## Modelling Flow in Fractures for Geothermal Energy

Pumping water through hot rocks can drive turbines to generate electricity. The system below has an injection well and a recovery well separated by 400 m. The two wells are 4.5 km deep and there are 3 fractures (A,B,C) lying between the wells. 5 intersecting junctions are labelled J1..J5 as well as the two boundary junctions J6,J7. The sections between each junction are labelled according to the fracture letter and the start/end junctions.
Hover mouse over sections or junctions to get info about the system. Junctions can be dragged interactively. Use the panel to change fracture properties. See 'Explore' for guidance.
First try to understand how the system is behaving. Once you see how it works, then you can build your own version in Excel.
Change the input pressure at the injection well [units m]
Pressure:
Show Labels:
Change the fracture width (into the page) [units m]
Frac A:
Frac B:
Frac C:
Change the fracture arpeture [units mm]
Frac A:
Frac B:
Frac C:
Constant information:
Boundary conditions:
Junction information:
Section information:
Note 1: Properties in these tables change depending on selections made when interaction with plot. Note 2: Injection well (I61) & recovery well (R57) are not fractures, hence have properties set to approximate borehole dimensions (gives negligible friction).
The linear system is defined by the equation:
CH=Q
where
C - is the [5x5] conductance matrix
H - is the [1x5] vector of unknown pressures
Q - is the [1x5] vector of system flows
Formulated Equation CH=Q:
If 'from' junction is 'j' and 'to' junction is 'k', a section's conductance c_jk is calculated as: $$c_{j,k}=c_{k,j}=\frac{ga_{j,k}^3}{12v}\frac{w_{j,k}}{L_{j,k}}$$ where
g - is gravity (m/s2)
v - is the kinematic viscosity (m2/s)
a_jk - is the arpeture (m)
w_jk - is the fracture width (m) from junction j to k
L_jk - is the fracture length (m) from junction j to k

Using the inputs it is possible to calculate conductances and populate the matrices
Intermediate calculation:
Populated Equation CH=Q:
Solving the linear system gives ...
Pressure at locations 1 to 5:
... and recall that h6 and h7 at the injection and recovery well are already known from the input.
A few additional calculations are needed to determine flow, but these are straight forward ...
Flow from junction 'k' to 'j' is related to the change in pressure: $$q_{j,k}=c_{j,k}(h_j-h_k)$$ Note 1: q will have units m3/s. To convert to L/s multiply by 1000 Note 2: Water flows from high pressure to low pressure. If you get a negative flow, it means that the direction will be opposite what you used.