Suppose a stream along an aquifer and waves produced in stream and we want to measure head changes in an aquifer at distance x from stream due to waves produced in stream. I want to know which equation will be suitable to solve to find head at x and t. 1D head equation or wave equation.
Just use the diffusion equation for 1-D ground water flow. Check Hantush's papers as he developed relationships between rise and fall of head (e.g aquifer adjacent to stream ) and corresponding water level changes in the aquifer
Additionally, you could use superposition of a sudden rise in water level in the stream and then sudden fall so stream water levels to simulate a wave action
For starters, see the classical work of Cooper & Rorabaugh, 1963, Ground-water Movements and Bank Storage due to Flood Stages in Surface Streams, Geol. Survey Water-Supply Paper 1536-J.
You may then want to consult the general, convolution-based analytical solution of the linearised 1-D Boussinesq eq. for sloping unconfined aquifers reported in the papers :
Akylas, E. and A. D. Koussis (2007) Response of sloping unconfined aquifer to stage changes in adjacent stream I. Theoretical analysis and derivation of system response functions, Journal of Hydrology, 338: 85-95.
Koussis A. D., E. Akylas and K. Mazi (2007) Response of sloping unconfined aquifer to stage changes in adjacent stream II. Applications, Journal of Hydrology, 338: 73-84.
Heat equation, also known as Boussinesq equation describing groundwater flow
For an accurate modelling the response of an unconfined aquifer to water level changes in the adjacent stream (i.e. h(x,t) head changes in aquifer at the time t at distance x from stream due to waves produced in stream) it is recommended to use the nonlinear partially differential equation known as Bussinesq equation
∂h/∂t ±( sinφa/n) ∂h/∂x = (k cosφa/n)∂/∂x(h∂h/∂x) (1)
in the case of aquifer which has an inclined base at an angle of ±φa and
∂h/∂t = (k/n)∂/∂x(h∂h/∂x) (1*)
when the aquifer is horizontal (i.e. φa = 0)
Notations:
• h(x,t) searched groundwater level (piezometric head) in aquifer
• t independent variable representing time
• x independent variable representing the 1D space (e.g for a semi-infinite interval x0≤x≤∞)
• n effective porosity of the aquifer
• k hydraulic conductivity of the aquifer
For an easier solve the nonlinear PDE a linearized form of this equation can be applied. The usually linearized form of the eq. (1) or (1*) is obtained through the introduction of an averaged piesometric head ĥ known as linearized Boussinesq equation for modelling 1D unsteady flow in an inclined aquifer:
∂h/∂t ±c ∂h/∂x = a2·∂2h/∂x2 (2)
where denoted
c= sinφa/n
a2 = (kcosφa ĥ)/n
The meaning of the other notations remains the same as for Equation (1).
The second term of the equation modeled the convective processes and the last term the diffusions processes.
Remarks
-When the eq. (2) is applied to heat transfer (Fourier’s second law) the dependent variable h is temperature and a2=D is the thermal diffusivity.
-When eq. (2) is applied to mass diffusion (Fick's second law) h is mass concentration and a2 =D is the coefficient of diffusion or the diffusivity.
-When eq. (2) is applied to calculate wave propagation in open channels than this represents the simplified form of the Saint-Venant equation which is known as diffusive wave equation describing non-inertia wave propagation. In this case c is the celerity of diffusion wave and D is diffusion coefficient depending from the open channel flow parameter. If in (2) a2=0 than we obtain the kinematic wave equation.
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