The Straight-Line Solution Method to the

MBE

MATERIAL BALANCE EXPRESSED AS A LINEAR EQUATION

Havlena and Odeh described techniques that in certain situations allow the material balance Equation to be written as a linear equation and even solved graphically.

They further stated that with their analysis methods, the otherwise static material balance relation attained a dynamic meaning.

This discussion has been adapted from Dake's treatment of the work of Havlena and Odeh.

When the pore volume expressed in terms of oil saturation only.

The most general form of Material Balance Equation is Np [Bt + (Rp - Rsi) Bg] = N (Bt – Bti ) + mNBoi(Bg - Bgi) + (We-WpBw) + Bgi + (Cf +CwSw) (NBoi ) (Pi-P) ……(1) (1-Swi )

Another form of general Material Balance Equation is:

)2(---

When considering both the oil and gas saturations in calculating the pore volume.

A more convenient form of the MBE can be determined by introducing the concept of the total (two-phase) formation volume factor Bt into the equation. When considering the gas cap volume in the saturation equation the previous equation becomes:

There are essentially three unknowns in the above Equation :a. The original oil in place Nb. The cumulative water influx Wec. The original size of the gas cap as compared to the oil zone size m

) ----3(

In developing a methodology for determining the above three unknowns, Havlena and Odeh expressed Equation (2) in the following form:

) ----4(

Havlena and Odeh further expressed Equation (4) in a more condensed form as:

F = N [Eo + m Eg + Ef,w] + (We + Winj Bw + Ginj Bginj)

Assuming, for the purpose of simplicity, that no pressure maintenance by gas or water injection is being considered, the above relationship can be further simplified and written as:

F = N [ Eo +m Eg + Ef,w ] + We ---- (5)

In which the terms F, Eo, Eg, and Ef w are defined by the followingrelationships:

F represents the underground withdrawal and given by:

F = Np [Bo + (Rp - Rs) Bg] + Wp Bw ----- (6)

In terms of the two-phase formation volume factor Bt, the underground withdrawal F can be written as:

F = Np [Bt + (Rp - Rsi) Bg] + WpBw ----- (7) Eo describes the expansion of oil and its originally dissolved gas and

isexpressed in terms of the oil formation volume factor as:

Eo = (Bo - Boi) + (Rsi - Rs) Bg ---- (8) Or equivalently, in terms of Bt:

Eo = Bt-Bti ----- (9)

Eg is the term describing the expansion of the gas-cap gas and is defined by the following expression:

Eg = Boi [(Bg/Bgi) - 1] ------- (10)In terms of the two-phase formation volume factor Bt,

essentially Bti = Boi , or

Eg = Bti [(Bg/Bgi) - 1]

Ef, w represents the expansion of the initial water and the reduction in the pore volume and is given by:

) --------11(

Havlena and Odeh examined several cases of varying reservoir types with Equation (5) and pointed out that the relationship can be rearranged into the form of a straight line. For example, in the case of a reservoir which has:

no initial gas cap m = 0 no water influx We = 0 and negligible formation and water compressibilities cf and cw = 0 Equation (5) reduces to: F = N Eo ……………(12) The above expression suggests that a plot of the

parameter F as a function of the oil expansion parameter Eo would yield a straight line with a slope N and intercept equal to zero.

 

The Straight-Line Solution Method to the MBE

The straight-line solution method requires the plotting of a variable group versus another variable group, with the variable group selection depending on the mechanism of production under which the reservoir is producing.

The most important aspect of this method of solution is that it attaches significance the sequence of the plotted points, the direction in which they plot, and to the shape of the resulting plot.

The significance of the straight-line approach is that the sequence of plotting is important and if the plotted data deviates from this straight line, there is some reason for it. This significant observation will provide the engineer with valuable information that can be used in determining the following unknowns:

Initial oil in place N Size of the gas cap m Water influx We Driving mechanism

The applications of the straight-line form of the MBE in solving reservoir engineering problems are presented next to illustrate the usefulness of this particular form. Six cases of applications are presented:

Case 1: Determination of N in volumetric under-saturated reservoirsCase 2: Determination of N in volumetric saturated reservoirsCase 3: Determination of N and m in gas cap drive reservoirsCase 4: Determination of N and We” in water drive reservoirsCase 5: Determination of N, m, and We in combination drive

reservoirsCase 6: Determination of average reservoir pressure, p

Another case of interest is a reservoir with an initial gas cap but with negligible water encroachment. Because of the initial gas cap, the connate water and rock compressibilities may be neglected with little error incurred. Equation (5) becomes:

……………..(13)

For the case of a water drive reservoir with no gas cap and negligible connate water and rock compressibilities, Equation (5) becomes:

In this case, F/E0 should plot as a linear function of We / E0. The y-intercept will be N. A suitable function describing the water influx is normally used in the analysis.

--------(14)

which concerns the expansion of the gas cap gas.

Here , the rock and connate water expansion is considered.

Then, the general material balance Equation may be written as:

……………….(1)

---------(15)

For the reservoir with no initial gas cap, negligible water influx, and negligible compressibilities of connate water and rock, equation 8-36 reduces to:

 

Another case of interest is a reservoir with an initial gas cap but with negligible water encroachment. Because of the initial gas cap, the connate water and rock compressibilities may be neglected with little error incurred. Equation (1) becomes:

For the case of a water drive reservoir with no gas cap and negligible connate water and rock compressibilities, Equation (1) becomes:

In this case, F/E0 should plot as a linear function of We / E0. The y-intercept will be N. A suitable function describing the water influx is normally used in the analysis.