Numerical and Computational Methods in Petroleum Engineering Assignment -
All questions need to be answered with complete and clear working.
Q1. The continuity equation for the radial one-dimensional horizontal single-phase fluid flow in a porous medium can be written as:
(1/r) ∂/∂r (uρr) = ∂/∂t (ρφ)
Show that under certain assumptions this equation can be linearized to the following diffusivity equation. Show all steps and state all the assumptions while deriving the equation.
(1/r) ∂/∂r (r ∂P/∂r) = (φμct/k) ∂P/∂t
Q2. a) Use the horizontal one-dimensional continuity equation to derive the following partial differential equation for an undersaturated oil reservoir (show all steps):
∂/∂x (k/μB ∂P/∂x) = φ(cr/B + d(1/B)/dP) ∂P/∂t
Hint: Assume ρ = Constant/B
b) Reduce the equation in (a) to the following simple diffusivity equation:
∂2P/∂x2 = φμc/k ∂P/∂t
Q3. The pressure distribution in Nameless reservoir is given by: P(r) = 1800 + 88.28 ln(r/0.25), where P is pressure in psi and r is radial distance from a drilled well in ft.
a) Estimate the pressure gradient (dP/dr) at distances 1, 10 and 100 ft from the well using central finite difference formula and h = 0.2. Work to 2 decimal places.
b) Determine the exact value of pressure gradient at these points and calculate the relative percentage error of the estimated values.
c) According to your calculations, where in the reservoir the pressure gradient is larger, closer to the well or further away from it? Explain why?
Q4. The Kappa Saphir suite is used for pressure transient analysis i.e. well testing. The foundation on well testing is based on the diffusivity equation. Answer the following questions related to well testing and diffusivity equation.
a) What is wellbore storage in well testing? Explain using a plot of production rate vs time.
b) Using a simple relationship explain the three flow regimes with regards to pressure propagation within the reservoir.
c) The diffusivity constant in well testing controls how fast or slow the pressure change can travel or diffuse through the formation. Write mathematical relationship for the diffusivity constant and explain the conditions at which the pressure change will be faster.
d) Identify and explain the different solutions that exist for the diffusivity equation based on the boundary conditions.
Q5. Suppose that y = y(t) is the solution to the following differential equation:
dy/dt = t2y + y2, y(0) = -1
a) Carry out two time steps of Euler's method with a step size of h = 0.25 so as to obtain approximations to y(0.25) and y(0.5). Round the values to three decimal places.
b) Carry out one time step of the Runge-Kutta method RK4 with a step size of h = 0.25 so as to obtain an approximation to y(0.25). Round the values to three decimal places.