1[t] S4[t] S3[t] S2[t] S1[t] t (years) t (years) Concentration (%) (%) Concentration Density (%) C4[t] M4[t] C3[t] M3[t] 0.10,, C2[t] M2[t] C1[t] M1[t] ρ4[t] S4[t] ρ3[t] S3[t] ρ2[t] S2[t] ρ1[t] S1[t] t 300 t (years) t (years) 500 t (years) Concentration (%) Density 0.5 Figure: Concentration of Methane (left). Density of each layer (right) C4[t] C3[t], C2[t] C1[t] ρ4[t] ρ3[t] ρ2[t] ρ1[t] t (years) t (years) Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 18 / 41 o

19 M4[t] M3[t], M2[t] M1[t] S4[t] S3[t] S2[t] S1[t] t (years) t (years) Concentration (%) Concentration 0.5 (%) Density Concentration (%) C4[t] C3[t] M4[t], M3[t] 0.10 C2[t], C1[t] M2[t] M1[t] ρ4[t] ρ3[t] S4[t] ρ2[t] S3[t] ρ1[t] S2[t] S1[t] t (years) t (years) t (years) t (years) Concentration (%) Density 0.5 Figure: Concentration of Methane (left) Density of each layer (right). 0.4 C4[t] C3[t], C2[t] 0.2 C1[t] ρ4[t] ρ3[t] ρ2[t] ρ1[t] t (years) t (years) Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 19 / 41 o

20 Model of Methane Volume No extraction: dm dt = κ κ is the inflow rate of Methane from underground, M is the volume of Methane and t is time With extraction: dm dt = κ C T C T is the total extraction rate from all processing plants A more realistic model would be dm dt = κ C T M However, data is provided in such a way that the extraction rate is constant, regardless of the volume of methane in the lake Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 20 / 41 o

21 Solution We have data from 2004 relating the methane volume in 2004 to M(2004)=1.15 M(1975) M(2004)= We find κ = Mm 3 yr 1 According to literature, the value of κ ranges from Mm 3 yr Mm 3 yr 1 Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 21 / 41 o

22 Need for more plants If we ignore the Gisenyi plant, the extraction rate of the KP1 plant is Mm 3 yr 1 Therefore if we had 2 plants like this, it would be Mm 3 yr 1 C 3 = Mm 3 yr 1 C 4 = Mm 3 yr 1 C 5 = Mm 3 yr 1 etc It is clear to see that in the worst case scenario of error margins, 4 KP1-esque plants would give us the safe and stable scenario of κ C T. Even 5 plants may still be stable Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 22 / 41 o

23 Graph No Extraction Out[7]= Plant 2 Plants 3 Plants 4 Plants 5 Plants Beginning of Danger Zone Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 23 / 41 o

24 Bubbles: Creation, movement and the effect of eruptions Bubble Physics Henry s Law : C l = HP Laplaces Law : P b = P l + 2γ R Surface tension P l P b δ n Critical radius : R c = 2γ C l H P l Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 24 / 41 o

25 Bubble Growth dr ρ b dt = k(c l(z) H(P l (z)) + 2γ R )) The higher the pressure, the larger the critical radius Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 25 / 41 o

26 Bubble Movement Force due to buoyancy: 4 3 πr3 gρ b Force due to Stokes drag on a bubble: 6πµR(v-ż) ż v = ( 4 3 πr3 gρ b ) 6πµR = 2R2 g 9ν Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 26 / 41 o

27 Bubble Movement P l = P a + h 2 gρ 2 + gρ 1 (h 1 z) Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 27 / 41 o

28 Scaling R = R c R t = τt ρ b = ρ 0 b ρ b z = (h 1 + h 2 )z P l = P a P l C l = Cl 0 C l T = T 0 T Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 28 / 41 o

29 Scaled Equations ρ b dr dt = C l(z) ζ(p l (z) + ζ R ) ρ b = P l dz dt = ζv + Γρ b R 2 The dimensionless parameters are: ζ, ζ, Γ Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 29 / 41 o

30 Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 30 / 41 o

31 Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 31 / 41 o

32 Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 32 / 41 o

33 Rayleigh-Taylor instability y j η 2 (x, t) ρ 2, p 2, φ 2 h 2 x i η1(x, t) ρ 1, p 1, φ 1 h 1 Base 2 φ r x φ r = 0, r = 1, 2 y 2 Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 33 / 41 o

