Consider singlephase fluid flow in a 2D horizontal reservoir

Consider single-phase fluid flow in a 2D horizontal reservoir. For the well located at the intersection of gridblocks 1,2,3, and 4 (red dot) calculate the production rate if the FBHP is 1200 psia. Given: h = 50 ft, r_w = 7 in, B = 1.2 RBBL/STB, mu = 1.5 cp

Solution

The gas reservoir model is based on Al-Hussainy and Ramey\'s (1966) expressions modified to account for changes in flow rate and pressure. To calculate the wellbore pressure in an infinite gas reservoir produced at a constant flow rate, including skin and the non-Darcy effects, the following expression is used: [ ( ) ( )] [ ( ) ( )] m P m P khT P T Q t S DQ RES BHP SC SC D - = + + 0 367 2 245 0 87 . log . . ................(1) where the real gas pseudo pressure is defined as: m P P z dP g P ( ) = ................................... 2ò 0 m (2) and the dimensionless time by: ( ) t kt c r D g t i w = (3) f m 2 ................................... where the total compressibility is approximated as:

c c (1 S ).................................... t = g - wi .(4) The non-Darcy factor is calculated with: D MP k R hr T g sc w sc = 0.159 ... ........ ......... ........ .. b m (5) where the velocity coefficient for consolidated sandstone was determined by Geertsma (1974) as: b f g k = 0005 55 . ........................................ . (6) The gas compressibility and the gas viscosity in these equations are evaluated at the reservoir pressure. The bottom hole pressure and gas flow rate vary with time in a case of underground blowout. Due to this fact, the solution for the wellbore pressure can be found by applying the principle of the super-position for different flow rates in equation (1). The result after that is: [ ( ) ( )] ( ) [ ( )] ( ) m P m P khT P T Q Q t t Q S DQ RES BHP SC SC j j j n D Dj n n - = å - - - + + = - 0 367 1 2 245 0 87 1 1 . log . . ........(7) After algebraic manipulation the solution for the flow rate can be obtained from the following quadratic equation: ( ( )) ( ( )) Q S t t D Q B A Q t t D n D Dn n 2 087 2 245 1 n 1 D Dn 1 087 2 245 0 87 + 0 + - + + - - - = - - - . log . . log . . .......(8) where: [ ] A m P m P khT P T RES BHP sc sc = ( )- ( ) . .................. 0 367 (9) and [( ) ( ( ))] B Q Q t t j j D Dj j n = - - - - = - å 1 1 1 1 log 2.2454 ..(10) The flow rate for each time step in the computer model is calculated through equation (8) using the value of the bottom hole pressure in that time step