Estimation of Pore Pressure Gradient and Fracture Gradient from Well Logs: A theoretical analysis of techniques in use
*
Suman Paul*, Rima Chatterjee* and Ashani Kundan** Indian School of Mines University, Dhanbad, ** ONGC, Mumbai E mail id:
[email protected]
Abstract The current paper concerns itself with the theoretical analysis of techniques in use for estimating Pore Pressure Gradient and Fracture Gradient. The work presented in this paper brings out important insights into the necessary conditions that must be satisfied by rock properties, for the different techniques to be usable, and thereby gives physical meaning of the different parameters empirically adjusted, while making predictions of these gradients and validating them. The study presented indicates that these parameters can be forward modelled. An example of this is Eaton’s equation, which has been analysed by the authors and contrasted with Miller and other equations that concern these gradients. The relation between effective stress and porosity has been analysed and the approach allows for forward modelling the log properties of mud supported rocks under conditions of normal compaction. Studies such as these help appreciate the uncertainty and reasons thereof and lead to better appreciation of the confidence level of the predictions. This in turn is important as both pre-drill as well as syn-drill analyses have significant impact on drilling a safe and useful well since the projections guide the development of the mud schedule, the casing program, rig selection and even wellhead ratings. Detailed post-drill analysis of data if integrated with results of the studies, have the potential to lead to the correct lessons learnt, thereby minimising risk and uncertainty in future drilling.
Introduction Pore Pressure Gradient and Fracture Gradient considerations impact the technical merits as well as the financial aspect of the well plan (Chennakrishnan, 2008). In areas where elevated Pore Pressure Gradients are known to cause difficulty for drillers, having an accurate pressure prediction at the proposed location is critical to a successful drilling operation. Pre-drill estimation of Pore Pressure Gradient is the standard practice for major oil companies. This should come as no surprise, as the reasons are quite compelling. Pore Pressure Gradient and Fracture Gradient information guides the development of the mud schedule, the casing program, rig selection and wellhead ratings. Each of these aspects of well planning is capital intensive and benefit from having a good pre-drill estimate of Pore Pressure Gradient (Mukherjee et al., 2009). In fact, it is hard to imagine budgeting a multi-million dollar well without this crucial information. Pore Pressure Gradient analysis can be useful in understanding geological influences on hydrocarbon accumulation. It is better to drill the flank of a structure rather than its highest point where higher pressure within the gas cap present more difficult drilling problems. Further, hydrocarbon accumulation favours slightly lowered Pore Pressure within zones of elevated pressures. Identification of these zones, aids in the overall exploration of petroleum reserves. Gas, due its buoyancy, can induce abnormally high formation pressures at very shallow depths. Shallow gas hazards present an important risk while drilling. Pore Pressure from seismic, together with lithology discrimination, can often identify these zones. The economic impact of having even one Pore Pressure related event such as a kick or stuck pipe far exceeds the cost of preparing a detailed analysis of the Pore Pressure for a planned well. Anticipating the pressure ramp and its magnitude will prepare the driller for what lies ahead. This is far more desirable than having contingency plans to respond to unscheduled pressure events. It is well known that drilling rates improve as the mud weight approaches the actual Pore Pressure Gradient. An accurate pressure prediction will aid in achieving this goal and thus lower the overall cost of drilling the well thorough improved drilling rate. Knowing the depth for the high pressure allows casing program optimization, increasing the probability of reaching the objective formations. When geologic uncertainty is reduced a casing string may sometimes be omitted, resulting in tremendous cost savings. Formation damage and underground blowouts are very costly events. Having a good pre-drill estimate of Pore Pressure Gradient and Fracture Gradient can significantly decrease the probability of these serious situations. Pore Pressures can be either normal or abnormal or subnormal.
Normal Pore Pressure will be the hydrostatic pressure due to the average density and vertical depth of the column of fluids above a particular point in the geological section, that is, to the water table or sea level. The convention is that abnormal pressures are higher than normal and subnormal pressures are lower. The main objective of this paper is to analyse (a) the different equations such as: Eaton’s equation, Miller’s equation, Bowers’s equation and Gassmann’s equation used for estimation of Pore Pressure Gradient and (b) the different equations such as; Matthew and Kelly’s equation and Pennebaker’s equation used for estimation of Fracture Gradient.
Methodology From the well log data, Pore Pressure Gradient and Fracture Gradient have been calculated as described in the flow chart (Figure 1). Seismic data (Velocity data) / Log data (LWD / Density log / Porosity log data) Density (ρ) Calculation Overburden Gradient (OBG) Calculation Normal Compaction (NCT) Trend / Compaction Trend (CT) Calculation Pore Pressure (PP) Calculation Fracture Gradient (FG) Calculation
Figure 1: Flow Chart for Pore Pressure and Fracture Gradient Estimation
Analysis of Pore Pressure Gradient Equations Eaton’s Equation and Analysis The Eaton’s equation (Eaton, 1972 and Eaton, 1975) is, (1) where, P = (PP)observed = Observed Pore Pressure S = P0 = Over burden pressure Pn = Normal Compaction Y = Resistivity or Δt etc. Yn = Resistivity or Δt at normal condition k = An exponent (Eaton’s exponent)
Analysis: Eaton’s equation can be written as, (PP)observed = P0 - (P0 - Pn) or, P0 - (PP)observed = (P0 - Pn)
(2)
or,
,
where, σe = P0 – PP and (σe)n = Effective Stress at normal compaction. or, or,
, where, Rn = Resistivity at normal compaction.
