HW11Key - Homework 11 Key Two Dimensional Simulation 10...

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Unformatted text preview: Homework 11 Key Two Dimensional Simulation 10 points This question deals with the simulation of an understaurated oil in two spatial dimensions (x, y). You may ignore the presence of wells. a. Write the material balance equation for an arbitrary cell (i. j) in difference (e.g. Tp) form. This was the form discussed and given in class. = ( ) + ( ) ( ) Use the 6-cell model shown below for the remainder of this problem. w (i,j) (2,1) (2,2) (2,3) i or y j or x (1,1) (1,2) (1,3) b. Write the explicit form of material balance equation for cell (2, 2) in index (e.g. using i and j) form. The above equation in general index form is = -1/ 2, , - -1, - +1/ 2, , - +1, + , 2 ( , - , - , 2 ( , - , ) ( ) ( ) ( ) ) -1/ -1 +1/ -1 The explicit form for cell (2, 2) is w ( )22 +1 2,2 - 2,2 3/ 2,22,2 - 1,2 5/ 2,2 3,2 - 2,2 2,3/ 2 2,2 - 2,1 2,5 / 2 2,3 - 2,2 - + - = Since this cell is at a boundary, the last term is zero. The final form is w +1 2,2 - 2,2 ) = ( 22 3/ 2,22,2 - 1,2 5/ 2,2 3,2 - 2,2 2,3/ 2 2,2 - 2,1 - + w c. Write the expression of the y-direction cell face transmissibility (e.g. Ty3/2,2 ) in terms of the properties of the relevant cells. Note that the cells are not of equal size. 1 The cell face transmissibility is the harmonic average of the transmissibilities on either side of the face in the y-direction -1 + 2,3/ 2 = 2 2,1 2,2 w d. Write the implicit form of the material balance equation for cell (2, 2) in index (e.g. using i and j) form. Start with = ( ) + ( ) ( ) Expanding w ( c t Vp ) +1 - = ( +1 ) + ( +1 ) ( c t Vp ) +1 22 - 22 +1 +1 +1 +1 = 3/2,2 ( 12 - 22 ) - 5/2,2 ( 22 - 32 ) + +1 +1 +1 +1 2,1/2 ( 21 - 22 ) - 2,3/ 2 ( 22 - 32 ) Because the upper boundary is sealed ( c t Vp ) +1 22 - 22 +1 +1 +1 +1 +1 +1 = 3/2,2 ( 12 - 22 ) - 5/2,2 ( 22 - 32 ) - 2,3/ 2 ( 22 - 32 ) e. The implicit for of the material balance equations can be written in matrix form A pn+1 = B. But this form depends on the ordering of the unknowns. Indicate the form of the matrix A for the following orderings: 0 0 0 0 0 0 0 0 0 0 0 0 0 +1 0 11 0 12 13 0 21 22 23 0 0 0 0 0 0 0 0 0 0 0 0 0 +1 0 11 0 21 0 12 22 13 23 0 0 0 0 0 0 0 0 0 0 0 0 0 +1 0 23 0 12 0 13 11 22 21 Note all matrices have non-zero diagonals and all are symmetric. w w You need not put in values.w Indicate the zero entries with a "0" and the nonzero entries with an "X". 2 3 ...
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