Gas reservoir material balance

Midway through the study, the company asks Lena's team for a quick review of the Birch Coulee gas pool, bought last year from another operator with five years of production history. 'The seller's reserves report used the volumetric gas in place,' Lena tells Ethan. 'Our own volumetric estimate in lesson 2 came to about 120 Bcf. Before we book anything, I want the material balance. For gas it is the most trusted method there is, if the data are good.'

Gas reservoirs are simpler than oil reservoirs in one important way: there is usually one phase in the reservoir, and gas is so compressible that its expansion dominates every other energy source. That makes the material balance for a gas reservoir a nearly perfect straight line when the assumptions hold, and a sensitive detector of water influx or other energy when they do not. This lesson develops the gas material balance, the p/z plot, and its main complications.

The volumetric gas material balance

For a gas reservoir with no water influx and negligible rock and water expansion, the pore volume occupied by gas stays constant. The gas initially in place, G standard volumes, occupied the pore volume at the initial formation volume factor Bgi. After producing Gp standard volumes, the remaining G − Gp occupies the same pore volume at the current Bg.

Because Bg is proportional to z/p at constant temperature, the equation rearranges into a straight line in p/z.

A plot of p/z against cumulative gas production Gp is therefore a straight line that starts at pi/zi and reaches zero at Gp = G. Extrapolating the line to p/z = 0 gives the gas in place. Extrapolating it to the abandonment p/z gives the recoverable gas. This is the p/z plot, the best known diagnostic in reservoir engineering.

Building the p/z plot

Each point needs an average reservoir pressure, usually from a shut-in test corrected to a datum depth, its z-factor from the gas composition (lesson 3), and the cumulative gas produced to that date, including any gas used as fuel or flared upstream of the sales meter. The z-factor is evaluated at reservoir temperature, which is assumed constant.

# Gas material balance: the p/z plot for a volumetric gas reservoir, with Hall-Yarborough z factors
import math
SI = False  
def z_hy(Tpr, Ppr):
    t = 1.0 / Tpr; A = 0.06125 * t * math.exp(-1.2 * (1 - t) ** 2)
    B = t * (14.76 - 9.76 * t + 4.58 * t * t); Cc = t * (90.7 - 242.2 * t + 42.4 * t * t); D = 2.18 + 2.82 * t
    y = 0.001
    for _ in range(100):
        f = -A * Ppr + (y + y*y + y**3 - y**4) / (1 - y) ** 3 - B * y * y + Cc * y ** D
        df = (1 + 4*y + 4*y*y - 4*y**3 + y**4) / (1 - y) ** 4 - 2 * B * y + Cc * D * y ** (D - 1)
        y -= f / df
    return A * Ppr / y
gg = 0.65; Tpc = 169.2 + 349.5 * gg - 74.0 * gg ** 2; Ppc = 756.8 - 131.0 * gg - 3.6 * gg ** 2   # Sutton, psia and R
T = 167 + 459.67
if SI:   # pressure kPa, cumulative gas 10^6 m3
    hist = [(21000, 0.0), (19560, 192.0), (18100, 416.0), (16580, 640.0), (15200, 864.0), (13800, 1088.0)]
    psi = lambda p: p / 6.895; gu, pu, p_ab = "10^6 m3", "kPa", 2000.0
else:    # pressure psia, cumulative gas Bcf
    hist = [(3050, 0.0), (2835, 6.9), (2612, 14.95), (2422, 23.0), (2198, 31.05), (1994, 39.1)]
    psi = lambda p: p; gu, pu, p_ab = "Bcf", "psia", 300.0
xs, ys = [], []
print(f" pressure    z     p/z ({pu})   Gp ({gu})")
for p, gp in hist:
    zz = z_hy(T / Tpc, psi(p) / Ppc)
    xs.append(gp); ys.append(p / zz)
    print(f" {p:7,.0f}  {zz:.3f}  {p/zz:9,.0f}   {gp:7,.1f}")
n = len(xs); mx = sum(xs) / n; my = sum(ys) / n
b = sum((x - mx) * (y - my) for x, y in zip(xs, ys)) / sum((x - mx) ** 2 for x in xs); a = my - b * mx
G = -a / b
print(f"straight line: p/z = {a:,.0f} {b:+.3f} Gp; gas in place G = {G:,.0f} {gu}")
z_ab = z_hy(T / Tpc, psi(p_ab) / Ppc)
gp_ab = G * (1 - (p_ab / z_ab) / a)
print(f"at an abandonment pressure of {p_ab:,.0f} {pu}: recovery {gp_ab:,.0f} {gu} = {gp_ab/G:.0%} of gas in place")

 pressure    z     p/z (psia)   Gp (Bcf)
   3,050  0.876      3,481       0.0
   2,835  0.871      3,256       6.9
   2,612  0.867      3,012      14.9
   2,422  0.866      2,796      23.0
   2,198  0.867      2,534      31.1
   1,994  0.871      2,289      39.1
straight line: p/z = 3,474 -30.222 Gp; gas in place G = 115 Bcf
at an abandonment pressure of 300 psia: recovery 105 Bcf = 91% of gas in place

The six points from the Birch Coulee gas pool fall on a straight line within the scatter expected from pressure tests. The line extrapolates to a gas in place of about 115 Bcf, about 4 percent below the volumetric 120 Bcf. Recovery to an abandonment pressure of 300 psia is about 105 Bcf, or 91 percent. Lena considers the two estimates consistent: the small difference could be a slightly smaller connected area or a slightly higher water saturation than the maps assume, and it is well within the volumetric range.

