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- A formal report is required for this project. Follow these guidelines for the report:
- The report must be typed and printed on single-sided, lettersized white paper.
*If*one cannot type equations or symbols, use a black/blue pen to write equations/symbols in the right places “neatly” and “cleanly.” Writings, marks and notes with pencils are*not*acceptable anywhere in the report.- Bulk of the report must be printed in black; occasional color texts for highlights and emphasis are allowed. Also, use italic text sparsely.
- Each graph must have a title (caption) and labels for the horizontal and vertical axes.
- If several curves are plotted in the same graph, each curve must be labeled clearly. A curve must use a unique line style (color, thickness, solid, dashed, etc.) so the curve can be identified.

A long and narrow oil reservoir is approximated by a rectangular prism shown below. The length, width and thickness of the prism are L=4200 ft, Δy=200 ft and Δz=50 ft, respectively. Two production wells, Well-PA and Well-PB, are drilled in the reservoir first, and their locations are x_{w} = 1300 ft, and 2900 ft, respectively. Three injection wells are drilled later and their locations are x_{w} = 50, 2100 and 3700 ft. Reservoir and oil properties are: porosity φ=0.20, permeability k= 10 mD, oil saturation S_{o} = 1.0, oil viscosity μ_{o} = 5 cp, oil compressibility c_{o} = 1.0x10^{-5} psi^{-1} , total compressibility c_{t} = 1.5x10^{-5} psi^{-1} , formation volume factor B_{o}=1.20 RB/STB at 3000 psia. All wells have the same wellbore radius r_{w }= 0.25 ft. All wells also have the same skin factor S = 0 except Well-PB, which has a skin factor of 2. The oil bubble point pressure is 800 psia.

- Use numerical simulation to study the oil reservoir and assume that the reservoir is 1-D and flow is single phase only.
- Divide the reservoir into 21 equal-interval blocks to have 21 simulation cells. All five wells are located at the centers of five separate simulation cells.
- Use block-centered grid system to discretize the PDE
*implicitly*.**(5 points)** - Make a stability analysis using von Newman method (Fourier Analysis) to find the largest time step ∆t possible to maintain numerical stability. Do not include wells in the stability analysis. (
**10 points**) - Incorporate the no-flow boundary conditions at both ends.
**(5 points)** - Include the five wells (two production wells and three injection wells) in the discretized finite difference equations. Note that the wells will be modelled by two methods: “specified-rate” and “specified-flowing pressure.”
**(5 points)** - Use Peaceman’s method to calculate productivity index Jw for each well.
**(5 points)** - Perform a “leakproof” test by setting well flow rate to zero for the five wells, and make a 100-day long simulation with Δt=10 days. Pressure everywhere in the reservoir must remain unchanged (equal to the initial pressure). Plot p(x, t=0), p(x, 50) and p(x, 100) vs x in the same graph.
**(10 points)** - Perform a “symmetric” test following these steps:
**(15 points)**- Deactivate the three injections wells by excluding them from the discretized equations.
- Set skin factors in both wells to zero.
- Use same production rate q= 50 STB/D for both production wells.
- Make a 100-day long simulation with Δt=10 days. The pressure distribution must be symmetric with respect to the center point (x=2100 ft).
- Plot p(x, t=0), p(x, 50) and p(x, 100) vs x in the same graph.

- Perform accuracy tests:
**(15 points)**- Set production rate to 50 STB/D for both production wells, and calculate the pressure distribution in the reservoir for t = 100 days, using Δt=1 day, 10 days and 50 days.
- Plot pressure distribution p(x, t=100) vs x for Δt=1 day, 10 days and 50 days in the same graph.

Instructions (Cont’d)

- Conduct the following studies:
- Calculate OIIP (oil initial in place), in STB (
**5 points**) - Set production rate of both production wells to 50 STB/D, and use Δt=10 days. Exclude the three injection wells.
- Calculate reservoir pressure distribution for t=0, 10, 20, …, 250 days
**(10 points)** - Plot p(x, t) vs x for t=0, 50, 100, 150, 200, 250 days
**(5 points)** - Calculate well flowing pressure Pwf of both wells for for t=0, 10, 20, …, 250 days
**(10 points)** - Plot Pwf(t) vs time for both wells in the same graph
**(5 points)** - Plot average pressure vs. time of the two simulation blocks with production wells, in the same graph.
**(5 points)**

- Calculate reservoir pressure distribution for t=0, 10, 20, …, 250 days
- Set flowing pressure Pwf to 1000 psia constant for both production wells,

- Calculate OIIP (oil initial in place), in STB (

and calculate production rates q(t) for t=1, 2, …, until q(t) < 0.01 q(t=1), i.e., 1% of the initial rate, for both wells. Formulate the rate calculation *implicitly*, using Δt=1 day. Plot rate q(t) vs t for both wells in the same graph. Exclude the three injection wells. **(20 points)**

- Cumulative production and recovery:
- Calculate and plot cumulative production from each well, Q(t) vs time in 10-3.
**(5 points)** - Calculate and plot the recovery fraction for the whole reservoir, RF(t) vs time.
**(5 points)**

- Calculate and plot cumulative production from each well, Q(t) vs time in 10-3.
- Repeat steps 10-3 but
- Activate the three injection wells (Well-i1, i2 and i3) when the average cell pressure in the cells with production wells falls below 2000 psi. Set Pwf = 2900 psia for all three injectors. Stop simulation if the injection rate is (almost) stabilized. For convenience assume the injection fluid has the same property as the production fluid in this study.
**(15 points)** - Plot production rates vs time of both wells in the same graph 3. Plot injection rate vs time of Well-i1.

- Activate the three injection wells (Well-i1, i2 and i3) when the average cell pressure in the cells with production wells falls below 2000 psi. Set Pwf = 2900 psia for all three injectors. Stop simulation if the injection rate is (almost) stabilized. For convenience assume the injection fluid has the same property as the production fluid in this study.
- Write the final report to include the following items:
- PDE with boundary conditions
**(5 points)** - Discretization of the PDE with boundary conditions and wells
**(5 points)** - All required graphs
- Brief summary of 10-1, 10-2, 10-3 and 10-4
**(10 points)**Observations, thoughts and suggestions about this project

- PDE with boundary conditions

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