(3)
or, (σe)observed фobservedmk = (σe)n фnmk [As we know, R0 = 1 / фm ]
(4)
Let, mk = n, then (σe . фn) is a constant, then σe = C ф–n, where, C = Constant
(5)
Eaton’s equation can be written in this form also. This equation relates Effective Stress with Porosity which indicates Effective Stress increases Porosity decreases. Miller’s Equation and Analysis The Miller’s equation (Miller, 1995) is, V = Vmatrix - (Vmatrix – Vmudline) e – λ (σe)n
(6)
where, V = Velocity Vmatrix = Velocity at matrix (14000 ft / sec to 17000 ft / sec for most shales) Vmudline = Velocity at mudline (5000 ft / sec) λ = It is an empirical parameter that yields the best fit for the relation between velocity and effective stress for the location of interest (σe)n = Effective Stress assuming normal pore pressure
Analysis: At mudline, P0 = Water pressure, PP = Water pressure and σe = 0. If we assume the velocity transform plays (7) V = Vmatrix (1 – ф) + ф Vfluid or, V = Vmatrix – ф Vmatrix + ф Vfluid or, Vmatrix - V = ф (Vmatrix - Vfluid) or, ф =
(8)
Now, we can write Miller’s equation as, Vmatrix - V = (Vmatrix – Vmudline) e – λ (σe)n
(9)
or, or, ф = ф0 – λ (σe)n , where, ф = Porosity and ф0 = Porosity at mudline. ф = ф0 e – λ (σe)n
(10)
Miller’s equation can be written in this form also and this equation is known as Porosity Decline Equation. Let, assume that a small change in effective stress is Δσe and so, the small change in porosity is Δф. So, we can write, = Kф, where Kф = Pore Elastic Constant. or, Δσe =
Kф
As we know the Miller’s equation is,
(11)
ф = ф0 e – λ (σe)n or, Δф = ф0 e – λ (σe)n Δσe – λ or, Δф = – λ (ф0 e – λ (σe)n) Δσe or, Δф = – λ ф Δσe
(12)
[As we know, ф = ф0 e – λ (σe)n]
or, Δσe = or, Δσe =
Kф
[As we know, Δσe =
Kф]
Now the questions are, 1) How does Kф behave with respect to applied stress? and 2) Is Kф independent of applied stress? If the answer is yes, then ф = ф0 e – λ (σe)n, when λ = So we can write,
(13) (14)
We know that, Kframe = Kgrain (1 – ф)C or,
(1 – ф)C
= =
or,
+
or,
=
or,
+
or,
=
+
=
+ (15) is a constant and ф = ф0 e – λ (σe)n holds.
If C is independent of ф, then
Bowers’s Equation and Analysis The Bowers’s equation (Bowers, 1993), V = Vmudline + A (σe)n B
(16)
where, V = Velocity (ft / sec) Vmudline = Velocity at mudline (≈ 5000 ft / sec) (σe)n = Effective Stress assuming at normal pressure (psi) A and B are impirical values that yielded the best fit for the relation between velocity and effective stress based on the location of where the data was taken.
Analysis: We know that, λ=
and
=
. So, we can write, λ =
(17)
Here, at mudline Kgrain tends to Kfluid as grains are in suspension and Kfluid is the Bulk Modulus of fluid.
But
may be very low, since grains are not in contact with one another in the sense of grains of a rock. is low, σe = 0 and Ф = Ф0 (Say). So, any relation for λ to Ф should be of
So, when we are at mudline, this form, λ = A (Ф0 - Ф)ζ + or, λ = A (Ф0 - Ф)ζ
[As
is very small and σe = 0]
(18)
translates the equation, dσe = - A (Ф0 - Ф)ζ
The basic equation dσe = -
a range of values such that (Ф0 - Ф)ζ is sufficiently near (Ф0 - Ф)ζ – 1, i.e., Ф is near
, when Ф is in such .
Then, dσe ≈ - A (Ф0 - Ф)ζ – 1 dФ , where is Constant of Integration.
(19)
When Ф = Ф0, σe = 0, then = 0 i.e., or,
(20) (21)
Therefore,
=
or,
=
[Let, B = is mudline velocity.
, where
Therefore,
=
or,
=
or, V =
+U
or, V =
+
]
, where (
is Constant.
(Say)
, where
is a Constant includes both constant U and constant B.
(22)
(23)
This form leads to Bowers’s Equation. Its unique feature is that σe is related not directly to Ф but to (Ф0 - Ф) in a power relationship which type of relationships characterises of Eaton’s relationship.
Gassmann’s Equation and Analysis The Gassmann’s equation relates the bulk modulus of a rock to its pore, frame and fluid properties. The bulk modulus of a saturated rock is given by the low frequency Gassmann theory (Gassmann, 1951) as,
(24) where, Ksat = Bulk modulus of the saturated rock Kframe = Bulk modulus of the porous rock frame (drained of any pore fluid) Kgrain = Bulk modulus of the grain Kfluid = Bulk modulus of the pore fluid Ф = Porosity (as fraction) Analysis: is independent of Ф where G is Frame Shear Modulus of Elasticity. Since Gframe can
It is known that
be written as [Ggrain (1 – Ф)c], where C is independent of Ggrain and Kframe can be written as [Kgrain (1 – Ф)c]. (More of Kgrain and C are discussed later.)
General Considerations - Relation between Effective Stress and Porosity We know that for saturated frame, (25)
Let, stress changes to Δσ and let ф changes to Δф. So, (26) or,
(27)
Now the substitution in the above equation leads to
or, or,
(28)
Gassmann’s equation states that,
From this equation we get, =
+
(29)
and also Kframe and Kgrain can be assume to be related through Kframe = Kgrain (1 – ф)C,
Let, λ =
where C is a constant and its value was between 1 to 3. Now we can get, =
or,
or,
=
(30)
Hence, Δф = or, Δф =
(31)
Here, λ is in the order of 100’s to 1000’s, so,
