Determination of Average Reservoir Pressure From Build-Up Surveys

Abstract

DIETZ, D.N., KONINKLIJKE/SHELL EXPLORATIE EN PRODUKTIE LABORATORIUM RIJSWIJK, THE NETHERLANDS MEMBER AIME Abstract A method for determining average reservoir pressure is presented, which is simpler to apply than that devised by Matthews, Brons and Hazebroek. For bounded reservoirs, identical results are obtained if stabilized-flow conditions prevail. The present method yields inferior results in the transient state. The method can, with a slight modification, also be used for water-drive reservoirs. Introduction In the method proposed by Matthews, Brons and Hazebroek for determining average reservoir pressure in a multi-well reservoir, the cumulative production (the production time of each well) enters twice: once when the buildup is plotted against In (t + deltat)/deltat to arrive at p*, and a second time when the correction, p*-p is determined with one of the several formulas for differently shaped drainage areas. Once a steady state has been attained, however, the previous production history should be immaterial. The same pressure distribution could have been arrived at after different cumulative productions of the individual wells. In principle, therefore, it should be possible to determine average pressures without referring to cumulative productions. In the following paragraphs an expression is presented for the difference between the pressure in a producing well and the average pressure of its drainage area. Then the build-up time needed to overcome this difference is indicated. CASE OF A CIRCULAR DRAINAGE AREA AND A CENTRAL WELL PRESSURE DISTRIBUTION BEFORE SHUT-IN The general differential equation of radial flow (see Muskat," Eq. 10.2) may be written (1) In the steady state, the rate of production of the well is equal to the rate of expansion of the fluid contained in the the drainage area; thus (2) and therefore, combining Eqs. 1 and 2 we have (3) which can be integrated to (4) The boundary condition can be introduced as (5) so that Eq. 4 can be rewritten (6) A second integration gives (7) AVERAGE PRESSURE OF THE DRAINAGE AREA The average pressure can be expressed as (8) Substitution of Eq. 7 and integration yields (9) BUILD-UP If Eq. 7 is applied to the wellbore at rw, and if C is eliminated by the combination with Eq. 9, we have for the pressure at the well under steady-state conditions (10) After shut-in, and as long as the physical boundaries of the entire reservoir have no influence, the pressure increases according to the well-known expression (11) JPT P. 955ˆ