What the shape of the p/z plot tells you

A straight p/z line means the reservoir behaves volumetrically: no significant influx and negligible rock compressibility. Departures from a straight line are diagnostic, and the most important is upward curvature. If water influx is supporting the pressure, p/z falls more slowly than the volumetric line, the points bend upward, and an early straight-line extrapolation through them overestimates G. Unfortunately, influx often looks linear for a long time before the curvature becomes obvious, so engineers check for it by other means too, such as the trend of the apparent gas in place, G calculated point by point, which rises with time when influx is acting.

Downward curvature can mean poorer communication than assumed, with parts of the reservoir depleting more slowly, so that measured pressures from wells in the depleted part are lower than the true average. It can also mean that the pressures were measured before stabilizing, which is common in low-permeability reservoirs.

Overpressured gas reservoirs

In abnormally pressured gas reservoirs, the rock and connate water compressibilities are no longer negligible, because the initial pressure is high and the pore volume shrinks noticeably as pressure falls. The p/z plot then shows two straight lines: an early, flatter segment where compaction supports pressure, and a later, steeper segment. Extrapolating the early segment overestimates G, sometimes by a large factor. Methods such as that of Ramagost and Farshad add a correction term for the effective compressibility.

Here the bar on ce indicates the average effective compressibility of rock and water over the pressure interval. Engineers working with overpressured gas use this corrected form, or a full material balance, before booking reserves from p/z extrapolation.

Water influx in gas reservoirs

With water influx, the gas material balance becomes G (Bg − Bgi) + We = Gp Bg + Wp Bw. Rearranged in the Havlena-Odeh form, F/Eg = G + We/Eg, where F = Gp Bg + Wp Bw and Eg = Bg − Bgi. Plotting F/Eg against Gp shows a horizontal line for a volumetric reservoir and a rising curve with influx, the gas equivalent of lesson 5's diagnostic. Once an aquifer model is chosen, a plot of F/Eg against We/Eg should be a straight line with intercept G and slope one.

Water-drive gas reservoirs need careful management. Producing them faster can reduce the gas trapped by the advancing water, because the water has less time to advance before the reservoir reaches abandonment pressure. Some operators also produce water from down-dip wells to slow the advance. Both strategies must be weighed against the extra cost of water handling.

Gas in place and reserves

Because gas recoveries are high and the p/z method is robust in volumetric reservoirs, gas reserves are often booked largely from material balance once enough history exists. State regulators such as the North Dakota Industrial Commission (NDIC) collect pressure and production data, which can provide independent pressure points. Under SEC rules and PRMS, the evaluator documents the method and the uncertainty, and the proved category is used only where the history supports reasonable certainty.

For tight and shale gas, where pressure does not stabilize and the drained volume grows with time, p/z extrapolation is unreliable, and engineers rely instead on decline analysis and rate-transient methods (lesson 12 and later courses).

Abandonment pressure and recovery

Gas recovery depends strongly on the abandonment pressure, the reservoir pressure at which the wells can no longer deliver gas economically into the sales system. Lowering it, by installing field compression to reduce wellhead pressure or by removing liquids that load up the wells, adds reserves directly. A compression project that lowers the abandonment p/z from 20 percent to 10 percent of its initial value adds 10 percent of the gas in place to recovery in a volumetric reservoir, which is why compression is one of the most common gas field investments.

Gas condensate and wet gas reservoirs

The p/z method applies to single-phase gas. For a wet gas, which stays gas in the reservoir but yields liquid at surface, the produced gas volume Gp must include the gas equivalent of the produced condensate and water vapor, converted from liquid volumes using their molecular weights and densities. Leaving the liquids out understates the withdrawal and overstates the gas in place. For a gas condensate below its dew point, liquid drops out in the reservoir, and the single-phase z-factor must be replaced by a two-phase z-factor measured in a constant volume depletion experiment (PET-260). The deep Tamarack condensate pool is analyzed that way.

Monitoring a gas reservoir

Because the p/z method depends on good average pressures, gas operators plan regular shut-in surveys, typically once a year in key wells, timed so the wells are shut in long enough to stabilize. Permanent downhole gauges reduce the need for shut-ins. Engineers update the p/z plot after each survey, recompute the apparent gas in place, and watch for any trend. A rising trend is the earliest reliable sign of water influx, often appearing years before water reaches the wells.

For the Birch Coulee gas pool, Lena asks Ethan to recommend one change to the operator's monitoring, which had surveyed only two wells each year. He suggests adding the down-dip well to the survey program: if an aquifer exists, its pressure will show the influence first.