Geothermal reservoir models:  crack plane model

McFarland, R.D.

doi:10.2172/5055525

Cited by 10 publications

(10 citation statements)

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“…From W2~6a~ + Wzz6az = 0 (7), the eigenvector which corresponds to the critical state aK#Oa2 = 0 is obtained in the form 3a2>0, 6a~ = 0 (at constant D) (9) which is admissible according to Eqn. From W2~6a~ + Wzz6az = 0 (7), the eigenvector which corresponds to the critical state aK#Oa2 = 0 is obtained in the form 3a2>0, 6a~ = 0 (at constant D) (9) which is admissible according to Eqn.…”

Section: Stability Conditions For a Crack Systemmentioning

confidence: 99%

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Stability and post-critical growth of a system of cooling or shrinkage cracks

1979

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When a system of parallel equidistant cooling cracks propagates into an elastic halfspace, it reaches at a certain depth of the cracks a critical point of instability, and the equilibrium path of the system bifurcates. Further extension of equally long cracks is unstable and impossible. The stable post-critical path consists of extension of every other crack upon further cooling, initially with a crack jump at constant temperature, while the intermediate cracks stop growing and gradually diminish their stress intensity factor until it becomes zero. This represents a second critical state at which these intermediate cracks suddenly close over a finite length at no change in temperature and at constant length of the leading cracks. Subsequently, as the cooling front further advances, the leading cracks grow at equal length until they again reach a critical state, at which every other crack stops growing, and the process in which the crack spacing doubles is repeated. In this manner, the spacing of the opened cooling cracks fluctuates around roughly the one-half value of the cooling penetration depth.The instability is determined by the sign of the second variation of the work needed to create the cracks, which leads to positive definiteness conditions for a matrix consisting of partial derivatives of the stress intensity factors with regard to crack lengths, subjected to admissibility conditions for the eigenvector of crack length increments. The first initial state of crack arrest is characterized by the vanishing of the diagonal element of the matrix, while the second critical state of crack closing is characterized by the vanishing of the determinant of this matrix.The critical states and the postcritical crack growth are calculated numerically by finite elements. The solution is applied to the cooling of a hot granite mass, the cracking of which is important for one recently proposed geothermal heat extraction scheme. The solution is also of interest for drying shrinkage cracks, especially in concrete. 0376-9429/79/050443-14500.20/0 Int. Journ. o[ Fracture, 15 (1979) 443-456

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Section: Stability Conditions For a Crack Systemmentioning

confidence: 99%

“…(8), (9)). First, the parallel cracks of equal length reach a critical state of crack arrest, which corresponds to vanishing of the diagonal terms in matrix W~ (Eqns.…”

Section: Stability Conditions For a Crack Systemmentioning

confidence: 99%

Stability and post-critical growth of a system of cooling or shrinkage cracks

1979

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“…The penny-shaped vertical crack has a diameter of 1,000 m. The water inlet is on the vertical line of symmetry, 150 m below the centre of the crack, and the water outlet is also on this line of symmetry, 150 m below the top of the ciack. 4 The initial rock temperature, Tn is assumed to vary linearly with depth and be 200°C at the top of the crack and 300°C at the bottom of the crack. The outlet pressure, which must be initially at least equal to the original tectonic stress, is considered to be 30 MN/m2 at all times.…”

Section: Numerical Results and Discussionmentioning

confidence: 99%

“…Thus, the associated averaged net force (WSik),k dx dy has to be added to the term -WP,i dx dy in equation (4 …”

Section: Balance Of Linear Momentummentioning

confidence: 99%

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Geothermal heat extraction by water circulation through a large crack in dry hot rock mass

Bažant

Ohtsubo

1978

Num Anal Meth Geomechanics

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SUMMARYOne proposed geothermal heat extraction scheme relies on water circulation in a large vertical crack created by hydraulic fracturing in a hot dry impermeable rock mass. Water flow, heat convection and crack opening widths are analysed by finite elements. Governing field equations of the problem are first set up rigorously and then various small terms are identified and neglected, retaining the effects of pressure gradient, buoyancy, velocity head (kinetic energy) and head loss due to viscous friction in the water flow equation, and the effects of heat convection in water and heat conduction in rock in the heat transfer equation. The finite element scheme for water flow is based on a variational principle that is typical for diffusion problems, and for heat transfer it is based on the method of least-square residuals. The system of differential equations is highly non-linear. The non-linear terms and coefficients are treated in the finite element analysis as constant; the finite element analysis of the steady-state pressures, fluxes and temperatures is then iterated, evaluating all non-linear terms and coefficients on the basis of the solution obtained in the previous iteration. Numerically calculated fields at various times after the start of cooling are presented. They indicate some features favourable for the geothermal scheme, such as formation of eddy currents, and downward flux of water toward hotter rock. However, other important questions would have to be solved to gain full understanding of this proposed geothermal scheme. INTRODUCflONIn one proposed scheme of heat extraction from hot dry rock geothermal reservoirs, a large vertical crack is to be induced by hydraulic fracturing from a borehole (Figure 1). The upper part of the crack is then intersected by a second borehole, and heat is extracted by circulating water through the crack. Influx of cold water is expected to cause secondary fracturing and large expansion of the region from which the heat can be withdrawn.Considerable interest has been recently shown in this scheme and various aspects of the problem have been studied. 1-3 One problem of particular interest is the circulation of water, heat convection and opening width of a single, main crack created by hydraulic fracture. This problem was analysed by McFarland 4 using a standard finite difference program for diffusion flows. The present paper deals with finite element analysis of the problem. Variations of the crack thickness due to temperature changes are taken into account, and differential equations describing the flow and heat convection in the crack more accurately than those used before are developed.

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Instability of Parallel Thermal Cracks and its Consequences for Hot-Dry Rock Geothermal Energy

Bažant

1980

Thermal Stresses in Severe Environments

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Geothermal reservoir models: crack plane model

Cited by 10 publications

References 0 publications

Stability and post-critical growth of a system of cooling or shrinkage cracks

Stability and post-critical growth of a system of cooling or shrinkage cracks

Geothermal heat extraction by water circulation through a large crack in dry hot rock mass

Instability of Parallel Thermal Cracks and its Consequences for Hot-Dry Rock Geothermal Energy

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