Material Balance Depletion Models For Two-Phase Geothermal Reservoirs
Authors: Economides M.J. 
Published: December 2007

Summary
Material balance calculations, often referred to as "zero-dimensional" or "lumped element" reservoir evaluations, have been used extensively in petroleum and geothermal engineering. This paper presents a Havlena and Odeh-type material balance depletion model for two-phase reservoirs incorporating adsorption phenomena. A straight line formed between groups of thermodynamic and adsorption properties of water and the cumulative production history provides the initial fluid in place production history provides the initial fluid in place and the size of the vapor-dominated "steam cap." Adsorption phenomena were found to be the controlling mechanism phenomena were found to be the controlling mechanism in the depletion of vapor-dominated geothermal reservoirs. A material balance for vapor-dominated geothermal reservoirs, demonstrating the importance of adsorption phenomena, is presented also. A straight line provides the initial fluid in place. provides the initial fluid in place.

Introduction
The general approach to the material balance depletion model was first suggested by Schilthuis. The equation provides a volumetric balance between the expansion of provides a volumetric balance between the expansion of reservoir fluids as a result of the cumulative production.
Havlena and Odeh, using the Schilthuis approach, developed linear expressions of the material balance equation for a variety of cases including undersaturated, gas-cap-drive and solution-gas-drive reservoirs. I present a similar development for the two-phase geothermal reservoir.
Studies of reservoir and production behavior of vapor-dominated geothermal systems have focused on estimates of resource size. Whiting and Ramey presented an application of material and energy balances to geothermal steam production. Ramey applying conventional techniques for natural gas reservoirs, attempted to estimate the reserves of steam in place.
Economides and Miller using the experimental results of Hsieh, in which the importance and magnitude of adsorption were demonstrated, introduced a new approach for material balance calculation of vapor-dominated systems. As I will show, desorption is the controlling mechanism in the depletion of vapor-dominated geothermal systems and the major variable in their material balance calculations.
The Form of the Material Balance Equation for a Two-Phase Geothermal Reservoir As in the Havlena and Odeh approach, a volume balance in reservoir cubic feet [m3] may be written (in their work they used res bbl [res m3]): underground withdrawal equals expansion of liquid water, plus boiled-off water, plus expansion of steam cap, plus expansion of desorbed water, plus reduction in the PV.
The reduction in PV will be neglected in the case of two-phase and vapor-dominated systems, as has been demonstrated for saturated reservoirs. Fig. 1 is a schematic of the two-phase geothermal model used in this analysis.

Expansion of Liquid Water and Boiled-Off Water
This term contains two components:liquid water expansion andboiled-off steam expansion.
suppose that Wi lbm [kg] were present originally; then WiVli would be the initial volume while Wivl would be the volume at a lower pressure. Hence, the difference would provide the liquid expansion
W i (vl-vli)................................(1)
The boiled-off steam is taken here as the difference between the product of the initial liquid in place and the corresponding evaporation specific volumes at the given pressure interval. This is analogous to the product of the pressure interval. This is analogous to the product of the initial oil in place, the solution GOR, and the gas FVF in the Havlena and Odeh analysis. In both cases the difference depicts the expansion of the evolved gas.
Wi (vlg-vlgi)...............................(2)

Expansion of Steam Cap
The total mass in the steam cap is taken as mWi, where m is expressed as the ratio of the mass of water in the vapor-dominated zone to the mass in the liquid-dominated zone. Then the expansion of the steam is given by
mWi(vg-vgj).................................(3)

Expansion of Desorbed Steam
As presented by Economides and Miller, the volume occupied by desorbed fluid is given by
mrM(Xi-X)vg.................................(4)
where mr is the mass of rock, M is the molecular weight (18 in the case that no noncondensable gases exist in appreciable quantities), and Xj and X are the masses of adsorbed water in lbm-mol/lbm [gmol/g] of rock in the pressure increment. pressure increment. JPT
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