Bifurcations and stability of phase transition fronts in geothermal reservoirs

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The stability of a layer of water above a layer of vapor, separated by a boiling or condensation surface, in a geothermal reservoir is considered. In an unperturbed state in lowpermeability rocks, there is one interface, which can be either a water boiling surface or a vapor condensation surface. At relatively large permeability values, two new solutions can be formed, corresponding to other positions of the interface. The conditions for the existence and merging of stationary solutions depending on the parameters of the physical system are studied numerically. The stability of stationary positions of interfaces was studied using the normal mode method. It was found that the transition to instability precedes bifurcations of solutions and can occur both at finite and infinite wave numbers.

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G. Tsypkin

Ishlinsky Institute for Problems in Mechanics RAS

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Email: tsypkin@ipmnet.ru
俄罗斯联邦, Moscow

参考

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2. Fig. 1. Positions of the phase transition surface depending on the pressure PL in the lower highly permeable layer. L = 400 m, ϕ = 0.2, k = 1.84⋅10–17 m2, TL = 540 K, P0 = 1 MPa: 1–2 – T0 = 440, 450 K. P = PL/PM, PM = 1 MPa.

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3. Fig. 2. Bifurcations of the basic solution. Positions of the interface depending on the pressure P0 in the upper highly permeable layer. L = 40 m, ϕ = 0.2, k = 10–16 m2, T0 = 450 K, TL = 468 K, PL = 1.175 MPa. P = P0/PM, PM = 1 MPa.

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4. Fig. 3. Stable (solid line) and unstable (dashed line) positions of the phase transition surface. T0 = 450 K. Other parameters as in Fig. 1.

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5. Fig. 4. Transition to instability with decreasing pressure in the lower permeable layer. Rate of growth (dashed line) and decay (solid line) at PL = 2.99 and 3 MPa, respectively. Parameters as in Fig. 1.

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6. Fig. 5. Stable (solid lines) and unstable (dashed lines) solutions. Parameters as in Fig. 2.

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7. Fig. 6. Transition to instability of the first solution with increasing pressure P0 at the upper boundary. The solid line is the stable state at P0 = 0.998464 MPa and the dotted line is the unstable state at P0 = 0.998463 MPa. The remaining parameters are as in Fig. 2.

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8. Fig. 7. Transition to instability of the third solution with decreasing pressure P0 at the upper boundary. Dispersion curves 1–3: P0 = 1.067, 1.06, 1.053 MPa. Other parameters as in Fig. 2

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