34 Solution Forms and Boundary Conditions φ r = e ikx iwt (A 1 cosh(ky) + A 2 sinh(ky)); r = 1, 2 η r = η 0r e ikx iwt ; r = 1, 2 y = 0 : φ r y (x, 0, t) = η 1 (x, t), r = 1, 2 t y = 0 : p 1 (x, 0, t) = p 2 (x, 0, t) y = h 2 : p 2 (x, h 2, t) = 0 φ 1 y (x, h 1, t) = 0 φ 2 y (x, h 2, t) = η 2 (x, t) t Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 34 / 41 o

35 Solutions Approximating for long wavelengths: φ 1 = iωη 01 e i(kx ωt) (coth( kh 1 ) cosh(ky) sinh(ky)) k φ 2 = iω k ei(kx ωt) (η 01 coth(kh 2 ) cosh(ky) η 02 cosech(kh 2 ) cosh(ky) η 01 sinh(ky)) Wave dispersion relation: ω = ±k g (h 1 h 2 ) h 1 + h 2 ± 2 ρ 1 + 4h 1 h 2 ρ 2 2 ρ 1 Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 35 / 41 o

36 Useful Results Using the dispersion relation, this is always stable if ρ 1 > ρ 2, so if ρ 1 drops below ρ 2 catastrophe Now, by observing a wave on the surface, we can get a critical radius: η 01 = η 02 [1 h 2 g R c = c l Hρ 2 g 2γH ( ), h 2ρ 1g(η 01 η 02 ) ρ 1 +ρ 2 ( ( )) ] g h 1 + h 2 ± (h 1 h 2 ) 2 2 ρ h 1 h 2, ρ 1 where η 02 is the amplitude of the wave on the surface, and η 01 is the amplitude of the wave at the isopycnic, both caused by an eruption. Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 36 / 41 o

37 A Chimney Model of the Pipe Flow lake surface p a rho 2 H rho 1 P=p a + rho_w g H inflow Slugs of liquid go up the pipe There are several flow regimes depending on the bubble/liquid volume ratio, but we will assume plug flow. Gas flows up the chimney/pipe length L because the density ρ p inside the pipe is less than the density ρ w outside the chimney/pipe. Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 37 / 41 o

38 Pipe flow The pressure drop driving the flow is given by (ρ w ρ p )gl and Bernoulli s principle determines the flow velocity v as (ρ w ρ p )gl = 1 2 ρ w v 2 so v = Pipe Flux = 2(ρw ρ p)gl ρ o 2(ρw ρ p)gl ρ w A, where A is the pipe area. Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 38 / 41 o

39 Effective Density in Pipe We use the perfect gas law to determine the density in the pipe assuming all the gas is released. This gives ρ p = ρ w p a (p a + ρ w gl) Note: Pipe flux increases in proportion to the square root of density difference times the pipe length. The proportion of the flux of methane and carbon dioxide is the same as at the depth of pipe. Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 39 / 41 o

40 Conclusions Recycling The experience of stratified flow studies suggest that the water should be reintroduced at a depth with matching density. we understand the relevant processes determining methane release under normal and under eruption conditions and we have made estimates. I think we understand the basic engineering processes associated with methane removal and water recycling. More data is needed to determine the methane source strength. Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 40 / 41 o

41 References Kling, G. W., et al. Lake Kivu gas extraction-report on lake stability. Report to the World Bank (2006). Ross, Kelly Ann, et al. Characterisation of the subaquatic groundwater discharge that maintains the permanent stratification within Lake Kivu; East Africa. PloS one 10.3 (2015): e Schmid, Martin, et al. How hazardous is the gas accumulation in Lake Kivu? Arguments for a risk assesment in light of the Nyiragongo volcano eruption of Acta vulcanologica (2002): Schmid, Martin, et al. Weak mixing in Lake Kivu: new insights indicate increasing risk of uncontrolled gas eruption. Geochemistry, Geophysics, Geosystems 6.7 (2005). Schmid, Martin, KellyAnn Ross, and Alfred West. Comment on An additional challenge of Lake Kivu in Central Africaupward movement of the chemoclines by Finn Hirslund. Journal of Limnology 71.2 (2012): 35. Moderators: N Fowkes, G Hocking, AMathematical Hutchinson, model D Mason[2em] of methaneindustry gas extraction Representative: from Lake Kivu 19DJanuary Ndanguza, 2018 University 41 / 41 o